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warn.inf = warn.inf), poisson = fbvppot(x = x, u = threshold, model = model, start = start, ..., sym = sym, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf)) structure(c(ft, call = call), class = c("bvpot", "evd")) } fbvcpot <- function(x, u, model = c("log", "bilog", "alog", "neglog", "negbilog", "aneglog", "ct", "hr", "amix"), start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { model <- match.arg(model) if(sym && !(model %in% c("alog","aneglog","ct"))) warning("Argument `sym' was ignored") switch(model, log = fbvclog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), bilog = fbvcbilog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), alog = fbvcalog(x = x, u = u, start = start, ..., sym = sym, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), neglog = fbvcneglog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), negbilog = fbvcnegbilog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), aneglog = fbvcaneglog(x = x, u = u, start = start, ..., sym = sym, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), ct = fbvcct(x = x, u = u, start = start, ..., sym = sym, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), hr = fbvchr(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), amix = fbvcamix(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf)) } fbvppot <- function(x, u, model = c("log", "bilog", "alog", "neglog", "negbilog", "aneglog", "ct", "hr", "amix"), start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { model <- match.arg(model) if(model %in% c("alog","aneglog","amix")) stop("This model is not appropriate for poisson likelihood") if(sym && (model != "ct")) warning("Argument `sym' was ignored") switch(model, log = fbvplog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), bilog = fbvpbilog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), neglog = fbvpneglog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), negbilog = fbvpnegbilog(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), ct = fbvpct(x = x, u = u, start = start, ..., sym = sym, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf), hr = fbvphr(x = x, u = u, start = start, ..., sym = FALSE, cshape = cshape, cscale = cscale, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf)) } ### Censored Likelihood Fitting ### fbvclog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvclog <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvclog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "log") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvclog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvclog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvclog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvclog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "log") } fbvcbilog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcbilog <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcbilog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "bilog") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, TRUE) f <- formals(nllbvcbilog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcbilog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcbilog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcbilog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "bilog") } fbvcalog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcalog <- function(scale1, shape1, scale2, shape2, asy1, asy2, dep) { if(sym) asy2 <- asy1 if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcalog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, dep, asy1, asy2, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "asy1", "asy2", "dep") else param <- c(param, "asy1", "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "alog") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, !sym, TRUE) f <- formals(nllbvcalog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcalog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcalog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcalog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "alog") } fbvcneglog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcneglog <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcneglog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "neglog") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvcneglog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcneglog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcneglog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcneglog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "neglog") } fbvcnegbilog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcnegbilog <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcnegbilog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "negbilog") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, TRUE) f <- formals(nllbvcnegbilog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcnegbilog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcnegbilog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcnegbilog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "negbilog") } fbvcaneglog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcaneglog <- function(scale1, shape1, scale2, shape2, asy1, asy2, dep) { if(sym) asy2 <- asy1 if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcaneglog, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, dep, asy1, asy2, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "asy1", "asy2", "dep") else param <- c(param, "asy1", "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "aneglog") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, !sym, TRUE) f <- formals(nllbvcaneglog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcaneglog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcaneglog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcaneglog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "aneglog") } fbvcct <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcct <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(sym) beta <- alpha if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcct, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "alpha", "beta") else param <- c(param, "alpha") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "ct") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, !sym) f <- formals(nllbvcct)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcct) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcct(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcct(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "ct") } fbvchr <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvchr <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvchr, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "hr") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvchr)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvchr) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvchr(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvchr(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "hr") } fbvcamix <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvcamix <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvcamix, spx$x1, spx$x2, spx$nn, spx$n, spx$thdi, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "amix") spx <- sep.bvdata(x = x, method = "cpot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, TRUE) f <- formals(nllbvcamix)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvcamix) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvcamix(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvcamix(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "censored", model = "amix") } ### Poisson Likelihood Fitting ### fbvplog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvplog <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvplog, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "log") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvplog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvplog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvplog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvplog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "log") } fbvpneglog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvpneglog <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvpneglog, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "neglog") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvpneglog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvpneglog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvpneglog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvpneglog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "neglog") } fbvpct <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvpct <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(sym) beta <- alpha if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvpct, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "alpha", "beta") else param <- c(param, "alpha") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "ct") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, !sym) f <- formals(nllbvpct)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvpct) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvpct(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvpct(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "ct") } fbvpbilog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvpbilog <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvpbilog, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "bilog") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, TRUE) f <- formals(nllbvpbilog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvpbilog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvpbilog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvpbilog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "bilog") } fbvpnegbilog <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvpnegbilog <- function(scale1, shape1, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvpnegbilog, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, alpha, beta, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "negbilog") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE, TRUE) f <- formals(nllbvpnegbilog)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvpnegbilog) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvpnegbilog(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvpnegbilog(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "negbilog") } fbvphr <- function(x, u, start, ..., sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nllbvphr <- function(scale1, shape1, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 .C(C_nllbvphr, spx$x1, spx$x2, spx$nn, spx$thdi, spx$r1, spx$r2, spx$lambda, dep, scale1, shape1, scale2, shape2, dns = double(1))$dns } param <- c("scale1", "shape1") if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, method = "pot", u = u, model = "hr") spx <- sep.bvdata(x = x, method = "ppot", u = u) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- c(TRUE, TRUE, !cscale, !cshape, TRUE) f <- formals(nllbvphr)[prind] names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nllbvphr) <- c(f[m], f[-m]) nll <- function(p, ...) nllbvphr(p, ...) if(l > 1) { body(nll) <- parse(text = paste("nllbvphr(", paste("p[", 1:l, "]", collapse=", "), ", ...)")) } start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nll", start.arg) == 1e+06) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nll, hessian = std.err, ..., method = method) cmar <- c(cscale, cshape); nat <- spx$nat bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, method = "pot", u = u, nat = nat, likelihood = "poisson", model = "hr") } ### Method Function ### "print.bvpot" <- function(x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:", deparse(x$call), "\n") cat("Likelihood:", x$likelihood, "\n") cat("Deviance:", deviance(x), "\n") cat("AIC:", AIC(x), "\n") if(!is.null(x$dep.summary)) cat("Dependence:", x$dep.summary, "\n") cat("\nThreshold:", round(x$threshold, digits), "\n") cat("Marginal Number Above:", x$nat[1:2], "\n") cat("Marginal Proportion Above:", round(x$nat[1:2]/x$n, digits), "\n") cat("Number Above:", x$nat[3], "\n") cat("Proportion Above:", round(x$nat[3]/x$n, digits), "\n") cat("\nEstimates\n") print.default(format(fitted(x), digits = digits), print.gap = 2, quote = FALSE) if(!is.null(std.errors(x))) { cat("\nStandard Errors\n") print.default(format(std.errors(x), digits = digits), print.gap = 2, quote = FALSE) } if(!is.null(x$corr)) { cat("\nCorrelations\n") print.default(format(x$corr, digits = digits), print.gap = 2, quote = FALSE) } cat("\nOptimization Information\n") cat(" Convergence:", x$convergence, "\n") cat(" Function Evaluations:", x$counts["function"], "\n") if(!is.na(x$counts["gradient"])) cat(" Gradient Evaluations:", x$counts["gradient"], "\n") if(!is.null(x$message)) cat(" Message:", x$message, "\n") cat("\n") invisible(x) } evd/R/profile.R0000644000175100001440000003174214260534716013051 0ustar hornikusers "profile.evd" <- function(fitted, which = names(fitted$estimate), conf = 0.999, mesh = fitted$std.err[which]/4, xmin = rep(-Inf, length(which)), xmax = rep(Inf, length(which)), convergence = FALSE, method = "BFGS", control = list(maxit = 500), ...) { if(!inherits(fitted, "evd")) stop("Use only with `evd' objects") if(inherits(fitted, "extreme")) stop("profiles not implemented for this model") if(length(xmin) != length(which)) stop("`xmin' and `which' must have the same length") if(length(xmax) != length(which)) stop("`xmax' and `which' must have the same length") if(length(fitted$estimate) < 2) stop("cannot profile one dimensional likelihood") if(!is.character(which)) stop("`which' must be a character vector") if(!all(which %in% names(fitted$estimate))) stop("`which' contains unrecognized or unestimated parameters") if(is.null(fitted$std.err) && missing(mesh)) stop("fitted model must contain standard errors") prof.list <- as.list(numeric(length(which))) names(xmin) <- names(xmax) <- names(prof.list) <- which if(is.null(names(mesh))) names(mesh) <- which mles <- fitted$estimate[which] for(j in which) { print(paste("profiling",j)) prof1 <- prof2 <- matrix(nrow = 0, ncol = length(fitted$estimate) + 1) npmles <- fitted$estimate[!names(fitted$estimate) %in% j] start <- as.list(npmles) call.args <- c(list(fitted$data, start, 0), as.list(fitted$fixed), list(FALSE, FALSE, method, FALSE, control)) names(call.args) <- c("x", "start", j, names(fitted$fixed), "std.err", "corr", "method", "warn.inf", "control") dimnames(prof1) <- dimnames(prof2) <- list(NULL, c(j, "deviance", names(start))) call.fn <- paste("f", class(fitted)[1], sep="") if(inherits(fitted, "gev")) { call.args$nsloc <- fitted$nsloc call.args$prob <- fitted$prob } if(inherits(fitted, "gumbelx")) { call.args$nsloc1 <- fitted$nsloc1 call.args$nsloc2 <- fitted$nsloc2 } if(inherits(fitted, "pot")) { call.args$threshold <- fitted$threshold call.args$npp <- fitted$npp call.args$period <- fitted$period call.args$cmax <- fitted$cmax call.args$r <- fitted$r call.args$ulow <- fitted$ulow call.args$rlow <- fitted$rlow call.args$mper <- fitted$mper } if(inherits(fitted, "bvevd")) { call.args$model <- fitted$model call.args$nsloc1 <- fitted$nsloc1 call.args$nsloc2 <- fitted$nsloc2 call.args$sym <- fitted$sym call.args$cloc <- fitted$cmar[1] call.args$cscale <- fitted$cmar[2] call.args$cshape <- fitted$cmar[3] } if(inherits(fitted, "bvpot")) { call.args$threshold <- fitted$threshold call.args$likelihood <- fitted$likelihood call.args$model <- fitted$model call.args$sym <- fitted$sym call.args$cscale <- fitted$cmar[1] call.args$cshape <- fitted$cmar[2] } lcnt <- TRUE; ppar <- mles[j] while(lcnt) { ppar <- as.vector(ppar + mesh[j]) if(ppar >= xmax[j]) ppar <- as.vector(xmax[j]) call.args[[j]] <- ppar fit.mod <- do.call(call.fn, call.args) if(convergence) print(fit.mod$convergence) call.args[["start"]] <- as.list(fit.mod$estimate) rop <- c(ppar, fit.mod$deviance, fit.mod$estimate) prof1 <- rbind(prof1, rop) ddf <- fit.mod$deviance - fitted$deviance lcnt <- (ddf <= qchisq(conf, 1)) && (ppar != xmax[j]) } call.args[["start"]] <- as.list(npmles) lcnt <- TRUE; ppar <- mles[j] while(lcnt) { ppar <- as.vector(ppar - mesh[j]) if(ppar <= xmin[j]) ppar <- as.vector(xmin[j]) call.args[[j]] <- ppar fit.mod <- do.call(call.fn, call.args) if(convergence) print(fit.mod$convergence) call.args[["start"]] <- as.list(fit.mod$estimate) rop <- c(ppar, fit.mod$deviance, fit.mod$estimate) prof2 <- rbind(prof2, rop) ddf <- fit.mod$deviance - fitted$deviance lcnt <- (ddf <= qchisq(conf, 1)) && (ppar != xmin[j]) } rop <- c(mles[j], fitted$deviance, npmles) prof2 <- prof2[nrow(prof2):1, ,drop = FALSE] prof <- rbind(prof2, rop, prof1) rownames(prof) <- NULL rdev <- qchisq(conf, 1) + fitted$deviance if(prof[1, "deviance"] == 2e6) { prof <- prof[-1, ,drop = FALSE] if(prof[1,"deviance"] <= rdev) warning(paste("If", j, "is to satisfy `conf',", "`mesh' must be smaller")) } if(prof[nrow(prof), "deviance"] == 2e6) { prof <- prof[-nrow(prof), ,drop = FALSE] if(prof[nrow(prof),"deviance"] <= rdev) warning(paste("If", j, "is to satisfy `conf',", "`mesh' must be smaller")) } prof.list[[j]] <- prof } structure(prof.list, deviance = fitted$deviance, xmin = xmin, xmax = xmax, class = "profile.evd") } profile2d <- function (fitted, ...) { UseMethod("profile2d") } "profile2d.evd" <- function(fitted, prof, which, pts = 20, convergence = FALSE, method = "Nelder-Mead", control = list(maxit = 5000), ...) { if(!inherits(fitted, "evd")) stop("Use only with `evd' objects") if(inherits(fitted, "extreme")) stop("profiles not implemented for this model") if (!inherits(prof, "profile.evd")) stop("`prof' must be a `profile.evd' object") if(length(fitted$estimate) < 3) stop("Cannot profile two dimensional likelihood") if(missing(which) || !is.character(which) || length(which) != 2) stop("`which' must be a character vector of length two") if(!all(which %in% names(fitted$estimate))) stop("`which' contains unrecognized or unestimated parameters") if(!all(which %in% names(prof))) stop("`which' contains unprofiled parameters") if(is.null(fitted$std.err)) stop("fitted model must contain standard errors") prof.list <- as.list(numeric(3)) names(prof.list) <- c("trace", which) limits1 <- range(prof[[which[1]]][,1]) limits2 <- range(prof[[which[2]]][,1]) mles <- fitted$estimate[which] prof <- matrix(NA, nrow = pts^2, ncol = length(fitted$estimate) + 1) parvec1 <- seq(limits1[1], limits1[2], length = pts) prof.list[[which[1]]] <- parvec1 parvec2 <- seq(limits2[1], limits2[2], length = pts) prof.list[[which[2]]] <- parvec2 pars <- expand.grid(parvec1, parvec2) start <- as.list(fitted$estimate[!names(fitted$estimate) %in% which]) # if method unspecified supress optim 1d warnings if(missing(method) && length(start) == 1) oldopt <- options(warn = -1) call.args <- c(list(fitted$data, start, 0, 0), as.list(fitted$fixed), list(FALSE, FALSE, method, FALSE, control)) names(call.args) <- c("x", "start", which[1], which[2], names(fitted$fixed), "std.err", "corr", "method", "warn.inf", "control") dimnames(prof) <- list(NULL, c(which, "deviance", names(start))) call.fn <- paste("f", class(fitted)[1], sep="") if(inherits(fitted, "gev")) { call.args$nsloc <- fitted$nsloc call.args$prob <- fitted$prob } if(inherits(fitted, "gumbelx")) { call.args$nsloc1 <- fitted$nsloc1 call.args$nsloc2 <- fitted$nsloc2 } #if(inherits(fitted, "pot")) { # call.args$threshold <- fitted$threshold # call.args$npp <- fitted$npp # call.args$period <- fitted$period # call.args$cmax <- fitted$cmax # call.args$r <- fitted$r # call.args$ulow <- fitted$ulow # call.args$rlow <- fitted$rlow # call.args$mper <- fitted$mper #} if(inherits(fitted, "bvevd")) { call.args$nsloc1 <- fitted$nsloc1 call.args$nsloc2 <- fitted$nsloc2 call.args$model <- fitted$model call.args$sym <- fitted$sym call.args$cloc <- fitted$cmar[1] call.args$cscale <- fitted$cmar[2] call.args$cshape <- fitted$cmar[3] } if(inherits(fitted, "bvpot")) { call.args$threshold <- fitted$threshold call.args$likelihood <- fitted$likelihood call.args$model <- fitted$model call.args$sym <- fitted$sym call.args$cscale <- fitted$cmar[1] call.args$cshape <- fitted$cmar[2] } for(i in 1:pts^2) { call.args[[which[1]]] <- pars[i,1] call.args[[which[2]]] <- pars[i,2] fit.mod <- do.call(call.fn, call.args) if(convergence) print(fit.mod$convergence) prof[i,1] <- pars[i,1] prof[i,2] <- pars[i,2] prof[i,3] <- fit.mod$deviance prof[i,-(1:3)] <- fit.mod$estimate } prof.list[["trace"]] <- prof if(missing(method) && length(start) == 1) oldopt <- options(oldopt) if(any(prof[,"deviance"] == 2e6)) warning("non-convergence present in profile2d object") structure(prof.list, deviance = fitted$deviance, class = "profile2d.evd") } "plot.profile.evd" <- function(x, which = names(x), main = NULL, ask = nb.fig < length(which) && dev.interactive(), ci = 0.95, clty = 2, ...) { if (!inherits(x, "profile.evd")) stop("Use only with `profile.evd' objects") if(!is.character(which)) stop("`which' must be a character vector") if(!all(which %in% names(x))) stop("`which' contains unprofiled parameters") nb.fig <- prod(par("mfcol")) if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if(is.null(main)) { fls <- toupper(substr(which, 1, 1)) ols <- substr(which, 2, nchar(which)) cwhich <- paste(fls, ols, sep = "") main <- paste("Profile Log-likelihood of", cwhich) } for(i in which) { plot(spline(x[[i]][,1], -x[[i]][,2]/2, n = 75), type = "l", xlab = i, ylab = "profile log-likelihood", main = main[match(i,which)], ...) cdist <- -(attributes(x)$deviance + qchisq(ci, df = 1))/2 abline(h = cdist, lty = clty) } invisible(x) } "confint.profile.evd" <- function(object, parm, level = 0.95, ...) { if(missing(parm)) parm <- names(object) if(!all(parm %in% names(object))) stop("`parm' contains unprofiled parameters") rdev <- attributes(object)$deviance + qchisq(level, df = 1) pct <- c("lower", "upper") ci <- array(NA, dim = c(length(parm), 2), dimnames = list(parm, pct)) # Assumes profile trace is unimodal for(i in parm) { x <- object[[i]] n <- nrow(x) th.l <- (x[1, 1] == attributes(object)$xmin[i]) th.u <- (x[n, 1] == attributes(object)$xmax[i]) halves <- c(diff(x[,"deviance"]) < 0, FALSE) if(x[1,"deviance"] <= rdev && !th.l) { warning(paste("cannot calculate lower confidence limit for", i)) ci[i,1] <- NA } if(x[1,"deviance"] <= rdev && th.l) ci[i,1] <- x[1, 1] if(x[1,"deviance"] > rdev) ci[i,1] <- approx(x[halves,2], x[halves,1], xout = rdev)$y if(x[n,"deviance"] <= rdev && !th.u) { warning(paste("cannot calculate upper confidence limit for", i)) ci[i,2] <- NA } if(x[n,"deviance"] <= rdev && th.u) ci[i,2] <- x[n, 1] if(x[n,"deviance"] > rdev) ci[i,2] <- approx(x[!halves,2], x[!halves,1], xout = rdev)$y } ci } "plot.profile2d.evd" <- function(x, main = NULL, ci = c(0.5,0.8,0.9,0.95,0.975, 0.99, 0.995), col = heat.colors(8), intpts = 75, xaxs = "r", yaxs = "r", ...) { if (!inherits(x, "profile2d.evd")) stop("Use only with `profile2d.evd' objects") which <- names(x)[2:3] if(is.null(main)) { fls <- toupper(substr(which, 1, 1)) ols <- substr(which, 2, nchar(which)) cwhich <- paste(fls, ols, sep = "") main <- paste("Profile Log-likelihood of", cwhich[1], "and", cwhich[2]) } br.pts <- attributes(x)$deviance + qchisq(c(0,ci), df = 2) prof <- x$trace[,"deviance"] if(any(prof == 2e6)) warning("non-convergence present in profile2d object") prof <- -prof/2 br.pts <- (-br.pts/2)[length(br.pts):1] col <- col[length(col):1] if(!requireNamespace("interp", quietly = TRUE)) { image(x[[which[1]]], x[[which[2]]], matrix(prof, nrow = length(x[[which[1]]])), col = col, breaks = c(-1e6+1, br.pts), main = main, xlab = which[1], ylab = which[2], xaxs = xaxs, yaxs = yaxs, ...) } else { lim1 <- range(x[[which[1]]]) lim2 <- range(x[[which[2]]]) prof.interp <- interp::interp(x$trace[,1], x$trace[,2], prof, xo = seq(lim1[1], lim1[2], length = intpts), yo = seq(lim2[1], lim2[2], length = intpts)) image(prof.interp, col = col, breaks = c(min(prof), br.pts), main = main, xlab = which[1], ylab = which[2], xaxs = xaxs, yaxs = yaxs, ...) } invisible(x) } evd/R/nonpar.R0000644000175100001440000002753214212017116012673 0ustar hornikusers "abvnonpar"<- function(x = 0.5, data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL, method = c("cfg","pickands","tdo","pot"), k = nrow(data)/4, convex = FALSE, rev = FALSE, madj = 0, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "t", ylab = "A(t)", ...) { if(mode(x) != "numeric" || any(x < 0, na.rm=TRUE) || any(x > 1, na.rm=TRUE)) stop("invalid argument for `x'") method <- match.arg(method) # Empirical transform to exponential epdata <- apply(data, 2, rank, na.last = "keep") nasm <- apply(data, 2, function(x) sum(!is.na(x))) epdata <- epdata / rep(nasm+1, each = nrow(data)) epdata <- -log(epdata) if(epmar) data <- epdata # End empirical transform if(!epmar) { if(method == "pot") { # Parametric pot transform to exponential if(any(k >= nasm)) stop("k is too large") u1 <- sort(data[,1], decreasing = TRUE)[k+1] u2 <- sort(data[,2], decreasing = TRUE)[k+1] d1ab <- (data[,1] > u1) & !is.na(data[,1]) d2ab <- (data[,2] > u2) & !is.na(data[,2]) if(!is.null(kmar)) { data[d1ab,1] <- mtransform(data[d1ab,1], c(u1, kmar)) data[d2ab,2] <- mtransform(data[d2ab,2], c(u2, kmar)) } else { mle.m1 <- c(u1, fitted(fpot(data[d1ab,1], threshold = u1))) mle.m2 <- c(u2, fitted(fpot(data[d2ab,2], threshold = u2))) data[d1ab,1] <- mtransform(data[d1ab,1], mle.m1) data[d2ab,2] <- mtransform(data[d2ab,2], mle.m2) } data[d1ab,1] <- -log(1 - k * data[d1ab,1] / nasm[1]) data[d2ab,2] <- -log(1 - k * data[d2ab,2] / nasm[2]) data[!d1ab, 1] <- epdata[!d1ab, 1] data[!d2ab, 2] <- epdata[!d2ab, 2] # End parametric pot transform } if(method != "pot") { # Parametric gev transform to exponential if(!is.null(kmar)) { data <- mtransform(data, kmar) } else { if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(data)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(data)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } mle.m1 <- fitted(fgev(data[,1], nsloc = nsloc1, std.err = FALSE)) loc.mle.m1 <- mle.m1[grep("^loc", names(mle.m1))] if(is.null(nsloc1)) loc.mle.m1 <- rep(loc.mle.m1, nrow(data)) else loc.mle.m1 <- nslocmat1 %*% loc.mle.m1 mle.m1 <- cbind(loc.mle.m1, mle.m1["scale"], mle.m1["shape"]) mle.m2 <- fitted(fgev(data[,2], nsloc = nsloc2, std.err = FALSE)) loc.mle.m2 <- mle.m2[grep("^loc", names(mle.m2))] if(is.null(nsloc2)) loc.mle.m2 <- rep(loc.mle.m2, nrow(data)) else loc.mle.m2 <- nslocmat2 %*% loc.mle.m2 mle.m2 <- cbind(loc.mle.m2, mle.m2["scale"], mle.m2["shape"]) data <- mtransform(data, list(mle.m1, mle.m2)) } # End parametric gev transform } } if(rev) data <- data[,2:1] data <- na.omit(data) if(plot || add) x <- seq(0, 1, length = 100) d1 <- data[,1]; d2 <- data[,2] sum1 <- sum(d1); slm1 <- sum(log(d1)) sum2 <- sum(d2); slm2 <- sum(log(d2)) nn <- nrow(data) nx <- length(x) mpmin <- function(a,b) { a[a > b] <- b[a > b] a } mpmax <- function(a,b) { a[a < b] <- b[a < b] a } if(method == "cfg") { if(!convex) { a <- numeric(nx) for(i in 1:nx) a[i] <- sum(log(mpmax((1-x[i]) * d1, x[i] * d2))) a <- (a - (1-x) * slm1 - x * slm2)/nn a <- pmin(1, pmax(exp(a), x, 1-x)) } else { x2 <- seq(0, 1, length = 250) a <- numeric(250) for(i in 1:250) a[i] <- sum(log(mpmax((1-x2[i]) * d1, x2[i] * d2))) a <- (a - (1-x2) * slm1 - x2 * slm2)/nn a <- pmin(1, pmax(exp(a), x2, 1-x2)) inch <- chull(x2, a) a <- a[inch] ; x2 <- x2[inch] a <- approx(x2, a, xout = x, method="linear")$y } } if(method == "pickands") { if(!convex) { a <- numeric(nx) if(madj == 2) { d1 <- d1/mean(d1) d2 <- d2/mean(d2) } for(i in 1:nx) a[i] <- sum(mpmin(d1/x[i], d2/(1-x[i]))) if(madj == 1) a <- a - x * sum1 - (1-x) * sum2 + nn a <- nn / a a <- pmin(1, pmax(a, x, 1-x)) } else { x2 <- seq(0, 1, length = 250) a <- numeric(250) if(madj == 2) { d1 <- d1/mean(d1) d2 <- d2/mean(d2) } for(i in 1:250) a[i] <- sum(mpmin(d1/x2[i], d2/(1-x2[i]))) if(madj == 1) a <- a - x2 * sum1 - (1-x2) * sum2 + nn a <- nn / a a <- pmin(1, pmax(a, x2, 1-x2)) inch <- chull(x2, a) a <- a[inch] ; x2 <- x2[inch] a <- approx(x2, a, xout = x, method="linear")$y } } # Undocumented method: Tiago de Oliveira (1997) if(method == "tdo") { if(!convex) { a <- numeric(nx) for(i in 1:nx) a[i] <- sum(mpmin(x[i]/(1 + nn*d1), (1-x[i])/(1 + nn*d2))) a <- 1 - a/(1 + log(nn)) a <- pmin(1, pmax(a, x, 1-x)) } else { x2 <- seq(0, 1, length = 250) a <- numeric(250) for(i in 1:250) a[i] <- sum(mpmin(x2[i]/(1 + nn*d1), (1-x2[i])/(1 + nn*d2))) a <- 1 - a/(1 + log(nn)) a <- pmin(1, pmax(a, x2, 1-x2)) inch <- chull(x2, a) a <- a[inch] ; x2 <- x2[inch] a <- approx(x2, a, xout = x, method="linear")$y } } # Undocumented pot method: Beirlant et al (2004) if(method == "pot") { a <- numeric(nx) rr <- rowSums(1/data) rrk <- sort(rr, decreasing = TRUE)[k+1] for(i in 1:nx) a[i] <- sum(mpmax(x[i]/(d1 * rr), (1 - x[i])/(d2 * rr))[rr > rrk]) a <- 2/k * a a0 <- 2/k * sum(1/(d2 * rr)[rr > rrk]) a1 <- 2/k * sum(1/(d1 * rr)[rr > rrk]) a <- a + 1 - (1-x) * a0 - x * a1 a <- pmin(1, pmax(a, x, 1-x)) } if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "qcbvnonpar"<- function(p = seq(0.75, 0.95, 0.05), data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL, mint = 1, method = c("cfg","pickands","tdo"), convex = FALSE, madj = 0, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, xlim = range(data[,1], na.rm = TRUE), ylim = range(data[,2], na.rm = TRUE), xlab = colnames(data)[1], ylab = colnames(data)[2], ...) { if(mode(p) != "numeric" || any(p <= 0) || any(p >= 1)) stop("`p' must be a vector of probabilities") method <- match.arg(method) nxv <- 100 x <- seq(0, 1, length = nxv) ax <- abvnonpar(x = x, data = data, epmar = epmar, nsloc1 = nsloc1, nsloc2 = nsloc2, method = method, convex = convex, madj = madj, kmar = kmar, plot = FALSE) np <- length(p) qct <- list() p <- p^mint if(add) { xlim <- par("usr")[1:2] ylim <- par("usr")[1:2] if(par("xlog")) xlim <- 10^xlim if(par("ylog")) ylim <- 10^ylim } for(i in 1:np) { qct[[i]] <- -cbind(x/ax * log(p[i]), (1-x)/ax * log(p[i])) if(epmar) { qct[[i]] <- cbind(quantile(data[,1], probs = exp(-qct[[i]][,1]), na.rm = TRUE), quantile(data[,2], probs = exp(-qct[[i]][,2]), na.rm = TRUE)) } else { if(is.null(kmar)) { # Transform from exponential margins mle.m1 <- fitted(fgev(data[,1], nsloc = nsloc1, std.err = FALSE)) mle.m2 <- fitted(fgev(data[,2], nsloc = nsloc2, std.err = FALSE)) mle.m1 <- mle.m1[c("loc","scale","shape")] mle.m2 <- mle.m2[c("loc","scale","shape")] qct[[i]] <- mtransform(qct[[i]], list(mle.m1, mle.m2), inv = TRUE) } else { if(!is.null(nsloc1) || !is.null(nsloc2)) warning("ignoring `nsloc1' and `nsloc2' arguments") qct[[i]] <- mtransform(qct[[i]], kmar, inv = TRUE) } } qct[[i]][1,1] <- 1.5 * xlim[2] qct[[i]][nxv,2] <- 1.5 * ylim[2] } if((!is.null(nsloc1) || !is.null(nsloc2)) && !epmar && is.null(kmar)) { data <- fbvevd(data, model = "log", dep = 1, nsloc1 = nsloc1, nsloc2 = nsloc2, std.err = FALSE)$tdata } if(plot) { plot(data, xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) return(invisible(qct)) } if(add) { for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) return(invisible(qct)) } qct } "amvnonpar"<- function(x = rep(1/d,d), data, d = 3, epmar = FALSE, nsloc = NULL, madj = 0, kmar = NULL, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) { if(!plot) { if(is.vector(x)) x <- as.matrix(t(x)) if(!is.matrix(x) || ncol(x) != d) stop("`x' must be a vector/matrix with `d' elements/columns") if(any(x < 0, na.rm = TRUE)) stop("`x' must be non-negative") rs <- rowSums(x) if(any(rs <= 0, na.rm = TRUE)) stop("row(s) of `x' must have a positive sum") if(max(abs(rs[!is.na(rs)] - 1)) > 1e-6) warning("row(s) of `x' will be rescaled") x <- x/rs } if(missing(data) || ncol(data) != d) stop("data must have `d' columns") if(plot) { if(d == 2) stop("use abvnonpar for bivariate plots") if(d >= 4) stop("cannot plot in high dimensions") } if(epmar) { # Empirical transform to exponential data <- apply(data, 2, rank, na.last = "keep") nasm <- apply(data, 2, function(x) sum(!is.na(x))) data <- data / rep(nasm+1, each = nrow(data)) data <- -log(data) # End empirical transform } if(!epmar) { # Parametric gev transform to exponential if(!is.null(kmar)) { data <- mtransform(data, kmar) } else { if(!is.null(nsloc)) { nslocmat <- list() if(!is.list(nsloc)) nsloc <- rep(list(nsloc), d) if(length(nsloc) != d) stop("`nsloc' should have `d' elements") for(k in 1:d) { if(!is.null(nsloc[[k]])) { if(is.vector(nsloc[[k]])) nsloc[[k]] <- data.frame(trend = nsloc[[k]]) if(nrow(nsloc[[k]]) != nrow(data)) stop("`nsloc' and data are not compatible") nslocmat[[k]] <- cbind(1, as.matrix(nsloc[[k]])) } } } mles <- list() for(k in 1:d) { mle.m <- fitted(fgev(data[,k], nsloc = nsloc[[k]], std.err = FALSE)) loc.mle.m <- mle.m[grep("^loc", names(mle.m))] if(is.null(nsloc[[k]])) loc.mle.m <- rep(loc.mle.m, nrow(data)) else loc.mle.m <- nslocmat[[k]] %*% loc.mle.m mles[[k]] <- cbind(loc.mle.m, mle.m["scale"], mle.m["shape"]) } data <- mtransform(data, mles) } # End parametric gev transform } data <- na.omit(data) depfn <- function(x, data, madj) { # quicker apply(am, 2, min) mpmin <- function(am,nr) { a <- am[1,] for(i in 2:nr) a[a > am[i,]] <- am[i, a > am[i,]] a } nn <- nrow(data) nx <- nrow(x) csum <- colSums(data) a <- numeric(nx) if(madj == 2) data <- nn * sweep(data, 2, csum, "/") for(i in 1:nx) { a[i] <- sum(mpmin(t(data)/x[i,], d)) } if(madj == 1) a <- a - colSums(t(x) * csum) + nn a <- nn / a xrmax <- apply(x, 1, max) pmin(1, pmax(a, xrmax)) } if(plot) { mz <- tvdepfn(depfn = depfn, col = col, blty = blty, grid = grid, lower = lower, ord = ord, lab = lab, lcex = lcex, data = data, madj = madj) return(invisible(mz)) } depfn(x = x, data = data, madj = madj) } evd/R/stocproc.R0000644000175100001440000002037713264361331013241 0ustar hornikusers "evmc" <- function(n, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), margins = c("uniform","rweibull","frechet","gumbel")) { model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, c("log", "hr", "neglog")) m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") nn <- as.integer(1) if(!(model %in% m1)) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if((model %in% c("log", "alog")) && dep > 1) stop("`dep' must be in the interval (0,1]") dep <- as.double(dep) } if(!(model %in% m2)) { if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(model == "alog" && (dep == 1 || any(asy == 0))) { asy <- c(0,0) dep <- 1 } asy <- as.double(asy[c(2,1)]) } if(!(model %in% m3)) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(model != "amix" && (alpha <= 0 || beta <= 0)) stop("`alpha' and `beta' must be positive") if(model == "bilog" && any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") if(model == "amix") { if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") alpha <- as.double(alpha + 3*beta) beta <- as.double(-beta) } else { alpha <- as.double(beta) beta <- as.double(alpha) } } evmc <- runif(n) for(i in 2:n) { evmc[c(i,i-1)] <- switch(model, log = .C(C_rbvlog, nn, dep, sim = evmc[c(i,i-1)])$sim, alog = .C(C_rbvalog, nn, dep, asy, sim = evmc[c(i,i-1)])$sim, hr = .C(C_rbvhr, nn, dep, sim = evmc[c(i,i-1)])$sim, neglog = .C(C_rbvneglog, nn, dep, sim = evmc[c(i,i-1)])$sim, aneglog = .C(C_rbvaneglog, nn, dep, asy, sim = evmc[c(i,i-1)])$sim, bilog = .C(C_rbvbilog, nn, alpha, beta, sim = evmc[c(i,i-1)])$sim, negbilog = .C(C_rbvnegbilog, nn, alpha, beta, sim = evmc[c(i,i-1)])$sim, ct = .C(C_rbvct, nn, alpha, beta, sim = evmc[c(i,i-1)])$sim, amix = .C(C_rbvamix, nn, alpha, beta, sim = evmc[c(i,i-1)])$sim) } switch(match.arg(margins), frechet = -1/log(evmc), uniform = evmc, rweibull = log(evmc), gumbel = -log(-log(evmc))) } "marma" <- function(n, p = 0, q = 0, psi, theta, init = rep(0, p), n.start = p, rand.gen = rfrechet, ...) { if(missing(psi)) psi <- numeric(0) if(missing(theta)) theta <- numeric(0) if(length(psi) != p || !is.numeric(psi) || any(psi < 0)) stop("`par' must be a non-negative vector of length `p'") if(length(theta) != q || !is.numeric(theta) || any(theta < 0)) stop("`theta' must be a non-negative vector of length `q'") if(length(init) != p || !is.numeric(init) || any(init < 0)) stop("`init' must be a non-negative vector of length `p'") marma <- c(init, numeric(n + n.start - p)) theta <- c(1, theta) innov <- rand.gen(n.start + n + q, ...) for(i in 1:(n + n.start - p)) marma[i+p] <- max(c(psi * marma[(i+p-1):i], theta * innov[(i+q):i])) if(n.start) marma <- marma[-(1:n.start)] marma } "mma" <- function(n, q = 1, theta, rand.gen = rfrechet, ...) { marma(n = n, q = q, theta = theta, rand.gen = rand.gen, ...) } "mar" <- function(n, p = 1, psi, init = rep(0, p), n.start = p, rand.gen = rfrechet, ...) { marma(n = n, p = p, psi = psi, init = init, n.start = n.start, rand.gen = rand.gen, ...) } "clusters"<- function(data, u, r = 1, ulow = -Inf, rlow = 1, cmax = FALSE, keep.names = TRUE, plot = FALSE, xdata = seq(along = data), lvals = TRUE, lty = 1, lwd = 1, pch = par("pch"), col = if(n > 250) NULL else "grey", xlab = "Index", ylab = "Data", ...) { n <- length(data) if(length(u) != 1) u <- rep(u, length.out = n) if(length(ulow) != 1) ulow <- rep(ulow, length.out = n) if(any(ulow > u)) stop("`u' cannot be less than `ulow'") if(is.null(names(data)) && keep.names) names(data) <- 1:n if(!keep.names) names(data) <- NULL high <- as.double((data > u) & !is.na(data)) high2 <- as.double((data > ulow) | is.na(data)) clstrs <- .C(C_clusters, high, high2, n, as.integer(r), as.integer(rlow), clstrs = double(3*n))$clstrs clstrs <- matrix(clstrs, n, 3) start <- clstrs[,2] ; end <- clstrs[,3] splvec <- clstrs[,1] start <- as.logical(start) end <- as.logical(end) clstrs <- split(data, splvec) names(clstrs) <- paste("cluster", names(clstrs), sep = "") if(any(!splvec)) clstrs <- clstrs[-1] nclust <- length(clstrs) acs <- sum(high)/nclust if(plot) { if(length(xdata) != length(data)) stop("`xdata' and `data' have different lengths") if(any(is.na(xdata))) stop("`xdata' cannot contain missing values") if(any(duplicated(xdata))) stop("`xdata' cannot contain duplicated values") eps <- min(diff(xdata))/2 start <- xdata[start] - eps end <- xdata[end] + eps plot(xdata, data, xlab = xlab, ylab = ylab, type = "n", ...) if(!is.null(col) && nclust > 0.5) { for(i in 1:nclust) { xvl <- c(start[i], end[i], end[i], start[i]) polygon(xvl, rep(par("usr")[3:4], each = 2), col = col) } } if(length(u) == 1) abline(h = u, lty = lty, lwd = lwd) else lines(xdata, u, lty = lty, lwd = lwd) if(lvals) { if(length(ulow) == 1) abline(h = ulow, lty = lty, lwd = lwd) else lines(xdata, ulow, lty = lty, lwd = lwd) } else { high <- as.logical(high) xdata <- xdata[high] data <- data[high] } points(xdata, data, pch = pch) } if(cmax) { if(keep.names) nmcl <- unlist(lapply(clstrs, function(x) names(x)[which.max(x)])) clstrs <- as.numeric(unlist(lapply(clstrs, max, na.rm = TRUE))) if(keep.names) names(clstrs) <- nmcl } attributes(clstrs)$acs <- acs if(plot) return(invisible(clstrs)) clstrs } "exi"<- function (data, u, r = 1, ulow = -Inf, rlow = 1) { n <- length(data) if (length(u) != 1) u <- rep(u, length.out = n) if (length(ulow) != 1) ulow <- rep(ulow, length.out = n) if (any(ulow > u)) stop("`u' cannot be less than `ulow'") if(r > 0.5) { clstrs <- clusters(data, u = u, r = r, ulow = ulow, rlow = rlow, keep.names = FALSE) exindex <- 1/attributes(clstrs)$acs } else { extms <- which(data > u) nn <- length(extms) if(nn == 0) return(NaN) if(nn == 1) return(1) iextms <- extms[-1] - extms[-nn] if(max(iextms) > 2.5) { den <- log(nn - 1) + log(sum((iextms - 1) * (iextms - 2))) exindex <- log(2) + 2*log(sum(iextms - 1)) - den exindex <- min(1, exp(exindex)) } else { den <- log(nn - 1) + log(sum(iextms^2)) exindex <- log(2) + 2*log(sum(iextms)) - den exindex <- min(1, exp(exindex)) } } exindex } exiplot <- function (data, tlim, r = 1, ulow = -Inf, rlow = 1, add = FALSE, nt = 100, lty = 1, xlab = "Threshold", ylab = "Ext. Index", ylim = c(0,1), ...) { nn <- length(data) if (all(data <= tlim[2])) stop("upper limit for threshold is too high") u <- seq(tlim[1], tlim[2], length = nt) x <- numeric(nt) for (i in 1:nt) { x[i] <- exi(data, u = u[i], r = r, ulow = ulow, rlow = rlow) } if(add) { lines(u, x, lty = lty, ...) } else { plot(u, x, type = "l", lty = lty, xlab = xlab, ylab = ylab, ylim = ylim, ...) } invisible(list(x = u, y = x)) } evd/R/mdiag.R0000644000175100001440000012171314225015120012450 0ustar hornikusers### Univariate GEV and POT Models ### "plot.uvevd" <- function(x, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, adjust = 1, jitter = FALSE, nplty = 2, ...) { if (!inherits(x, "uvevd")) stop("Use only with uvevd objects") if (!is.numeric(which) || any(which < 1) || any(which > 4)) stop("`which' must be in 1:4") nb.fig <- prod(par("mfcol")) show <- rep(FALSE, 4) show[which] <- TRUE if(missing(main)) { main <- c("Probability Plot", "Quantile Plot", "Density Plot", "Return Level Plot") } else { if(length(main) != length(which)) stop("number of plot titles is not correct") main2 <- character(4) main2[show] <- main main <- main2 } if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if (show[1]) { pp(x, ci = ci, cilwd = cilwd, a = a, main = main[1], xlim = c(0,1), ylim = c(0,1), ...) } if (show[2]) { qq(x, ci = ci, cilwd = cilwd, a = a, main = main[2], ...) } if (show[3]) { dens(x, adjust = adjust, nplty = nplty, jitter = jitter, main = main[3], ...) } if (show[4]) { rl(x, ci = ci, cilwd = cilwd, a = a, main = main[4], ...) } invisible(x) } "plot.gumbelx" <- function(x, interval, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, adjust = 1, jitter = FALSE, nplty = 2, ...) { if (!inherits(x, "gumbelx")) stop("Use only with gumbelx objects") if (!is.numeric(which) || any(which < 1) || any(which > 4)) stop("`which' must be in 1:4") nb.fig <- prod(par("mfcol")) show <- rep(FALSE, 4) show[which] <- TRUE if(missing(main)) { main <- c("Probability Plot", "Quantile Plot", "Density Plot", "Return Level Plot") } else { if(length(main) != length(which)) stop("number of plot titles is not correct") main2 <- character(4) main2[show] <- main main <- main2 } if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if (show[1]) { pp(x, ci = ci, cilwd = cilwd, a = a, main = main[1], xlim = c(0,1), ylim = c(0,1), ...) } if (show[2]) { qq(x, interval = interval, ci = ci, cilwd = cilwd, a = a, main = main[2], ...) } if (show[3]) { dens(x, adjust = adjust, nplty = nplty, jitter = jitter, main = main[3], ...) } if (show[4]) { rl(x, interval = interval, ci = ci, cilwd = cilwd, a = a, main = main[4], ...) } invisible(x) } "qq" <- function (x, ...) UseMethod("qq") "pp" <- function (x, ...) UseMethod("pp") "rl" <- function (x, ...) UseMethod("rl") "dens" <- function (x, ...) UseMethod("dens") "qq.gev" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Quantile Plot", xlab = "Model", ylab = "Empirical", ...) { quant <- qgev(ppoints(x$tdata, a = a), loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) if(!ci) { plot(quant, sort(x$tdata), main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgev(x$n*99, loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$tdata) matplot(quant, cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], quant, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(quant-smidge, env[,1], quant+smidge, env[,1], lwd = cilwd) segments(quant-smidge, env[,2], quant+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = quant, y = sort(x$tdata))) } "pp.gev" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Probability Plot", xlab = "Empirical", ylab = "Model", ...) { ppx <- ppoints(x$n, a = a) probs <- pgev(sort(x$tdata), loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) if(!ci) { plot(ppx, probs, main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgev(x$n*99, loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) env[,1] <- pgev(env[,1], loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) env[,2] <- pgev(env[,2], loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) matplot(ppx, cbind(probs, env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], ppx, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(ppx-smidge, env[,1], ppx+smidge, env[,1], lwd = cilwd) segments(ppx-smidge, env[,2], ppx+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = ppx, y = probs)) } "rl.gev" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Return Level Plot", xlab = "Return Period", ylab = "Return Level", ...) { ppx <- ppoints(x$tdata, a = a) rps <- c(1.001,10^(seq(0,3,len=200))[-1]) p.upper <- 1/rps rlev <- qgev(p.upper, loc = x$loc, scale = x$param["scale"], shape = x$param["shape"], lower.tail = FALSE) if(!ci) { plot(-1/log(ppx), sort(x$tdata),log = "x", main = main, xlab = xlab, ylab = ylab, ...) lines(-1/log(1-p.upper), rlev) } else { samp <- rgev(x$n*99, loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$tdata) matplot(-1/log(ppx), cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, log = "x", ...) lines(-1/log(1-p.upper), rlev) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], log10(-1/log(ppx)), xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments((-1/log(ppx))*exp(-smidge), env[,1], (-1/log(ppx))*exp(smidge), env[,1], lwd = cilwd) segments((-1/log(ppx))*exp(-smidge), env[,2], (-1/log(ppx))*exp(smidge), env[,2], lwd = cilwd) } invisible(list(x = -1/log(1-p.upper), y = rlev)) } "dens.gev" <- function(x, adjust = 1, nplty = 2, jitter = FALSE, main = "Density Plot", xlab = "Quantile", ylab = "Density", ...) { xlimit <- range(x$tdata) xlimit[1] <- xlimit[1] - diff(xlimit) * 0.075 xlimit[2] <- xlimit[2] + diff(xlimit) * 0.075 xvec <- seq(xlimit[1], xlimit[2], length = 100) dens <- dgev(xvec, loc = x$loc, scale = x$param["scale"], shape = x$param["shape"]) plot(spline(xvec, dens), main = main, xlab = xlab, ylab = ylab, type = "l", ...) if(jitter) rug(jitter(x$tdata)) else rug(x$tdata) lines(density(x$tdata, adjust = adjust), lty = nplty) invisible(list(x = xvec, y = dens)) } "qq.gumbelx" <- function(x, interval, ci = TRUE, cilwd = 1, a = 0, main = "Quantile Plot", xlab = "Model", ylab = "Empirical", ...) { quant <- qgumbelx(ppoints(x$data, a = a), interval = interval, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) if(!ci) { plot(quant, sort(x$data), main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgumbelx(x$n*99, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$data) matplot(quant, cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], quant, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(quant-smidge, env[,1], quant+smidge, env[,1], lwd = cilwd) segments(quant-smidge, env[,2], quant+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = quant, y = sort(x$data))) } "pp.gumbelx" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Probability Plot", xlab = "Empirical", ylab = "Model", ...) { ppx <- ppoints(x$n, a = a) probs <- pgumbelx(sort(x$data), loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) if(!ci) { plot(ppx, probs, main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgumbelx(x$n*99, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) env[,1] <- pgumbelx(env[,1], loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) env[,2] <- pgumbelx(env[,2], loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) matplot(ppx, cbind(probs, env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], ppx, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(ppx-smidge, env[,1], ppx+smidge, env[,1], lwd = cilwd) segments(ppx-smidge, env[,2], ppx+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = ppx, y = probs)) } "rl.gumbelx" <- function(x, interval, ci = TRUE, cilwd = 1, a = 0, main = "Return Level Plot", xlab = "Return Period", ylab = "Return Level", ...) { ppx <- ppoints(x$data, a = a) rps <- c(1.001,10^(seq(0,3,len=200))[-1]) p.upper <- 1/rps rlev <- qgumbelx(p.upper, interval = interval, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"], lower.tail = FALSE) if(!ci) { plot(-1/log(ppx), sort(x$data),log = "x", main = main, xlab = xlab, ylab = ylab, ...) lines(-1/log(1-p.upper), rlev) } else { samp <- rgumbelx(x$n*99, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) samp <- matrix(samp, x$n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$data) matplot(-1/log(ppx), cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, log = "x", ...) lines(-1/log(1-p.upper), rlev) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], log10(-1/log(ppx)), xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments((-1/log(ppx))*exp(-smidge), env[,1], (-1/log(ppx))*exp(smidge), env[,1], lwd = cilwd) segments((-1/log(ppx))*exp(-smidge), env[,2], (-1/log(ppx))*exp(smidge), env[,2], lwd = cilwd) } invisible(list(x = -1/log(1-p.upper), y = rlev)) } "dens.gumbelx" <- function(x, adjust = 1, nplty = 2, jitter = FALSE, main = "Density Plot", xlab = "Quantile", ylab = "Density", ...) { xlimit <- range(x$data) xlimit[1] <- xlimit[1] - diff(xlimit) * 0.075 xlimit[2] <- xlimit[2] + diff(xlimit) * 0.075 xvec <- seq(xlimit[1], xlimit[2], length = 100) dens <- dgumbelx(xvec, loc1 = x$param["loc1"], scale1 = x$param["scale1"], loc2 = x$param["loc2"], scale2 = x$param["scale2"]) plot(spline(xvec, dens), main = main, xlab = xlab, ylab = ylab, type = "l", ...) if(jitter) rug(jitter(x$data)) else rug(x$data) lines(density(x$data, adjust = adjust), lty = nplty) invisible(list(x = xvec, y = dens)) } "qq.pot" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Quantile Plot", xlab = "Model", ylab = "Empirical", ...) { quant <- qgpd(ppoints(x$nhigh, a = a), loc = x$threshold, scale = x$scale, shape = x$param["shape"]) if(!ci) { plot(quant, sort(x$exceedances), main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgpd(x$nhigh*99, loc = x$threshold, scale = x$scale, shape = x$param["shape"]) samp <- matrix(samp, x$nhigh, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$exceedances) matplot(quant, cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], quant, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(quant-smidge, env[,1], quant+smidge, env[,1], lwd = cilwd) segments(quant-smidge, env[,2], quant+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = quant, y = sort(x$exceedances))) } "pp.pot" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Probability Plot", xlab = "Empirical", ylab = "Model", ...) { ppx <- ppoints(x$nhigh, a = a) probs <- pgpd(sort(x$exceedances), loc = x$threshold, scale = x$scale, shape = x$param["shape"]) if(!ci) { plot(ppx, probs, main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- rgpd(x$nhigh*99, loc = x$threshold, scale = x$scale, shape = x$param["shape"]) samp <- matrix(samp, x$nhigh, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) env[,1] <- pgpd(env[,1], loc = x$threshold, scale = x$scale, shape = x$param["shape"]) env[,2] <- pgpd(env[,2], loc = x$threshold, scale = x$scale, shape = x$param["shape"]) matplot(ppx, cbind(probs, env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], ppx, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(ppx-smidge, env[,1], ppx+smidge, env[,1], lwd = cilwd) segments(ppx-smidge, env[,2], ppx+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = ppx, y = probs)) } "rl.pot" <- function(x, ci = TRUE, cilwd = 1, a = 0, main = "Return Level Plot", xlab = "Return Period", ylab = "Return Level", ...) { rpstmfc <- c(1.001,10^(seq(0,3,len=200))[-1]) rlev <- qgpd(1/rpstmfc, loc = x$threshold, scale = x$scale, shape = x$param["shape"], lower.tail = FALSE) mfc <- x$npp * x$nhigh/length(x$data) rps <- rpstmfc/mfc ppx <- 1/(mfc * (1 - ppoints(x$nhigh, a = a))) if(!ci) { plot(ppx, sort(x$exceedances), log = "x", main = main, xlab = xlab, ylab = ylab, ...) lines(rps, rlev) } else { samp <- rgpd(x$nhigh*99, loc = x$threshold, scale = x$scale, shape = x$param["shape"]) samp <- matrix(samp, x$nhigh, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) rs <- sort(x$exceedances) matplot(ppx, cbind(rs,env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, log = "x", ...) lines(rps, rlev) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], log10(ppx), xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(ppx*exp(-smidge), env[,1], ppx*exp(smidge), env[,1], lwd = cilwd) segments(ppx*exp(-smidge), env[,2], ppx*exp(smidge), env[,2], lwd = cilwd) } invisible(list(x = rps, y = rlev)) } "dens.pot" <- function(x, adjust = 1, nplty = 2, jitter = FALSE, main = "Density Plot", xlab = "Quantile", ylab = "Density", ...) { xlimit <- range(x$exceedances) xlimit[2] <- xlimit[2] + diff(xlimit) * 0.075 xvec <- seq(xlimit[1], xlimit[2], length = 100) dens <- dgpd(xvec, loc = x$threshold, scale = x$scale, shape = x$param["shape"]) plot(spline(xvec, dens), main = main, xlab = xlab, ylab = ylab, type = "l", ...) if(jitter) rug(jitter(x$exceedances)) else rug(x$exceedances) flipexceed <- c(x$exceedances, 2*x$threshold - x$exceedances) flipdens <- density(flipexceed, adjust = adjust, from = xlimit[1], to = xlimit[2]) flipdens$y <- 2*flipdens$y lines(flipdens, lty = nplty) invisible(list(x = xvec, y = dens)) } ### Bivariate EVD Models ### "plot.bvevd" <- function(x, mar = 0, which = 1:6, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, grid = 50, legend = TRUE, nplty = 2, blty = 3, method = "cfg", convex = FALSE, rev = FALSE, p = seq(0.75, 0.95, 0.05), mint = 1, half = FALSE, ...) { if (!inherits(x, "bvevd")) stop("Use only with `bvevd' objects") nb.fig <- prod(par("mfcol")) if(mar == 1 || mar == 2) { indx <- paste(c("loc","scale","shape"), as.character(mar), sep="") tdata <- na.omit(x$tdata[, mar]) n <- length(tdata) param <- x$param[indx] names(param) <- c("loc","scale","shape") gev.mar <- structure(list(param = param, tdata = tdata, n = n, loc = param["loc"]), class = c("gev", "uvevd", "evd")) if(missing(which)) which <- 1:4 plot(gev.mar, which = which, main = main, ask = ask, ci = ci, cilwd = cilwd, ...) return(invisible(x)) } if (!is.numeric(which) || any(which < 1) || any(which > 6)) stop("`which' must be in 1:6") show <- rep(FALSE, 6) show[which] <- TRUE if(missing(main)) { main <- c("Conditional Plot One", "Conditional Plot Two", "Density Plot", "Dependence Function", "Quantile Curves", "Spectral Density") } else { if(length(main) != length(which)) stop("number of plot titles is not correct") main2 <- character(6) main2[show] <- main main <- main2 } if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if (show[1]) { bvcpp(x, mar = 1, ci = ci, cilwd = cilwd, a = a, main = main[1], xlim = c(0,1), ylim = c(0,1), ...) } if (show[2]) { bvcpp(x, mar = 2, ci = ci, cilwd = cilwd, a = a, main = main[2], xlim = c(0,1), ylim = c(0,1), ...) } if (show[3]) { bvdens(x, grid = grid, legend = legend, main = main[3], ...) } if (show[4]) { bvdp(x, nplty = nplty, blty = blty, method = method, convex = convex, rev = rev, main = main[4], ...) } if (show[5]) { bvqc(x, p = p, mint = mint, legend = legend, main = main[5], ...) } if (show[6]) { bvh(x, half = half, main = main[6], ...) } invisible(x) } "bvcpp" <- function (x, ...) UseMethod("bvcpp") "bvdens" <- function (x, ...) UseMethod("bvdens") "bvdp" <- function (x, ...) UseMethod("bvdp") "bvqc" <- function (x, ...) UseMethod("bvqc") "bvh" <- function (x, ...) UseMethod("bvh") "bvcpp.bvevd" <- function(x, mar = 2, ci = TRUE, cilwd = 1, a = 0, main = "Conditional Probability Plot", xlab = "Empirical", ylab = "Model", ...) { data <- x$tdata mle.m1 <- x$param[c("loc1","scale1","shape1")] mle.m2 <- x$param[c("loc2","scale2","shape2")] data[,1:2] <- exp(-mtransform(data[,1:2], list(mle.m1, mle.m2))) narow <- is.na(data[,1]) | is.na(data[,2]) data <- data[!narow,, drop=FALSE] n <- nrow(data) ppx <- ppoints(n, a = a) if(x$model %in% c("log","hr","neglog")) { probs <- ccbvevd(data, mar = mar, dep = x$param["dep"], model = x$model)} if(x$model %in% c("alog","aneglog")) probs <- ccbvevd(data, mar = mar, dep = x$param["dep"], asy = x$param[c("asy1","asy2")], model = x$model) if(x$model %in% c("bilog","negbilog","ct","amix")) probs <- ccbvevd(data, mar = mar, alpha = x$param["alpha"], beta = x$param["beta"], model = x$model) probs <- sort(probs) if(!ci) { plot(ppx, probs, main = main, xlab = xlab, ylab = ylab, ...) abline(0, 1) } else { samp <- runif(n*99) samp <- matrix(samp, n, 99) samp <- apply(samp, 2, sort) samp <- apply(samp, 1, sort) env <- t(samp[c(3,97),]) matplot(ppx, cbind(probs, env), main = main, xlab = xlab, ylab = ylab, type = "pnn", pch = 4, ...) xyuser <- par("usr") smidge <- min(diff(c(xyuser[1], ppx, xyuser[2])))/2 smidge <- max(smidge, (xyuser[2] - xyuser[1])/200) segments(ppx-smidge, env[,1], ppx+smidge, env[,1], lwd = cilwd) segments(ppx-smidge, env[,2], ppx+smidge, env[,2], lwd = cilwd) abline(0, 1) } invisible(list(x = ppx, y = probs)) } "bvdens.bvevd" <- function(x, grid = 50, legend = TRUE, pch = 1, main = "Density Plot", xlab = colnames(x$data)[1], ylab = colnames(x$data)[2], ...) { xlimit <- range(x$tdata[,1], na.rm = TRUE) ylimit <- range(x$tdata[,2], na.rm = TRUE) xlimit[1] <- xlimit[1] - diff(xlimit) * 0.1 xlimit[2] <- xlimit[2] + diff(xlimit) * 0.1 ylimit[1] <- ylimit[1] - diff(ylimit) * 0.1 ylimit[2] <- ylimit[2] + diff(ylimit) * 0.1 xvec <- seq(xlimit[1], xlimit[2], length = grid) yvec <- seq(ylimit[1], ylimit[2], length = grid) xyvals <- expand.grid(xvec, yvec) mar1 <- x$param[c("loc1","scale1","shape1")] mar2 <- x$param[c("loc2","scale2","shape2")] if(x$model %in% c("log","hr","neglog")) dfunargs <- list(dep = x$param["dep"], mar1 = mar1, mar2 = mar2) if(x$model %in% c("alog","aneglog")) dfunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")], mar1 = mar1, mar2 = mar2) if(x$model %in% c("bilog","negbilog","ct","amix")) dfunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"], mar1 = mar1, mar2 = mar2) dfunargs <- c(list(x = xyvals, model = x$model), dfunargs) dens <- do.call("dbvevd", dfunargs) dens <- matrix(dens, nrow = grid, ncol = grid) contour(xvec, yvec, dens, main = main, xlab = xlab, ylab = ylab, ...) data <- x$tdata if(ncol(data) == 2) points(data, pch = pch) if(ncol(data) == 3) { si <- data[,3] ; data <- data[,1:2] points(data[is.na(si),], pch = 4) points(data[si & !is.na(si),], pch = 16) points(data[!si & !is.na(si),], pch = 1) legwrd <- c("True","False","Unknown") ; legpch <- c(16,1,4) if(!any(is.na(si))) {legwrd <- legwrd[1:2] ; legpch <- legpch[1:2]} if(legend) legend(xlimit[1], ylimit[2], legwrd, pch = legpch) } invisible(list(x = xyvals, y = dens)) } "bvdp.bvevd" <- function(x, method = "cfg", convex = FALSE, rev = FALSE, add = FALSE, lty = 1, nplty = 2, blty = 3, main = "Dependence Function", xlab = "t", ylab = "A(t)", ...) { if(ncol(x$data) == 3) nplty <- 0 abvnonpar(data = x$data[,1:2], nsloc1 = x$nsloc1, nsloc2 = x$nsloc2, epmar = FALSE, method = method, convex = convex, rev = rev, plot = TRUE, lty = nplty, blty = blty, main = main, xlab = xlab, ylab = ylab, add = add, ...) if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(rev = rev, add = TRUE, lty = lty, model = x$model), afunargs) do.call("abvevd", afunargs) invisible(x) } "bvh.bvevd" <- function(x, half = FALSE, add = FALSE, lty = 1, main = "Spectral Density", xlab = "t", ylab = "h(t)", ...) { if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(half = half, add = add, plot = TRUE, lty = lty, main = main, xlab = xlab, ylab = ylab, model = x$model), afunargs) do.call("hbvevd", afunargs) invisible(x) } "bvqc.bvevd"<- function(x, p = seq(0.75, 0.95, 0.05), mint = 1, add = FALSE, legend = TRUE, lty = 1, lwd = 1, col = 1, xlim = range(x$tdata[,1], na.rm = TRUE), ylim = range(x$tdata[,2], na.rm = TRUE), xlab = colnames(x$data)[1], ylab = colnames(x$data)[2], ...) { if(mode(p) != "numeric" || any(p <= 0) || any(p >= 1)) stop("`p' must be a vector of probabilities") nom <- 100 om <- seq(0, 1, length = nom) # Calculate A(t) if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(x = om, plot = FALSE, model = x$model), afunargs) aom <- do.call("abvevd", afunargs) # End Calculate A(t) np <- length(p) qct <- list() p <- p^mint if(add) { xlim <- par("usr")[1:2] ylim <- par("usr")[1:2] if(par("xlog")) xlim <- 10^xlim if(par("ylog")) ylim <- 10^ylim } for(i in 1:np) { qct[[i]] <- -cbind(om/aom * log(p[i]), (1-om)/aom * log(p[i])) mar1 <- x$param[c("loc1","scale1","shape1")] mar2 <- x$param[c("loc2","scale2","shape2")] qct[[i]] <- mtransform(qct[[i]], list(mar1, mar2), inv = TRUE) qct[[i]][1,1] <- 1.5 * xlim[2] qct[[i]][nom,2] <- 1.5 * ylim[2] } if(!add) { if(is.null(xlab)) xlab <- "" if(is.null(ylab)) ylab <- "" if(ncol(x$tdata) == 2) { plot(x$tdata[,1:2], xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) } if(ncol(x$tdata) == 3) { plot(x$tdata[,1:2], xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, type = "n", ...) si <- x$tdata[,3] ; data <- x$tdata[,1:2] points(data[is.na(si),], pch = 4) points(data[si & !is.na(si),], pch = 16) points(data[!si & !is.na(si),], pch = 1) legwrd <- c("True","False","Unknown") ; legpch <- c(16,1,4) if(!any(is.na(si))) {legwrd <- legwrd[1:2] ; legpch <- legpch[1:2]} if(legend) legend(xlim[1], ylim[2], legwrd, pch = legpch) } for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) } else { for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) } return(invisible(qct)) } ### Bivariate POT Models ### "plot.bvpot" <- function(x, mar = 0, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), grid = 50, above = FALSE, levels = NULL, tlty = 1, blty = 3, rev = FALSE, p = seq(0.75, 0.95, 0.05), half = FALSE, ...) { if (!inherits(x, "bvpot")) stop("Use only with `bvpot' objects") nb.fig <- prod(par("mfcol")) if(mar == 1 || mar == 2) { indx <- paste(c("scale","shape"), as.character(mar), sep="") param <- x$param[indx] names(param) <- c("scale","shape") mdata <- x$data[, mar] mth <- x$threshold[mar] mexceed <- mdata[mdata > mth & !is.na(mdata)] pot.mar <- structure(list(param = param, data = mdata, threshold = mth, exceedances = mexceed, nhigh = length(mexceed), npp = length(mdata), scale = param["scale"]), class = c("pot", "uvevd", "evd")) if(missing(which)) which <- 1:4 plot(pot.mar, which = which, main = main, ask = ask, ...) return(invisible(x)) } if (!is.numeric(which) || any(which < 1) || any(which > 4)) stop("`which' must be in 1:4") show <- rep(FALSE, 4) show[which] <- TRUE if(missing(main)) { main <- c("Density Plot", "Dependence Function", "Quantile Curves", "Spectral Density") } else { if(length(main) != length(which)) stop("number of plot titles is not correct") main2 <- character(4) main2[show] <- main main <- main2 } if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if (show[1]) { bvdens(x, grid = grid, above = above, levels = levels, tlty = tlty, main = main[1], ...) } if (show[2]) { bvdp(x, blty = blty, rev = rev, main = main[2], ...) } if (show[3]) { bvqc(x, p = p, above = above, tlty = tlty, main = main[3], ...) } if (show[4]) { bvh(x, half = half, main = main[4], ...) } invisible(x) } "bvdens.bvpot" <- function(x, grid = 50, above = FALSE, tlty = 1, levels = NULL, main = "Density Plot", pch = 1, xlab = colnames(x$data)[1], ylab = colnames(x$data)[2], xlim, ylim, ...) { xlimit <- range(x$data[,1], na.rm = TRUE) ylimit <- range(x$data[,2], na.rm = TRUE) xlimit[1] <- xlimit[1] - diff(xlimit) * 0.1 xlimit[2] <- xlimit[2] + diff(xlimit) * 0.1 ylimit[1] <- ylimit[1] - diff(ylimit) * 0.1 ylimit[2] <- ylimit[2] + diff(ylimit) * 0.1 if(missing(xlim)) xlim <- xlimit if(missing(ylim)) ylim <- ylimit u1 <- x$threshold[1] u2 <- x$threshold[2] if((xlimit[2] <= u1) || (ylimit[2] <= u2)) stop("x and y limits must contain thresholds") xvec <- seq(u1, xlimit[2], length = grid) yvec <- seq(u2, ylimit[2], length = grid) xyvals <- txyvals <- fxyvals <- expand.grid(xvec, yvec) mar1 <- x$param[c("scale1","shape1")] mar2 <- x$param[c("scale2","shape2")] # Transform exceedance grid to frechet margins txyvals[,1] <- mtransform(xyvals[,1], c(u1, mar1)) txyvals[,2] <- mtransform(xyvals[,2], c(u2, mar2)) lambda <- x$nat[1:2]/(nrow(x$data) + 1) fxyvals[,1] <- -1/log(1 - lambda[1] * txyvals[,1]) fxyvals[,2] <- -1/log(1 - lambda[2] * txyvals[,2]) # End transform if(x$model %in% c("log","hr","neglog")) dfunargs <- list(dep = x$param["dep"], mar1 = c(1,1,1), mar2 = c(1,1,1)) if(x$model %in% c("alog","aneglog")) dfunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")], mar1 = c(1,1,1), mar2 = c(1,1,1)) if(x$model %in% c("bilog","negbilog","ct","amix")) dfunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"], mar1 = c(1,1,1), mar2 = c(1,1,1)) dfunargs <- c(list(x = fxyvals, model = x$model), dfunargs) # Jacobian terms txyvals[,1] <- fxyvals[,1]^2 * txyvals[,1]^(1 + mar1[2]) / (1 - lambda[1] * txyvals[,1]) txyvals[,1] <- lambda[1] * txyvals[,1] / mar1[1] txyvals[,2] <- fxyvals[,2]^2 * txyvals[,2]^(1 + mar2[2]) / (1 - lambda[2] * txyvals[,2]) txyvals[,2] <- lambda[2] * txyvals[,2] / mar2[1] # End jacobian terms dens <- do.call("dbvevd", dfunargs) dens <- dens * txyvals[,1] * txyvals[,2] dens <- matrix(dens, nrow = grid, ncol = grid) if(is.null(levels)) { levels <- seq(10, 40, length = 4) levels <- dens[cbind(levels, levels)] levels <- signif(levels, 1) } contour(xvec, yvec, dens, main = main, xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, levels = levels, ...) abline(v = u1, lty = tlty); abline(h = u2, lty = tlty) data <- x$data if(above) { above <- (data[,1] > u1) & (data[,2] > u2) data <- data[above,] } points(data, pch = pch) invisible(list(x = xyvals, y = dens)) } "bvdp.bvpot" <- function(x, rev = FALSE, add = FALSE, lty = 1, blty = 3, main = "Dependence Function", xlab = "t", ylab = "A(t)", ...) { if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(rev = rev, add = add, plot = TRUE, lty = lty, blty = blty, main = main, xlab = xlab, ylab = ylab, model = x$model), afunargs) do.call("abvevd", afunargs) invisible(x) } "bvqc.bvpot"<- function(x, p = seq(0.75, 0.95, 0.05), above = FALSE, tlty = 1, add = FALSE, lty = 1, lwd = 1, col = 1, xlim = range(x$data[,1], na.rm = TRUE), ylim = range(x$data[,2], na.rm = TRUE), xlab = colnames(x$data)[1], ylab = colnames(x$data)[2], ...) { if(mode(p) != "numeric" || any(p <= 0) || any(p >= 1)) stop("`p' must be a vector of probabilities") nom <- 100 om <- seq(0, 1, length = nom) # Calculate A(t) if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(x = om, plot = FALSE, model = x$model), afunargs) aom <- do.call("abvevd", afunargs) # End Calculate A(t) np <- length(p) qct <- list() if(add) { xlim <- par("usr")[1:2] ylim <- par("usr")[1:2] if(par("xlog")) xlim <- 10^xlim if(par("ylog")) ylim <- 10^ylim } u1 <- x$threshold[1] u2 <- x$threshold[2] lambda <- x$nat[1:2]/(nrow(x$data) + 1) for(i in 1:np) { qct[[i]] <- cbind((1 - p[i]^(om/aom))/lambda[1], (1 - p[i]^((1-om)/aom))/lambda[2]) mar1 <- c(u1, x$param[c("scale1","shape1")]) mar2 <- c(u2, x$param[c("scale2","shape2")]) qct[[i]] <- mtransform(qct[[i]], list(mar1, mar2), inv = TRUE) qct[[i]][1,1] <- 1.5 * xlim[2] qct[[i]][nom,2] <- 1.5 * ylim[2] } data <- x$data if(above) { above <- (data[,1] > u1) & (data[,2] > u2) data <- data[above,] } if(!add) { if(is.null(xlab)) xlab <- "" if(is.null(ylab)) ylab <- "" plot(data, xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) abline(v = u1, lty = tlty); abline(h = u2, lty = tlty) for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) } else { for(i in 1:np) lines(qct[[i]], lty = lty, lwd = lwd, col = col) } return(invisible(qct)) } "bvh.bvpot" <- function(x, half = FALSE, add = FALSE, lty = 1, main = "Spectral Density", xlab = "t", ylab = "h(t)", ...) { if(x$model %in% c("log","hr","neglog")) afunargs <- list(dep = x$param["dep"]) if(x$model %in% c("alog","aneglog")) afunargs <- list(dep = x$param["dep"], asy = x$param[c("asy1","asy2")]) if(x$model %in% c("bilog","negbilog","ct","amix")) afunargs <- list(alpha = x$param["alpha"], beta = x$param["beta"]) afunargs <- c(list(half = half, add = add, plot = TRUE, lty = lty, main = main, xlab = xlab, ylab = ylab, model = x$model), afunargs) do.call("hbvevd", afunargs) invisible(x) } ### Documented Ancillary Functions ### "mtransform"<- function(x, p, inv = FALSE, drp = FALSE) { if(is.list(p)) { if(is.null(dim(x)) && length(x) != length(p)) stop(paste("`p' must have", length(x), "elements")) if(!is.null(dim(x)) && ncol(x) != length(p)) stop(paste("`p' must have", ncol(x), "elements")) if(is.null(dim(x))) dim(x) <- c(1, length(p)) for(i in 1:length(p)) x[,i] <- Recall(x[,i], p[[i]], inv = inv) if(ncol(x) == 1 || (nrow(x) == 1 && drp)) x <- drop(x) return(x) } if(is.null(dim(x))) dim(x) <- c(length(x), 1) p <- matrix(t(p), nrow = nrow(x), ncol = 3, byrow = TRUE) if(min(p[,2]) <= 0) stop("invalid marginal scale") gumind <- (p[,3] == 0) nzshapes <- p[!gumind,3] if(!inv) { x <- (x - p[,1])/p[,2] x[gumind, ] <- exp(-x[gumind, ]) if(any(!gumind)) x[!gumind, ] <- pmax(1 + nzshapes*x[!gumind, ], 0)^(-1/nzshapes) } else { x[gumind, ] <- p[gumind,1] - p[gumind,2] * log(x[gumind, ]) x[!gumind, ] <- p[!gumind,1] + p[!gumind,2] * (x[!gumind, ]^(-nzshapes) - 1)/nzshapes } if(ncol(x) == 1 || (nrow(x) == 1 && drp)) x <- drop(x) x } "ccbvevd" <- function(x, mar = 2, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), lower.tail = TRUE) { if(min(x[,1:2]) <= 0 || max(x[,1:2]) >= 1) stop("x must contain values in (0,1)") model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") if(model %in% m1) dep <- 1 if(model %in% m3) alpha <- beta <- 1 imodel <- match(model, c("log","alog","hr","neglog","aneglog", "bilog","negbilog","ct","amix")) n <- nrow(x) if(ncol(x) == 2) { ccop <- .C(C_ccop, as.double(x[,1]), as.double(x[,2]), as.integer(mar), as.double(dep), as.double(asy[1]), as.double(asy[2]), as.double(alpha), as.double(beta), as.integer(n), as.integer(imodel), ccop = double(n))$ccop } if(ncol(x) == 3) { "dbvevd.case" <- function(x1, x2, case, mar, dep, asy, alpha, beta) { n <- max(length(x1), length(x2)) x1 <- rep(-1/log(x1), length = n) x2 <- rep(-1/log(x2), length = n) case <- rep(case, length = n) mpar <- as.double(1) split <- as.integer(1) if(mar == 1) { tmp <- x1; x1 <- x2; x2 <- tmp if(model %in% c("alog","aneglog")) asy <- rev(asy) if(model %in% c("bilog","negbilog","ct")) { tmp <- alpha; alpha <- beta; beta <- tmp } if(model == "amix") { alpha <- alpha + 3*beta; beta <- -beta } } nl <- switch(model, log = .C(C_nlbvlog, as.double(x1), as.double(x2), n, case, as.double(dep), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, alog = .C(C_nlbvalog, as.double(x1), as.double(x2), n, case, as.double(dep), as.double(asy[1]), as.double(asy[2]), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, hr = .C(C_nlbvhr, as.double(x1), as.double(x2), n, case, as.double(dep), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, neglog = .C(C_nlbvneglog, as.double(x1), as.double(x2), n, case, as.double(dep), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, aneglog = .C(C_nlbvaneglog, as.double(x1), as.double(x2), n, case, as.double(dep), as.double(asy[1]), as.double(asy[2]), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, bilog = .C(C_nlbvbilog, as.double(x1), as.double(x2), n, case, as.double(alpha), as.double(beta), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, negbilog = .C(C_nlbvnegbilog, as.double(x1), as.double(x2), n, case, as.double(alpha), as.double(beta), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, ct = .C(C_nlbvct, as.double(x1), as.double(x2), n, case, as.double(alpha), as.double(beta), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns, amix = .C(C_nlbvamix, as.double(x1), as.double(x2), n, case, as.double(alpha), as.double(beta), rep(mpar,n), mpar, mpar, rep(mpar,n), mpar, mpar, split, dns = double(n))$dns) jac.alt <- 1/x1 + 1/x2 + 2*log(x1 * x2) exp(jac.alt - nl) } ccop <- numeric(n) case <- as.integer(x[,3]) eps <- .Machine$double.eps^0.5 if(mar == 2) { fm <- x[,1] ; cm <- x[,2] } if(mar == 1) { fm <- x[,2] ; cm <- x[,1] } for(i in 1:n) { if(is.na(case[i])) { ccop[i] <- .C(C_ccop, as.double(x[i,1]), as.double(x[i,2]), as.integer(mar), as.double(dep), as.double(asy[1]), as.double(asy[2]), as.double(alpha), as.double(beta), as.integer(1), as.integer(imodel), ccop = double(1))$ccop } else { den <- integrate("dbvevd.case", eps, 1-eps, x2 = cm[i], case = case[i], mar=mar, dep=dep, asy=asy, alpha=alpha, beta=beta)$value num <- integrate("dbvevd.case", eps, fm[i], x2 = cm[i], case = case[i], mar=mar, dep=dep, asy=asy, alpha=alpha, beta=beta)$value ccop[i] <- num/den } } } if(!lower.tail) ccop <- 1 - ccop ccop } evd/R/mvdist.R0000644000175100001440000003370714611644157012723 0ustar hornikusers "rmvevd" <- function(n, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0)) { model <- match.arg(model) if(model == "log" && !missing(asy)) warning("ignoring `asy' argument") switch(model, log = rmvlog(n = n, dep = dep, d = d, mar = mar), alog = rmvalog(n = n, dep = dep, asy = asy, d = d, mar = mar)) } "rmvlog"<- # Uses Algorithm 2.1 in Stephenson(2003) function(n, dep, d = 2, mar = c(0,1,0)) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") sim <- .C(C_rmvlog_tawn, as.integer(n), as.integer(d), as.double(dep), sim = double(d*n))$sim mtransform(matrix(1/sim, ncol=d, byrow=TRUE), mar, inv = TRUE, drp = TRUE) } "rmvalog"<- # Uses Algorithm 2.2 in Stephenson(2003) function(n, dep, asy, d = 2, mar = c(0,1,0)) { nb <- 2^d-1 dep <- rep(dep, length.out = nb-d) asy <- mvalog.check(asy, dep, d = d) dep <- c(rep(1,d), dep) sim <- .C(C_rmvalog_tawn, as.integer(n), as.integer(d), as.integer(nb), as.double(dep), as.double(t(asy)), sim = double(n*d))$sim mtransform(matrix(1/sim, ncol=d, byrow=TRUE), mar, inv = TRUE, drp = TRUE) } "pmvevd" <- function(q, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0), lower.tail = TRUE) { model <- match.arg(model) if(model == "log" && !missing(asy)) warning("ignoring `asy' argument") switch(model, log = pmvlog(q = q, dep = dep, d = d, mar = mar, lower.tail = lower.tail), alog = pmvalog(q = q, dep = dep, asy = asy, d = d, mar = mar, lower.tail = lower.tail)) } "pmvlog"<- function (q, dep, d = 2, mar = c(0, 1, 0), lower.tail = TRUE) { if (length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if (is.null(dim(q))) dim(q) <- c(1, d) if (ncol(q) != d) stop("`q' and `d' are not compatible") if (lower.tail) { q <- mtransform(q, mar) pp <- exp(-apply(q^(1/dep), 1, sum)^dep) } else { pp <- numeric(1) ss <- c(list(numeric(0)), subsets(d)) ssl <- d - sapply(ss, length) for (i in 1:(2^d)) { tmpq <- q tmpq[, ss[[i]]] <- Inf pp <- (-1)^ssl[i] * Recall(tmpq, dep, d, mar) + pp } } pp } "pmvalog"<- function (q, dep, asy, d = 2, mar = c(0, 1, 0), lower.tail = TRUE) { if (is.null(dim(q))) dim(q) <- c(1, d) if (ncol(q) != d) stop("`q' and `d' are not compatible") nb <- 2^d - 1 dep2 <- rep(dep, length.out = nb - d) asy2 <- mvalog.check(asy, dep2, d = d) dep2 <- c(rep(1, d), dep2) if (lower.tail) { q <- mtransform(q, mar) inner <- function(par) apply((rep(par[1:d], rep(nrow(q), d)) * q)^(1/par[d + 1]), 1, sum)^par[d + 1] comps <- apply(cbind(asy2, dep2), 1, inner) if (is.null(dim(comps))) dim(comps) <- c(1, nb) pp <- exp(-apply(comps, 1, sum)) } else { pp <- numeric(1) ss <- c(list(numeric(0)), subsets(d)) ssl <- d - sapply(ss, length) for (i in 1:(2^d)) { tmpq <- q tmpq[, ss[[i]]] <- Inf pp <- (-1)^ssl[i] * Recall(tmpq, dep, asy, d, mar) + pp } } pp } "dmvevd" <- function(x, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0), log = FALSE) { model <- match.arg(model) if(model == "log" && !missing(asy)) warning("ignoring `asy' argument") switch(model, log = dmvlog(x = x, dep = dep, d = d, mar = mar, log = log), alog = dmvalog(x = x, dep = dep, asy = asy, d = d, mar = mar, log = log)) } "dmvlog"<- function(x, dep, d = 2, mar = c(0,1,0), log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(is.null(dim(x))) dim(x) <- c(1,d) if(ncol(x) != d) stop("`x' and `d' are not compatible") if(is.list(mar)) { if(length(mar) != d) stop("`mar' and `d' are not compatible") for(i in 1:d) mar[[i]] <- matrix(t(mar[[i]]), nrow = nrow(x), ncol = 3, byrow = TRUE) } else mar <- matrix(t(mar), nrow = nrow(x), ncol = 3, byrow = TRUE) dns <- numeric(nrow(x)) x <- mtransform(x, mar) ext <- apply(x, 1, function(z) any(z %in% c(0,Inf))) dns[ext] <- -Inf idep <- 1/dep cf <- matrix(0, nrow = d, ncol = d) diag(cf) <- dep^(1:d - 1) cf[,1] <- exp(lgamma(1:d - dep) - lgamma(1:d) - lgamma(1 - dep)) if(d >= 3) { for(i in 3:d) { for(j in 2:(d-1)) cf[i,j] <- ((i - 1 - j*dep) * cf[i-1,j] + dep*(j-1)*cf[i-1,j-1])/ (i - 1) } } cf <- log(cf[d,]) - lgamma(1:d) if(any(!ext)) { x <- x[!ext, ,drop=FALSE] if(is.list(mar)) { marscl <- marshp <- matrix(NA, nrow = nrow(x), ncol = d) for(i in 1:d) { mar[[i]] <- mar[[i]][!ext, ,drop=FALSE] marscl[,i] <- mar[[i]][,2] marshp[,i] <- mar[[i]][,3] } } else { mar <- mar[!ext, ,drop=FALSE] marscl <- mar[,2] marshp <- mar[,3] } z <- rowSums(x^(idep))^dep lx <- log(x) .expr1 <- rowSums(lx * (idep + marshp) - log(marscl)) dns[!ext] <- .expr1 + (1 - d*idep) * log(z) - z lz <- matrix(log(z), nrow = d, ncol = nrow(x), byrow = TRUE) lz <- (0:(d-1)) * lz + cf dns[!ext] <- dns[!ext] + log(colSums(exp(lz))) + lgamma(d) - (d-1)*log(dep) } if(!log) dns <- exp(dns) dns } "dmvalog"<- function(x, dep, asy, d = 2, mar = c(0,1,0), log = FALSE) { nb <- 2^d-1 dep <- rep(dep, length.out = nb-d) ss <- mvalog.check(asy, dep, d = d, ss = TRUE) asy <- ss$asy ; ss <- ss$ss dep <- c(rep(1,d), dep) if(is.null(dim(x))) dim(x) <- c(1,d) if(ncol(x) != d) stop("`x' and `d' are not compatible") nn <- nrow(x) if(is.list(mar)) { if(length(mar) != d) stop("`mar' and `d' are not compatible") for(i in 1:d) mar[[i]] <- matrix(t(mar[[i]]), nrow = nn, ncol = 3, byrow = TRUE) } else mar <- matrix(t(mar), nrow = nn, ncol = 3, byrow = TRUE) x <- mtransform(x, mar) ext <- apply(x, 1, function(z) any(z %in% c(0,Inf))) dns <- numeric(nn) dns[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] if(is.list(mar)) { marscl <- marshp <- matrix(NA, nrow = nn, ncol = d) for(i in 1:d) { mar[[i]] <- mar[[i]][!ext, ,drop=FALSE] marscl[,i] <- mar[[i]][,2] marshp[,i] <- mar[[i]][,3] } } else { mar <- mar[!ext, ,drop=FALSE] marscl <- mar[,2] marshp <- mar[,3] } cfinit <- function(dep) { if(dep==1) return(diag(d)) cf <- matrix(0, nrow = d, ncol = d) diag(cf) <- dep^(1:d - 1) cf[,1] <- exp(lgamma(1:d - dep) - lgamma(1:d) - lgamma(1 - dep)) if(d >= 3) { for(i in 3:d) { for(j in 2:(d-1)) cf[i,j] <- ((i - 1 - j*dep) * cf[i-1,j] + dep * (j-1) * cf[i-1,j-1])/(i - 1) } } cf } cfs <- paste("cf", 1:nb, sep = "") for(i in 1:nb) assign(cfs[i], cfinit(dep[i])) qfn <- function(lz, dep, p, cf) { if(p == 1) return(rep(0, nn)) cf <- log(cf[p,1:p]) - lgamma(1:p) lz <- matrix(lz, nrow = p, ncol = nn, byrow = TRUE) lz <- ((1-p):0) * lz + cf log(colSums(exp(lz))) + lgamma(p) - (p-1)*log(dep) } indm <- matrix(NA, nrow = 2^(d-1), ncol = d) for(i in 1:d) indm[,i] <- which(sapply(ss, function(x) i %in% x)) indm <- as.matrix(do.call("expand.grid", as.data.frame(indm))) inner <- function(par) { int <- (rep(par[1:d], each = nn) * x)^(1/par[d+1]) rowSums(int)^par[d+1] } zm <- log(apply(cbind(asy,dep), 1, inner)) if(is.null(dim(zm))) dim(zm) <- c(1,nb) vv <- rowSums(exp(zm)) lx <- log(x) mexpr <- rowSums(lx * marshp - log(marscl)) tot1 <- matrix(0, nrow = nn, ncol = 2^(d*(d-1))) tot2 <- matrix(0, nrow = nn, ncol = d) for(i in 1:(2^(d*(d-1)))) { indmi <- indm[i,] pp <- tabulate(indmi, 2^d-1)[indmi] if(all(asy[indmi + (2^d-1)*(0:(d-1))] != 0)) { for(j in 1:d) tot2[,j] <- 1/dep[indmi[j]] * (log(asy[indmi[j],j]) + lx[,j]) + (1 - 1/dep[indmi[j]]) * zm[,indmi[j]] + qfn(zm[,indmi[j]], dep[indmi[j]], pp[j], get(cfs[indmi[j]])) / pp[j] tot1[,i] <- rowSums(tot2) } else tot1[,i] <- -Inf } dns[!ext] <- log(rowSums(exp(tot1))) - vv + mexpr } if(!log) dns <- exp(dns) dns } "amvevd" <- function(x = rep(1/d,d), dep, asy, model = c("log", "alog"), d = 3, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) { model <- match.arg(model) if(model == "log" && !missing(asy)) warning("ignoring `asy' argument") if(!plot) { if(is.vector(x)) x <- as.matrix(t(x)) if(!is.matrix(x) || ncol(x) != d) stop("`x' must be a vector/matrix with `d' elements/columns") if(any(x < 0, na.rm = TRUE)) stop("`x' must be non-negative") rs <- rowSums(x) if(any(rs <= 0, na.rm = TRUE)) stop("row(s) of `x' must have a positive sum") if(max(abs(rs[!is.na(rs)] - 1)) > 1e-6) warning("row(s) of `x' will be rescaled") x <- x/rs } if(plot) { if(d == 2) stop("use abvnonpar for bivariate plots") if(d >= 4) stop("cannot plot in high dimensions") } switch(model, log = amvlog(x = x, dep = dep, d = d, plot = plot, col = col, blty = blty, grid = grid, lower = lower, ord = ord, lab = lab, lcex = lcex), alog = amvalog(x = x, dep = dep, d = d, asy = asy, plot = plot, col = col, blty = blty, grid = grid, lower = lower, ord = ord, lab = lab, lcex = lcex)) } "amvlog"<- function(x, dep, d = 3, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") depfn <- function(x, dep) rowSums(x^(1/dep))^dep if(plot) { mz <- tvdepfn(depfn = depfn, col = col, blty = blty, grid = grid, lower = lower, ord = ord, lab = lab, lcex = lcex, dep = dep) return(invisible(mz)) } depfn(x = x, dep = dep) } "amvalog"<- function(x, dep, asy, d = 3, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) { dep <- rep(dep, length.out = 2^d-d-1) asy <- mvalog.check(asy, dep, d) depfn <- function(x, dep, asy) { d <- ncol(x) dep <- c(rep(1,d), dep) tot <- matrix(0, nrow = nrow(x), ncol = 2^d-1) idep <- 1/dep x <- t(x) for(k in 1:(2^d-1)) tot[,k] <- colSums((asy[k,] * x)^idep[k])^dep[k] rowSums(tot) } if(plot) { mz <- tvdepfn(depfn = depfn, col = col, blty = blty, grid = grid, lower = lower, ord = ord, lab = lab, lcex = lcex, dep = dep, asy = asy) return(invisible(mz)) } depfn(x = x, dep = dep, asy = asy) } ### Ancillary Functions ### "tvdepfn" <- # Plots Dependence Function On Simplex S3 function(depfn, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1, ...) { oldpar <- par(pty="s") on.exit(par(oldpar)) s3 <- sqrt(1/3) x <- seq(-s3, s3, len = 2*grid) y <- seq(0, 1, len = 2*grid) a <- function(x,y) { w1 <- y w2 <- (x/s3 + 1 - y)/2 vals <- numeric(length(w1)) tpts <- cbind(w1,w2,w3=1-w1-w2)[, ord] nv <- apply(tpts, 1, function(b) any(b < 0)) is.na(vals) <- nv vals[!nv] <- depfn(tpts[!nv,], ...) vals } z <- outer(x,y,a) mz <- min(z, na.rm = TRUE) mxz <- max(z, na.rm = TRUE) if(mz < 1/3 || mxz > 1 || any(is.nan(z))) warning("parameters may be too near edge of parameter space") else if(mz < lower) warning("`lower' is greater than calculated values") plot(c(-s3, s3), c(0.5-s3, 0.5+s3), type="n", axes=FALSE, xlab="", ylab="") if(!is.null(lab)) { lab <- lab[ord] eps <- 0.025 text(0, 1+eps, lab[1], adj = c(0.5, 0), cex = lcex) text(s3, -eps, lab[2], adj = c(1, 1), cex = lcex) text(-s3, -eps, lab[3], adj = c(0, 1), cex = lcex) } image(x, y, z, zlim = c(lower,1), xlim = c(-s3,s3), ylim = c(0.5-s3, 0.5+s3), col = col, add = TRUE) polygon(c(-s3, s3, 0), c(0, 0, 1), lty = blty) invisible(mz) } "mvalog.check" <- # Checks And Transforms Arguments For Asymmetric Logistic function(asy, dep, d, ss = FALSE) { if(mode(dep) != "numeric" || any(dep <= 0) || any(dep > 1)) stop("invalid argument for `dep'") nb <- 2^d-1 if(mode(asy) != "list" || length(asy) != nb) stop(paste("`asy' should be a list of length", nb)) tasy <- function(theta, b) { trans <- matrix(0, nrow=nb, ncol=d) for(i in 1:nb) trans[i,(1:d %in% b[[i]])] <- theta[[i]] trans } b <- subsets(d) if(any(sapply(asy, length) != sapply(b, length))) stop("`asy' is not of the correct form") asy <- tasy(asy, b) if(!is.matrix(asy) || mode(asy) != "numeric") stop("`asy' is not of the correct form") if(min(asy) < 0 || max(asy) > 1) stop("`asy' must contain parameters in [0,1]") if(any(apply(asy,2,sum) != 1) || any(asy[c(rep(FALSE,d),dep==1),] != 0) || any(apply(asy[-(1:d),,drop=FALSE],1,function(x) sum(x!=0)) == 1)) stop("`asy' does not satisfy the appropriate constraints") if(ss) return(list(asy = asy, ss = b)) asy } "subsets" <- # Lists All Subsets Of 1:d function(d) { x <- 1:d k <- NULL for(m in x) k <- rbind(cbind(TRUE, k), cbind(FALSE, k)) pset <- apply(k, 1, function(z) x[z]) pset[sort.list(unlist(lapply(pset,length)))[-1]] } evd/R/bvfit.R0000644000175100001440000014300613264357514012522 0ustar hornikusers "fbvevd" <- function(x, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { call <- match.call() model <- match.arg(model) if(!is.matrix(x) && !is.data.frame(x)) stop("`x' must be a matrix or data frame") if(ncol(x) != 2) { if(ncol(x) != 3) stop("`x' has incorrect number of columns") if(!is.logical(x[,3])) stop("third column of `x' must be logical") } if(sym && !(model %in% c("alog","aneglog","ct"))) warning("Argument `sym' was ignored") ft <- switch(model, log = fbvlog(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), alog = fbvalog(x=x, start=start, ..., sym=sym, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), hr = fbvhr(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2= nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), neglog = fbvneglog(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), aneglog = fbvaneglog(x=x, start=start, ..., sym=sym, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), bilog = fbvbilog(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), negbilog = fbvnegbilog(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr= corr, method=method, warn.inf=warn.inf), ct = fbvct(x=x, start=start, ..., sym=sym, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf), amix = fbvamix(x=x, start=start, ..., sym=FALSE, nsloc1=nsloc1, nsloc2=nsloc2, cshape=cshape, cscale=cscale, cloc=cloc, std.err=std.err, corr=corr, method=method, warn.inf=warn.inf)) structure(c(ft, call = call), class = c("bvevd","evd")) } "fbvlog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvlog <- function(loc1, scale1, shape1, loc2, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || dep < 0.1 || dep > 1) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvlog, spx$x1, spx$x2, spx$n, spx$si, dep, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1, as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "log") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:7)[c(!cscale, !cshape, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvlog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvlog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvlog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvlog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvlog(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "log") } "fbvalog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvalog <- function(loc1, scale1, shape1, loc2, scale2, shape2, asy1, asy2, dep) { if(sym) asy2 <- asy1 if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || any(c(dep,asy1,asy2) > 1) || any(c(asy1,asy2) < 0.001) || dep < 0.1) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvalog, spx$x1, spx$x2, spx$n, spx$si, dep, asy1, asy2, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "asy1", "asy2", "dep") else param <- c(param, "asy1", "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "alog") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:9)[c(!cscale, !cshape, TRUE, !sym, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvalog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvalog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvalog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvalog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvalog(", paste("p[",1:l,"]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "alog") } "fbvhr"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvhr <- function(loc1, scale1, shape1, loc2, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || dep < 0.2 || dep > 10) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvhr, spx$x1, spx$x2, spx$n, spx$si, dep, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "hr") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:7)[c(!cscale, !cshape, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvhr)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvhr)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvhr) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvhr(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvhr(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "hr") } "fbvneglog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvneglog <- function(loc1, scale1, shape1, loc2, scale2, shape2, dep) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || dep < 0.05 || dep > 5) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvneglog, spx$x1, spx$x2, spx$n, spx$si, dep, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "neglog") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:7)[c(!cscale, !cshape, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvneglog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvneglog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvneglog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvneglog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvneglog(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "neglog") } "fbvaneglog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvaneglog <- function(loc1, scale1, shape1, loc2, scale2, shape2, asy1, asy2, dep) { if(sym) asy2 <- asy1 if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || any(c(asy1,asy2) > 1) || any(c(asy1,asy2) < 0.001) || dep < 0.05 || dep > 5) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvaneglog, spx$x1, spx$x2, spx$n, spx$si, dep, asy1, asy2, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "asy1", "asy2", "dep") else param <- c(param, "asy1", "dep") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "aneglog") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:9)[c(!cscale, !cshape, TRUE, !sym, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvaneglog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvaneglog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvaneglog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvaneglog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvaneglog(", paste("p[",1:l,"]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "aneglog") } "fbvbilog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvbilog <- function(loc1, scale1, shape1, loc2, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || any(c(alpha,beta) < 0.1) || any(c(alpha,beta) > 0.999)) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvbilog, spx$x1, spx$x2, spx$n, spx$si, alpha, beta, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "bilog") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:8)[c(!cscale, !cshape, TRUE, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvbilog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvbilog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvbilog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvbilog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvbilog(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "bilog") } "fbvnegbilog"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvnegbilog <- function(loc1, scale1, shape1, loc2, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || any(c(alpha,beta) < 0.1) || any(c(alpha,beta) > 20)) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvnegbilog, spx$x1, spx$x2, spx$n, spx$si, alpha, beta, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "negbilog") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:8)[c(!cscale, !cshape, TRUE, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvnegbilog)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvnegbilog)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvnegbilog) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvnegbilog(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvnegbilog(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "negbilog") } "fbvct"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvct <- function(loc1, scale1, shape1, loc2, scale2, shape2, alpha, beta) { if(sym) beta <- alpha if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01) || any(c(alpha,beta) < 0.001) || any(c(alpha,beta) > 30)) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvct, spx$x1, spx$x2, spx$n, spx$si, alpha, beta, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") if(!sym) param <- c(param, "alpha", "beta") else param <- c(param, "alpha") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "ct") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:8)[c(!cscale, !cshape, TRUE, !sym)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvct)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvct)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvct) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvct(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvct(", paste("p[",1:l,"]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "ct") } "fbvamix"<- function(x, start, ..., sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlbvamix <- function(loc1, scale1, shape1, loc2, scale2, shape2, alpha, beta) { if(cshape) shape2 <- shape1 if(cscale) scale2 <- scale1 if(any(c(scale1,scale2) < 0.01)) return(1e6) if(alpha < 0 || (alpha + 3*beta) < 0) return(1e6) if((alpha + beta) > 1 || (alpha + 2*beta) > 1) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = nrow(x)) if(cloc) loc2 <- loc1 else { if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = nrow(x)) } if(spx$n.m1) m1l <- .C(C_nlgev, spx$x.m1, spx$n.m1, loc1[spx$na == 2], scale1, shape1, dns = double(1))$dns else m1l <- 0 if(spx$n.m2) m2l <- .C(C_nlgev, spx$x.m2, spx$n.m2, loc2[spx$na == 1], scale2, shape2, dns = double(1))$dns else m2l <- 0 bvl <- .C(C_nlbvamix, spx$x1, spx$x2, spx$n, spx$si, alpha, beta, loc1[spx$na == 0], scale1, shape1, loc2[spx$na == 0], scale2, shape2, cfalse, dns = double(1))$dns if(any(is.nan(c(m1l,m2l,bvl)))) { warning("NaN returned in likelihood") return(1e6) } if(any(c(m1l,m2l,bvl) == 1e6)) return(1e6) else return(m1l + m2l + bvl) } if(cloc && !identical(nsloc1, nsloc2)) stop("nsloc1 and nsloc2 must be identical") if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != nrow(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1,as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != nrow(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", "shape1") if(!cloc) param <- c(param, loc.param2) else loc.param2 <- NULL if(!cscale) param <- c(param, "scale2") if(!cshape) param <- c(param, "shape2") param <- c(param, "alpha", "beta") nmdots <- names(list(...)) start <- bvstart.vals(x = x, start = start, nmdots = nmdots, param = param, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "amix") spx <- sep.bvdata(x = x) cfalse <- as.integer(0) nm <- names(start) l <- length(nm) fixed.param <- list(...)[nmdots %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") prind <- (5:8)[c(!cscale, !cshape, TRUE, TRUE)] f <- c(as.list(numeric(length(loc.param1))), formals(nlbvamix)[2:3], as.list(numeric(length(loc.param2))), formals(nlbvamix)[prind]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlbvamix) <- c(f[m], f[-m]) nllh <- function(p, ...) nlbvamix(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlbvamix(", paste("p[",1:l,"]", collapse = ", "), ", ...)")) start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm cmar <- c(cloc, cscale, cshape) bvpost.optim(x = x, opt = opt, nm = nm, fixed.param = fixed.param, std.err = std.err, corr = corr, sym = sym, cmar = cmar, nsloc1 = nsloc1, nsloc2 = nsloc2, model = "amix") } ### Method Function ### "print.bvevd" <- function(x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:", deparse(x$call), "\n") cat("Deviance:", deviance(x), "\n") cat("AIC:", AIC(x), "\n") if(!is.null(x$dep.summary)) cat("Dependence:", x$dep.summary, "\n") cat("\nEstimates\n") print.default(format(fitted(x), digits = digits), print.gap = 2, quote = FALSE) if(!is.null(std.errors(x))) { cat("\nStandard Errors\n") print.default(format(std.errors(x), digits = digits), print.gap = 2, quote = FALSE) } if(!is.null(x$corr)) { cat("\nCorrelations\n") print.default(format(x$corr, digits = digits), print.gap = 2, quote = FALSE) } cat("\nOptimization Information\n") cat(" Convergence:", x$convergence, "\n") cat(" Function Evaluations:", x$counts["function"], "\n") if(!is.na(x$counts["gradient"])) cat(" Gradient Evaluations:", x$counts["gradient"], "\n") if(!is.null(x$message)) cat(" Message:", x$message, "\n") cat("\n") invisible(x) } ### Ancillary Functions ### "bvstart.vals" <- # Calculate Starting Values For Bivariate Models function(x, start, nmdots, param, method = c("evd","pot"), nsloc1 = NULL, nsloc2 = NULL, u = NULL, model) { method <- match.arg(method) if(method == "evd") { loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") } if(method == "pot") loc.param1 <- loc.param2 <- NULL if(missing(start)) { start <- as.list(numeric(length(param))) names(start) <- param if(method == "evd") { loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") st1 <- fitted(fgev(x[,1], nsloc = nsloc1, std.err = FALSE)) st2 <- fitted(fgev(x[,2], nsloc = nsloc2, std.err = FALSE)) st1 <- as.list(st1); st2 <- as.list(st2) } if(method == "pot") { st1 <- fitted(fpot(x[,1], u[1], std.err = FALSE)) st2 <- fitted(fpot(x[,2], u[2], std.err = FALSE)) st1 <- as.list(st1); st2 <- as.list(st2) } start[c(loc.param1, "scale1", "shape1")] <- st1 tmp2 <- loc.param2 if("scale2" %in% param) tmp2 <- c(tmp2, "scale2") if("shape2" %in% param) tmp2 <- c(tmp2, "shape2") tmp <- sub("2", "", tmp2) start[tmp2] <- st2[tmp] if(model == "log") start[["dep"]] <- 0.75 if(model == "alog") { start[["asy1"]] <- 0.75 if("asy2" %in% param) start[["asy2"]] <- 0.75 start[["dep"]] <- 0.65 } if(model == "hr") start[["dep"]] <- 1 if(model == "neglog") start[["dep"]] <- 0.6 if(model == "aneglog") { start[["asy1"]] <- 0.75 if("asy2" %in% param) start[["asy2"]] <- 0.75 start[["dep"]] <- 0.8 } if(model == "bilog") start[["alpha"]] <- start[["beta"]] <- 0.75 if(model == "negbilog") start[["alpha"]] <- start[["beta"]] <- 1/0.6 if(model == "ct") { start[["alpha"]] <- 0.6 if("beta" %in% param) start[["beta"]] <- 0.6 } if(model == "amix") { start[["alpha"]] <- 0.75 start[["beta"]] <- 0 } start <- start[!(param %in% nmdots)] } if(any(!is.na(match(names(start),c("mar1","mar2","asy"))))) { if(("mar1" %in% names(start)) && (length(start$mar1) != (2+length(loc.param1)))) stop("mar1 in `start' has incorrect length") if(("mar2" %in% names(start)) && (length(start$mar2) != (2+length(loc.param2)))) stop("mar2 in `start' has incorrect length") if(("asy" %in% names(start)) && (length(start$asy) != 2)) stop("asy in `start' should have length two") start <- unlist(start) names(start)[grep("mar1",names(start))] <- c(loc.param1,"scale1","shape1") names(start)[grep("mar2",names(start))] <- c(loc.param2,"scale2","shape2") start <- as.list(start) } if(!is.list(start)) stop("`start' must be a named list") start } "sep.bvdata" <- # Separate Bivariate Data For Bivariate Models function(x, method = c("evd","cpot","ppot"), u = NULL) { method <- match.arg(method) if(method == "evd") { na <- rowSums(cbind(is.na(x[,1]), 2*is.na(x[,2]))) if(!any(na == 0)) stop("`x' must have at least one complete observation") x.m1 <- as.double(x[na == 2, 1]) n.m1 <- as.integer(length(x.m1)) x.m2 <- as.double(x[na == 1, 2]) n.m2 <- as.integer(length(x.m2)) x.full <- x[na == 0, , drop = FALSE] x1 <- as.double(x.full[,1]) x2 <- as.double(x.full[,2]) n <- as.integer(nrow(x.full)) if(ncol(x) == 3) { si <- x.full[,3] si[is.na(si)] <- 2 } else si <- rep(2, n) si <- as.integer(si) spx <- list(x.m1 = x.m1, n.m1 = n.m1, x.m2 = x.m2, n.m2 = n.m2, x1 = x1, x2 = x2, n = n, si = si, na = na) } else { x1 <- x[,1] x2 <- x[,2] n <- length(x1) r1 <- r2 <- NULL iau1 <- (x1 > u[1]) & !is.na(x1) iau2 <- (x2 > u[2]) & !is.na(x2) nat <- c(sum(iau1), sum(iau2), sum(iau1 & iau2)) lambda1 <- sum(iau1) / (n + 1) lambda2 <- sum(iau2) / (n + 1) lambda <- c(lambda1, lambda2) if(method == "ppot") { x1[is.na(x1)] <- mean(x1[!iau1], na.rm = TRUE) x2[is.na(x2)] <- mean(x2[!iau2], na.rm = TRUE) r1 <- 1 - rank(x1) / (n + 1) r2 <- 1 - rank(x2) / (n + 1) r1[iau1] <- lambda1 r2[iau2] <- lambda2 } x1 <- x1 - u[1] x2 <- x2 - u[2] x1[!iau1] <- 0 x2[!iau2] <- 0 i0 <- iau1 | iau2 x1 <- x1[i0] ; x2 <- x2[i0] if(method == "ppot") { r1 <- r1[i0] ; r2 <- r2[i0] } nn <- length(x1) thdi <- as.logical(x1) + 2*as.logical(x2) spx <- list(x1 = x1, x2 = x2, nn = nn, n = n, thdi = thdi, lambda = lambda, r1 = r1, r2 = r2, nat = nat) } spx } "bvpost.optim" <- # Post-optimization Processing function(x, opt, nm, fixed.param, std.err, corr, sym, cmar, method = c("evd","pot"), nsloc1 = NULL, nsloc2 = NULL, u = NULL, nat = NULL, likelihood = NULL, model) { method <- match.arg(method) if(opt$convergence != 0) { warning(paste("optimization for", model, "may not have succeeded"), call. = FALSE) if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop(paste("observed information matrix for", model, "is singular; use std.err = FALSE")) var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop(paste("observed information matrix for", model, "is singular; use std.err = FALSE")) std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL fixed <- unlist(fixed.param) param <- c(opt$par, fixed) fixed2 <- NULL if(method == "evd") { if(cmar[1]) { loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") fixed2 <- c(fixed2, param[loc.param1]) } if(cmar[2]) fixed2 <- c(fixed2, param["scale1"]) if(cmar[3]) fixed2 <- c(fixed2, param["shape1"]) } if(method == "pot") { if(cmar[1]) fixed2 <- c(fixed2, param["scale1"]) if(cmar[2]) fixed2 <- c(fixed2, param["shape1"]) } if(sym) { if(model %in% c("alog","aneglog")) fixed2 <- c(fixed2, param["asy1"]) if(model == "ct") fixed2 <- c(fixed2, param["alpha"]) } if(!is.null(fixed2)) { names(fixed2) <- sub("1", "2", names(fixed2)) names(fixed2) <- sub("alpha", "beta", names(fixed2)) } param <- c(param, fixed2) # Transform to stationarity x2 <- x if(!is.null(nsloc1)) { trend <- param[paste("loc1", names(nsloc1), sep="")] trend <- drop(as.matrix(nsloc1) %*% trend) x2[,1] <- x[,1] - trend } if(!is.null(nsloc2)) { trend <- param[paste("loc2", names(nsloc2), sep="")] trend <- drop(as.matrix(nsloc2) %*% trend) x2[,2] <- x[,2] - trend } # End transform # Dependence chi if(model %in% c("log", "hr", "neglog")) { dep <- param["dep"] dep.sum <- 2*(1 - abvevd(dep = dep, model = model)) } if(model %in% c("alog", "aneglog")) { dep <- param["dep"] asy <- param[c("asy1", "asy2")] dep.sum <- 2*(1 - abvevd(dep = dep, asy = asy, model = model)) } if(model %in% c("bilog", "negbilog", "ct", "amix")) { alpha <- param["alpha"] beta <- param["beta"] dep.sum <- 2*(1-abvevd(alpha = alpha, beta = beta, model = model)) } # End dependence chi out <- list(estimate = opt$par, std.err = std.err, fixed = fixed, fixed2 = fixed2, param = param, deviance = 2*opt$value, dep.summary = dep.sum, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, data = x) if(method == "evd") out <- c(out, list(tdata = x2, nsloc1 = nsloc1, nsloc2 = nsloc2)) if(method == "pot") out <- c(out, list(threshold = u, nat = nat, likelihood = likelihood)) c(out, list(n = nrow(x), sym = sym, cmar = cmar, model = model)) } evd/R/uvdist.R0000644000175100001440000003317414225005362012717 0ustar hornikusers "rfrechet"<- function(n, loc = 0, scale = 1, shape = 1) { if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") loc + scale * rexp(n)^(-1/shape) } "rgumbel"<- function(n, loc = 0, scale = 1) { rgev(n, loc = loc, scale = scale, shape = 0) } "rrweibull"<- function(n, loc = 0, scale = 1, shape = 1) { if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") loc - scale * rexp(n)^(1/shape) } "rnweibull"<- function(n, loc = 0, scale = 1, shape = 1) { if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") loc - scale * rexp(n)^(1/shape) } "rgev"<- function(n, loc = 0, scale = 1, shape = 0) { if(min(scale) < 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") if(shape == 0) return(loc - scale * log(rexp(n))) else return(loc + scale * (rexp(n)^(-shape) - 1)/shape) } "rgumbelx"<- function(n, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1) { if(min(scale1) < 0 || min(scale2) < 0) stop("invalid scale") if(any(loc1 > loc2)) stop("loc1 cannot be greater than loc2") pmax(rgumbel(n = n, loc = loc1, scale = scale1), rgumbel(n = n, loc = loc2, scale = scale2)) } "rgpd"<- function(n, loc = 0, scale = 1, shape = 0) { if(min(scale) < 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") if(shape == 0) return(loc + scale*rexp(n)) else return(loc + scale * (runif(n)^(-shape) - 1) / shape) } "rextreme"<- function(n, quantfun, ..., distn, mlen = 1, largest = TRUE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(missing(quantfun)) quantfun <- get(paste("q", distn, sep=""), mode="function") if(largest) quantfun(rbeta(n, mlen, 1), ...) else quantfun(rbeta(n, 1, mlen), ...) } "rorder"<- function(n, quantfun, ..., distn, mlen = 1, j = 1, largest = TRUE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(!is.numeric(j) || length(j) != 1 || j < 1 || j %% 1 != 0) stop("`j' must be a non-negative integer") if(j > mlen) stop("`j' cannot be greater than `mlen'") if(!largest) j <- mlen+1-j if(missing(quantfun)) quantfun <- get(paste("q", distn, sep=""), mode="function") quantfun(rbeta(n, mlen+1-j, j), ...) } "qfrechet"<- function(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") if(!lower.tail) p <- 1 - p loc + scale * (-log(p))^(-1/shape) } "qgumbel"<- function(p, loc = 0, scale = 1, lower.tail = TRUE) { qgev(p, loc = loc, scale = scale, shape = 0, lower.tail = lower.tail) } "qrweibull"<- function(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") if(!lower.tail) p <- 1 - p loc - scale * (-log(p))^(1/shape) } "qnweibull"<- function(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale) < 0 || min(shape) <= 0) stop("invalid arguments") if(!lower.tail) p <- 1 - p loc - scale * (-log(p))^(1/shape) } "qgev"<- function(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale) < 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") if(!lower.tail) p <- 1 - p if(shape == 0) return(loc - scale * log(-log(p))) else return(loc + scale * ((-log(p))^(-shape) - 1)/shape) } "qgumbelx"<- function(p, interval, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1, lower.tail = TRUE, ...) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale1) < 0 || min(scale2) < 0) stop("invalid scale") if(any(loc1 > loc2)) stop("loc1 cannot be greater than loc2") if(!lower.tail) p <- 1 - p n <- length(p) out <- numeric(n) for(i in 1:n) { tmpfn <- function(z) exp(-(z - loc1)/scale1) + exp(-(z - loc2)/scale2) + log(p[i]) out[i] <- uniroot(tmpfn, interval = interval, ...)$root } out } "qgpd"<- function(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(min(scale) < 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") if(lower.tail) p <- 1 - p if(shape == 0) return(loc - scale*log(p)) else return(loc + scale * (p^(-shape) - 1) / shape) } "qextreme"<- function(p, quantfun, ..., distn, mlen = 1, largest = TRUE, lower.tail = TRUE) { if(min(p, na.rm = TRUE) <= 0 || max(p, na.rm = TRUE) >=1) stop("`p' must contain probabilities in (0,1)") if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(missing(quantfun)) quantfun <- get(paste("q", distn, sep=""), mode="function") if(!lower.tail) p <- 1 - p if(largest) quantfun(p^(1/mlen), ...) else quantfun(1-(1-p)^(1/mlen), ...) } "pfrechet"<- function(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") q <- pmax((q - loc)/scale,0) p <- exp(-q^(-shape)) if(!lower.tail) p <- 1 - p p } "pgumbel"<- function(q, loc = 0, scale = 1, lower.tail = TRUE) { pgev(q, loc = loc, scale = scale, shape = 0, lower.tail = lower.tail) } "prweibull"<- function(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") q <- pmin((q - loc)/scale,0) p <- exp(-(-q)^shape) if(!lower.tail) p <- 1 - p p } "pnweibull"<- function(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") q <- pmin((q - loc)/scale,0) p <- exp(-(-q)^shape) if(!lower.tail) p <- 1 - p p } "pgev"<- function(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) { if(min(scale) <= 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") q <- (q - loc)/scale if(shape == 0) p <- exp(-exp(-q)) else p <- exp( - pmax(1 + shape * q, 0)^(-1/shape)) if(!lower.tail) p <- 1 - p p } "pgumbelx"<- function(q, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1, lower.tail = TRUE) { if(min(scale1) < 0 || min(scale2) < 0) stop("invalid scale") if(any(loc1 > loc2)) stop("loc1 cannot be greater than loc2") q1 <- (q - loc1)/scale1 q2 <- (q - loc2)/scale2 p <- exp(-exp(-q1)) * exp(-exp(-q2)) if(!lower.tail) p <- 1 - p p } "pgpd" <- function(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) { if(min(scale) <= 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") q <- pmax(q - loc, 0)/scale if(shape == 0) p <- 1 - exp(-q) else { p <- pmax(1 + shape * q, 0) p <- 1 - p^(-1/shape) } if(!lower.tail) p <- 1 - p p } "pextreme"<- function(q, distnfun, ..., distn, mlen = 1, largest = TRUE, lower.tail = TRUE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") distn <- distnfun(q, ...) if(!largest) distn <- 1-distn p <- distn^mlen if(largest != lower.tail) p <- 1 - p p } "porder"<- function(q, distnfun, ..., distn, mlen = 1, j = 1, largest = TRUE, lower.tail = TRUE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(!is.numeric(j) || length(j) != 1 || j < 1 || j %% 1 != 0) stop("`j' must be a non-negative integer") if(j > mlen) stop("`j' cannot be greater than `mlen'") lachooseb <- function(a,b) lgamma(a+1) - lgamma(b+1) - lgamma(a-b+1) if(largest) svec <- (mlen+1-j):mlen else svec <- 0:(j-1) if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") distn <- distnfun(q, ...) store <- matrix(0,nrow=length(q),ncol=j) for(k in 1:j) store[,k] <- exp(lachooseb(mlen,svec[k]) + svec[k]*log(distn) + (mlen-svec[k])*log(1-distn)) p <- apply(store,1,sum) if(largest != lower.tail) p <- 1 - p p } "dfrechet"<- function(x, loc = 0, scale = 1, shape = 1, log = FALSE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") x <- (x - loc)/scale xpos <- x[x>0 | is.na(x)] nn <- length(x) scale <- rep(scale, length.out = nn)[x>0 | is.na(x)] shape <- rep(shape, length.out = nn)[x>0 | is.na(x)] d <- numeric(nn) d[x>0 | is.na(x)] <- log(shape/scale) - (1+shape) * log(xpos) - xpos^(-shape) d[x<=0 & !is.na(x)] <- -Inf if(!log) d <- exp(d) d } "dgumbel"<- function(x, loc = 0, scale = 1, log = FALSE) { dgev(x, loc = loc, scale = scale, shape = 0, log = log) } "drweibull"<- function(x, loc = 0, scale = 1, shape = 1, log = FALSE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") x <- (x - loc)/scale xneg <- x[x<0 | is.na(x)] nn <- length(x) scale <- rep(scale, length.out = nn)[x<0 | is.na(x)] shape <- rep(shape, length.out = nn)[x<0 | is.na(x)] d <- numeric(nn) d[x<0 | is.na(x)] <- log(shape/scale) + (shape-1) * log(-xneg) - (-xneg)^shape d[x>=0 & !is.na(x)] <- -Inf if(!log) d <- exp(d) d } "dnweibull"<- function(x, loc = 0, scale = 1, shape = 1, log = FALSE) { if(min(scale) <= 0 || min(shape) <= 0) stop("invalid arguments") x <- (x - loc)/scale xneg <- x[x<0 | is.na(x)] nn <- length(x) scale <- rep(scale, length.out = nn)[x<0 | is.na(x)] shape <- rep(shape, length.out = nn)[x<0 | is.na(x)] d <- numeric(nn) d[x<0 | is.na(x)] <- log(shape/scale) + (shape-1) * log(-xneg) - (-xneg)^shape d[x>=0 & !is.na(x)] <- -Inf if(!log) d <- exp(d) d } "dgev"<- function(x, loc = 0, scale = 1, shape = 0, log = FALSE) { if(min(scale) <= 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") x <- (x - loc)/scale if(shape == 0) d <- log(1/scale) - x - exp(-x) else { nn <- length(x) xx <- 1 + shape*x xxpos <- xx[xx>0 | is.na(xx)] scale <- rep(scale, length.out = nn)[xx>0 | is.na(xx)] d <- numeric(nn) d[xx>0 | is.na(xx)] <- log(1/scale) - xxpos^(-1/shape) - (1/shape + 1)*log(xxpos) d[xx<=0 & !is.na(xx)] <- -Inf } if(!log) d <- exp(d) d } "dgumbelx"<- function(x, loc1 = 0, scale1 = 1, loc2 = 0, scale2 = 1, log = FALSE) { if(min(scale1) < 0 || min(scale2) < 0) stop("invalid scale") if(any(loc1 > loc2)) stop("loc1 cannot be greater than loc2") x1 <- (x - loc1)/scale1 x2 <- (x - loc2)/scale2 d <- exp(-exp(-x1) + log(1/scale2) - x2 - exp(-x2)) + exp(-exp(-x2) + log(1/scale1) - x1 - exp(-x1)) if(log) d <- log(d) d } "dgpd"<- function(x, loc = 0, scale = 1, shape = 0, log = FALSE) { if(min(scale) <= 0) stop("invalid scale") if(length(shape) != 1) stop("invalid shape") d <- (x - loc)/scale nn <- length(d) scale <- rep(scale, length.out = nn) index <- (d > 0 & ((1 + shape * d) > 0)) | is.na(d) if(shape == 0) { d[index] <- log(1/scale[index]) - d[index] d[!index] <- -Inf } else { d[index] <- log(1/scale[index]) - (1/shape + 1) * log(1 + shape * d[index]) d[!index] <- -Inf } if(!log) d <- exp(d) d } "dextreme"<- function(x, densfun, distnfun, ..., distn, mlen = 1, largest = TRUE, log = FALSE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(missing(densfun)) densfun <- get(paste("d", distn, sep=""), mode="function") if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") dens <- densfun(x, ..., log = TRUE) distn <- distnfun(x, ...)[!is.infinite(dens)] if(!largest) distn <- 1 - distn distn <- (mlen-1) * log(distn) d <- numeric(length(x)) d[!is.infinite(dens)] <- log(mlen) + dens[!is.infinite(dens)] + distn d[is.infinite(dens)] <- -Inf if(!log) d <- exp(d) d } "dorder"<- function(x, densfun, distnfun, ..., distn, mlen = 1, j = 1, largest = TRUE, log = FALSE) { if(!is.numeric(mlen) || length(mlen) != 1 || mlen < 1 || mlen %% 1 != 0) stop("`mlen' must be a non-negative integer") if(!is.numeric(j) || length(j) != 1 || j < 1 || j %% 1 != 0) stop("`j' must be a non-negative integer") if(j > mlen) stop("`j' cannot be greater than `mlen'") if(!largest) j <- mlen + 1 - j if(missing(densfun)) densfun <- get(paste("d", distn, sep=""), mode="function") if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") dens <- densfun(x, ..., log = TRUE) distn <- distnfun(x, ...)[!is.infinite(dens)] distn <- (mlen-j) * log(distn) + (j-1) * log(1-distn) comb <- lgamma(mlen+1) - lgamma(j) - lgamma(mlen-j+1) d <- numeric(length(x)) d[!is.infinite(dens)] <- comb + dens[!is.infinite(dens)] + distn d[is.infinite(dens)] <- -Inf if(!log) d <- exp(d) d } evd/R/bvdist.R0000644000175100001440000014542413264357327012713 0ustar hornikusers "rbvevd" <- function(n, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0,1,0), mar2 = mar1) { model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") switch(model, log = rbvlog(n = n, dep = dep, mar1 = mar1, mar2 = mar2), alog = rbvalog(n = n, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2), hr = rbvhr(n = n, dep = dep, mar1 = mar1, mar2 = mar2), neglog = rbvneglog(n = n, dep = dep, mar1 = mar1, mar2 = mar2), aneglog = rbvaneglog(n = n, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2), bilog = rbvbilog(n = n, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2), negbilog = rbvnegbilog(n = n, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2), ct = rbvct(n = n, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2), amix = rbvamix(n = n, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2)) } "rbvlog"<- # Uses Algorithm 1.1 in Stephenson(2003) function(n, dep, mar1 = c(0,1,0), mar2 = mar1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") sim <- .C(C_rbvlog_shi, as.integer(n), as.double(dep), sim = double(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(1/sim, list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvalog"<- # Uses Algorithm 1.2 in Stephenson(2003) function(n, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(dep == 1 || any(asy == 0)) { asy <- c(0,0) dep <- 1 } sim <- .C(C_rbvalog_shi, as.integer(n), as.double(dep), as.double(asy), sim = double(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(1/sim, list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvhr" <- function(n, dep, mar1 = c(0,1,0), mar2 = mar1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") sim <- .C(C_rbvhr, as.integer(n), as.double(dep), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvneglog"<- function(n, dep, mar1 = c(0,1,0), mar2 = mar1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") sim <- .C(C_rbvneglog, as.integer(n), as.double(dep), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvaneglog"<- function(n, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") sim <- .C(C_rbvaneglog, as.integer(n), as.double(dep), as.double(asy), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvbilog"<- function(n, alpha, beta, mar1 = c(0,1,0), mar2 = mar1) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0) || any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") sim <- .C(C_rbvbilog, as.integer(n), as.double(alpha), as.double(beta), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvnegbilog"<- function(n, alpha, beta, mar1 = c(0,1,0), mar2 = mar1) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") sim <- .C(C_rbvnegbilog, as.integer(n), as.double(alpha), as.double(beta), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvct" <- function(n, alpha, beta, mar1 = c(0,1,0), mar2 = mar1) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") sim <- .C(C_rbvct, as.integer(n), as.double(alpha), as.double(beta), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "rbvamix" <- function(n, alpha, beta, mar1 = c(0,1,0), mar2 = mar1) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") sim <- .C(C_rbvamix, as.integer(n), as.double(alpha), as.double(beta), sim = runif(2*n))$sim sim <- matrix(sim, nrow = n, ncol = 2, byrow = TRUE) mtransform(-log(sim), list(mar1, mar2), inv = TRUE, drp = TRUE) } "pbvevd" <- function(q, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") switch(model, log = pbvlog(q = q, dep = dep, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), alog = pbvalog(q = q, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), hr = pbvhr(q = q, dep = dep, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), neglog = pbvneglog(q = q, dep = dep, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), aneglog = pbvaneglog(q = q, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), bilog = pbvbilog(q = q, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), negbilog = pbvnegbilog(q = q, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), ct = pbvct(q = q, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail), amix = pbvamix(q = q, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, lower.tail = lower.tail)) } "pbvlog"<- function(q, dep, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) v <- apply(q^(1/dep),1,sum)^dep pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvalog"<- function(q, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) asy <- rep(asy,rep(nrow(q),2)) v <- apply((asy*q)^(1/dep),1,sum)^dep + apply((1-asy)*q,1,sum) pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvhr" <- function(q, dep, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) fn <- function(x1,x2) x1*pnorm(1/dep + dep * log(x1/x2) / 2) v <- fn(q[,1],q[,2]) + fn(q[,2],q[,1]) pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvneglog"<- function(q, dep, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) v <- apply(q,1,sum) - apply(q^(-dep),1,sum)^(-1/dep) pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvaneglog"<- function(q, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) asy <- rep(asy,rep(nrow(q),2)) v <- apply(q,1,sum) - apply((asy*q)^(-dep),1,sum)^(-1/dep) pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvbilog"<- function(q, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0) || any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) gma <- numeric(nrow(q)) for(i in 1:nrow(q)) { gmafn <- function(x) (1-alpha) * q[i,1] * (1-x)^beta - (1-beta) * q[i,2] * x^alpha if(any(is.na(q[i,]))) gma[i] <- NA else if(any(is.infinite(q[i,]))) gma[i] <- 0.5 else if(q[i,1] == 0) gma[i] <- 0 else if(q[i,2] == 0) gma[i] <- 1 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } v <- q[,1] * gma^(1-alpha) + q[,2] * (1 - gma)^(1-beta) pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvnegbilog"<- function(q, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) gma <- numeric(nrow(q)) for(i in 1:nrow(q)) { gmafn <- function(x) (1+alpha) * q[i,1] * x^alpha - (1+beta) * q[i,2] * (1-x)^beta if(any(is.na(q[i,]))) gma[i] <- NA else if(any(is.infinite(q[i,]))) gma[i] <- Inf else if(q[i,1] == 0) gma[i] <- 1 else if(q[i,2] == 0) gma[i] <- 0 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } v <- q[,1] + q[,2] - q[,1] * gma^(1+alpha) - q[,2] * (1 - gma)^(1+beta) v[is.infinite(gma)] <- Inf pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvct" <- function(q, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) u <- (alpha * q[,2]) / (alpha * q[,2] + beta * q[,1]) v <- q[,2] * pbeta(u, shape1 = alpha, shape2 = beta + 1) + q[,1] * pbeta(u, shape1 = alpha + 1, shape2 = beta, lower.tail = FALSE) v[is.infinite(q[,1]) || is.infinite(q[,2])] <- Inf v[(q[,1] + q[,2]) == 0] <- 0 pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "pbvamix"<- function(q, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, lower.tail = TRUE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") if(is.null(dim(q))) dim(q) <- c(1,2) q <- mtransform(q, list(mar1, mar2)) qsum <- apply(q, 1, sum) v <- qsum - (alpha + beta) * q[,1] + alpha * (q[,1]^2)/qsum + beta * (q[,1]^3)/(qsum^2) v[is.infinite(q[,1]) || is.infinite(q[,2])] <- Inf v[(q[,1] + q[,2]) == 0] <- 0 pp <- exp(-v) if(!lower.tail) { pp <- 1 - pgev(-log(q[,1])) - pgev(-log(q[,2])) + pp } pp } "abvevd" <- function(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), rev = FALSE, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "t", ylab = "A(t)", ...) { if(any(x < 0, na.rm = TRUE) || any(x > 1, na.rm = TRUE)) stop("invalid argument for `x'") model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") if(rev && (model %in% c("aneglog", "alog"))) asy <- asy[2:1] if(rev && (model %in% c("bilog", "negbilog", "ct"))) { tmpalpha <- alpha alpha <- beta beta <- tmpalpha } if(rev && (model == "amix")) { tmpalpha <- alpha alpha <- alpha + 3*beta beta <- -beta } switch(model, log = abvlog(x = x, dep = dep, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), alog = abvalog(x = x, dep = dep, asy = asy, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), hr = abvhr(x = x, dep = dep, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), neglog = abvneglog(x = x, dep = dep, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), aneglog = abvaneglog(x = x, dep = dep, asy = asy, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), bilog = abvbilog(x = x, alpha = alpha, beta = beta, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), negbilog = abvnegbilog(x = x, alpha = alpha, beta = beta, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), ct = abvct(x = x, alpha = alpha, beta = beta, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...), amix = abvamix(x = x, alpha = alpha, beta = beta, plot = plot, add = add, lty = lty, lwd = lwd, col = col, blty = blty, blwd = blwd, xlim = xlim, ylim = ylim, xlab = xlab, ylab = ylab, ...)) } "abvlog"<- function(x = 0.5, dep, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) idep <- 1/dep a <- (x^idep + (1-x)^idep)^dep if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvalog"<- function(x = 0.5, dep, asy = c(1,1), plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(plot || add) x <- seq(0, 1, length = 100) idep <- 1/dep a <- ((asy[1]*x)^idep + (asy[2]*(1-x))^idep)^dep + (1-asy[1])*x + (1-asy[2])*(1-x) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvhr" <- function(x = 0.5, dep, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) fn <- function(z) z*pnorm(1/dep + dep * log(z/(1-z)) / 2) a <- fn(x) + fn(1-x) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvneglog"<- function(x = 0.5, dep, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) a <- 1 - (x^(-dep) + (1-x)^(-dep))^(-1/dep) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvaneglog"<- function(x = 0.5, dep, asy = c(1,1), plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") a <- 1 - ((asy[1]*x)^(-dep) + (asy[2]*(1-x))^(-dep))^(-1/dep) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvbilog"<- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0) || any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") if(plot || add) x <- seq(0, 1, length = 100) gma <- numeric(length(x)) for(i in 1:length(x)) { gmafn <- function(z) (1-alpha) * x[i] * (1-z)^beta - (1-beta) * (1-x[i]) * z^alpha if(is.na(x[i])) gma[i] <- NA else if(x[i] == 0) gma[i] <- 0 else if(x[i] == 1) gma[i] <- 1 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } a <- x * gma^(1-alpha) + (1-x) * (1 - gma)^(1-beta) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvnegbilog"<- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) gma <- numeric(length(x)) for(i in 1:length(x)) { gmafn <- function(z) (1+alpha) * x[i] * z^alpha - (1+beta) * (1-x[i]) * (1-z)^beta if(is.na(x[i])) gma[i] <- NA else if(x[i] == 0) gma[i] <- 1 else if(x[i] == 1) gma[i] <- 0 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } a <- 1 - x * gma^(1+alpha) - (1-x) * (1 - gma)^(1+beta) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvct" <- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) u <- (alpha * (1-x)) / (alpha * (1-x) + beta * x) a <- (1-x) * pbeta(u, shape1 = alpha, shape2 = beta + 1) + x * pbeta(u, shape1 = alpha + 1, shape2 = beta, lower.tail = FALSE) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "abvamix" <- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) a <- 1 - (alpha + beta) * x + alpha * (x^2) + beta * (x^3) if(plot || add) { if(!add) { plot(x, a, type="n", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, ...) polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty = blty, lwd = blwd) } lines(x, a, lty = lty, lwd = lwd, col = col) return(invisible(list(x = x, y = a))) } a } "hbvevd" <- function(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), half = FALSE, plot = FALSE, add = FALSE, lty = 1, ...) { if(any(x < 0, na.rm = TRUE) || any(x > 1, na.rm = TRUE)) stop("invalid argument for `x'") model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") switch(model, log = hbvlog(x = x, dep = dep, plot = plot, add = add, half = half, lty = lty, ...), alog = hbvalog(x = x, dep = dep, asy = asy, plot = plot, add = add, half = half, lty = lty, ...), hr = hbvhr(x = x, dep = dep, plot = plot, add = add, half = half, lty = lty, ...), neglog = hbvneglog(x = x, dep = dep, plot = plot, add = add, half = half, lty = lty, ...), aneglog = hbvaneglog(x = x, dep = dep, asy = asy, plot = plot, add = add, half = half, lty = lty, ...), bilog = hbvbilog(x = x, alpha = alpha, beta = beta, plot = plot, add = add, half = half, lty = lty, ...), negbilog = hbvnegbilog(x = x, alpha = alpha, beta = beta, plot = plot, add = add, half = half, lty = lty, ...), ct = hbvct(x = x, alpha = alpha, beta = beta, plot = plot, add = add, half = half, lty = lty, ...), amix = hbvamix(x = x, alpha = alpha, beta = beta, plot = plot, add = add, half = half, lty = lty, ...)) } "hbvlog"<- function(x = 0.5, dep, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) idep <- 1/dep h <- (idep - 1) * (x * (1-x))^(-1-idep) * (x^(-idep) + (1-x)^(-idep))^(dep-2) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvalog"<- function(x = 0.5, dep, asy = c(1,1), plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(plot || add) x <- seq(0, 1, length = 100) idep <- 1/dep h <- (idep - 1) * (asy[1] * asy[2])^idep * (x * (1-x))^(-1-idep) * ((asy[1]/x)^idep + (asy[2]/(1-x))^idep)^(dep-2) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvhr" <- function(x = 0.5, dep, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) h <- dep * dnorm(1/dep + dep * log(x/(1-x)) / 2) h <- h / (2 * x * (1-x)^2) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvneglog"<- function(x = 0.5, dep, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) h <- (1 + dep) * (x * (1-x))^(dep-1) * (x^dep + (1-x)^dep)^(-1/dep-2) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvaneglog"<- function(x = 0.5, dep, asy = c(1,1), plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(plot || add) x <- seq(0, 1, length = 100) if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") h <- (1 + dep) * (asy[1] * asy[2])^(-dep) * (x * (1-x))^(dep-1) * ((x/asy[1])^dep + ((1-x)/asy[2])^dep)^(-1/dep-2) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvbilog"<- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0) || any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") if(plot || add) x <- seq(0, 1, length = 100) gma <- numeric(length(x)) for(i in 1:length(x)) { gmafn <- function(z) (1-alpha) * (1-x[i]) * (1-z)^beta - (1-beta) * x[i] * z^alpha if(is.na(x[i])) gma[i] <- NA else if(x[i] == 0) gma[i] <- 0 else if(x[i] == 1) gma[i] <- 1 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } a <- x * gma^(1-alpha) + (1-x) * (1 - gma)^(1-beta) h <- exp(log(1-alpha) + log(beta) + (beta - 1)*log(1-gma) + log(1-x)) + exp(log(1-beta) + log(alpha) + (alpha - 1)*log(gma) + log(x)) h <- exp(log(1-alpha) + log(1-beta) - log(x * (1-x)) - log(h)) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvnegbilog"<- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) gma <- numeric(length(x)) for(i in 1:length(x)) { gmafn <- function(z) (1+alpha) * (1-x[i]) * z^alpha - (1+beta) * x[i] * (1-z)^beta if(is.na(x[i])) gma[i] <- NA else if(x[i] == 0) gma[i] <- 1 else if(x[i] == 1) gma[i] <- 0 else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } h <- exp(log(1+alpha) + log(alpha) + (alpha - 1)*log(gma) + log(1-x)) + exp(log(1+beta) + log(beta) + (beta - 1)*log(1-gma) + log(x)) h <- exp(log(1+alpha) + log(1+beta) + alpha * log(gma) + beta * log(1-gma) - log(x * (1-x)) - log(h)) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvct" <- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) u <- (alpha * x) / (alpha * x + beta * (1-x)) c1 <- alpha * beta / (alpha + beta + 1) h <- dbeta(u, shape1 = alpha + 1, shape2 = beta + 1) / (alpha * x^2 * (1-x) + beta * x * (1-x)^2) * c1 if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "hbvamix" <- function(x = 0.5, alpha, beta, plot = FALSE, add = FALSE, half = FALSE, lty = 1, xlab = "t", ylab = "h(t)", xlim = c(0,1), ylim = c(0, max(h, na.rm = TRUE)), ...) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") if(plot || add) x <- seq(0, 1, length = 100) h <- 2 * alpha + 6 * beta * (1-x) if(half) h <- h/2 if(plot || add) { if(!add) { plot(x, h, type = "l", xlab = xlab, ylab = ylab, xlim = xlim, ylim = ylim, lty = lty, ...) } lines(x, h, lty = lty) return(invisible(list(x = x, y = h))) } h } "dbvevd" <- function(x, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { model <- match.arg(model) m1 <- c("bilog", "negbilog", "ct", "amix") m2 <- c(m1, "log", "hr", "neglog") m3 <- c("log", "alog", "hr", "neglog", "aneglog") if((model %in% m1) && !missing(dep)) warning("ignoring `dep' argument") if((model %in% m2) && !missing(asy)) warning("ignoring `asy' argument") if((model %in% m3) && !missing(alpha)) warning("ignoring `alpha' argument") if((model %in% m3) && !missing(beta)) warning("ignoring `beta' argument") switch(model, log = dbvlog(x = x, dep = dep, mar1 = mar1, mar2 = mar2, log = log), alog = dbvalog(x = x, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2, log = log), hr = dbvhr(x = x, dep = dep, mar1 = mar1, mar2 = mar2, log = log), neglog = dbvneglog(x = x, dep = dep, mar1 = mar1, mar2 = mar2, log = log), aneglog = dbvaneglog(x = x, dep = dep, asy = asy, mar1 = mar1, mar2 = mar2, log = log), bilog = dbvbilog(x = x, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, log = log), negbilog = dbvnegbilog(x = x, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, log = log), ct = dbvct(x = x, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, log = log), amix = dbvamix(x = x, alpha = alpha, beta = beta, mar1 = mar1, mar2 = mar2, log = log)) } "dbvlog"<- function(x, dep, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] idep <- 1/dep z <- apply(x^idep,1,sum)^dep lx <- log(x) .expr1 <- (idep+mar1[,3])*lx[,1] + (idep+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) d[!ext] <- .expr1 + (1-2*idep)*log(z) + log(idep-1+z) - z } if(!log) d <- exp(d) d } "dbvalog"<- function(x, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0 || dep > 1) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] asy <- matrix(asy, ncol = 2, nrow = nrow(x), byrow = TRUE) idep <- 1/dep z <- apply((asy*x)^idep,1,sum)^dep v <- z + apply((1-asy)*x,1,sum) f1asy <- (idep)*log(asy) f2asy <- log(1-asy) lx <- log(x) fx <- (idep-1)*lx jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .expr1 <- apply(f2asy,1,sum) .expr2 <- f2asy[,1] + f1asy[,2] + fx[,2] .expr3 <- f2asy[,2] + f1asy[,1] + fx[,1] .expr4 <- (1-idep)*log(z) + log(exp(.expr2)+exp(.expr3)) .expr5 <- apply(cbind(f1asy,fx),1,sum) + (1-2*idep)*log(z) + log(idep-1+z) d[!ext] <- log(exp(.expr1)+exp(.expr4)+exp(.expr5))-v+jac } if(!log) d <- exp(d) d } "dbvhr" <- function(x, dep, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] fn <- function(x1, x2, nm = pnorm) x1 * nm(1/dep + dep * log(x1/x2) / 2) v <- fn(x[,1], x[,2]) + fn(x[,2], x[,1]) .expr1 <- fn(x[,1], x[,2]) * fn(x[,2], x[,1]) + dep * fn(x[,1], x[,2], nm = dnorm) / 2 lx <- log(x) jac <- mar1[,3]*lx[,1] + mar2[,3]*lx[,2] - log(mar1[,2]*mar2[,2]) d[!ext] <- log(.expr1)+jac-v } if(!log) d <- exp(d) d } "dbvneglog"<- function(x, dep, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] idep <- 1/dep z <- apply(x^(-dep),1,sum)^(-idep) v <- apply(x,1,sum) - z lx <- log(x) fx <- (-dep-1)*lx jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .expr1 <- (1+dep)*log(z) + log(exp(fx[,1])+exp(fx[,2])) .expr2 <- fx[,1] + fx[,2] + (1+2*dep)*log(z) + log(1+dep+z) d[!ext] <- log(1-exp(.expr1)+exp(.expr2))-v+jac } if(!log) d <- exp(d) d } "dbvaneglog"<- function(x, dep, asy = c(1,1), mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(dep) != 1 || mode(dep) != "numeric" || dep <= 0) stop("invalid argument for `dep'") if(length(asy) != 2 || mode(asy) != "numeric" || min(asy) < 0 || max(asy) > 1) stop("invalid argument for `asy'") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] asy <- matrix(asy, ncol = 2, nrow = nrow(x), byrow = TRUE) idep <- 1/dep z <- apply((asy*x)^(-dep),1,sum)^(-idep) v <- apply(x,1,sum) - z fasy <- (-dep)*log(asy) lx <- log(x) fx <- (-dep-1)*lx jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .expr1 <- fasy[,1] + fx[,1] .expr2 <- fasy[,2] + fx[,2] .expr3 <- (1+dep)*log(z) + log(exp(.expr1)+exp(.expr2)) .expr4 <- apply(cbind(fasy,fx),1,sum) + (1+2*dep)*log(z) + log(1+dep+z) d[!ext] <- log(1-exp(.expr3)+exp(.expr4))-v+jac } if(!log) d <- exp(d) d } "dbvbilog"<- function(x, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0) || any(c(alpha,beta) >= 1)) stop("`alpha' and `beta' must be in the open interval (0,1)") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] gma <- numeric(nrow(x)) for(i in 1:nrow(x)) { gmafn <- function(z) (1-alpha) * x[i,1] * (1-z)^beta - (1-beta) * x[i,2] * z^alpha if(any(is.na(x[i,]))) gma[i] <- NA else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } v <- x[,1] * gma^(1-alpha) + x[,2] * (1 - gma)^(1-beta) lx <- log(x) jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .expr1 <- exp((1-alpha)*log(gma) + (1-beta)*log(1-gma)) .expr2 <- exp(log(1-alpha) + log(beta) + (beta - 1)*log(1-gma) + lx[,1]) + exp(log(1-beta) + log(alpha) + (alpha - 1)*log(gma) + lx[,2]) d[!ext] <- log(.expr1 + (1-alpha)*(1-beta)/.expr2) - v + jac } if(!log) d <- exp(d) d } "dbvnegbilog"<- function(x, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] gma <- numeric(nrow(x)) for(i in 1:nrow(x)) { gmafn <- function(z) (1+alpha) * x[i,1] * z^alpha - (1+beta) * x[i,2] * (1-z)^beta if(any(is.na(x[i,]))) gma[i] <- NA else gma[i] <- uniroot(gmafn, lower = 0, upper = 1, tol = .Machine$double.eps^0.5)$root } v <- x[,1] + x[,2] - x[,1] * gma^(1+alpha) - x[,2] * (1 - gma)^(1+beta) lx <- log(x) jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .expr1 <- (1-gma^(1+alpha)) * (1 - (1-gma)^(1+beta)) .expr2 <- exp(log(1+alpha) + log(1+beta) + alpha*log(gma) + beta*log(1-gma)) .expr3 <- exp(log(1+alpha) + log(alpha) + (alpha - 1)*log(gma) + lx[,1]) + exp(log(1+beta) + log(beta) + (beta - 1)*log(1-gma) + lx[,2]) d[!ext] <- log(.expr1 + .expr2/.expr3) - v + jac } if(!log) d <- exp(d) d } "dbvct"<- function(x, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(any(c(alpha,beta) <= 0)) stop("`alpha' and `beta' must be non-negative") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] u <- (alpha * x[,2]) / (alpha * x[,2] + beta * x[,1]) v <- x[,2] * pbeta(u, shape1 = alpha, shape2 = beta + 1) + x[,1] * pbeta(u, shape1 = alpha + 1, shape2 = beta, lower.tail = FALSE) lx <- log(x) jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) .c1 <- alpha * beta / (alpha + beta + 1) .expr1 <- pbeta(u, shape1 = alpha, shape2 = beta + 1) * pbeta(u, shape1 = alpha + 1, shape2 = beta, lower.tail = FALSE) .expr2 <- dbeta(u, shape1 = alpha + 1, shape2 = beta + 1) / (alpha * x[,2] + beta * x[,1]) d[!ext] <- log(.expr1 + .c1 * .expr2) - v + jac } if(!log) d <- exp(d) d } "dbvamix"<- function(x, alpha, beta, mar1 = c(0,1,0), mar2 = mar1, log = FALSE) { if(length(alpha) != 1 || mode(alpha) != "numeric") stop("invalid argument for `alpha'") if(length(beta) != 1 || mode(beta) != "numeric") stop("invalid argument for `beta'") if(alpha < 0) stop("`alpha' must be non-negative") if((alpha + beta) > 1) stop("`alpha' + `beta' cannot be greater than one") if((alpha + 2*beta) > 1) stop("`alpha' + `2*beta' cannot be greater than one") if((alpha + 3*beta) < 0) stop("`alpha' + `3*beta' must be non-negative") if(is.null(dim(x))) dim(x) <- c(1,2) mar1 <- matrix(t(mar1), nrow = nrow(x), ncol = 3, byrow = TRUE) mar2 <- matrix(t(mar2), nrow = nrow(x), ncol = 3, byrow = TRUE) d <- numeric(nrow(x)) x <- mtransform(x, list(mar1, mar2)) ext <- apply(x,1,function(z) any(z %in% c(0,Inf))) d[ext] <- -Inf if(any(!ext)) { x <- x[!ext, ,drop=FALSE] mar1 <- mar1[!ext, ,drop=FALSE] mar2 <- mar2[!ext, ,drop=FALSE] xsum <- apply(x, 1, sum) v <- xsum - (alpha + beta) * x[,1] + alpha * (x[,1]^2)/xsum + beta * (x[,1]^3)/(xsum^2) lx <- log(x) jac <- (1+mar1[,3])*lx[,1] + (1+mar2[,3])*lx[,2] - log(mar1[,2]*mar2[,2]) x1a <- x[,1]/xsum; x2a <- x[,2]/xsum v1 <- 1 - alpha * (x2a)^2 - beta * (3 * x2a^2 - 2 * x2a^3) v2 <- 1 - alpha * (x1a)^2 - 2 * beta * x1a^3 v12 <- (-2 * alpha * x1a * x2a - 6 * beta * x1a^2 * x2a) / xsum d[!ext] <- log(v1 * v2 - v12) - v + jac } if(!log) d <- exp(d) d } evd/R/uvfit.R0000644000175100001440000010573114224760602012541 0ustar hornikusers "fextreme"<- function(x, start, densfun, distnfun, ..., distn, mlen = 1, largest = TRUE, std.err = TRUE, corr = FALSE, method = "Nelder-Mead") { if (missing(x) || length(x) == 0 || !is.numeric(x)) stop("`x' must be a non-empty numeric object") if(any(is.na(x))) stop("`x' must not contain missing values") if (!is.list(start)) stop("`start' must be a named list") call <- match.call() if(missing(densfun)) densfun <- get(paste("d", distn, sep=""), mode="function") if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") nllh <- function(p, ...) { dvec <- dens(p, ..., log = TRUE) if(any(is.infinite(dvec))) return(1e6) else return(-sum(dvec)) } nm <- names(start) l <- length(nm) f1 <- formals(densfun) f2 <- formals(distnfun) args <- names(f1) mtch <- match(nm, args) if (any(is.na(mtch))) stop("`start' specifies unknown arguments") formals(densfun) <- c(f1[c(1, mtch)], f1[-c(1, mtch)]) formals(distnfun) <- c(f2[c(1, mtch)], f2[-c(1, mtch)]) dens <- function(p, x, densfun, distnfun, ...) dextreme(x, densfun, distnfun, p, ...) if(l > 1) body(dens) <- parse(text = paste("dextreme(x, densfun, distnfun,", paste("p[",1:l,"]", collapse = ", "), ", ...)")) opt <- optim(start, nllh, x = x, hessian = TRUE, ..., densfun = densfun, distnfun = distnfun, mlen = mlen, largest = largest, method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if (var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL structure(list(estimate = opt$par, std.err = std.err, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, call = call, data = x, n = length(x)), class = c("extreme", "evd")) } "forder"<- function(x, start, densfun, distnfun, ..., distn, mlen = 1, j = 1, largest = TRUE, std.err = TRUE, corr = FALSE, method = "Nelder-Mead") { if (missing(x) || length(x) == 0 || !is.numeric(x)) stop("`x' must be a non-empty numeric object") if(any(is.na(x))) stop("`x' must not contain missing values") if (!is.list(start)) stop("`start' must be a named list") call <- match.call() if(missing(densfun)) densfun <- get(paste("d", distn, sep=""), mode="function") if(missing(distnfun)) distnfun <- get(paste("p", distn, sep=""), mode="function") nllh <- function(p, ...) { dvec <- dens(p, ..., log = TRUE) if(any(is.infinite(dvec))) return(1e6) else return(-sum(dvec)) } nm <- names(start) l <- length(nm) f1 <- formals(densfun) f2 <- formals(distnfun) args <- names(f1) mtch <- match(nm, args) if (any(is.na(mtch))) stop("`start' specifies unknown arguments") formals(densfun) <- c(f1[c(1, mtch)], f1[-c(1, mtch)]) formals(distnfun) <- c(f2[c(1, mtch)], f2[-c(1, mtch)]) dens <- function(p, x, densfun, distnfun, ...) dorder(x, densfun, distnfun, p, ...) if(l > 1) body(dens) <- parse(text = paste("dorder(x, densfun, distnfun,", paste("p[",1:l,"]", collapse = ", "), ", ...)")) opt <- optim(start, nllh, x = x, hessian = TRUE, ..., densfun = densfun, distnfun = distnfun, mlen = mlen, j = j, largest = largest, method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if (var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL names(std.err) <- nm structure(list(estimate = opt$par, std.err = std.err, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, call = call, data = x, n = length(x)), class = c("extreme", "evd")) } "fgev"<- function(x, start, ..., nsloc = NULL, prob = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { call <- match.call() if(missing(x) || length(x) == 0 || !is.numeric(x)) stop("`x' must be a non-empty numeric vector") if(is.null(prob)) { ft <- fgev.norm(x = x, start = start, ..., nsloc = nsloc, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) } else { if(length(prob) != 1 || !is.numeric(prob) || prob < 0 || prob > 1) stop("`prob' should be a probability in [0,1]") ft <- fgev.quantile(x = x, start = start, ..., nsloc = nsloc, prob = prob, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) } structure(c(ft, call = call), class = c("gev", "uvevd", "evd")) } "fgev.norm"<- function(x, start, ..., nsloc = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlgev <- function(loc, scale, shape) { if(scale <= 0) return(1e6) if(!is.null(nsloc)) { ns <- numeric(length(loc.param)) for(i in 1:length(ns)) ns[i] <- get(loc.param[i]) loc <- drop(nslocmat %*% ns) } else loc <- rep(loc, length.out = length(x)) .C(C_nlgev, x, n, loc, scale, shape, dns = double(1))$dns } if(!is.null(nsloc)) { if(is.vector(nsloc)) nsloc <- data.frame(trend = nsloc) if(nrow(nsloc) != length(x)) stop("`nsloc' and data are not compatible") nsloc <- nsloc[!is.na(x), ,drop = FALSE] nslocmat <- cbind(1,as.matrix(nsloc)) } x <- as.double(x[!is.na(x)]) n <- as.integer(length(x)) loc.param <- paste("loc", c("",names(nsloc)), sep="") param <- c(loc.param, "scale", "shape") if(missing(start)) { start <- as.list(numeric(length(param))) names(start) <- param start$scale <- sqrt(6 * var(x))/pi start$loc <- mean(x) - 0.58 * start$scale start <- start[!(param %in% names(list(...)))] } if(!is.list(start)) stop("`start' must be a named list") if(!length(start)) stop("there are no parameters left to maximize over") nm <- names(start) l <- length(nm) f <- c(as.list(numeric(length(loc.param))), formals(nlgev)[2:3]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlgev) <- c(f[m], f[-m]) nllh <- function(p, ...) nlgev(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlgev(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) fixed.param <- list(...)[names(list(...)) %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL param <- c(opt$par, unlist(fixed.param)) if(!is.null(nsloc)) { trend <- param[paste("loc", names(nsloc), sep="")] trend <- drop(as.matrix(nsloc) %*% trend) x2 <- x - trend } else x2 <- x list(estimate = opt$par, std.err = std.err, fixed = unlist(fixed.param), param = param, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, data = x, tdata = x2, nsloc = nsloc, n = length(x), prob = NULL, loc = param["loc"]) } "fgev.quantile"<- function(x, start, ..., nsloc = NULL, prob, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlgev <- function(quantile, scale, shape) { if(scale <= 0) return(1e6) quantile <- rep(quantile, length.out = length(x)) if(prob == 0 && shape >= 0) return(1e6) if(prob == 1 && shape <= 0) return(1e6) if(shape == 0) loc <- quantile + scale * log(-log(1-prob)) else loc <- quantile + scale/shape * (1 - (-log(1-prob))^(-shape)) if(!is.null(nsloc)) { ns <- numeric(length(loc.param) - 1) for(i in 1:length(ns)) ns[i] <- get(loc.param[i+1]) loc <- drop(nslocmat %*% ns) + loc } if(any(is.infinite(loc))) return(1e6) .C(C_nlgev, x, n, loc, scale, shape, dns = double(1))$dns } if(is.null(nsloc)) loc.param <- "quantile" else loc.param <- c("quantile", paste("loc", names(nsloc), sep="")) param <- c(loc.param, "scale", "shape") if(missing(start)) { start <- as.list(numeric(length(param))) names(start) <- param start$scale <- sqrt(6 * var(x, na.rm = TRUE))/pi start.loc <- mean(x, na.rm = TRUE) - 0.58 * start$scale start$quantile <- start.loc - start$scale * log(-log(1-prob)) if(prob == 0) { fpft <- fgev(x = x, ..., nsloc = nsloc, prob = 0.001, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) start <- as.list(fitted(fpft)) } if(prob == 1) { fpft <- fgev(x = x, ..., nsloc = nsloc, prob = 0.999, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) start <- as.list(fitted(fpft)) } start <- start[!(param %in% names(list(...)))] } if(!is.list(start)) stop("`start' must be a named list") if(!length(start)) stop("there are no parameters left to maximize over") if(!is.null(nsloc)) { if(is.vector(nsloc)) nsloc <- data.frame(trend = nsloc) if(nrow(nsloc) != length(x)) stop("`nsloc' and data are not compatible") nsloc <- nsloc[!is.na(x), ,drop = FALSE] nslocmat <- as.matrix(nsloc) } x <- as.double(x[!is.na(x)]) n <- as.integer(length(x)) nm <- names(start) l <- length(nm) f <- c(as.list(numeric(length(loc.param))), formals(nlgev)[2:3]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlgev) <- c(f[m], f[-m]) nllh <- function(p, ...) nlgev(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlgev(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) fixed.param <- list(...)[names(list(...)) %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm .mat <- diag(1/std.err, nrow = length(std.err)) if(corr) { corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else { std.err <- var.cov <- corr <- NULL } param <- c(opt$par, unlist(fixed.param)) if(!is.null(nsloc)) { trend <- param[paste("loc", names(nsloc), sep="")] trend <- drop(as.matrix(nsloc) %*% trend) x2 <- x - trend } else x2 <- x if(param["shape"] == 0) loc <- param["quantile"] + param["scale"] * log(-log(1-prob)) else loc <- param["quantile"] + param["scale"]/param["shape"] * (1 - (-log(1-prob))^(-param["shape"])) list(estimate = opt$par, std.err = std.err, fixed = unlist(fixed.param), param = param, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, data = x, tdata = x2, nsloc = nsloc, n = length(x), prob = prob, loc = loc) } "fgumbel"<- function(x, start, ..., nsloc = NULL, prob = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { fgev(x = x, ..., shape = 0, nsloc = nsloc, prob = prob, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) } "fgumbelx"<- function(x, start, ..., nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { call <- match.call() if(missing(x) || length(x) == 0 || !is.numeric(x)) stop("`x' must be a non-empty numeric vector") nlgumbelx <- function(loc1, scale1, loc2, scale2) { if(scale1 <= 0 || scale2 <= 0) return(1e6) if(!is.null(nsloc1)) { ns <- numeric(length(loc.param1)) for(i in 1:length(ns)) ns[i] <- get(loc.param1[i]) loc1 <- drop(nslocmat1 %*% ns) } else loc1 <- rep(loc1, length.out = length(x)) if(!is.null(nsloc2)) { ns <- numeric(length(loc.param2)) for(i in 1:length(ns)) ns[i] <- get(loc.param2[i]) loc2 <- drop(nslocmat2 %*% ns) } else loc2 <- rep(loc2, length.out = length(x)) if(any(loc1 > loc2)) return(1e6) .C(C_nlgumbelx, x, n, loc1, scale1, loc2, scale2, dns = double(1))$dns } if(!is.null(nsloc1)) { if(is.vector(nsloc1)) nsloc1 <- data.frame(trend = nsloc1) if(nrow(nsloc1) != length(x)) stop("`nsloc1' and data are not compatible") nslocmat1 <- cbind(1, as.matrix(nsloc1)) } if(!is.null(nsloc2)) { if(is.vector(nsloc2)) nsloc2 <- data.frame(trend = nsloc2) if(nrow(nsloc2) != length(x)) stop("`nsloc2' and data are not compatible") nslocmat2 <- cbind(1,as.matrix(nsloc2)) } x <- as.double(x[!is.na(x)]) n <- as.integer(length(x)) loc.param1 <- paste("loc1", c("",names(nsloc1)), sep="") loc.param2 <- paste("loc2", c("",names(nsloc2)), sep="") param <- c(loc.param1, "scale1", loc.param2, "scale2") if(missing(start)) { start <- as.list(numeric(length(param))) names(start) <- param emc <- -digamma(1) b0 <- mean(x) b1 <- sum((1:n-1)/(n-1) * sort(x))/n start$scale1 <- (2*b1-b0)/log(2) start$loc1 <- b0 - start$scale1*emc - start$scale1*log(2) start$scale2 <- start$scale1 start$loc2 <- start$loc1 + 1e-02 start <- start[!(param %in% names(list(...)))] } if(!is.list(start)) stop("`start' must be a named list") if(!length(start)) stop("there are no parameters left to maximize over") nm <- names(start) l <- length(nm) f <- c(as.list(numeric(length(loc.param1))), formals(nlgumbelx)[2], as.list(numeric(length(loc.param2))), formals(nlgumbelx)[4]) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlgumbelx) <- c(f[m], f[-m]) nllh <- function(p, ...) nlgumbelx(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlgumbelx(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) fixed.param <- list(...)[names(list(...)) %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL param <- c(opt$par, unlist(fixed.param)) ft <- list(estimate = opt$par, std.err = std.err, fixed = unlist(fixed.param), param = param, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, data = x, nsloc1 = nsloc1, nsloc2 = nsloc2, n = length(x)) structure(c(ft, call = call), class = c("gumbelx", "evd")) } "fpot"<- function(x, threshold, model = c("gpd", "pp"), start, npp = length(x), cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, mper = NULL, ..., std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { call <- match.call() model <- match.arg(model) if(missing(x) || length(x) == 0 || mode(x) != "numeric") stop("`x' must be a non-empty numeric vector") if(missing(threshold) || length(threshold) != 1 || mode(threshold) != "numeric") stop("`threshold' must be a numeric value") threshold <- as.double(threshold) if(is.null(mper)) { ft <- fpot.norm(x = x, threshold = threshold, model = model, start = start, npp = npp, cmax = cmax, r = r, ulow = ulow, rlow = rlow, ..., std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) } else { if(model == "pp") stop("`mper' cannot be specified in point process models") ft <- fpot.quantile(x = x, threshold = threshold, start = start, npp = npp, cmax = cmax, r = r, ulow = ulow, rlow = rlow, ..., mper = mper, std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) } structure(c(ft, call = call), class = c("pot", "uvevd", "evd")) } "fpot.norm"<- function(x, threshold, model, start, npp = length(x), cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, ..., std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { if(model == "gpd") { nlpot <- function(loc, scale, shape) { .C(C_nlgpd, exceed, nhigh, threshold, scale, shape, dns = double(1))$dns } # Avoids note produced by R CMD check formals(nlpot) <- formals(nlpot)[2:3] } if(model == "pp") { nlpot <- function(loc, scale, shape) { .C(C_nlpp, exceed, nhigh, loc, scale, shape, threshold, nop, dns = double(1))$dns } } nn <- length(x) nop <- as.double(nn/npp) if(cmax) { exceed <- clusters(x, u = threshold, r = r, ulow = ulow, rlow = rlow, cmax = TRUE, keep.names = FALSE) extind <- attributes(exceed)$acs exceed <- as.double(exceed) nhigh <- length(exceed) ; nat <- as.integer(nhigh * extind) extind <- 1/extind } else { extind <- r <- NULL high <- (x > threshold) & !is.na(x) exceed <- as.double(x[high]) nhigh <- nat <- length(exceed) } if(!nhigh) stop("no data above threshold") pat <- nat/nn param <- c("scale", "shape") if(model == "pp") param <- c("loc", param) if(missing(start)) { if(model == "gpd") { start <- list(scale = 0, shape = 0) start$scale <- mean(exceed) - threshold } if(model == "pp") { start <- list(loc = 0, scale = 0, shape = 0) start$scale <- sqrt(6 * var(x))/pi start$loc <- mean(x) + (log(nop) - 0.58) * start$scale } start <- start[!(param %in% names(list(...)))] } if(!is.list(start)) stop("`start' must be a named list") if(!length(start)) stop("there are no parameters left to maximize over") nm <- names(start) l <- length(nm) f <- formals(nlpot) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlpot) <- c(f[m], f[-m]) nllh <- function(p, ...) nlpot(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlpot(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) fixed.param <- list(...)[names(list(...)) %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL param <- c(opt$par, unlist(fixed.param)) if(model == "gpd") scale <- param["scale"] if(model == "pp") scale <- param["scale"] + param["shape"] * (threshold - param["loc"]) list(estimate = opt$par, std.err = std.err, fixed = unlist(fixed.param), param = param, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, threshold = threshold, cmax = cmax, r = r, ulow = ulow, rlow = rlow, npp = npp, nhigh = nhigh, nat = nat, pat = pat, extind = extind, data = x, exceedances = exceed, mper = NULL, scale = scale) } "fpot.quantile"<- function(x, threshold, start, npp = length(x), cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, mper, ..., std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) { nlpot <- function(rlevel, shape) { if(is.infinite(mper) && shape >= 0) return(1e6) rlevel <- rlevel - threshold if(shape == 0) scale <- rlevel / log(adjmper) else scale <- shape * rlevel / (adjmper^shape - 1) .C(C_nlgpd, exceed, nhigh, threshold, scale, shape, dns = double(1))$dns } nn <- length(x) if(cmax) { exceed <- clusters(x, u = threshold, r = r, ulow = ulow, rlow = rlow, cmax = TRUE, keep.names = FALSE) extind <- attributes(exceed)$acs exceed <- as.double(exceed) nhigh <- length(exceed) ; nat <- as.integer(nhigh * extind) extind <- 1/extind } else { extind <- r <- NULL high <- (x > threshold) & !is.na(x) exceed <- as.double(x[high]) nhigh <- nat <- length(exceed) } if(!nhigh) stop("no data above threshold") pat <- nat/nn adjmper <- mper * npp * nhigh/nn if(adjmper <= 1) stop("`mper' is too small") param <- c("rlevel", "shape") if(missing(start)) { start <- list(rlevel = 0, shape = 0) stscale <- mean(exceed) - threshold start$rlevel <- threshold + stscale*log(adjmper) if(is.infinite(mper)) { stmp <- 100/(npp * nhigh/nn) fpft <- fpot(x = x, threshold = threshold, npp = npp, cmax = cmax, r = r, ulow = ulow, rlow = rlow, mper = stmp, ..., std.err = std.err, corr = corr, method = method, warn.inf = warn.inf) start <- as.list(fitted(fpft)) } start <- start[!(param %in% names(list(...)))] } if(!is.list(start)) stop("`start' must be a named list") if(!length(start)) stop("there are no parameters left to maximize over") nm <- names(start) l <- length(nm) f <- formals(nlpot) names(f) <- param m <- match(nm, param) if(any(is.na(m))) stop("`start' specifies unknown arguments") formals(nlpot) <- c(f[m], f[-m]) nllh <- function(p, ...) nlpot(p, ...) if(l > 1) body(nllh) <- parse(text = paste("nlpot(", paste("p[",1:l, "]", collapse = ", "), ", ...)")) fixed.param <- list(...)[names(list(...)) %in% param] if(any(!(param %in% c(nm,names(fixed.param))))) stop("unspecified parameters") start.arg <- c(list(p = unlist(start)), fixed.param) if(warn.inf && do.call("nllh", start.arg) == 1e6) warning("negative log-likelihood is infinite at starting values") opt <- optim(start, nllh, hessian = TRUE, ..., method = method) if(is.null(names(opt$par))) names(opt$par) <- nm if (opt$convergence != 0) { warning("optimization may not have succeeded") if(opt$convergence == 1) opt$convergence <- "iteration limit reached" } else opt$convergence <- "successful" if(std.err) { tol <- .Machine$double.eps^0.5 var.cov <- qr(opt$hessian, tol = tol) if(var.cov$rank != ncol(var.cov$qr)) stop("observed information matrix is singular; use std.err = FALSE") var.cov <- solve(var.cov, tol = tol) std.err <- diag(var.cov) if(any(std.err <= 0)) stop("observed information matrix is singular; use std.err = FALSE") std.err <- sqrt(std.err) names(std.err) <- nm if(corr) { .mat <- diag(1/std.err, nrow = length(std.err)) corr <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm)) diag(corr) <- rep(1, length(std.err)) } else corr <- NULL } else std.err <- var.cov <- corr <- NULL param <- c(opt$par, unlist(fixed.param)) rlevel <- param["rlevel"] - threshold if(param["shape"] == 0) scale <- rlevel / log(adjmper) else scale <- param["shape"] * rlevel / (adjmper^param["shape"] - 1) list(estimate = opt$par, std.err = std.err, fixed = unlist(fixed.param), param = param, deviance = 2*opt$value, corr = corr, var.cov = var.cov, convergence = opt$convergence, counts = opt$counts, message = opt$message, threshold = threshold, cmax = cmax, r = r, ulow = ulow, rlow = rlow, npp = npp, nhigh = nhigh, nat = nat, pat = pat, extind = extind, data = x, exceedances = exceed, mper = mper, scale = scale) } ### Method Functions ### "print.evd" <- function(x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:", deparse(x$call), "\n") cat("Deviance:", x$deviance, "\n") cat("\nEstimates\n") print.default(format(x$estimate, digits = digits), print.gap = 2, quote = FALSE) if(!is.null(x$std.err)) { cat("\nStandard Errors\n") print.default(format(x$std.err, digits = digits), print.gap = 2, quote = FALSE) } if(!is.null(x$corr)) { cat("\nCorrelations\n") print.default(format(x$corr, digits = digits), print.gap = 2, quote = FALSE) } cat("\nOptimization Information\n") cat(" Convergence:", x$convergence, "\n") cat(" Function Evaluations:", x$counts["function"], "\n") if(!is.na(x$counts["gradient"])) cat(" Gradient Evaluations:", x$counts["gradient"], "\n") if(!is.null(x$message)) cat(" Message:", x$message, "\n") cat("\n") invisible(x) } "confint.evd" <- function (object, parm, level = 0.95, ...) { cf <- fitted(object) pnames <- names(cf) if (missing(parm)) parm <- seq(along = pnames) else if (is.character(parm)) parm <- match(parm, pnames, nomatch = 0) if(any(!parm)) stop("`parm' contains unknown parameters") a <- (1 - level)/2 a <- c(a, 1 - a) pct <- paste(round(100 * a, 1), "%") ci <- array(NA, dim = c(length(parm), 2), dimnames = list(pnames[parm], pct)) ses <- std.errors(object)[parm] ci[] <- cf[parm] + ses %o% qnorm(a) ci } "anova.evd" <- function (object, object2, ..., half = FALSE) { if(missing(object)) stop("model one must be specified") if(missing(object2)) stop("model two must be specified") dots <- as.list(substitute(list(...)))[-1] dots <- sapply(dots,function(x) deparse(x)) if(!length(dots)) dots <- NULL model1 <- deparse(substitute(object)) model2 <- deparse(substitute(object2)) models <- c(model1, model2, dots) narg <- length(models) for(i in 1:narg) { if(!inherits(get(models[i], envir = parent.frame()), "evd")) stop("Use only with 'evd' objects") } for(i in 1:(narg-1)) { a <- get(models[i], envir = parent.frame()) b <- get(models[i+1], envir = parent.frame()) if((!all(names(fitted(b)) %in% names(fitted(a)))) && (!identical(c("bilog","log"), c(a$model, b$model))) && (!identical(c("negbilog","neglog"), c(a$model, b$model)))) { warning("models may not be nested") } } dv <- npar <- numeric(narg) for(i in 1:narg) { evmod <- get(models[i], envir = parent.frame()) dv[i] <- evmod$deviance npar[i] <- length(evmod$estimate) } df <- -diff(npar) if(any(df <= 0)) stop("models are not nested") dvdiff <- diff(dv) if(any(dvdiff < 0)) stop("negative deviance difference") if(half) dvdiff <- 2*dvdiff pval <- pchisq(dvdiff, df = df, lower.tail = FALSE) table <- data.frame(npar, dv, c(NA,df), c(NA,dvdiff), c(NA,pval)) dimnames(table) <- list(models, c("M.Df", "Deviance", "Df", "Chisq", "Pr(>chisq)")) structure(table, heading = c("Analysis of Deviance Table\n"), class = c("anova", "data.frame")) } "fitted.evd" <- function (object, ...) object$estimate "std.errors" <- function (object, ...) UseMethod("std.errors") "std.errors.evd" <- function (object, ...) object$std.err "vcov.evd" <- function (object, ...) object$var.cov "logLik.evd" <- function(object, ...) { val <- -deviance(object)/2 attr(val, "df") <- length(fitted(object)) class(val) <- "logLik" val } "print.pot" <- function(x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:", deparse(x$call), "\n") cat("Deviance:", x$deviance, "\n") cat("\nThreshold:", round(x$threshold, digits), "\n") cat("Number Above:", x$nat, "\n") cat("Proportion Above:", round(x$pat, digits), "\n") if(!is.null(x$extind)) { cat("\nClustering Interval:", x$r, "\n") if(is.finite(x$ulow)) { cat("Lower Threshold:", round(x$ulow, digits), "\n") cat("Lower Clustering Interval:", x$rlow, "\n") } cat("Number of Clusters:", x$nhigh, "\n") cat("Extremal Index:", round(x$extind, digits), "\n") } cat("\nEstimates\n") print.default(format(x$estimate, digits = digits), print.gap = 2, quote = FALSE) if(!is.null(x$std.err)) { cat("\nStandard Errors\n") print.default(format(x$std.err, digits = digits), print.gap = 2, quote = FALSE) } if(!is.null(x$corr)) { cat("\nCorrelations\n") print.default(format(x$corr, digits = digits), print.gap = 2, quote = FALSE) } cat("\nOptimization Information\n") cat(" Convergence:", x$convergence, "\n") cat(" Function Evaluations:", x$counts["function"], "\n") if(!is.na(x$counts["gradient"])) cat(" Gradient Evaluations:", x$counts["gradient"], "\n") if(!is.null(x$message)) cat(" Message:", x$message, "\n") cat("\n") invisible(x) } evd/R/pmdiag.R0000644000175100001440000002671012637167310012647 0ustar hornikusers "mrlplot"<- function(data, tlim, pscale = FALSE, nt = max(100, length(data)), lty = c(2,1,2), col = 1, conf = 0.95, main = "Mean Residual Life Plot", xlab = "Threshold", ylab = "Mean Excess", ...) { data <- sort(data[!is.na(data)]) nn <- length(data) if(nn <= 5) stop("`data' has too few non-missing values") if(missing(tlim)) { tlim <- c(data[1], data[nn - 4]) tlim <- tlim - .Machine$double.eps^0.5 } if(all(data <= tlim[2])) stop("upper limit for threshold is too high") u <- seq(tlim[1], tlim[2], length = nt) if(pscale) { tlim[1] <- mean(data <= tlim[1], na.rm = TRUE) tlim[2] <- mean(data <= tlim[2], na.rm = TRUE) pvec <- seq(tlim[1], tlim[2], length = nt) u <- quantile(data, probs = pvec, na.rm = TRUE) } x <- matrix(NA, nrow = nt, ncol = 3, dimnames = list(NULL, c("lower", "mrl", "upper"))) for(i in 1:nt) { data <- data[data > u[i]] x[i,2] <- mean(data - u[i]) sdev <- sqrt(var(data)) sdev <- (qnorm((1 + conf)/2) * sdev)/sqrt(length(data)) x[i,1] <- x[i,2] - sdev x[i,3] <- x[i,2] + sdev } if(pscale) { u <- pvec if(missing(xlab)) xlab <- "Threshold probability" } matplot(u, x, type = "l", lty = lty, col = col, main = main, xlab = xlab, ylab = ylab, ...) invisible(list(x = u, y = x)) } "tcplot"<- function(data, tlim, model = c("gpd", "pp"), pscale = FALSE, cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, nt = 25, which = 1:npar, conf = 0.95, lty = 1, lwd = 1, type = "b", cilty = 1, vci = TRUE, xlab, xlim, ylabs, ylims, ask = nb.fig < length(which) && dev.interactive(), ...) { model <- match.arg(model) u <- seq(tlim[1], tlim[2], length = nt) if(pscale) { tlim[1] <- mean(data <= tlim[1], na.rm = TRUE) tlim[2] <- mean(data <= tlim[2], na.rm = TRUE) pvec <- seq(tlim[1], tlim[2], length = nt) u <- quantile(data, probs = pvec, na.rm = TRUE) } locs <- scls <- shps <- matrix(NA, nrow = nt, ncol = 3) dimnames(locs) <- list(round(u,2), c("lower", "loc", "upper")) dimnames(shps) <- list(round(u,2), c("lower", "shape", "upper")) if(model == "gpd") { pname <- "mscale" npar <- 2 } if(model == "pp") { pname <- "scale" npar <- 3 } dimnames(scls) <- list(round(u,2), c("lower", pname, "upper")) z <- fpot(data, u[1], model = model, cmax = cmax, r = r, ulow = ulow, rlow = rlow, corr = TRUE, ...) stvals <- as.list(round(fitted(z), 3)) for(i in 1:nt) { z <- fpot(data, u[i], model = model, start = stvals, cmax = cmax, r = r, ulow = ulow, rlow = rlow, corr = TRUE, ...) stvals <- as.list(fitted(z)) mles <- fitted(z) stderrs <- std.errors(z) cnst <- qnorm((1 + conf)/2) shp <- mles["shape"] scl <- mles["scale"] shpse <- stderrs["shape"] sclse <- stderrs["scale"] if(model == "pp") { loc <- mles["loc"] locse <- stderrs["loc"] locs[i,] <- c(loc - cnst*locse, loc, loc + cnst*locse) } if(model == "gpd") { scl <- scl - shp*u[i] covar <- z$corr[1,2] * prod(stderrs) sclse <- sqrt(sclse^2 - 2*u[i]*covar + (u[i]*shpse)^2) } scls[i,] <- c(scl - cnst*sclse, scl, scl + cnst*sclse) shps[i,] <- c(shp - cnst*shpse, shp, shp + cnst*shpse) } show <- rep(FALSE, npar) show[which] <- TRUE nb.fig <- prod(par("mfcol")) if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if(pscale) u <- pvec if(missing(xlim)) xlim <- tlim if(missing(xlab)) { xlab <- "Threshold" if(pscale) xlab <- "Threshold probability" } if(model == "pp") { ylab <- c("Location","Scale","Shape") if(!missing(ylabs)) ylab[show] <- ylabs ylim <- rbind(range(locs), range(scls), range(shps)) if(!missing(ylims)) ylim[show,] <- ylims if(show[1]) { matplot(u, locs, type = "n", xlab = xlab, ylab = ylab[1], xlim = xlim, ylim = ylim[1,]) lines(u, locs[,2], lty = lty, lwd = lwd, type = type) if(vci) segments(u, locs[,1], u, locs[,3], lty = cilty) else { lines(u, locs[,1], lty = cilty) lines(u, locs[,3], lty = cilty) } } if(show[2]) { matplot(u, scls, type = "n", xlab = xlab, ylab = ylab[2], xlim = xlim, ylim = ylim[2,]) lines(u, scls[,2], lty = lty, lwd = lwd, type = type) if(vci) segments(u, scls[,1], u, scls[,3], lty = cilty) else { lines(u, scls[,1], lty = cilty) lines(u, scls[,3], lty = cilty) } } if(show[3]) { matplot(u, shps, type = "n", xlab = xlab, ylab = ylab[3], xlim = xlim, ylim = ylim[3,]) lines(u, shps[,2], lty = lty, lwd = lwd, type = type) if(vci) segments(u, shps[,1], u, shps[,3], lty = cilty) else { lines(u, shps[,1], lty = cilty) lines(u, shps[,3], lty = cilty) } } rtlist <- list(locs = locs, scales = scls, shapes = shps) } if(model == "gpd") { ylab <- c("Modified Scale","Shape") if(!missing(ylabs)) ylab[show] <- ylabs ylim <- rbind(range(scls), range(shps)) if(!missing(ylims)) ylim[show,] <- ylims if(show[1]) { matplot(u, scls, type = "n", xlab = xlab, ylab = ylab[1], xlim = xlim, ylim = ylim[1,]) lines(u, scls[,2], lty = lty, lwd = lwd, type = type) if(vci) segments(u, scls[,1], u, scls[,3], lty = cilty) else { lines(u, scls[,1], lty = cilty) lines(u, scls[,3], lty = cilty) } } if(show[2]) { matplot(u, shps, type = "n", xlab = xlab, ylab = ylab[2], xlim = xlim, ylim = ylim[2,]) lines(u, shps[,2], lty = lty, lwd = lwd, type = type) if(vci) segments(u, shps[,1], u, shps[,3], lty = cilty) else { lines(u, shps[,1], lty = cilty) lines(u, shps[,3], lty = cilty) } } rtlist <- list(scales = scls, shapes = shps) } invisible(rtlist) } "chiplot"<- function(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, trunc = TRUE, spcases = FALSE, lty = 1, cilty = 2, col = 1, cicol = 1, xlim = c(0,1), ylim1 = c(-1,1), ylim2 = c(-1,1), main1 = "Chi Plot", main2 = "Chi Bar Plot", xlab = "Quantile", ylab1 = "Chi", ylab2 = "Chi Bar", ask = nb.fig < length(which) && dev.interactive(), ...) { data <- na.omit(data) n <- nrow(data) data <- cbind(rank(data[, 1])/(n + 1), rank(data[, 2])/(n + 1)) rowmax <- apply(data, 1, max) rowmin <- apply(data, 1, min) eps <- .Machine$double.eps^0.5 qlim2 <- c(min(rowmax) + eps, max(rowmin) - eps) if(!is.null(qlim)) { if(qlim[1] < qlim2[1]) stop("lower quantile limit is too low") if(qlim[2] > qlim2[2]) stop("upper quantile limit is too high") if(qlim[1] > qlim[2]) stop("lower quantile limit is less than upper quantile limit") } else qlim <- qlim2 u <- seq(qlim[1], qlim[2], length = nq) cu <- cbaru <- numeric(nq) for(i in 1:nq) cu[i] <- mean(rowmax < u[i]) for(i in 1:nq) cbaru[i] <- mean(rowmin > u[i]) chiu <- 2 - log(cu)/log(u) chibaru <- (2 * log(1 - u))/log(cbaru) - 1 cnst <- qnorm((1 + conf)/2) varchi <- ((1/log(u)^2 * 1)/cu^2 * cu * (1 - cu))/n varchi <- cnst * sqrt(varchi) varchibar <- (((4 * log(1 - u)^2)/(log(cbaru)^4 * cbaru^2)) * cbaru * ( 1 - cbaru))/n varchibar <- cnst * sqrt(varchibar) chiu <- cbind(chilow = chiu-varchi, chi = chiu, chiupp = chiu+varchi) chibaru <- cbind(chiblow = chibaru-varchibar, chib = chibaru, chibupp = chibaru+varchibar) chiulb <- 2-log(pmax(2*u-1,0))/log(u) chibarulb <- 2*log(1-u)/log(1-2*u+pmax(2*u-1,0)) - 1 if(trunc) { chiu[chiu > 1] <- 1 chibaru[chibaru > 1] <- 1 chiu <- apply(chiu, 2, function(x) pmax(x, chiulb)) chibaru <- apply(chibaru, 2, function(x) pmax(x, chibarulb)) } show <- logical(2) show[which] <- TRUE lty <- c(cilty, lty, cilty) col <- c(cicol, col, cicol) nb.fig <- prod(par("mfcol")) if (ask) { op <- par(ask = TRUE) on.exit(par(op)) } if(show[1]) { matplot(u, chiu, type = "l", lty = lty, col = col, xlim = xlim, ylim = ylim1, main = main1, xlab = xlab, ylab = ylab1, ...) if(spcases) { segments(qlim[1],0,qlim[2],0, lty = 5, col = "grey") segments(qlim[1],1,qlim[2],1, lty = 5, col = "grey") lines(u, chiulb, lty = 5, col = "grey") } } if(show[2]) { matplot(u, chibaru, type = "l", lty = lty, col = col, xlim = xlim, ylim = ylim2, main = main2, xlab = xlab, ylab = ylab2, ...) if(spcases) { segments(qlim[1],0,qlim[2],0, lty = 5, col = "grey") segments(qlim[1],1,qlim[2],1, lty = 5, col = "grey") lines(u, chibarulb, lty = 5, col = "grey") } } plvals <- list(quantile = u, chi = chiu, chibar = chibaru) if(!show[1]) plvals$chi <- NULL if(!show[2]) plvals$chib <- NULL invisible(plvals) } ## Bivariate Threshold Choice ## "bvtcplot"<- function(x, spectral = FALSE, xlab, ylab, ...) { if(!is.matrix(x) && !is.data.frame(x)) stop("`x' must be a matrix or data frame") if(ncol(x) != 2) stop("`x' has incorrect number of columns") x <- x[complete.cases(x),] nn <- nrow(x) ula <- apply(x, 2, rank)/(nn + 1) fla <- -1/log(ula) rr <- rowSums(fla); ww <- fla/rr rro <- sort(rr, decreasing = TRUE)[-1] k <- 1:(nn-1) k0 <- max(which(rro*k/nn > 2)) if(!spectral) { if(missing(xlab)) xlab <- "k" if(missing(ylab)) ylab <- "H([0,1])" plot(k, rro*k/nn, xlab = xlab, ylab = ylab, ...) abline(h = 2, v = k0) return(invisible(list(x = k, y = rro*k/nn, k0 = k0))) } xx <- yy <- seq(0, 1, len = 100) for(k in 1:100) yy[k] <- sum(rr > rro[k0] & ww[,1] <= xx[k]) if(missing(xlab)) xlab <- "w" if(missing(ylab)) ylab <- "H([0,w])" plot(xx, 2/k0 * yy, type = "l", xlab = xlab, ylab = ylab, ...) abline(h = c(0,2)) return(invisible(list(x = xx, y = 2/k0 * yy, k0 = k0))) } ## Hypothesis test for independence ## "evind.test"<- function(x, method = c("ratio", "score"), verbose = FALSE) { method <- match.arg(method) if(!is.matrix(x) && !is.data.frame(x)) stop("`x' must be a matrix or data frame") if(ncol(x) != 2) stop("`x' has incorrect number of columns") dname <- paste(deparse(substitute(x))) if(method == "ratio") { meth <- "Likelihood Ratio Test Of Independence" fobj1 <- fbvevd(x, model = "log") estimate <- fitted(fobj1) if(!verbose) estimate <- estimate["dep"] fobj2 <- fbvevd(x, model = "log", dep = 1) lrt <- anova(fobj1, fobj2, half = TRUE) stat <- c(norm.llhratio = lrt[["Chisq"]][2]) pval <- c(p.value = lrt[["Pr(>chisq)"]][2]) } if(method == "score") { meth <- "Score Test Of Independence" fobj1 <- fbvevd(x, model = "log") estimate <- fitted(fobj1) if(!verbose) estimate <- estimate["dep"] fobj2 <- fbvevd(x, model = "log", dep = 1) ft <- fitted(fobj2) mmles <- list(ft[c("loc1","scale1","shape1")], ft[c("loc2","scale2","shape2")]) xtr <- mtransform(x, mmles) xtr <- xtr[complete.cases(xtr),] nn <- nrow(xtr) rsm <- rowSums(xtr) rsl <- rowSums(xtr * log(xtr)) tawn <- rsl - log(apply(xtr, 1, prod)) - (rsm - 2) * log(rsm) - 1/rsm tawn <- (nn/2 * log(nn))^(-1/2) * sum(tawn) stat <- c(norm.score = tawn) pval <- c(p.value = pnorm(tawn)) } rval <- list(statistic = stat, p.value = pval, estimate = estimate, null.value = c(dependence = "independence"), alternative = "greater", method = meth, data.name = dname) class(rval) <- "htest" return(rval) } evd/vignettes/0000755000175100001440000000000014673500175013066 5ustar hornikusersevd/vignettes/Multivariate_Extremes.Rnw0000644000175100001440000004631212637167310020104 0ustar hornikusers\documentclass[11pt,a4paper]{article} \usepackage{amsmath,amssymb} \pagestyle{plain} \setlength{\parindent}{0in} \setlength{\parskip}{1.5ex plus 0.5ex minus 0.5ex} \setlength{\oddsidemargin}{0in} \setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.3in} \setlength{\textheight}{9.8in} %\VignetteIndexEntry{Statistics Of Extremes: Chapter 9} \begin{document} \title{Statistics of Multivariate Extremes} \author{Alec Stephenson} \maketitle \begin{center} \LARGE \textbf{Summary} \\ \end{center} \normalsize \vspace{0.5cm} This vignette uses the \textbf{evd} package to reproduce the figures, tables and analysis in Chapter 9 of Beirlant et al.\ (2001). The chapter was written by Segers and Vandewalle (2004). The code reproduces almost all figures, but for space reasons only some are shown. Deviations from the book are given as footnotes. Differences will inevitably exist due to numerical optimization and random number generation. \normalsize \section{Introduction} \label{Intro} The methods used here are illustrated using the \texttt{lossalae} dataset, which contains observations on $1500$ liability claims. The indemnity payment (loss) and the allocated loss adjustment expense (ALAE) is recorded in USD for each claim. The ALAE is the additional expenses associated with the settlement of the claim (e.g.\ claims investigation expenses and legal fees). The dataset also has an attribute called \texttt{capped}, which gives the row names of the indemnity payments that were capped at their policy limit. We first scale the data so that one unit corresponds to $100\,000$ USD. Putting the data on a sensible scale assists with the numerical optimization involved in maximum likelihood estimation\footnote{The book reports an unsatisfactory fit of the GEV model to the margins. It therefore uses only empirical marginal distributions. This was perhaps due to not scaling the data. In this document we use either fully nonparametric or fully parametric methods.}. The code below plots the raw data using the log scale for both axes (see Figure \ref{rawdata}), and plots the data transformed to uniform $(0,1)$ margins using an empirical transform. <>= options(show.signif.stars=FALSE) library(evd); nn <- nrow(lossalae) loss <- lossalae/1e+05; lts <- c(1e-04, 100) plot(loss, log = "xy", xlim = lts, ylim = lts) @ <<>>= ula <- apply(loss, 2, rank)/(nn + 1) plot(ula) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Scatterplot of ALAE verses Loss: original data (log-scale).} \label{rawdata} \end{figure} \section{Parametric Models} Any bivariate extreme value distribution function can be represented in the form \begin{equation*} G(z_1,z_2) = \exp\left\{ - (y_1 + y_2)A\left(\frac{y_1}{y_1+y_2}\right)\right\}, \label{bvdepfn} \end{equation*} where \begin{equation*} y_j = y_j(z_j) = \{1+\xi_j(z_j-\mu_j)/\sigma_j\}_{+}^{-1/\xi_j} \label{mtrans} \end{equation*} for $\sigma_j > 0$ and $j=1,2$, and where \begin{equation*} A(\omega)=-\log\{G(y_1^{-1}(\omega),y_2^{-1}(1-\omega))\}, \label{dep} \end{equation*} defined on $0\leq\omega\leq1$ is called the dependence function\footnote{The book uses the definition $B(\omega) = A(1-\omega)$.}. The marginal distributions are generalized extreme value (GEV), given by $G_j(z_j) = \exp(-y_j)$. It follows that $A(0)=A(1)=1$, and that $A(\cdot)$ is a convex function with $\max(\omega,1-\omega) \leq A(\omega) \leq 1$ for all $0\leq\omega\leq1$. At independence $A(1/2) = 1$. At complete dependence $A(1/2) = 0.5$. The dependence function represents only the dependence structure of the distribution, and hence only the dependence parameters of parametric models need to be specified in order to produce dependence function plots. The code below plots dependence functions for four different parametric models. The first of these is given in Figure \ref{asylogdfn}. <>= abvevd(dep = 0.5, asy = c(1,1), model = "alog", plot = TRUE) abvevd(dep = 0.5, asy = c(0.6,0.9), model = "alog", add = TRUE, lty = 2) abvevd(dep = 0.5, asy = c(0.8,0.5), model = "alog", add = TRUE, lty = 3) @ <<>>= abvevd(dep = -1/(-2), model = "neglog", plot = TRUE) abvevd(dep = -1/(-1), model = "neglog", add = TRUE, lty = 2) abvevd(dep = -1/(-0.5), model = "neglog", add = TRUE, lty = 3) @ <<>>= abvevd(alpha = 1, beta = -0.2, model = "amix", plot = TRUE) abvevd(alpha = 0.6, beta = 0.1, model = "amix", add = TRUE, lty = 2) abvevd(alpha = 0.2, beta = 0.2, model = "amix", add = TRUE, lty = 3) @ <<>>= abvevd(dep = 1/1.25, model = "hr", plot = TRUE) abvevd(dep = 1/0.83, model = "hr", add = TRUE, lty = 2) abvevd(dep = 1/0.5, model = "hr", add = TRUE, lty = 3) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Dependence functions: asymmetric logistic model.} \label{asylogdfn} \end{figure} \section{Componentwise Maxima} For demonstration purposes we use the data introduced in Section \ref{Intro} to create a dataset of componentwise block maxima by randomly taking $k=50$ groups of size $m=30$, producing $k$ componentwise maxima taken over $m$ observations\footnote{The data may be completely different to the book due to random selection.}. Bivariate extreme value distributions are typically used to model data of this type. The code below creates the componentwise maxima data \texttt{cml} and produces two data plots, the first showing the original data and the componentwise maxima, and the second showing the componentwise maxima data transformed to standard exponential margins. <<>>= set.seed(131); cml <- loss[sample(nn),] xx <- rep(1:50, each = 30); lts <- c(1e-04, 100) cml <- cbind(tapply(cml[,1], xx, max), tapply(cml[,2], xx, max)) colnames(cml) <- colnames(loss) plot(loss, log = "xy", xlim = lts, ylim = lts, col = "grey") points(cml) ecml <- -log(apply(cml,2,rank)/51) plot(ecml) @ The following code estimates and plots the dependence function $A(\cdot)$ from the componentwise maxima data. The first code chunk uses various nonparametric estimates of the dependence function, and also uses empirical (i.e.\ nonparametric) estimation of the margins, as specified by \texttt{epmar = TRUE}. The four different estimates are shown in Figure \ref{nonpardfn}. The second code chunk uses maximum likelihood estimation for parametric models. The call to \texttt{fbvevd} fits the model, and the call to \texttt{plot} plots the parametric dependence function estimates. The argument specification \texttt{asy1 = 1} in the first call to \texttt{fbvevd} constrains the model fit so that the first asymmetry parameter of the model is fixed at the value one. <>= pp <- "pickands"; cc <- "cfg" abvnonpar(data = cml, epmar = TRUE, method = pp, plot = TRUE, lty = 3) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 1, lty = 2) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 2, lty = 4) abvnonpar(data = cml, epmar = TRUE, method = cc, add = TRUE, lty = 1) @ <<>>= m1 <- fbvevd(cml, asy1 = 1, model = "alog") m2 <- fbvevd(cml, model = "log") m3 <- fbvevd(cml, model = "bilog") plot(m1, which = 4, nplty = 3) plot(m2, which = 4, nplty = 3, lty = 2, add = TRUE) plot(m3, which = 4, nplty = 3, lty = 4, add = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Nonparametric dependence function estimates by Pickands (dotted line), Deheuvels (dashed line), Hall-Tajvidi (dot-dashed line) and Cap\'{e}r\`{a}a-Foug\`{e}res-Genest (solid line) based on componentwise block maxima data and using empirical marginal estimation.} \label{nonpardfn} \end{figure} The objects produced by \texttt{fbvevd} contain information about the parametric fit of the bivariate extreme value distribution. For example, \texttt{m2} contains information on the fit of a (symmetric) logistic extreme value distribution, which has a single dependence parameter and three parameters on each of the GEV margins. Using \texttt{plot(m2)} produces several diagnostic plots, including quantile curves and spectral densities. Using \texttt{deviance(m2)} produces the deviance, which is equal to twice the negative log-likelihood. The following shows the parameter estimates and their standard errors, and gives an analysis of deviance table for testing \texttt{m2} verses \texttt{m3}, which is possible since the models are nested, with \texttt{m3} having one additional dependence parameter. The call to \texttt{exind.test} produces a score test for independence, following Tawn (1988). Omitting the \texttt{method} argument gives a likelihood ratio test, also from Tawn (1988), which is typically more accurate. <<>>= round(rbind(fitted(m2), std.errors(m2)), 3) anova(m3, m2) evind.test(cml, method = "score") @ The code below uses the function \texttt{qcbvnonpar} to plot quantile curves using nonparametric dependence function estimates. Quantile curves are defined as \begin{equation*} Q(F, p) = \{(z_1,z_2): F(z_1,z_2) = p\}, \end{equation*} where $F$ is a distribution function and $p$ is a probability. We use the default nonparametric estimation method and we again use empirical estimation of the margins\footnote{Using parametric marginal estimates tends to produce more sensible quantile curve plots, but we follow the book here. Unlike the book, the quantile curves in Figure \ref{nonparqc} are not step functions because the empirical marginal transforms include interpolation.}, as specified by \texttt{epmar = TRUE}. For parametric dependence models similar plots can be produced using e.g.\ \texttt{plot(m2, which = 5)}. Note that because we plot curves corresponding to the distribution of the original dataset rather than the componentwise maxima, we pass the argument \texttt{mint = 30}. <>= lts <- c(0.01,100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) points(cml); pp <- c(0.98,0.99,0.995) qcbvnonpar(pp, data = cml, epmar = TRUE, mint = 30, add = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Estimated quantile curves $Q(\hat{F},p)$ for $p=0.98,0.99,0.995$ based on the componentwise block maxima data shown as black circles, using the Cap\'{e}r\`{a}a-Foug\`{e}res-Genest nonparametric estimate of the dependence function and using empirical marginal estimation.} \label{nonparqc} \end{figure} \section{Excesses Over A Threshold} We now consider all the $1500$ observations on liability claims. We assume that the data are distributed according to the distribution function $F$, and we are interested in $F(z)$ where $z=(z_1,z_2)$ is in some sense large. The methods we use assume that $F$ is in the domain of attraction of some bivariate extreme value distribution $G$, and we focus on large data points to estimate features of $G$, and hence of $F(z)$ for large $z$. Typically we focus on points $z$ that lie above a certain threshold. The functions \texttt{tcplot} and \texttt{mrlplot} can be used for producing plots on each margin to help determine thresholds $u_1$ and $u_2$ for methods that focus primarily on points $z$ such that $z_1 > u_1$ and $z_2 > u_2$. Alternatively, the function \texttt{bvtcplot} can be used to help determine a single threshold $u^{*}$ for methods that focus on points $z$ such that $r(z) > u^{*}$, where $r(z) = x_1(z_1) + x_2(z_2)$, and $x_j(z_j) = -1/\log \hat{F}_j(z_j)$ for $j=1,2$ where $F_j$ is estimated empirically. Following Segers and Vandewalle (2004), a sensible choice for threshold $u^{*}$ might be found from Figure \ref{bvtc} by taking the $k$th largest $r(z)$, where $k$ is the largest value for which the y-axis is close to two. Figure \ref{bvtc} is plotted below using \texttt{bvtcplot}. The value of $k$ is returned invisibly. Setting \texttt{spectral = TRUE} uses the $k$th largest points to plot a nonparametric estimate of $H([0,\omega])$ where $H$ is the spectral measure of $G$. <>= k0 <- bvtcplot(loss)$k0 bvtcplot(loss, spectral = TRUE) @ <>= k0 <- bvtcplot(loss)$k0 @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{A plot of $(k/n)r_{(n-k)}$ as a function of $k$, where $r_{(1)} \leq \dots \leq r_{(n)}$ are the ordered values of $r$. The y-axis provides an estimate of $H([0,1]) = 2$ for the spectral measure $H$ of $G$.} \label{bvtc} \end{figure} The parametric approach to the problem can employ models similar to those used for bivariate extreme value distributions. We first consider the margins separately by fitting a univariate generalized Pareto distribution to the excesses over the threshold $u_j$ on each margin $j=1,2$. We choose the thresholds so that the number of exceedances is roughly\footnote{The value is chosen so that the thresholds match exactly with those used in the book.} half of the value \texttt{k0}. <<>>= thresh <- apply(loss, 2, sort, decreasing = TRUE)[(k0+5)/2,] mar1 <- fitted(fpot(loss[,1], thresh[1])) mar2 <- fitted(fpot(loss[,2], thresh[2])) rbind(mar1,mar2) @ Parametric threshold models can be fitted using the function \texttt{fbvpot}, with the parametric model specified using the \texttt{model} argument. The default approach uses censored likelihood methodology, where a bivariate extreme value dependence structure is fitted to the data censored at the marginal thresholds $u_1$ and $u_2$. Alternatively, a Poisson process model can be employed using the \texttt{likelihood} argument, employing the methodology of Coles and Tawn (1991). Some examples of parametric fits are given below. Diagnostic plots for the fitted models can be produced using e.g.\ \texttt{plot(m2)}. <<>>= m1 <- fbvpot(loss, thresh, model = "alog", asy1 = 1) m2 <- fbvpot(loss, thresh, model = "bilog") m3 <- fbvpot(loss, thresh, model = "bilog", likelihood = "poisson") round(rbind(fitted(m2), std.errors(m2)), 3) @ The following code plots parametric and nonparametric estimates for the bivariate extreme value dependence structure fitted to the upper tail of $F$. The parametric estimates use the previously fitted models. The nonparametric estimate can be plotted using the \texttt{"pot"} method and takes the value \texttt{k0} to specify the threshold. <<>>= abvnonpar(data = loss, method = "pot", k = k0, epmar = TRUE, plot = TRUE, lty = 3) plot(m1, which = 2, add = TRUE) plot(m2, which = 2, add = TRUE, lty = 4) plot(m3, which = 2, add = TRUE, lty = 2) @ Figure \ref{qcthresh} uses our fitted asymmetric logistic model \texttt{m1} to plot quantile curves at probabilities $p=0.98,0.99,0.995$. The thresholds used for the censored likelihood model fit are also added to the plot. <>= lts <- c(1e-04, 100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) plot(m1, which = 3, p = c(0.95,0.975,0.99), tlty = 0, add = TRUE) abline(v=thresh[1], h=thresh[2]) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Quantile curves for probabilities $p=0.98,0.99,0.995$ for an asymmetric logistic model fit using censored likelihood estimation, with censoring at marginal thresholds given by the vertical and horizontal lines.} \label{qcthresh} \end{figure} Models based on bivariate extreme value distributions assume that the margins are either asymptotically dependent or are perfectly independent. They cannot account for situations where the dependence between the margins vanishes at increasingly extreme levels. The remainder of this section illustrates the estimation of dependence measures that can identify such cases. We consider three quantities as defined in Coles \textit{et al.} (1999). The coefficient of extremal dependence $\chi \in [0,1]$ is the tendency for one variable to be large given that the other is large. When $\chi = 0$ the variables are asymptotically independent, and when $\chi > 0$ they are asymptotically independent. The second measure $\bar{\chi}$ identifies the strength of dependence for asymptotically independent variables. When $\bar{\chi} = 1$ the variables are asymptotically dependent, and when $-1 \leq \bar{\chi} < 1$ they are asymptotically independent. The third measure is the coefficient of tail dependence $\eta$, which satisfies $\bar{\chi} = 2\eta - 1$. The following code produces Figure \ref{chiplot} which shows estimates of the functions $\chi(u)$ and $\bar{\chi}(u)$, as defined in Coles \textit{et al.} (1999), for $0 < u < 1$. The functions are defined so that $\chi = \lim_{u \rightarrow 1}\chi(u)$ and $\bar{\chi} = \lim_{u \rightarrow 1}\bar{\chi}(u)$. In this case $\chi(u) > 0 $ for all $u$ but there is little evidence that $\bar{\chi}$ is close to one, so it is difficult to specify the form of dependence on the basis of this plot. <>= old <- par(mfrow = c(2,1)) chiplot(loss, ylim1 = c(-0.25,1), ylim2 = c(-0.25,1), nq = 200, qlim = c(0.02,0.98), which = 1:2, spcases = TRUE) par(old) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{The dependence measures $\chi(u)$ and $\bar{\chi}(u)$. Estimates (solid line), 95\% pointwise confidence intervals (dot-dashed lines). The dashed lines represent the theoretical limits of the functions and the exact independence case at zero.} \label{chiplot} \end{figure} We now consider the coefficient of tail dependence $\eta$. We can estimate $\eta$ using univariate theory because of its relationship with $T = \min\{x_1(z_1),x_2(z_2)\}$. If we fit a generalized Pareto distribution to the data points in $T$ that exceed a large fixed threshold, then the estimated shape parameter of the fitted distribution provides an estimate of $\eta$. The call to \texttt{tcplot} plots estimates of $\eta$ at different thresholds in order to assist with threshold choice. The plot seems roughly linear after $u=0.8$, so we take the 80th percentile of $T$ as our threshold. Finally, we use \texttt{anova} to perform a likelihood ratio test for asymptotic dependence, with the null hypothesis $\eta = 1$ versus the alternative $\eta < 1$. <>= fla <- apply(-1/log(ula), 1, min) thresh <- quantile(fla, probs = c(0.025, 0.975)) tcplot(fla, thresh, nt = 100, pscale = TRUE, which = 2, vci = FALSE, cilty = 2, type = "l", ylim = c(-0.2,1.2), ylab = "Tail Dependence") abline(h = c(0,1)) @ <<>>= thresh <- quantile(fla, probs = 0.8) m1 <- fpot(fla, thresh = thresh) cat("Tail Dependence:", fitted(m1)["shape"], "\n") @ <<>>= m2 <- fpot(fla, thresh = thresh, shape = 1) anova(m1, m2, half = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Maximum likelihood estimates (solid line) and 95\% pointwise confidence intervals (dot-dashed lines) for $\eta$ at different threshold probabilities.} \label{etaplot} \end{figure} \section*{Bibliography} Beirlant, J., Goegebeur, Y., Segers, J and Teugels, J. (2004) \textit{Statistics of Extremes: Theory and Applications}. Wiley, U.K. Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analysis. \textit{Extremes}, \textbf{2}, 339--365. Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. \textit{J.\ R.\ Statist.\ Soc.\ B}, \textbf{53}, 377--392. Segers, J. and Vandewalle, B. (2004). Statistics of Multivariate Extremes. In Beirlant et al. (eds.), \textit{Statistics of Extremes: Theory and Applications}. Wiley, U.K. Tawn, J. A. (1988). 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r‰0âŠàb```b`aad`b2Y˜€# 'fËÉ,NÊϲø€XÎ!Ab t$*Сát4T<ª>?JG@õ…BåãP͉l@¥c âQ ú âaP̓šրƇҡhæDCh øRfTX%evd/src/0000755000175100001440000000000014673500176011646 5ustar hornikusersevd/src/evd_init.c0000644000175100001440000000427414673500101013606 0ustar hornikusers#include "header.h" #include static const R_CMethodDef CEntries[] = { {"nlgpd", (DL_FUNC) &nlgpd, 6}, {"nlpp", (DL_FUNC) &nlpp, 8}, {"clusters", (DL_FUNC) &clusters, 6}, {"ccop", (DL_FUNC) &ccop, 11}, {"rbvlog_shi", (DL_FUNC) &rbvlog_shi, 3}, {"rbvalog_shi", (DL_FUNC) &rbvalog_shi, 4}, {"rmvlog_tawn", (DL_FUNC) &rmvlog_tawn, 4}, {"rmvalog_tawn", (DL_FUNC) &rmvalog_tawn, 6}, {"rbvlog", (DL_FUNC) &rbvlog, 3}, {"rbvalog", (DL_FUNC) &rbvalog, 4}, {"rbvhr", (DL_FUNC) &rbvhr, 3}, {"rbvneglog", (DL_FUNC) &rbvneglog, 3}, {"rbvaneglog", (DL_FUNC) &rbvaneglog, 4}, {"rbvbilog", (DL_FUNC) &rbvbilog, 4}, {"rbvnegbilog", (DL_FUNC) &rbvnegbilog, 4}, {"rbvct", (DL_FUNC) &rbvct, 4}, {"rbvamix", (DL_FUNC) &rbvamix, 4}, {"nlgev", (DL_FUNC) &nlgev, 6}, {"nlgumbelx", (DL_FUNC) &nlgumbelx, 7}, {"nslmvalog", (DL_FUNC) &nslmvalog, 13}, {"nlbvlog", (DL_FUNC) &nlbvlog, 13}, {"nlbvalog", (DL_FUNC) &nlbvalog, 15}, {"nlbvhr", (DL_FUNC) &nlbvhr, 13}, {"nlbvneglog", (DL_FUNC) &nlbvneglog, 13}, {"nlbvaneglog", (DL_FUNC) &nlbvaneglog, 15}, {"nlbvbilog", (DL_FUNC) &nlbvbilog, 14}, {"nlbvnegbilog", (DL_FUNC) &nlbvnegbilog, 14}, {"nlbvct", (DL_FUNC) &nlbvct, 14}, {"nlbvamix", (DL_FUNC) &nlbvamix, 14}, {"nllbvclog", (DL_FUNC) &nllbvclog, 12}, {"nllbvcalog", (DL_FUNC) &nllbvcalog, 14}, {"nllbvchr", (DL_FUNC) &nllbvchr, 12}, {"nllbvcneglog", (DL_FUNC) &nllbvcneglog, 12}, {"nllbvcaneglog", (DL_FUNC) &nllbvcaneglog, 14}, {"nllbvcbilog", (DL_FUNC) &nllbvcbilog, 13}, {"nllbvcnegbilog", (DL_FUNC) &nllbvcnegbilog, 13}, {"nllbvcct", (DL_FUNC) &nllbvcct, 13}, {"nllbvcamix", (DL_FUNC) &nllbvcamix, 13}, {"nllbvplog", (DL_FUNC) &nllbvplog, 13}, {"nllbvphr", (DL_FUNC) &nllbvphr, 13}, {"nllbvpneglog", (DL_FUNC) &nllbvpneglog, 13}, {"nllbvpbilog", (DL_FUNC) &nllbvpbilog, 14}, {"nllbvpnegbilog", (DL_FUNC) &nllbvpnegbilog, 14}, {"nllbvpct", (DL_FUNC) &nllbvpct, 14}, {NULL, NULL, 0} }; void R_init_evd(DllInfo *dll) { R_registerRoutines(dll, CEntries, NULL, NULL, NULL); R_useDynamicSymbols(dll, FALSE); R_forceSymbols(dll, TRUE); } evd/src/pot.c0000644000175100001440000000476514673500101012614 0ustar hornikusers#include "header.h" void nlgpd(double *data, int *n, double *loc, double *scale, double *shape, double *dns) { int i; double *dvec, eps; dvec = (double *)R_alloc(*n, sizeof(double)); eps = R_pow(DBL_EPSILON, 0.3); if(*scale <= 0) { *dns = 1e6; return; } for(i=0;i<*n;i++) { data[i] = (data[i] - *loc) / *scale; if(fabs(*shape) <= eps) dvec[i] = log(1 / *scale) - data[i]; else { data[i] = 1 + *shape * data[i]; if(data[i] <= 0) { *dns = 1e6; return; } dvec[i] = log(1 / *scale) - (1 / *shape + 1) * log(data[i]); } } for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } void nlpp(double *exceed, int *nhigh, double *loc, double *scale, double *shape, double *thresh, double *nop, double *dns) { int i; double *dvec, d2, u, eps; dvec = (double *)R_alloc(*nhigh, sizeof(double)); eps = R_pow(DBL_EPSILON, 0.3); if(*scale <= 0) { *dns = 1e6; return; } for(i=0;i<*nhigh;i++) { exceed[i] = (exceed[i] - *loc) / *scale; if(fabs(*shape) <= eps) dvec[i] = log(1 / *scale) - exceed[i]; else { exceed[i] = 1 + *shape * exceed[i]; if(exceed[i] <= 0) { *dns = 1e6; return; } dvec[i] = log(1 / *scale) - (1 / *shape + 1) * log(exceed[i]); } } u = (*thresh - *loc) / *scale; if(fabs(*shape) <= eps) d2 = - *nop * exp(-u); else { u = 1 + *shape * u; if(u <= 0 && *shape > 0) { *dns = 1e6; return; } if(u <= 0 && *shape < 0) d2 = 0; else d2 = - *nop * R_pow(u, -1 / *shape); } *dns = -d2; for(i=0;i<*nhigh;i++) *dns = *dns - dvec[i]; } void clusters(double *high, double *high2, int *n, int *r, int *rlow, double *clstrs) { int i,j,rr; int incl = 0, clind = 0, shigh = 0, shigh2 = 0; for(i=0;i<*n;i++) { if(high[i] && incl) { clstrs[i + 0 * *n] = clind; } if(high[i] && !incl) { incl = 1; clstrs[i + 1 * *n] = 1; clind++; clstrs[i + 0 * *n] = clind; } if(!high[i] && incl) { if(*r > *n-i) rr = *n-i; else rr = *r; for(j=i;j<(i+rr);j++) { shigh = shigh + high[j]; } if(*rlow > *n-i) rr = *n-i; else rr = *rlow; for(j=i;j<(i+rr);j++) { shigh2 = shigh2 + high2[j]; } if(!shigh || !shigh2) { incl = 0; clstrs[i - 1 + 2 * *n] = 1; } else clstrs[i + 0 * *n] = clind; shigh = shigh2 = 0; } } if(incl) clstrs[*n - 1 + 2 * *n] = 1; } evd/src/fit.c0000644000175100001440000006042214673500101012564 0ustar hornikusers#include "header.h" void nlgev(double *data, int *n, double *loc, double *scale, double *shape, double *dns) { int i; double *dvec; dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { data[i] = (data[i] - loc[i]) / *scale; if(*shape == 0) dvec[i] = log(1 / *scale) - data[i] - exp(-data[i]); else { data[i] = 1 + *shape * data[i]; if(data[i] <= 0) { *dns = 1e6; return; } dvec[i] = log(1 / *scale) - R_pow(data[i], -1 / *shape) - (1 / *shape + 1) * log(data[i]); } } for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } void nlgumbelx(double *data, int *n, double *loc1, double *scale1, double *loc2, double *scale2, double *dns) { int i; double *dvec, *datam1, *datam2; dvec = (double *)R_alloc(*n, sizeof(double)); datam1 = (double *)R_alloc(*n, sizeof(double)); datam2 = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (data[i] - loc1[i]) / *scale1; datam2[i] = (data[i] - loc2[i]) / *scale2; dvec[i] = exp(-exp(-datam1[i]) + log(1 / *scale2) - datam2[i] - exp(-datam2[i])) + exp(-exp(-datam2[i]) + log(1 / *scale1) - datam1[i] - exp(-datam1[i])); } for(i=0;i<*n;i++) *dns = *dns - log(dvec[i]); } void nlbvlog(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double idep, *dvec, *z; dvec = (double *)R_alloc(*n, sizeof(double)); z = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } idep = 1 / *dep; for(i=0;i<*n;i++) { z[i] = R_pow(exp(idep * datam1[i]) + exp(idep * datam2[i]), *dep); dvec[i] = (idep + *shape1) * datam1[i] + (idep + *shape2) * datam2[i] - log(*scale1 * *scale2); dvec[i] = dvec[i] + (1-2*idep)*log(z[i]) - z[i]; if(si[i] == 0) dvec[i] = dvec[i] + log(z[i]); else if(si[i] == 1) dvec[i] = dvec[i] + log(idep-1); else dvec[i] = dvec[i] + log(idep-1+z[i]); } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvalog(double *datam1, double *datam2, int *n, int *si, double *dep, double *asy1, double *asy2, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double idep,c1,c2,c3,c4; double *e1,*e2,*e3,*e4,*z,*v,*jc,*dvec; e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); e3 = (double *)R_alloc(*n, sizeof(double)); e4 = (double *)R_alloc(*n, sizeof(double)); z = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } idep = 1 / *dep; c1 = log(1 - *asy1) + log(1 - *asy2); c2 = idep * log(*asy1) + idep * log(*asy2); c3 = log(1 - *asy1) + idep * log(*asy2); c4 = log(1 - *asy2) + idep * log(*asy1); for(i=0;i<*n;i++) { z[i] = R_pow(exp(idep * (log(*asy1) + datam1[i])) + exp(idep * (log(*asy2) + datam2[i])), *dep); v[i] = (1 - *asy1) * exp(datam1[i]) + (1 - *asy2) * exp(datam2[i]) + z[i]; jc[i] = (1 + *shape1) * datam1[i] + (1 + *shape2) * datam2[i] - log(*scale1 * *scale2); e1[i] = c3 + (idep - 1) * datam2[i]; e2[i] = c4 + (idep - 1) * datam1[i]; e3[i] = (1 - idep) * log(z[i]) + log(exp(e1[i]) + exp(e2[i])); e4[i] = c2 + (idep - 1) * datam1[i] + (idep - 1) * datam2[i] + (1 - 2*idep) * log(z[i]); dvec[i] = jc[i] - v[i]; if(si[i] == 0) { e4[i] = e4[i] + log(z[i]); dvec[i] = dvec[i] + log(exp(c1) + exp(e3[i]) + exp(e4[i])); } else if(si[i] == 1) { e4[i] = e4[i] + log(idep-1); dvec[i] = dvec[i] + e4[i]; } else { e4[i] = e4[i] + log(idep-1+z[i]); dvec[i] = dvec[i] + log(exp(c1) + exp(e3[i]) + exp(e4[i])); } } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvhr(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double idep; double *e1,*e2,*e3,*v,*jc,*dvec; e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); e3 = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } idep = 1 / *dep; for(i=0;i<*n;i++) { e1[i] = exp(datam1[i]) * pnorm(idep + *dep * (datam1[i] - datam2[i]) / 2, 0, 1, 1, 0); e2[i] = exp(datam2[i]) * pnorm(idep + *dep * (datam2[i] - datam1[i]) / 2, 0, 1, 1, 0); e3[i] = exp(datam1[i]) * dnorm(idep + *dep * (datam1[i] - datam2[i]) / 2, 0, 1, 0); v[i] = e1[i] + e2[i]; if(si[i] == 0) dvec[i] = e1[i] * e2[i]; else if(si[i] == 1) dvec[i] = *dep * e3[i] / 2; else dvec[i] = e1[i] * e2[i] + *dep * e3[i] / 2; jc[i] = *shape1 * datam1[i] + *shape2 * datam2[i] - log(*scale1 * *scale2); dvec[i] = log(dvec[i]) + jc[i] - v[i]; } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvneglog(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double idep; double *e1,*e2,*z,*v,*jc,*dvec; e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); z = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } idep = 1 / *dep; for(i=0;i<*n;i++) { z[i] = R_pow(exp(-*dep * datam1[i]) + exp(-*dep * datam2[i]), -idep); v[i] = exp(datam1[i]) + exp(datam2[i]) - z[i]; jc[i] = (1 + *shape1)*datam1[i] + (1 + *shape2)*datam2[i] - log(*scale1 * *scale2); e1[i] = (1 + *dep) * log(z[i]) + log(exp((-*dep-1) * datam1[i]) + exp((-*dep-1) * datam2[i])); e2[i] = (-*dep-1) * datam1[i] + (-*dep-1) * datam2[i] + (1 + 2 * *dep) * log(z[i]); dvec[i] = jc[i] - v[i]; if(si[i] == 0) { e2[i] = e2[i] + log(z[i]); dvec[i] = dvec[i] + log(1 - exp(e1[i]) + exp(e2[i])); } else if(si[i] == 1) { e2[i] = e2[i] + log(1 + *dep); dvec[i] = dvec[i] + e2[i]; } else { e2[i] = e2[i] + log(1 + *dep + z[i]); dvec[i] = dvec[i] + log(1 - exp(e1[i]) + exp(e2[i])); } } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvaneglog(double *datam1, double *datam2, int *n, int *si, double *dep, double *asy1, double *asy2, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double idep; double *e1,*e2,*e3,*e4,*z,*v,*jc,*dvec; e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); e3 = (double *)R_alloc(*n, sizeof(double)); e4 = (double *)R_alloc(*n, sizeof(double)); z = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } idep = 1 / *dep; for(i=0;i<*n;i++) { z[i] = R_pow(exp(-*dep * (log(*asy1) + datam1[i])) + exp(-*dep * (log(*asy2) + datam2[i])), -idep); v[i] = exp(datam1[i]) + exp(datam2[i]) - z[i]; jc[i] = (1 + *shape1)*datam1[i] + (1 + *shape2)*datam2[i] - log(*scale1 * *scale2); e1[i] = -*dep * log(*asy1) + (-*dep - 1) * datam1[i]; e2[i] = -*dep * log(*asy2) + (-*dep - 1) * datam2[i]; e3[i] = (1 + *dep) * log(z[i]) + log(exp(e1[i]) + exp(e2[i])); e4[i] = e1[i] + e2[i] + (1 + 2 * *dep) * log(z[i]); dvec[i] = jc[i] - v[i]; if(si[i] == 0) { e4[i] = e4[i] + log(z[i]); dvec[i] = dvec[i] + log(1 - exp(e3[i]) + exp(e4[i])); } else if(si[i] == 1) { e4[i] = e4[i] + log(1 + *dep); dvec[i] = dvec[i] + e4[i]; } else { e4[i] = e4[i] + log(1 + *dep + z[i]); dvec[i] = dvec[i] + log(1 - exp(e3[i]) + exp(e4[i])); } } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvbilog(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i,j; double *e1,*e2,*v,*jc,*dvec,*gma; double llim,midpt,ilen,lval,midval,uval,delta,eps; gma = (double *)R_alloc(*n, sizeof(double)); e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } delta = eps = R_pow(DBL_EPSILON, 0.5); for(i=0;i<*n;i++) { llim = 0; ilen = 1; lval = (1 - *alpha) * exp(datam1[i]); uval = (*beta - 1) * exp(datam2[i]); if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvnegbilog(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i,j; double *e1,*e2,*e3,*v,*jc,*dvec,*gma; double llim,midpt,ilen,lval,midval,uval,delta,eps; gma = (double *)R_alloc(*n, sizeof(double)); e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); e3 = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } delta = eps = R_pow(DBL_EPSILON, 0.5); for(i=0;i<*n;i++) { llim = 0; ilen = 1; uval = (1 + *alpha) * exp(datam1[i]); lval = - (1 + *beta) * exp(datam2[i]); if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvct(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double *e1,*e2,*u,*v,*jc,*dvec; double c; e1 = (double *)R_alloc(*n, sizeof(double)); e2 = (double *)R_alloc(*n, sizeof(double)); u = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } c = *alpha * *beta / (*alpha + *beta + 1); for(i=0;i<*n;i++) { u[i] = (*alpha * exp(datam2[i])) / (*alpha * exp(datam2[i]) + *beta * exp(datam1[i])); v[i] = exp(datam2[i]) * pbeta(u[i], *alpha, *beta + 1, 1, 0) + exp(datam1[i]) * pbeta(u[i], *alpha + 1, *beta, 0, 0); jc[i] = (1 + *shape1) * datam1[i] + (1 + *shape2) * datam2[i] - log(*scale1 * *scale2); e1[i] = pbeta(u[i], *alpha, *beta + 1, 1, 0) * pbeta(u[i], *alpha + 1, *beta, 0, 0); e2[i] = dbeta(u[i], *alpha + 1, *beta + 1, 0) / (*alpha * exp(datam2[i]) + *beta * exp(datam1[i])); if(si[i] == 0) dvec[i] = log(e1[i]) - v[i] + jc[i]; else if(si[i] == 1) dvec[i] = log(c * e2[i]) - v[i] + jc[i]; else dvec[i] = log(e1[i] + c * e2[i]) - v[i] + jc[i]; } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nlbvamix(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns) { int i; double *v,*v1,*v2,*v12,*u,*u1,*u2,*jc,*dvec; double apb; v1 = (double *)R_alloc(*n, sizeof(double)); v2 = (double *)R_alloc(*n, sizeof(double)); v12 = (double *)R_alloc(*n, sizeof(double)); u = (double *)R_alloc(*n, sizeof(double)); u1 = (double *)R_alloc(*n, sizeof(double)); u2 = (double *)R_alloc(*n, sizeof(double)); v = (double *)R_alloc(*n, sizeof(double)); jc = (double *)R_alloc(*n, sizeof(double)); dvec = (double *)R_alloc(*n, sizeof(double)); for(i=0;i<*n;i++) { datam1[i] = (datam1[i] - loc1[i]) / *scale1; datam2[i] = (datam2[i] - loc2[i]) / *scale2; if(*shape1 == 0) datam1[i] = -datam1[i]; else { datam1[i] = 1 + *shape1 * datam1[i]; if(datam1[i] <= 0) { *dns = 1e6; return; } else datam1[i] = -1 / *shape1 * log(datam1[i]); } if(*shape2 == 0) datam2[i] = -datam2[i]; else { datam2[i] = 1 + *shape2 * datam2[i]; if(datam2[i] <= 0) { *dns = 1e6; return; } else datam2[i] = -1 / *shape2 * log(datam2[i]); } } apb = *alpha + *beta; for(i=0;i<*n;i++) { jc[i] = (1 + *shape1) * datam1[i] + (1 + *shape2) * datam2[i] - log(*scale1 * *scale2); u[i] = exp(datam1[i]) + exp(datam2[i]); u1[i] = exp(datam1[i])/u[i]; u2[i] = exp(datam2[i])/u[i]; v[i] = u[i] - exp(datam1[i]) * (apb - *alpha * u1[i] - *beta * u1[i] * u1[i]); v1[i] = 1 - *alpha * u2[i] * u2[i] - *beta * (3 * u2[i]*u2[i] - 2 * u2[i]*u2[i]*u2[i]); v2[i] = 1 - *alpha * u1[i]*u1[i] - 2 * *beta * u1[i]*u1[i]*u1[i]; v12[i] = (-2 * *alpha * u1[i] * u2[i] - 6 * *beta * u1[i]*u1[i] * u2[i]) / u[i]; if(si[i] == 0) dvec[i] = log(v1[i] * v2[i]) - v[i] + jc[i]; else if(si[i] == 1) dvec[i] = log(- v12[i]) - v[i] + jc[i]; else dvec[i] = log(v1[i] * v2[i] - v12[i]) - v[i] + jc[i]; } if(*split > 0.5) { for(i=0;i<*n;i++) dns[i] = - dvec[i]; } else { for(i=0;i<*n;i++) *dns = *dns - dvec[i]; } } void nslmvalog(double *data, int *n, int *d, double *deps, double *thetas, double *mpar, double *psrvs, int *q, int *nslocid, double *nsloc, int *depindx, int *thetaindx, double *dns) { int i,j,k,l,dd,nn,qq,niinbm,ndepp,nmp; double iterm1, iterm2, term1, term2, eps; double thetasum, psrv, repdens; double dep, theta, loc; double *tdata, *dvec; int tmp1, tmp2; dd = *d; nn = *n; qq = *q; eps = R_pow(DBL_EPSILON, 0.3); ndepp = R_pow(2, dd) - 1 - dd; niinbm = R_pow(2, dd - 1) - 1; if(*nslocid) nmp = 4; else nmp = 3; *dns = 0; tdata = (double *)R_Calloc(nn * dd * sizeof(double), double); dvec = (double *)R_Calloc(nn * sizeof(double), double); for(i=0;i 1) { *dns = 1e6; return; } else { iterm1 = iterm1 + (1-thetasum)/tdata[i*dd+j]; iterm2 = iterm2 + (1-thetasum)/tdata[i*dd+j]; } repdens = repdens + log(iterm2) - iterm1 - log(mpar[nmp*j+1]) - mpar[nmp*j+2] * log(tdata[i*dd+j]); } else tdata[i*dd+j] = NA_REAL; } dvec[i] = dvec[i] + exp(repdens); } dvec[i] = log(dvec[i]) - log(qq); } for(i=0;i 1) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); lambda2[0] = R_pow(lambda2[0], -1 / *dep); lambda2[1] = R_pow(lambda2[1], -1 / *dep); zdn = R_pow(lambda2[0] + lambda2[1], *dep - 1); zdn = -zdn * (lambda2[0] + lambda2[1]); for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; v1[i] = R_pow(data1[i], -1 / *dep); v2[i] = R_pow(data2[i], -1 / *dep); v12[i] = R_pow(v1[i] + v2[i], *dep - 1); v[i] = v12[i] * (v1[i] + v2[i]); v1[i] = -(v1[i]/data1[i]) * v12[i]; v2[i] = -(v2[i]/data2[i]) * v12[i]; v12[i] = (1 - 1 / *dep) * v1[i] * v2[i] / v[i]; if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcbilog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i,j; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *q, *q1, *q2, *q12, *x1, *x2, *qa, *qb; double lambda2[2], lambda3[2], lambdaq, zdn; double llim,midpt,ilen,lval,midval,uval,delta,eps; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); q = (double *)R_alloc(*nn, sizeof(double)); qa = (double *)R_alloc(*nn, sizeof(double)); qb = (double *)R_alloc(*nn, sizeof(double)); q1 = (double *)R_alloc(*nn, sizeof(double)); q2 = (double *)R_alloc(*nn, sizeof(double)); q12 = (double *)R_alloc(*nn, sizeof(double)); x1 = (double *)R_alloc(*nn, sizeof(double)); x2 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0.1 || *beta < 0.1 || *alpha > 0.999 || *beta > 0.999) { *dns = 1e6; return; } delta = eps = R_pow(DBL_EPSILON, 0.8); lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); llim = 0; ilen = 1; lval = (1 - *alpha) / lambda2[0]; uval = (*beta - 1) / lambda2[1]; if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcalog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *asy1, double *asy2, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *x1, *x2, *x12; double lambda2[2], lambda3[2], zdn; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); x1 = (double *)R_alloc(*nn, sizeof(double)); x2 = (double *)R_alloc(*nn, sizeof(double)); x12 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.1 || *dep > 1 || *asy1 < 0.001 || *asy2 < 0.001 || *asy1 > 1 || *asy2 > 1) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); lambda3[0] = R_pow(*asy1 / lambda2[0], 1 / *dep); lambda3[1] = R_pow(*asy2 / lambda2[1], 1 / *dep); zdn = R_pow(lambda3[0] + lambda3[1], *dep - 1); zdn = (*asy1 - 1)/lambda2[0] + (*asy2 - 1)/lambda2[1] - zdn * (lambda3[0] + lambda3[1]); for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; x1[i] = R_pow(*asy1 / data1[i], 1 / *dep); x2[i] = R_pow(*asy2 / data2[i], 1 / *dep); x12[i] = R_pow(x1[i] + x2[i], *dep - 1); v[i] = (1 - *asy1)/data1[i] + (1 - *asy2)/data2[i] + x12[i] * (x1[i] + x2[i]); v1[i] = ((*asy1 - 1)/data1[i] - x1[i] * x12[i]) / data1[i]; v2[i] = ((*asy2 - 1)/data2[i] - x2[i] * x12[i]) / data2[i]; v12[i] = (1 - 1 / *dep) * x1[i]/data1[i] * x2[i]/data2[i] * x12[i] / (x1[i] + x2[i]); if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcneglog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *x1, *x2, *x12; double lambda2[2], lambda3[2], zdn; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); x1 = (double *)R_alloc(*nn, sizeof(double)); x2 = (double *)R_alloc(*nn, sizeof(double)); x12 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.05 || *dep > 5) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); lambda3[0] = R_pow(lambda2[0], *dep); lambda3[1] = R_pow(lambda2[1], *dep); zdn = R_pow(lambda3[0] + lambda3[1], -1 / *dep - 1); zdn = zdn * (lambda3[0] + lambda3[1]) - 1/lambda2[0] - 1/lambda2[1]; for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; x1[i] = R_pow(data1[i], *dep); x2[i] = R_pow(data2[i], *dep); x12[i] = R_pow(x1[i] + x2[i], -1 / *dep - 1); v[i] = 1/data1[i] + 1/data2[i] - x12[i] * (x1[i] + x2[i]); v1[i] = (x1[i] * x12[i] - 1/data1[i]) / data1[i]; v2[i] = (x2[i] * x12[i] - 1/data2[i]) / data2[i]; v12[i] = -(1 + *dep) * (x1[i]/data1[i]) * (x2[i]/data2[i]) * x12[i] / (x1[i] + x2[i]); if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcnegbilog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i,j; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *q, *q1, *q2, *q12, *x1, *x2, *qa, *qb; double lambda2[2], lambda3[2], lambdaq, zdn; double llim,midpt,ilen,lval,midval,uval,delta,eps; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); q = (double *)R_alloc(*nn, sizeof(double)); qa = (double *)R_alloc(*nn, sizeof(double)); qb = (double *)R_alloc(*nn, sizeof(double)); q1 = (double *)R_alloc(*nn, sizeof(double)); q2 = (double *)R_alloc(*nn, sizeof(double)); q12 = (double *)R_alloc(*nn, sizeof(double)); x1 = (double *)R_alloc(*nn, sizeof(double)); x2 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0.1 || *beta < 0.1 || *alpha > 20 || *beta > 20) { *dns = 1e6; return; } delta = eps = R_pow(DBL_EPSILON, 0.8); lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); llim = 0; ilen = 1; uval = (1 + *alpha) / lambda2[0]; lval = - (1 + *beta) / lambda2[1]; if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcaneglog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *asy1, double *asy2, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *x1, *x2, *x12; double lambda2[2], lambda3[2], zdn; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); x1 = (double *)R_alloc(*nn, sizeof(double)); x2 = (double *)R_alloc(*nn, sizeof(double)); x12 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.05 || *dep > 5 || *asy1 < 0.001 || *asy2 < 0.001 || *asy1 > 1 || *asy2 > 1) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); lambda3[0] = R_pow(lambda2[0] / *asy1, *dep); lambda3[1] = R_pow(lambda2[1] / *asy2, *dep); zdn = R_pow(lambda3[0] + lambda3[1], -1 / *dep - 1); zdn = zdn * (lambda3[0] + lambda3[1]) - 1/lambda2[0] - 1/lambda2[1]; for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; x1[i] = R_pow(data1[i] / *asy1, *dep); x2[i] = R_pow(data2[i] / *asy2, *dep); x12[i] = R_pow(x1[i] + x2[i], -1 / *dep - 1); v[i] = 1/data1[i] + 1/data2[i] - x12[i] * (x1[i] + x2[i]); v1[i] = (x1[i] * x12[i] - 1/data1[i]) / data1[i]; v2[i] = (x2[i] * x12[i] - 1/data2[i]) / data2[i]; v12[i] = -(1 + *dep) * x1[i]/data1[i] * x2[i]/data2[i] * x12[i] / (x1[i] + x2[i]); if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcct(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *x; double lambda2[2], zdn; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); x = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0.001 || *beta < 0.001 || *alpha > 30 || *beta > 30) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); zdn = *alpha * lambda2[0] / (*alpha * lambda2[0] + *beta * lambda2[1]); zdn = -pbeta(zdn, *alpha + 1, *beta, 0, 0) / lambda2[0] - pbeta(zdn, *alpha, *beta + 1, 1, 0) / lambda2[1]; for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; x[i] = *alpha * data1[i] / (*alpha * data1[i] + *beta * data2[i]); v[i] = pbeta(x[i], *alpha + 1, *beta, 0, 0) / data1[i] + pbeta(x[i], *alpha, *beta + 1, 1, 0) / data2[i]; v1[i] = -pbeta(x[i], *alpha + 1, *beta, 0, 0) / R_pow(data1[i], 2); v2[i] = -pbeta(x[i], *alpha, *beta + 1, 1, 0) / R_pow(data2[i], 2); v12[i] = -(*alpha * *beta) * dbeta(x[i], *alpha + 1, *beta, 0) / (data1[i] * R_pow(*alpha * data1[i] + *beta * data2[i], 2)); if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvchr(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double lambda2[2], zdn, idep; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.2 || *dep > 10) { *dns = 1e6; return; } idep = 1/ *dep; lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); zdn = -1/lambda2[0] * pnorm(idep + *dep * (log(lambda2[1]) - log(lambda2[0]))/2, 0, 1, 1, 0) - 1/lambda2[1] * pnorm(idep + *dep * (log(lambda2[0]) - log(lambda2[1]))/2, 0, 1, 1, 0); for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; idep = 1/ *dep; v[i] = 1/data1[i] * pnorm(idep + *dep * (log(data2[i]) - log(data1[i]))/2, 0, 1, 1, 0) + 1/data2[i] * pnorm(idep + *dep * (log(data1[i]) - log(data2[i]))/2, 0, 1, 1, 0); v1[i] = -1/R_pow(data1[i], 2) * pnorm(idep + *dep * (log(data2[i]) - log(data1[i]))/2, 0, 1, 1, 0); v2[i] = -1/R_pow(data2[i], 2) * pnorm(idep + *dep * (log(data1[i]) - log(data2[i]))/2, 0, 1, 1, 0); v12[i] = - *dep / (2 * data1[i] * data2[i]) / data1[i] * dnorm(idep + *dep * (log(data2[i]) - log(data1[i]))/2, 0, 1, 0); if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } void nllbvcamix(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *t1, *t2, *v, *v1, *v2, *v12; double *x; double lambda2[2], zdn; dvec = (double *)R_alloc(*nn, sizeof(double)); t1 = (double *)R_alloc(*nn, sizeof(double)); t2 = (double *)R_alloc(*nn, sizeof(double)); v = (double *)R_alloc(*nn, sizeof(double)); v1 = (double *)R_alloc(*nn, sizeof(double)); v2 = (double *)R_alloc(*nn, sizeof(double)); v12 = (double *)R_alloc(*nn, sizeof(double)); x = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0 || *alpha + 3 * *beta < 0 || *alpha + *beta > 1 || *alpha + 2 * *beta > 1) { *dns = 1e6; return; } lambda2[0] = -1/log(1 - lambda[0]); lambda2[1] = -1/log(1 - lambda[1]); lambda2[0] = 1/lambda2[0]; lambda2[1] = 1/lambda2[1]; zdn = lambda2[0]/(lambda2[0] + lambda2[1]); zdn = -lambda2[0] - lambda2[1] + (*alpha + *beta) * lambda2[0] - *alpha * lambda2[0] * zdn - *beta * lambda2[0] * zdn * zdn; for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) t1[i] = exp(-data1[i]); else { t1[i] = 1 + *shape1 * data1[i]; if(t1[i] <= 0) { *dns = 1e6; return; } t1[i] = R_pow(t1[i], -1 / *shape1); } data1[i] = -1/log(1 - lambda[0] * t1[i]); if(*shape2 == 0) t2[i] = exp(-data2[i]); else { t2[i] = 1 + *shape2 * data2[i]; if(t2[i] <= 0) { *dns = 1e6; return; } t2[i] = R_pow(t2[i], -1 / *shape2); } data2[i] = -1/log(1 - lambda[1] * t2[i]); t1[i] = R_pow(data1[i], 2) * R_pow(t1[i], 1 + *shape1) / (1 - lambda[0] * t1[i]); t1[i] = lambda[0] * t1[i] / *scale1; t2[i] = R_pow(data2[i], 2) * R_pow(t2[i], 1 + *shape2) / (1 - lambda[1] * t2[i]); t2[i] = lambda[1] * t2[i] / *scale2; x[i] = 1 / (data1[i] + data2[i]); v[i] = 1/data1[i] + 1/data2[i] - (*alpha + *beta) / data1[i] + *alpha * data2[i] * x[i] / data1[i] + *beta * data2[i] * data2[i] * x[i] * x[i] / data1[i]; v1[i] = -1 / (data1[i] * data1[i]) + *alpha * x[i] * x[i] + *beta * x[i] * x[i] * x[i] * (data1[i] + 3 * data2[i]); v2[i] = -1 / (data2[i] * data2[i]) + *alpha * x[i] * x[i] + 2 * *beta * x[i] * x[i] * x[i] * data2[i]; v12[i] = -2 * *alpha * x[i] * x[i] * x[i] - 6 * *beta * x[i] * x[i] * x[i] * x[i] * data2[i]; if(thid[i] < 1.5) dvec[i] = log(-v1[i]) + log(t1[i]) - v[i]; if(thid[i] >= 1.5 && thid[i] < 2.5) dvec[i] = log(-v2[i]) + log(t2[i]) - v[i]; if(thid[i] >= 2.5) dvec[i] = log(v1[i] * v2[i] - v12[i]) + log(t1[i]) + log(t2[i]) - v[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; *dns = *dns - (*n - *nn) * zdn; } /* Point Process Likelihood Routines */ void nllbvplog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *r, *w, *jac, *h; double idep, v, utt[2]; dvec = (double *)R_alloc(*nn, sizeof(double)); r = (double *)R_alloc(*nn, sizeof(double)); w = (double *)R_alloc(*nn, sizeof(double)); jac = (double *)R_alloc(*nn, sizeof(double)); h = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.1 || *dep > 1) { *dns = 1e6; return; } for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); idep = 1 / *dep; h[i] = log(idep - 1) - (1+idep) * log(w[i] * (1-w[i])) + (*dep - 2) * log(R_pow(w[i],-idep) + R_pow(1-w[i],-idep)); dvec[i] = jac[i] + h[i] - 3 * r[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; utt[0] = -1 / log(1 - p[0]); utt[1] = -1 / log(1 - p[1]); v = R_pow(R_pow(utt[0],-1 / *dep) + R_pow(utt[1],-1 / *dep), *dep); *dns = *dns + v; } void nllbvpneglog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *r, *w, *jac, *h; double v, utt[2]; dvec = (double *)R_alloc(*nn, sizeof(double)); r = (double *)R_alloc(*nn, sizeof(double)); w = (double *)R_alloc(*nn, sizeof(double)); jac = (double *)R_alloc(*nn, sizeof(double)); h = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *dep < 0.05 || *dep > 5) { *dns = 1e6; return; } for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); h[i] = 1 / (1 + R_pow(1 / w[i] - 1, *dep)); h[i] = log(*dep + 1) + log(1 - h[i]) + (1 + 1 / *dep) * log(h[i]) - log(1 - w[i]) - 2 * log(w[i]); dvec[i] = jac[i] + h[i] - 3 * r[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; utt[0] = -1 / log(1 - p[0]); utt[1] = -1 / log(1 - p[1]); v = 1 / utt[0] + 1 / utt[1] - R_pow(R_pow(utt[0], *dep) + R_pow(utt[1], *dep), -1 / *dep); *dns = *dns + v; } void nllbvpct(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i; double *dvec, *r, *w, *jac, *h; double v, utt[2]; dvec = (double *)R_alloc(*nn, sizeof(double)); r = (double *)R_alloc(*nn, sizeof(double)); w = (double *)R_alloc(*nn, sizeof(double)); jac = (double *)R_alloc(*nn, sizeof(double)); h = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0.001 || *beta < 0.001 || *alpha > 30 || *beta > 30) { *dns = 1e6; return; } for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); h[i] = (*alpha + *beta + 1) * log(*alpha * w[i] + *beta * (1-w[i])) + lgammafn(*alpha) + lgammafn(*beta); h[i] = lgammafn(*alpha + *beta + 1) + *alpha * log(*alpha) + *beta * log(*beta) + (*alpha - 1) * log(w[i]) + (*beta - 1) * log(1-w[i]) - h[i]; dvec[i] = jac[i] + h[i] - 3 * r[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; utt[0] = -1 / log(1 - p[0]); utt[1] = -1 / log(1 - p[1]); v = *alpha * utt[0] /(*alpha * utt[0] + *beta * utt[1]); v = pbeta(v, *alpha + 1, *beta, 0, 0) / utt[0] + pbeta(v, *alpha, *beta + 1, 1, 0) / utt[1]; *dns = *dns + v; } void nllbvpbilog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns) { int i,j; double *dvec, *r, *w, *jac, *h; double v, utt[2]; double llim,midpt,ilen,lval,midval,uval,delta,eps; dvec = (double *)R_alloc(*nn, sizeof(double)); r = (double *)R_alloc(*nn, sizeof(double)); w = (double *)R_alloc(*nn, sizeof(double)); jac = (double *)R_alloc(*nn, sizeof(double)); h = (double *)R_alloc(*nn, sizeof(double)); if(*scale1 < 0.01 || *scale2 < 0.01 || *alpha < 0.1 || *beta < 0.1 || *alpha > 0.999 || *beta > 0.999) { *dns = 1e6; return; } delta = eps = R_pow(DBL_EPSILON, 0.8); for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); llim = 0; ilen = 1; lval = (1 - *alpha) * (1 - w[i]); uval = (*beta - 1) * w[i]; if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j 20 || *beta > 20) { *dns = 1e6; return; } delta = eps = R_pow(DBL_EPSILON, 0.8); for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); llim = 0; ilen = 1; uval = (1 + *alpha) * (1 - w[i]); lval = - (1 + *beta) * w[i]; if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(j=0;j 10) { *dns = 1e6; return; } for(i=0;i<*nn;i++) { data1[i] = data1[i] / *scale1; data2[i] = data2[i] / *scale2; if(*shape1 == 0) data1[i] = exp(-data1[i]); else { data1[i] = 1 + *shape1 * data1[i]; if(data1[i] <= 0) { *dns = 1e6; return; } data1[i] = R_pow(data1[i], -1 / *shape1); } data1[i] = -1/log(1 - r1[i] * data1[i]); if(*shape2 == 0) data2[i] = exp(-data2[i]); else { data2[i] = 1 + *shape2 * data2[i]; if(data2[i] <= 0) { *dns = 1e6; return; } data2[i] = R_pow(data2[i], -1 / *shape2); } data2[i] = -1/log(1 - r2[i] * data2[i]); r[i] = log(data1[i] + data2[i]); w[i] = data1[i] / exp(r[i]); if(thid[i] < 1.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]); if(thid[i] >= 1.5 && thid[i] < 2.5) jac[i] = 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); if(thid[i] >= 2.5) jac[i] = 2 * log(data1[i]) + 1 / data1[i] + (1 + *shape1) * log(1 - exp(-1 / data1[i])) - log(*scale1) - *shape1 * log(p[0]) + 2 * log(data2[i]) + 1 / data2[i] + (1 + *shape2) * log(1 - exp(-1 / data2[i])) - log(*scale2) - *shape2 * log(p[1]); idep = 1 / *dep; h[i] = log(*dep / 2) - 2 * log(w[i]) - log(1-w[i]) + dnorm(idep + *dep * (log(1-w[i]) - log(w[i]))/2, 0, 1, 1); dvec[i] = jac[i] + h[i] - 3 * r[i]; } for(i=0;i<*nn;i++) *dns = *dns - dvec[i]; utt[0] = -1 / log(1 - p[0]); utt[1] = -1 / log(1 - p[1]); v = pnorm(1 / *dep + *dep * log(utt[1]/utt[0]) / 2, 0, 1, 1, 0) / utt[0] + pnorm(1 / *dep + *dep * log(utt[0]/utt[1]) / 2, 0, 1, 1, 0) / utt[1]; *dns = *dns + v; } evd/src/ccop.c0000644000175100001440000001524114673500101012725 0ustar hornikusers#include "header.h" /* Conditional copulas condition on 2nd margin. */ double ccbvlog(double m1, double m2, double oldm1, double dep) { double tm1,tm2,idep,u,v,fval; tm1 = -log(m1); tm2 = -log(m2); idep = 1/dep; u = R_pow(tm1, idep) + R_pow(tm2, idep); v = R_pow(u, dep); fval = exp(-v) * (1 / m2) * R_pow(tm2, idep-1) * R_pow(u, dep-1) - oldm1; return fval; } double ccbvalog(double m1, double m2, double oldm1, double dep, double asy1, double asy2) { double tm1,tm2,idep,u,v,fval; tm1 = -log(m1); tm2 = -log(m2); idep = 1/dep; u = R_pow(asy1*tm1, idep) + R_pow(asy2*tm2, idep); v = (1-asy1)*tm1 + (1-asy2)*tm2 + R_pow(u, dep); fval = exp(-v) * (1 / m2) * (1 - asy2 + R_pow(asy2, idep) * R_pow(tm2, idep-1) * R_pow(u, dep-1)) - oldm1; return fval; } double ccbvhr(double m1, double m2, double oldm1, double dep) { double tm1,tm2,v,idep,fval; tm1 = -log(m1); tm2 = -log(m2); idep = 1 / dep; v = tm2 * pnorm(idep + (log(tm2) - log(tm1)) * dep/2, 0, 1, 1, 0) + tm1 * pnorm(idep + (log(tm1) - log(tm2)) * dep/2, 0, 1, 1, 0); fval = pnorm(idep + (log(tm2) - log(tm1)) * dep/2, 0, 1, 1, 0) * exp(-v) / m2 - oldm1; return fval; } double ccbvneglog(double m1, double m2, double oldm1, double dep) { double tm1,tm2,v,idep,fval; tm1 = -log(m1); tm2 = -log(m2); idep = 1 / dep; v = R_pow((R_pow(tm2,-dep) + R_pow(tm1,-dep)),-idep); fval = exp(v) * m1 * (1-R_pow(1 + R_pow(tm2/tm1,dep), -1-idep)) - oldm1; return fval; } double ccbvaneglog(double m1, double m2, double oldm1, double dep, double asy1, double asy2) { double tm1,tm2,v,idep,fval; tm1 = -log(m1); tm2 = -log(m2); idep = 1 / dep; v = R_pow(asy1 * tm2, -dep) + R_pow(asy2 * tm1, -dep); fval = exp(R_pow(v, -idep)) * m1 * (1 - R_pow(asy1, -dep) * R_pow(tm2, -dep-1) * R_pow(v, -idep-1)) - oldm1; return fval; } double ccbvbilog(double m1, double m2, double oldm1, double alpha, double beta) { int i; double tm1,tm2,v,fval; double delta,eps,llim,midpt,ilen,lval,midval,uval; tm1 = -log(m1); tm2 = -log(m2); delta = eps = R_pow(DBL_EPSILON, 0.75); llim = 0; ilen = 1; lval = (1 - alpha) * tm1; uval = (beta - 1) * tm2; if(!(sign(lval) != sign(uval))) error("values at end points are not of opposite sign"); for(i=0;i #include #define RANDIN GetRNGstate() #define RANDOUT PutRNGstate() #define UNIF unif_rand() #define EXP exp_rand() /* from pot.c */ void nlgpd(double *data, int *n, double *loc, double *scale, double *shape, double *dns); void nlpp(double *exceed, int *nhigh, double *loc, double *scale, double *shape, double *thresh, double *nop, double *dns); void clusters(double *high, double *high2, int *n, int *r, int *rlow, double *clstrs); /* from ccop.c */ double ccbvlog(double m1, double m2, double oldm1, double dep); double ccbvalog(double m1, double m2, double oldm1, double dep, double asy1, double asy2); double ccbvhr(double m1, double m2, double oldm1, double dep); double ccbvneglog(double m1, double m2, double oldm1, double dep); double ccbvaneglog(double m1, double m2, double oldm1, double dep, double asy1, double asy2); double ccbvbilog(double m1, double m2, double oldm1, double alpha, double beta); double ccbvnegbilog(double m1, double m2, double oldm1, double alpha, double beta); double ccbvct(double m1, double m2, double oldm1, double alpha, double beta); double ccbvamix(double m1, double m2, double oldm1, double alpha, double beta); void ccop(double *m1, double *m2, int *cnd, double *dep, double *asy1, double *asy2, double *alpha, double *beta, int *n, int *model, double *ccop); /* from sim.c */ void rbvlog_shi(int *n, double *alpha, double *sim); void rbvalog_shi(int *n, double *alpha, double *asy, double *sim); void rmvlog_tawn(int *n, int *d, double *alpha, double *sim); void rmvalog_tawn(int *n, int *d, int *nb, double *alpha, double *asy, double *sim); double rpstable(double cexp); double maximum_n(int n, double *x); void rbvlog(int *n, double *dep, double *sim); void rbvalog(int *n, double *dep, double *asy, double *sim); void rbvhr(int *n, double *dep, double *sim); void rbvneglog(int *n, double *dep, double *sim); void rbvaneglog(int *n, double *dep, double *asy, double *sim); void rbvbilog(int *n, double *alpha, double *beta, double *sim); void rbvnegbilog(int *n, double *alpha, double *beta, double *sim); void rbvct(int *n, double *alpha, double *beta, double *sim); void rbvamix(int *n, double *alpha, double *beta, double *sim); /* from fit.c */ void nlgev(double *data, int *n, double *loc, double *scale, double *shape, double *dns); void nlgumbelx(double *data, int *n, double *loc1, double *scale1, double *loc2, double *scale2, double *dns); void nlbvalog(double *datam1, double *datam2, int *n, int *si, double *dep, double *asy1, double *asy2, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvlog(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvhr(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvneglog(double *datam1, double *datam2, int *n, int *si, double *dep, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvaneglog(double *datam1, double *datam2, int *n, int *si, double *dep, double *asy1, double *asy2, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvbilog(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvnegbilog(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvct(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nlbvamix(double *datam1, double *datam2, int *n, int *si, double *alpha, double *beta, double *loc1, double *scale1, double *shape1, double *loc2, double *scale2, double *shape2, int *split, double *dns); void nslmvalog(double *data, int *n, int *d, double *deps, double *thetas, double *mpar, double *psrvs, int *q, int *nslocid, double *nsloc, int *depindx, int *thetaindx, double *dns); /* from bvpot.c (censored) */ void nllbvclog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcbilog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcalog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *asy1, double *asy2, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcneglog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcnegbilog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcaneglog(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *asy1, double *asy2, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcct(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvchr(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvcamix(double *data1, double *data2, int *nn, int *n, double *thid, double *lambda, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); /* from bvpot.c (poisson) */ void nllbvplog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvpneglog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvpct(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvpbilog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvpnegbilog(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *alpha, double *beta, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); void nllbvphr(double *data1, double *data2, int *nn, double *thid, double *r1, double *r2, double *p, double *dep, double *scale1, double *shape1, double *scale2, double *shape2, double *dns); evd/NAMESPACE0000644000175100001440000000562714225014531012274 0ustar hornikusersuseDynLib(evd, .registration = TRUE, .fixes = "C_") export(rextreme,rorder,rrweibull,rnweibull,rfrechet,rgev,rgumbelx,rgpd,rgumbel,rbvevd,rbvalog,rbvamix,rbvaneglog,rbvbilog,rbvct,rbvhr,rbvlog,rbvnegbilog,rbvneglog,rmvevd,rmvlog,rmvalog, dextreme,dorder,drweibull,dnweibull,dfrechet,dgev,dgumbelx,dgpd,dgumbel,dbvevd,dbvalog,dbvamix,dbvaneglog,dbvbilog,dbvct,dbvhr,dbvlog,dbvnegbilog,dbvneglog,dmvevd,dmvlog,dmvalog, pextreme,porder,prweibull,pnweibull,pfrechet,pgev,pgumbelx,pgpd,pgumbel,pbvevd,pbvalog,pbvamix,pbvaneglog,pbvbilog,pbvct,pbvhr,pbvlog,pbvnegbilog,pbvneglog,pmvevd,pmvlog,pmvalog, qextreme,qrweibull,qnweibull,qfrechet,qgev,qgumbelx,qgpd,qgumbel,qcbvnonpar, abvalog,abvamix,abvaneglog,abvbilog,abvct,abvevd,abvhr,abvlog,abvnegbilog,abvneglog,abvnonpar,amvalog,amvevd,amvlog,amvnonpar, hbvalog,hbvamix,hbvaneglog,hbvbilog,hbvct,hbvevd,hbvhr,hbvlog,hbvnegbilog,hbvneglog,fbvalog,fbvamix,fbvaneglog,fbvbilog,fbvcalog,fbvcamix,fbvcaneglog,fbvcbilog,fbvcct,fbvchr,fbvclog,fbvcnegbilog,fbvcneglog,fbvcpot,fbvct,fbvevd,fbvhr,fbvlog,fbvnegbilog,fbvneglog,fbvpbilog,fbvpct,fbvphr,fbvplog,fbvpnegbilog,fbvpneglog,fbvpot,fbvppot,fextreme,fgev,fgev.norm,fgev.quantile,fgumbel,fgumbelx,forder,fpot,fpot.norm,fpot.quantile, bvcpp,bvdens,bvdp,bvh,bvpost.optim,bvqc,bvstart.vals,ccbvevd,evind.test,chiplot,clusters,dens,evmc,exi,exiplot,mar,marma,mma,mrlplot,mtransform,mvalog.check,pp,profile2d,qq,rl,sep.bvdata,std.errors,subsets,tcplot,bvtcplot,tvdepfn) S3method(print, bvevd) S3method(print, bvpot) S3method(print, evd) S3method(print, pot) S3method(plot, bvevd) S3method(plot, bvpot) S3method(plot, uvevd) S3method(plot, gumbelx) S3method(plot, profile.evd) S3method(plot, profile2d.evd) S3method(qq, gev) S3method(qq, pot) S3method(qq, gumbelx) S3method(rl, gev) S3method(rl, pot) S3method(rl, gumbelx) S3method(pp, gev) S3method(pp, pot) S3method(pp, gumbelx) S3method(dens, gev) S3method(dens, pot) S3method(dens, gumbelx) S3method(bvdens, bvevd) S3method(bvdens, bvpot) S3method(bvdp, bvevd) S3method(bvdp, bvpot) S3method(bvh, bvevd) S3method(bvh, bvpot) S3method(bvqc, bvevd) S3method(bvqc, bvpot) S3method(fitted, evd) S3method(logLik, evd) S3method(profile2d, evd) S3method(profile, evd) S3method(std.errors, evd) S3method(vcov, evd) S3method(confint, evd) S3method(anova, evd) S3method(confint, profile.evd) S3method(bvcpp, bvevd) importFrom("grDevices", "chull", "dev.interactive", "heat.colors") importFrom("graphics", "abline", "contour", "image", "lines", "matplot", "par", "plot", "points", "polygon", "rug", "segments", "text") importFrom("stats", "AIC", "anova", "approx", "complete.cases", "dbeta", "density", "deviance", "dnorm", "fitted", "integrate", "na.omit", "optim", "pbeta", "pchisq", "pnorm", "ppoints", "qchisq", "qnorm", "quantile", "rbeta", "rexp", "runif", "spline", "uniroot", "var") evd/inst/0000755000175100001440000000000013264344706012034 5ustar hornikusersevd/inst/README0000644000175100001440000000265512637167310012721 0ustar hornikusersThe evd package extends simulation, distribution, quantile and density functions to parametric extreme value distributions, and provides fitting functions which calculate maximum likelihood estimates for univariate and bivariate maxima models, and for univariate and bivariate threshold models. The file CHANGES documents changes from previous versions. The users guide (in pdf) is in the `doc' directory. A vignette on multivariate extremes is also included. A reference manual containing the help files can be downloaded from CRAN. The package contains the following (user-level) functions. All comments, criticisms and queries are gratefully received. Univariate Distributions: [rpqd]gev [rpqd]gumbel [rpqd]rweibull [rpqd]nweibull [rpqd]frechet [rpqd]gpd [rpqd]extreme [rpd]order Bivariate/Multivariate EVDs: [rpda]bvevd [rpda]mvevd Non-parametric Estimation: abvnonpar amvnonpar qcbvnonpar Stochastic Processes: evmc marma mar mma clusters exi Fitting Models: fbvevd fgev fpot fbvpot forder fextreme Pre-model Diagnostics: mrlplot tcplot chiplot bvtcplot evind.test Model Diagnostics: plot.uvevd plot.bvevd anova.evd Profile Likelihoods: profile.evd plot.profile.evd profile2d.evd plot.profile2d.evd The following datasets are also included: failure fox lisbon ocmulgee oldage oxford lossalae portpirie sask sealevel uccle venice sealevel2 venice2 evd/inst/CHANGES0000644000175100001440000005165314611655412013035 0ustar hornikusersCHANGES SINCE VERSION 2.3-3 ARE AS FOLLOWS Removed legacy S constants etc in the C code as required by CRAN Now uses interp package (if available) instead of akima for plot.profile2d.evd as required by CRAN due to licensing issues Fixed bug in dmvevd that gave an error when q was a vector and lower.tail was FALSE Changed CITATION file to use bibentry Removed some non-ASCII characters in an Rd file CHANGES FROM VERSION 2.3-2 ARE AS FOLLOWS Changed url in CITATION to canonical form. Registered C routines in evd_init.c file. Added a = 0 argument to plotting functions. CHANGES FROM VERSION 2.3-0 ARE AS FOLLOWS Minor edits to vignette. Added cmax to output of fpot to correct bug in profile call. Using requireNamespace in plot.profile2d.evd Added importFrom calls to NAMESPACE Removed dependency on stats Added Imports for stats, grDevices, graphics [Version 2.3-1 was not uploaded to CRAN.] CHANGES FROM VERSION 2.2-7 ARE AS FOLLOWS The demo soe9 has been replaced with a vignette on multivariate extremes. Added functions bvtcplot and evind.test. Added [dpqr]nweibull functions to the namespace! Implemented the possion likelihood for bivariate threshold models. CHANGES FROM VERSION 2.2-6 ARE AS FOLLOWS Documentation edited. In particular, \synopsis sections have been removed as this is no longer supported. Added venice2 dataset. Added [dpqr]nweibull functions. The lazy loading of datasets is now implemented. Some internal C and R code cleaning to avoid warnings. In particular, optimizations over one parameter allow consistent inclusion of the "Brent" method. CHANGES FROM VERSION 2.2-5 ARE AS FOLLOWS Included proper NAMESPACE file and removed zzz.R. CHANGES FROM VERSION 2.2-4 ARE AS FOLLOWS Added default NAMESPACE file to avoid note produced under R CMD check. Changed licence to GPL-3. Datasets are now binary .rda files instead of .R. Renamed doc/guide22.tex to doc/guide22.txt. Call to library in plot.profile2d.evd did not use full argument matching. Fixed. CHANGES FROM VERSION 2.2-3 ARE AS FOLLOWS Removed defunct url from description file. Internal coding change to fpot.norm in order to avoid note produced under R CMD check. CHANGES FROM VERSION 2.2-2 ARE AS FOLLOWS Added a Suggests field to the description file in order to avoid warnings under future versions of R CMD check. CHANGES FROM VERSION 2.2-1 ARE AS FOLLOWS Fitted objects now contain the variance covariance matrix, which can be accessed by the function vcov. New function confint (methods confint.profile.evd and confint.evd) for calculating Wald and profile confidence intervals. The internal function pcint has now been deleted, and plot.profile.evd no longer returns profile confidence intervals invisibly. The function amvnonpar now implements non-parametric estimators for dependence functions of extreme value distributions of any dimension. The function amvevd now implements parametric evaluation of dependence functions of logistic and asymmetric logistic models of any dimension. I have removed the bootstrap confidence intervals option from chiplot. This had problems with zero empirical cdf estimates for some replications. A solution would be to use smoothed estimates, but I decided against implementing this. Argument trunc added to chiplot to control whether estimates are truncated at theoretical upper and lower bounds. The old interfaces to abvnonpar and amvnonpar have been removed. I have included a CITATION file in the /inst directory so that typing citation("evd") returns an appropriate citation. The internal functions ccop and ccop.case have been converted into a new documented function ccbvevd. The recommended R package "stats" is now a required package for evd. The files INDEX and data/00Index are now unnecessary (the information is now automatically generated from documentation files) and have been removed. The defunct function abvpar has been removed. CHANGES FROM VERSION 2.2-0 ARE AS FOLLOWS New function hbvevd for plotting the spectral density of bivariate extreme value models. The spectral density is also now plotted from plot.bvevd. Plotting diagnostics are now implemented for bivariate threshold methods using the method function plot.bvpot. The fbvpot function is now consistent with fbvevd; the dsm argument has been removed, and the asymmetric mixed model has been added. Also, the Husler-Reiss model has been added for (undocumented) Poisson process likelihood fitting. The new argument boot in the function chiplot allows bootstrap confidence intervals. The new argument spcase plots lines corresponding to special cases for comparison. The functions mrlplot and tcplot have a new argument pscale allowing the x-axis to be the threshold exceedance probability rather than the threshold. The default plotting character in bivariate density plots is now a circle, to be consistent with quantile curves plots. Also, all plotting functions now produce all available plots by default. The function tcplot has a new argument vci allowing control over the plotting style of the confidence intervals. Other arguments allow more control over labels and limits. The clusters function with plot = TRUE would produce an error if no data points were above the threshold. This has been fixed. The demo soe9 has been expanded to include all of Chapter Nine of the text Statistics of Extremes. CHANGES FROM VERSION 2.1-7 ARE AS FOLLOWS Added the asymmetric mixed bivariate model to the function fbvevd. A demo soe9 has been included, giving examples from Chapter Nine of the book statistics of extremes. An associated file demo.txt has been added to the /inst directory. The atvnonpar and atvpar functions have been renamed to amvnonpar and amvevd. The abvpar function has been renamed to abvevd, but the former still runs, with a warning. An argument rev has been added to abvpar and abvnonpar, allowing the evaluation of A(1-t) rather than A(t). It can also be passed from plot.bvevd. The argument epmar has been added to abvnonpar and atvnonpar. It allows empirical estimation of the margins. The formerly internal function mtransform is now documented. It transforms to and from exponential distributions under the gev model. The profile plots now plot profile log-likelihoods rather than profile deviances. The plotting functions plot.uvevd and plot.bvevd have an additional argument cilwd to control the line width of confidence intervals. The functions abvpar and abvnonpar now have default axes labels and a border line width argument. The function abvnonpar is simpler; the "tdo" method is now undocumented, old methods "deheuvels" and "halltajvidi" can now be implemented using the argument madj. For back compatability, the old interface can still be used. The weight function option for method "cfg" has been removed, and the "cfg" definition now has a simpler equivalent representation. Similar changes have been made to atvnonpar (now renamed to amvnonpar). In fbvevd the dependence summaries, and hence the argument dsm, have been removed, though the former Dependence One value is now automatically given in the output. The argument warn.inf is now undocumented. The marginal option "exponential" to the function evmc has been changed to "rweibull" (negative exponential) because the former swaps the bivariate tails around. The function anova now has a logical argument half to deal with non-regular cases where the asymptotic distribution of the deviance difference is known to be one half of a chi-squared. Also, it no longer errors when testing bilog vs log or negbilog vs bilog. Added the dataset lossalae on general liability claims. CHANGES FROM VERSION 2.1-6 ARE AS FOLLOWS The exi function has been simplified; it now returns only a single estimate, but a new estimation method based on inter-exceedance times has been added. A new function exiplot plots estimates of the extremal index. The rlow argument (lower clustering interval) is now hidden in the documentation for fpot, clusters, exi and exiplot. Due to numerical problems, the function rmvevd could return Inf when the dependence parameters were small. This has been fixed. The maintainer thanks Mohammed Mehdi Gholam Rezaee for reporting this. The anova function errored when used inside another function. This is now corrected. The maintainer thanks William Valdar for reporting this. In function abvnonpar the method "hall" is now "halltajvidi", and an additional comment has been added to the help file, following a request from Nader Tajvidi. Typographic sign error in User's Guide corrected. The maintainer thanks C L Wong for reporting this. Spelling errors in uccle help file corrected. The maintainer thanks Tobias Verbeke for reporting this. Apologies to residents of Belgium! CHANGES FROM VERSION 2.1-5 ARE AS FOLLOWS The data passed to the bivariate fitting function fbvevd can now be a data frame with a third column of mode logical. See the corresponding documentation for more details. A new dataset called sealevel2 has been introduced. CHANGES FROM VERSION 2.1-4 ARE AS FOLLOWS The lower.tail argument for pbvevd and pmvevd did not properly produce survivor functions when set to FALSE. This has been fixed. CHANGES FROM VERSION 2.1-3 ARE AS FOLLOWS A file was accidently included in Version 2.1-3 which meant that the package would not easily install under C compilers other than gcc. This has been fixed. CHANGES FROM VERSION 2.1-2 ARE AS FOLLOWS The function dextreme errored when the argument largest was FALSE. This has been fixed. The maintainer thanks Brian Tolley for reporting this. CHANGES FROM VERSION 2.1-1 ARE AS FOLLOWS The function fbvevd now has arguments cshape, cscale and cloc for fitting common marginal parameters. The function fbvpot now has arguments cshape and cscale for fitting common marginal parameters. The Husler-Reiss model is now implemented in fbvpot. New function chiplot for plotting estimates of the dependence measures chi and chi-bar for bivariate data. The latex file for The Users' Guide is now included in the doc directory along with the pdf file. The default x-label "theshold" in mrlplot has been corrected to "threshold". CHANGES FROM VERSION 2.1-0 ARE AS FOLLOWS New function fbvpot for fitting bivariate threshold models. A plot method for these models will be implemented at a later date. New argument sym in function fbvevd, to allow the fitting of dependence structures under a symmetry constraint. The argument also exists in the new function fbvpot. CHANGES FROM VERSION 2.0-1 ARE AS FOLLOWS Univariate threshold models are now implemented. The main new function is fpot, which calculates maximum likelihood estimates under the generalized Pareto and point process representations. Density, distribution, quantile and simulation functions for the generalized Pareto distribution have been added. The new function clusters identifies extreme clusters. A related function exi calculates estimates of a quantity known as the extremal index. The new plotting functions mrlplot and tcplot aid threshold selection. The class structure has changed slightly. Models fitted using fextreme and forder still have class c("extreme", "evd"). Models fitted using fbvevd still have class c("bvevd", "evd"). Models fitted using the new function fpot have class c("pot", "uvevd", "evd"). Models fitted using fgev now have class c("gev", "uvevd", "evd"). The method function plot.gev is now plot.uvevd. This operates on both the gev and pot classes, due to the new class structure. The defunct functions fgumbel, frweibull, ffrechet (each defunct since version 1.2-0) and fbvall (defunct since version 2.0-0) have been removed. CHANGES FROM VERSION 2.0-0 ARE AS FOLLOWS Internal change to avoid warnings under R version 1.7.0 when calling the multivariate asymmetric logistic distribution, density, quantile and generation functions. The tests directory has been removed. The function evmc unintentionally reversed asymmetric dependence structures. This has been fixed. The maintainer thanks Chris Ferro for reporting this. CHANGES FROM VERSION 1.2-3 ARE AS FOLLOWS The function fgev.quantile is now internal; the functionality has been absorbed into fgev. It is now possible to parameterize gev model fits using the endpoint of the distribution by setting prob to zero or one in fgev. New functions for generating stochastic processes associated with extreme value theory. marma, mar and mma generate max autoregressive moving average processes. evmc generates first order Markov chains with bivariate extreme value dependence structures. The functions fbvlog, fbvalog, etc are now internal. The new function fbvevd should be used for the fitting of all bivariate models. The rbvevd, dbvevd and pbvevd functions replace individual functions for bivariate models. Similarly, rmvevd and dmvevd functions replace individual functions for multivariate models. The function abvpar replaces individual functions for plotting and calculating the dependence functions of parametric bivariate models. New functions atvnonpar and atvpar, which calculate and plot dependence functions of trivariate extreme value distributions, using non-parametric estimates and parametric models (at given parameter values). Fitted bivariate models (i.e. objects of class bvevd) now include Akaike's Information Criterion and, optionally, various dependence structure summaries. The argument dsm controls this option. New lower.tail argument in bivariate and multivariate distribution functions. The function pcint is now internal. Profile confidence intervals are now invisibly returned from plot.profile.evd. Also, the argument ci of plot.profile.evd can now be a vector. Functions called from plot.evd (dens, pp, qq, rl) and plot.bvevd (bvdens, bvdp, bvcpp) are now internal. The default labelling of the x-axis for the return level plot has been changed from the technically correct "-1/log(1-1/Return Period)" to the more widely used "Return Period". There now exists a density function for the multivariate asymmetric logistic model. The density function can be calculated by calling dmvevd with model = "alog". The argument mar of rmvevd, pmvevd and dmvevd can now be a list with d elements, where d is the dimension of the distribution. An extraction function logLik.evd has been added so that the function AIC.default in R base can be used. The function profile2d.evd has a new method argument (to be consistent with profile.evd) and new arguments xaxs and yaxs (to override the default behaviour of the function image). The function plot.bvevd has new arguments blty (border line type) and grid. In the function abvnonpar the logical argument convex replaces the numeric argument modify. The functions fext, rext, etc are now called fextreme, rextreme, etc. The objects formery of class "evd" are now of class c("gev","evd") or c("extreme","evd"). The plot.evd function is now plot.gev. The row names of all data.frame datasets are now the years of observation. Furthermore, the period of observation for the venice data has been corrected in the help file. The arguments mesh and conf in profile.evd now work as documented. The fbvall function is defunct. New datasets: failure, fox, lisbon, ocmulgee, oldage, sask and uccle. The CHANGES file has been moved to the top-level directory. CHANGES FROM VERSION 1.2-2 ARE AS FOLLOWS Two more datasets - venice and portpirie, intended for use in the evdbayes package. Fixed minor problem with extraction functions for bvall objects when only one model is fitted. Contours for bivariate density plots are now chosen by the contour function. Error messages in internal functions are more informative. CHANGES FROM VERSION 1.2-1 ARE AS FOLLOWS New methods hall and tdo for calculating non-parametric dependence function estimates. BUG fix: the non-parameteric dependence function estimator of Caperaa et al (the default) was plotting/calculating A(1-x) rather than A(x). The recommended citation for the package is now the article included in R-News Volume 2/2. Extra graphical arguments have been included in the abv[...] functions. CHANGES FROM VERSION 1.2-0 ARE AS FOLLOWS The profile.evd function now has a method argument to specify the optimization method. The default method is now BFGS. A BUG existed in profile.evd; a fatal error would result if the mesh argument did not have a names attribute. This has been fixed. Internal: General code cleaning/optimizing. New internal functions have been created. CHANGES FROM VERSION 1.1-0 ARE AS FOLLOWS Class orientated objects have been introduced, including print methods and extraction functions. Diagnostics plots can be implemented using plot, or the lower level functions dens, rl, pp, qq (univariate) and bvdens, bvcpp, bvdp (bivariate). Parameters can be profiled using the functions profile and profile2d. Profile deviance surfaces can be plotted using plot. Profile confidence intervals can be calculated using pcint. Fitted models can be compared using the function anova. New function fgev.quantile to fit the GEV distribution, re-parameterizing using a quantile. This allows profile deviances of extreme quantiles to be plotted. Extra argument corr for fitting functions. If corr is TRUE the correlation matrix is calculated. By default corr is FALSE. Extra argument warn.inf for fitting functions. When warn.inf is TRUE (the default) a warning is given if the negative log-likelihood is infinite at the starting values. The ffrechet, fgumbel and frweibull functions are defunct. Internal: Function ccop for calculating conditional copulas. CHANGES FROM VERSION 1.0-0 ARE AS FOLLOWS Automated starting values for fitting functions. Compiled code is now used within the bivariate fitting routines and all bivariate simulation functions. They are consequently much faster than those in Version 1.0-0. Simulation, distribution, density and fitting functions for the bivariate bilogistic, bivariate negative bilogistic and bivariate Coles-Tawn models. A new function fbvall which fits all bivariate models simultaneously. For every model it returns maximum likelihood estimates, standard errors, criteria for model comparisons based on the deviance (e.g. AIC), and summaries of the dependence structure (see the help file for details). A new function abvnonpar which calculates or plots non-parametric estimates of the dependence function. All functions explicitly allow for missing values. This includes bivariate fitting functions, where missing values can occur on either or both margins within any observation. Extra argument nsloc for univariate fitting functions, and arguments nsloc1 and nsloc2 for bivariate fitting functions. These allow non-stationary fitting using linear models for the location parameters. Extra argument std.err for fitting functions. If std.err is FALSE the standard errors are not calculated. By default std.err is TRUE. Extra argument method for fitting functions. The method argument explicitly specifies the optimization method to be passed to the function optim. The default method is now BFGS for all fitting functions except fext and forder, where the default method is still Nelder-Mead. The sealevel data frame has been expanded to include observations from 1912 to 1992. There are 39 missing values. Explicit error handling in fitting functions when the observed information matrix is singular. Artificial constraints are now placed on dependence parameters within bivariate fitting functions to prevent numerical problems. The asy argument for the bivariate asymmetric logistic and bivariate asymmetric negative logistic models now defaults to c(1,1). The default values of the marginal parameters for the bivariate and multivariate functions in Version 1.0-0 are different. This is counter intuitive, so the default value for each margin within the multivariate functions is now c(0,1,0) (Gumbel), which is consistent with the functions for bivariate models. The bivariate density functions contained a fairly minor BUG; if one of the marginal parameter arguments were passed a three column matrix and some of the rows of the matrix produced zero density, a fatal error would result. This has now been fixed. Internal: The .C interface is called with PACKAGE = "evd". Internal: R_alloc replaces malloc in C routines. evd/inst/CITATION0000644000175100001440000000107414611654516013173 0ustar hornikusersbibentry(bibtype ="Article", title = "evd: Extreme Value Distributions", author = c(person(c("A.", "G."), "Stephenson")), journal = "R News", year = "2002", volume = "2", number = "2", pages = "31-32", month = "June", url = "https://CRAN.R-project.org/doc/Rnews/", textVersion = paste("A. G. 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evd/inst/doc/Multivariate_Extremes.R0000644000175100001440000002272014673500175017250 0ustar hornikusers### R code from vignette source 'Multivariate_Extremes.Rnw' ################################################### ### code chunk number 1: rawdata ################################################### options(show.signif.stars=FALSE) library(evd); nn <- nrow(lossalae) loss <- lossalae/1e+05; lts <- c(1e-04, 100) plot(loss, log = "xy", xlim = lts, ylim = lts) ################################################### ### code chunk number 2: Multivariate_Extremes.Rnw:42-44 ################################################### ula <- apply(loss, 2, rank)/(nn + 1) plot(ula) ################################################### ### code chunk number 3: Multivariate_Extremes.Rnw:49-50 ################################################### options(show.signif.stars=FALSE) library(evd); nn <- nrow(lossalae) loss <- lossalae/1e+05; lts <- c(1e-04, 100) plot(loss, log = "xy", xlim = lts, ylim = lts) ################################################### ### code chunk number 4: asylogdfn ################################################### abvevd(dep = 0.5, asy = c(1,1), model = "alog", plot = TRUE) abvevd(dep = 0.5, asy = c(0.6,0.9), model = "alog", add = TRUE, lty = 2) abvevd(dep = 0.5, asy = c(0.8,0.5), model = "alog", add = TRUE, lty = 3) ################################################### ### code chunk number 5: Multivariate_Extremes.Rnw:88-91 ################################################### abvevd(dep = -1/(-2), model = "neglog", plot = TRUE) abvevd(dep = -1/(-1), model = "neglog", add = TRUE, lty = 2) abvevd(dep = -1/(-0.5), model = "neglog", add = TRUE, lty = 3) ################################################### ### code chunk number 6: Multivariate_Extremes.Rnw:94-97 ################################################### abvevd(alpha = 1, beta = -0.2, model = "amix", plot = TRUE) abvevd(alpha = 0.6, beta = 0.1, model = "amix", add = TRUE, lty = 2) abvevd(alpha = 0.2, beta = 0.2, model = "amix", add = TRUE, lty = 3) ################################################### ### code chunk number 7: Multivariate_Extremes.Rnw:100-103 ################################################### abvevd(dep = 1/1.25, model = "hr", plot = TRUE) abvevd(dep = 1/0.83, model = "hr", add = TRUE, lty = 2) abvevd(dep = 1/0.5, model = "hr", add = TRUE, lty = 3) ################################################### ### code chunk number 8: Multivariate_Extremes.Rnw:108-109 ################################################### abvevd(dep = 0.5, asy = c(1,1), model = "alog", plot = TRUE) abvevd(dep = 0.5, asy = c(0.6,0.9), model = "alog", add = TRUE, lty = 2) abvevd(dep = 0.5, asy = c(0.8,0.5), model = "alog", add = TRUE, lty = 3) ################################################### ### code chunk number 9: Multivariate_Extremes.Rnw:121-129 ################################################### set.seed(131); cml <- loss[sample(nn),] xx <- rep(1:50, each = 30); lts <- c(1e-04, 100) cml <- cbind(tapply(cml[,1], xx, max), tapply(cml[,2], xx, max)) colnames(cml) <- colnames(loss) plot(loss, log = "xy", xlim = lts, ylim = lts, col = "grey") points(cml) ecml <- -log(apply(cml,2,rank)/51) plot(ecml) ################################################### ### code chunk number 10: nonpardfn ################################################### pp <- "pickands"; cc <- "cfg" abvnonpar(data = cml, epmar = TRUE, method = pp, plot = TRUE, lty = 3) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 1, lty = 2) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 2, lty = 4) abvnonpar(data = cml, epmar = TRUE, method = cc, add = TRUE, lty = 1) ################################################### ### code chunk number 11: Multivariate_Extremes.Rnw:142-148 ################################################### m1 <- fbvevd(cml, asy1 = 1, model = "alog") m2 <- fbvevd(cml, model = "log") m3 <- fbvevd(cml, model = "bilog") plot(m1, which = 4, nplty = 3) plot(m2, which = 4, nplty = 3, lty = 2, add = TRUE) plot(m3, which = 4, nplty = 3, lty = 4, add = TRUE) ################################################### ### code chunk number 12: Multivariate_Extremes.Rnw:153-154 ################################################### pp <- "pickands"; cc <- "cfg" abvnonpar(data = cml, epmar = TRUE, method = pp, plot = TRUE, lty = 3) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 1, lty = 2) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 2, lty = 4) abvnonpar(data = cml, epmar = TRUE, method = cc, add = TRUE, lty = 1) ################################################### ### code chunk number 13: Multivariate_Extremes.Rnw:164-167 ################################################### round(rbind(fitted(m2), std.errors(m2)), 3) anova(m3, m2) evind.test(cml, method = "score") ################################################### ### code chunk number 14: nonparqc ################################################### lts <- c(0.01,100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) points(cml); pp <- c(0.98,0.99,0.995) qcbvnonpar(pp, data = cml, epmar = TRUE, mint = 30, add = TRUE) ################################################### ### code chunk number 15: Multivariate_Extremes.Rnw:185-186 ################################################### lts <- c(0.01,100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) points(cml); pp <- c(0.98,0.99,0.995) qcbvnonpar(pp, data = cml, epmar = TRUE, mint = 30, add = TRUE) ################################################### ### code chunk number 16: Multivariate_Extremes.Rnw:202-204 (eval = FALSE) ################################################### ## k0 <- bvtcplot(loss)$k0 ## bvtcplot(loss, spectral = TRUE) ################################################### ### code chunk number 17: bvtc ################################################### k0 <- bvtcplot(loss)$k0 ################################################### ### code chunk number 18: Multivariate_Extremes.Rnw:213-214 ################################################### k0 <- bvtcplot(loss)$k0 ################################################### ### code chunk number 19: Multivariate_Extremes.Rnw:224-228 ################################################### thresh <- apply(loss, 2, sort, decreasing = TRUE)[(k0+5)/2,] mar1 <- fitted(fpot(loss[,1], thresh[1])) mar2 <- fitted(fpot(loss[,2], thresh[2])) rbind(mar1,mar2) ################################################### ### code chunk number 20: Multivariate_Extremes.Rnw:233-237 ################################################### m1 <- fbvpot(loss, thresh, model = "alog", asy1 = 1) m2 <- fbvpot(loss, thresh, model = "bilog") m3 <- fbvpot(loss, thresh, model = "bilog", likelihood = "poisson") round(rbind(fitted(m2), std.errors(m2)), 3) ################################################### ### code chunk number 21: Multivariate_Extremes.Rnw:242-247 ################################################### abvnonpar(data = loss, method = "pot", k = k0, epmar = TRUE, plot = TRUE, lty = 3) plot(m1, which = 2, add = TRUE) plot(m2, which = 2, add = TRUE, lty = 4) plot(m3, which = 2, add = TRUE, lty = 2) ################################################### ### code chunk number 22: qcthresh ################################################### lts <- c(1e-04, 100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) plot(m1, which = 3, p = c(0.95,0.975,0.99), tlty = 0, add = TRUE) abline(v=thresh[1], h=thresh[2]) ################################################### ### code chunk number 23: Multivariate_Extremes.Rnw:261-262 ################################################### lts <- c(1e-04, 100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) plot(m1, which = 3, p = c(0.95,0.975,0.99), tlty = 0, add = TRUE) abline(v=thresh[1], h=thresh[2]) ################################################### ### code chunk number 24: chiplot ################################################### old <- par(mfrow = c(2,1)) chiplot(loss, ylim1 = c(-0.25,1), ylim2 = c(-0.25,1), nq = 200, qlim = c(0.02,0.98), which = 1:2, spcases = TRUE) par(old) ################################################### ### code chunk number 25: Multivariate_Extremes.Rnw:285-286 ################################################### old <- par(mfrow = c(2,1)) chiplot(loss, ylim1 = c(-0.25,1), ylim2 = c(-0.25,1), nq = 200, qlim = c(0.02,0.98), which = 1:2, spcases = TRUE) par(old) ################################################### ### code chunk number 26: etaplot ################################################### fla <- apply(-1/log(ula), 1, min) thresh <- quantile(fla, probs = c(0.025, 0.975)) tcplot(fla, thresh, nt = 100, pscale = TRUE, which = 2, vci = FALSE, cilty = 2, type = "l", ylim = c(-0.2,1.2), ylab = "Tail Dependence") abline(h = c(0,1)) ################################################### ### code chunk number 27: Multivariate_Extremes.Rnw:304-307 ################################################### thresh <- quantile(fla, probs = 0.8) m1 <- fpot(fla, thresh = thresh) cat("Tail Dependence:", fitted(m1)["shape"], "\n") ################################################### ### code chunk number 28: Multivariate_Extremes.Rnw:310-312 ################################################### m2 <- fpot(fla, thresh = thresh, shape = 1) anova(m1, m2, half = TRUE) ################################################### ### code chunk number 29: Multivariate_Extremes.Rnw:317-318 ################################################### fla <- apply(-1/log(ula), 1, min) thresh <- quantile(fla, probs = c(0.025, 0.975)) tcplot(fla, thresh, nt = 100, pscale = TRUE, which = 2, vci = FALSE, cilty = 2, type = "l", ylim = c(-0.2,1.2), ylab = "Tail Dependence") abline(h = c(0,1)) evd/inst/doc/guide22.txt0000644000175100001440000030006112637167310014600 0ustar hornikusers\documentclass[11pt,a4paper]{article} \usepackage{t1enc} \usepackage[latin1]{inputenc} \usepackage[english]{babel} \usepackage{amsmath,amssymb} \usepackage{graphics} \usepackage[round]{natbib} \bibliographystyle{jrss} \pagestyle{plain} \setlength{\parindent}{0in} \setlength{\parskip}{1.5ex plus 0.5ex minus 0.5ex} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.3in} \setlength{\textheight}{9.8in} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \begin{document} \sloppy \begin{center} \LARGE A User's Guide to the evd Package (Version 2.2) \\ \Large \vspace{0.2cm} Alec Stephenson \\ \normalsize Copyright \copyright 2006 \\ \vspace{0.2cm} Department of Statistics and Applied Probability,\\ National University of Singapore, \\ Singapore 117546. \\ \vspace{0.2cm} E-mail: alec\_stephenson@hotmail.com \\ 25th March 2006 \end{center} %\nocite{frantiag84} %\nocite{gumbgold64} %\nocite{gumbmust67} %\nocite{jenk55} %\nocite{joe97} %\nocite{kotzbala00} %\nocite{mont70} %\nocite{montotte78} %\nocite{smit86} %\nocite{sney77} %\nocite{step:sim} %\nocite{tawn93} \section{Introduction} \setcounter{footnote}{0} \subsection{What is the evd package?} \label{intro} The evd (extreme value distributions) package is an add-on package for the R \citep{R} statistical computing system. The package contains the following (user-level) functions. It also contains the demo \verb+soe9+, giving examples from Chapter Nine of \citet{beirgoeg04}. Univariate Distributions. Density, distribution, simulation and quantile (inverse distribution) functions for univariate parametric distributions.\\ \verb+ dgev dgpd dgumbel drweibull dfrechet dextreme dorder+\\ \verb+ pgev pgpd pgumbel prweibull pfrechet pextreme porder+\\ \verb+ rgev rgpd rgumbel rrweibull rfrechet rextreme rorder+\\ \verb+ qgev qgpd qgumbel qrweibull qfrechet qextreme+ Multivariate Distributions. Density, distribution, simulation and dependence functions for multivariate parametric extreme value models.\\ \verb+ dbvevd dmvevd pbvevd pmvevd rbvevd rmvevd abvevd amvevd+ Non-parametric Estimation. Calculate and plot non-parametric estimates of dependence functions and quantile curves.\\ \verb+ abvnonpar amvnonpar qcbvnonpar+ Stochastic Processes. Generate stochastic processes associated with extreme value theory, identify extreme clusters and estimate the extremal index.\\ \verb+ evmc marma mar mma clusters exi+ Fitting Models. Obtain maximum likelihood estimates and standard errors for univariate and bivariate models used in extreme value theory.\\ \verb+ fbvevd fgev fpot forder fextreme+ Pre-model Diagnostics. Threshold identification and dependence summaries.\\ \verb+ mrlplot tcplot chiplot+ Model Diagnostics. Model diagnostics for fitted models; diagnostic plots and analysis of deviance.\\ \verb+ plot.uvevd plot.bvevd anova.evd+ Profile likelihoods. Obtain profile traces, plot profile log-likelihoods and obtain profile confidence intervals from fitted models.\\ \verb+ profile.evd plot.profile.evd profile2d.evd plot.profile2d.evd+ The following datasets are also included in the package.\\ \verb+ failure fox lisbon ocmulgee oldage oxford lossalae+\\ \verb+ portpirie sask sealevel uccle venice sealevel2+ \subsection{Obtaining the package/guide} The evd package can be downloaded from CRAN (The Comprehensive R Archive Network) at \verb+http://cran.r-project.org/+. This guide (in pdf) will be in the directory \verb+evd/doc/+ underneath wherever the package is installed. \subsection{Contents} This guide contains examples\footnote{All of the examples presented in this guide are called with \texttt{options(digits = 4)}, and with the option \texttt{show.signif.stars} set to \texttt{FALSE}.} on the use of the evd package. The examples do not include any theoretical justification. See \citet{cole01} for an introduction to the statistics of extreme values, and \citet{beirgoeg04} for a more detailed treatment. Section \ref{uni} covers the standard (non-fitting) functions for univariate distributions. Sections \ref{biv} and \ref{mult} do the same for bivariate and multivariate extreme value models. Dependence functions of extreme value distributions are discussed in Section \ref{depfun}. Stochastic processes are discussed in Section \ref{stochproc}. Maximum likelihood fitting of univariate models, peaks over threshold models and bivariate extreme value models is discussed in Sections \ref{unifit}, \ref{potfit} and \ref{bivfit} respectively. Three practical examples using the data sets \verb+oxford+, \verb+rain+ and \verb+sealevel+ are given in Sections \ref{egoxford}, \ref{egrain} and \ref{egsealevel} respectively. This guide should not be viewed as an alternative to the documentation files included within the package. These remain the definitive source of information. A reference manual containing all the documentation files can be downloaded from CRAN. \subsection{Citing the package} Volume 2/2 of R-News (the newsletter of the R-project) contains an article that describes an earlier version of the evd package. To cite the package in publications please cite the R-News article. The article and the corresponding citation can be downloaded from \verb+http://www.cran.r-project.org/doc/Rnews/+. \subsection{Caveat} I have checked these functions as best I can but, as ever, they may contain bugs. If you find a bug or suspected bug in the code or the documentation please report it to me at \verb+alec_stephenson@hotmail.com+. Please include an appropriate subject line. \subsection{Legalese} This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. See the GNU General Public License for more details. A copy of the GNU General Public License can be obtained from \verb+http://www.gnu.org/copyleft/gpl.html+. You can also obtain it by writing to the Free Software Foundation, Inc., 59 Temple Place -- Suite 330, Boston, MA 02111-1307, USA. \section{Univariate Distributions} \setcounter{footnote}{0} \label{uni} The Gumbel, Fr\'{e}chet and (reversed) Weibull distribution functions are respectively given by \begin{align} &G(z) = \exp\left\{-\exp\left[-\left(\frac{z-a}{b}\right)\right]\right\}, \quad -\infty < z < \infty \label{gumbel} \\ &G(z) = \begin{cases} 0, & z \leq a, \\ \exp\left\{-\left(\frac{z-a}{b}\right)^{-\alpha}\right\}, & z > a, \end{cases} \label{frechet} \\ &G(z) = \begin{cases} \exp\left\{-\left[-\left(\frac{z-a}{b}\right)\right]^{\alpha}\right\}, & z < a, \\ 1, & z \geq a, \end{cases} \label{weibull} \end{align} where $a$ is a location parameter, $b > 0$ is a scale parameter and $\alpha > 0$ is a shape parameter. The distribution \eqref{weibull} is often referred to as the Weibull distribution. To avoid confusion I will call this the reversed Weibull, since it is related by a change of sign to the three parameter Weibull distribution used in survival analysis. The Generalised Extreme Value (GEV) distribution function is given by \begin{equation} G(z) = \exp \left\{ - \left[ 1+ \xi \left( z-\mu \right) /\sigma \right]_{+}^{-1/\xi} \right\}, \label{gev} \end{equation} where ($\mu,\sigma,\xi$) are the location, scale and shape parameters respectively, $\sigma > 0$ and $h_{+}=\max(h,0)$. When $\xi>0$ the GEV distribution has a finite lower end point, given by $\mu - \sigma/\xi$. When $\xi<0$ the GEV distribution has a finite upper end point, also given by $\mu - \sigma/\xi$. The parametric form of the GEV encompasses that of the Gumbel, Fr\`{e}chet and reversed Weibull distributions. The Gumbel distribution is obtained in the limit as $\xi\rightarrow0$. The Fr\'{e}chet and Weibull distributions are obtained when $\xi>0$ and $\xi<0$ respectively. To recover the parameterisation of the Fr\'{e}chet distribution \eqref{frechet} set $\xi=1/\alpha>0$, $\sigma=b/\alpha>0$ and $\mu=a+b$. To recover the parameterisation of the reversed Weibull distribution \eqref{weibull} set $\xi=-1/\alpha<0$, $\sigma=b/\alpha>0$ and $\mu=a-b$. The generalised Pareto distribution (GPD) function is given by \begin{equation*} G(z) = 1 - \left[1 + \xi \left( z-\mu \right) /\sigma \right]_{+}^{-1/\xi}, \end{equation*} for $z > \mu$, where ($\mu,\sigma,\xi$) are the location, scale and shape parameters respectively, $\sigma > 0$ and $h_{+}=\max(h,0)$. The GPD has a finite lower end point, given by $\mu$. When $\xi<0$ the GPD also has a finite upper end point, given by $\mu - \sigma/\xi$. A shifted exponential distribution is obtained in the limit as $\xi\rightarrow0$. It is standard practice within R to concatenate the letters r, p, q and d with an abbreviated distribution name to yield the names of the corresponding simulation, distribution, quantile (inverse distribution) and density functions respectively. The evd package follows this convention. Each of the five distributions defined above has an associated set of functions, as given in Section \ref{intro}. Some examples are given below. They should be familiar to those who have had previous experience with R. \begin{verbatim} > rgev(6, loc = c(20,1), scale = .5, shape = 1) [1] 23.7290 1.2492 19.6680 0.8662 19.7939 2.6512 > rgpd(3, loc = 2) [1] 2.483681 3.666805 2.837809 > qrweibull(seq(0.1, 0.4, 0.1), 2, 0.5, 1, lower.tail = FALSE) > qrweibull(seq(0.9, 0.6, -0.1), loc = 2, scale = 0.5, shape = 1) # Both give [1] 1.947 1.888 1.822 1.745 > pfrechet(2:6, 2, 0.5, 1) [1] 0.0000 0.6065 0.7788 0.8465 0.8825 > pfrechet(2:6, 2, 0.5, 1, low = FALSE) [1] 1.0000 0.3935 0.2212 0.1535 0.1175 > drweibull(-1:3, 2, 0.5, log = TRUE) [1] -5.307 -3.307 -1.307 -Inf -Inf > dgumbel(-1:3, 0, 1) [1] 0.17937 0.36788 0.25465 0.11820 0.04737 \end{verbatim} Let $F$ be an arbitrary distribution function, and let $X_1,\dots,X_m$ be a random sample from $F$. Define $U_m=\max\{X_1,\dots,X_m\}$ and $L_m=\min\{X_1,\dots,X_m\}$. The distributions of $U_m$ and $L_m$ are given by \begin{align} &\Pr(U_m \leq x) = [F(x)]^m \label{maxdens} \\ &\Pr(L_m \leq x) = 1 - [1 - F(x)]^m. \label{mindens} \end{align} Simulation, distribution, quantile and density functions for the distribution of $U_m$, given an integer $m$ and an arbitrary distribution function $F$, are provided by \verb+rextreme+, \verb+pextreme+, \verb+qextreme+ and \verb+dextreme+ respectively. The integer $m$ should be given to the argument \verb+mlen+. The distribution $F$ is most easily specified by passing an abbreviated distribution name to the argument \verb+distn+. If \verb+largest+ is \verb+FALSE+ the distribution of $L_m$ is used. Some examples are given below. \begin{verbatim} > rextreme(1, distn = "norm", sd = 2, mlen = 20, largest = FALSE) > min(rnorm(20, mean = 0, sd = 2)) # Both simulate from the same distribution [1] -2.612 > rextreme(4, distn = "exp", rate = 1, mlen = 5) > rextreme(4, distn = "exp", mlen = 5) # Both simulate from the same distribution [1] 2.2001 0.8584 4.5595 3.9397 > pextreme(c(.4, .5), distn = "norm", mean = 0.5, sd = c(1, 2), mlen = 4) [1] 0.04484 0.06250 > dextreme(c(1, 4), distn = "gamma", shape = 1, scale = 0.3, mlen = 100) [1] 0.3261328 0.0005398 \end{verbatim} Let $X_{(1)} \geq X_{(2)} \geq \dots \geq X_{(m)}$ be the order statistics of the random sample $X_1,\dots,X_m$. The distribution of the $j$th largest order statistic, for $j = 1,\dots,m$, is \begin{equation} \Pr(X_{(j)} \leq x) = \sum_{k=0}^{j-1} \binom{m}{k} [F(x)]^{m-k} [1 - F(x)]^k. \label{orderdens} \end{equation} The distribution of the $j$th smallest order statistic is obtained by setting $j = m + 1 - j$. Simulation, distribution and density functions for the distribution of $X_{(j)}$, for given integers $m$ and $j \in \{1,\dots,m\}$, and for an arbitrary distribution function $F$, are provided by \verb+rorder+, \verb+porder+ and \verb+dorder+ respectively. The integer $m$ should again be given to the argument \verb+mlen+. If \verb+largest+ is \verb+FALSE+ the distribution of the \verb+j+th smallest order statistic $X_{(m+j-1)}$ is used. Some examples are given below. \begin{verbatim} > rorder(1, distn = "norm", mlen = 20, j = 2) [1] 2.284 > porder(c(1, 2), distn = "gamma", shape = c(.5, .7), mlen = 10, j = 2) [1] 0.5177 0.8259 > dorder(c(1, 2), distn = "gamma", shape = c(.5, .7), mlen = 10, j = 2) [1] 0.7473 0.3081 \end{verbatim} \section{Bivariate Extreme Value Distributions} \setcounter{footnote}{0} \label{biv} The evd package contains functions associated with nine parametric bivariate extreme value distributions. The univariate marginal distributions in each case are GEV, with marginal parameters ($\mu_1,\sigma_1,\xi_1$) and ($\mu_2,\sigma_2,\xi_2$). There are three symmetric models, with distribution functions \begin{align} &G(z_1,z_2) = \exp\left\{- (y_1^{1/\alpha}+y_2^{1/\alpha})^\alpha \right\}, \quad 0<\alpha\leq1, \label{log} \\ &G(z_1,z_2) = \exp\left\{ - y_1 - y_2 + (y_1^{-r}+y_2^{-r})^{-1/r} \right\}, \quad r>0, \label{neglog} \\ &G(z_1,z_2) = \exp\left( - y_1\Phi\{\lambda^{-1}+{\textstyle\frac{1}{2}}\lambda[\log(y_1/y_2)]\} - y_2\Phi\{\lambda^{-1}+{\textstyle\frac{1}{2}}\lambda[\log(y_2/y_1)]\}\right), \quad \lambda>0, \notag \end{align} known as the logistic \citep{gumb60b}, negative logistic \citep{gala75} and H\"{u}sler-Reiss \citep{huslreis89} models respectively, where \begin{equation} y_j = y_j(z_j) = \{1+\xi_j(z_j-\mu_j)/\sigma_j\}_{+}^{-1/\xi_j} \label{mtrans} \end{equation} for $j=1,2$. Independence\footnote{ Independence occurs when $G(z_1,z_2) = \exp\{-(y_1+y_2)\}$.} is obtained when $\alpha=1$, $r\downarrow0$ or $\lambda\downarrow0$. Complete dependence\footnote{ Complete dependence occurs when $G(z_1,z_2) = \exp\{-\max(y_1,y_2)\}$.} is obtained when $\alpha\downarrow0$, $r\rightarrow\infty$ or $\lambda\rightarrow\infty$. The distributions functions \eqref{log} and \eqref{neglog} have asymmetric extensions, given by \begin{align} &G(z_1,z_2) = \exp\left\{ - (1-\theta_1)y_1 - (1-\theta_2)y_2 - [(\theta_1y_1)^{1/\alpha}+(\theta_2y_2)^{1/\alpha}]^\alpha\right\}, \quad 0<\alpha\leq1, \notag \\ &G(z_1,z_2) = \exp\left\{ - y_1 - y_2 + [(\theta_1y_1)^{-r}+(\theta_2y_2)^{-r}]^{-1/r}\right\}, \quad r>0, \notag \end{align} known as the asymmetric logistic \citep{tawn88} and asymmetric negative logistic \citep{joe90} models respectively, where the asymmetry parameters $0\leq\theta_1,\theta_2\leq1$. For the asymmetric logistic model independence is obtained when either $\alpha = 1$, $\theta_1 = 0$ or $\theta_2 = 0$. Different limits occur when $\theta_1$ and $\theta_2$ are fixed and $\alpha\downarrow0$. For the asymmetric negative logistic model independence is obtained when either $r\downarrow0$, $\theta_1\downarrow0$ or $\theta_2\downarrow0$. Different limits occur when $\theta_1$ and $\theta_2$ are fixed and $r\rightarrow\infty$. The remaining four bivariate models are defined in Appendix A. Density, distribution and simulation functions for each of the nine models are provided by \verb+dbvevd+, \verb+pbvevd+ and \verb+rbvevd+ respectively. The argument \verb+model+ denotes the specified model, which must be either \verb+"log"+ (the default), \verb+"alog"+, \verb+"hr"+, \verb+"neglog"+, \verb+"aneglog"+, \verb+"bilog"+, \verb+"negbilog"+, \verb+"ct"+ or \verb+"amix"+ (or any unique partial match). The first argument in \verb+pbvevd+ and \verb+dbvevd+ should be a vector of length two or a matrix with two columns, so that each row specifies a value for $(z_1,z_2)$. The parameters of the specified model can be passed using one or more of the arguments \verb+dep+, \verb+asy+, \verb+alpha+ and \verb+beta+. The marginal parameters ($\mu_1,\sigma_1,\xi_1$) and ($\mu_2,\sigma_2,\xi_2$) can be passed using the arguments \verb+mar1+ and \verb+mar2+ respectively. Gumbel marginal distributions are used by default. The arguments \verb+mar1+ and \verb+mar2+ can also be matrices with three columns, in which case each column represents a vector of values to be passed to the corresponding marginal parameter. Some examples are given below. \begin{verbatim} > rbvevd(3, dep = .8, asy = c(.4, 1), model = "alog") [,1] [,2] [1,] 0.07876 -0.7971 [2,] 0.01091 -0.8113 [3,] -0.10491 -0.8831 > rbvevd(3, alpha = .5, beta = 1.2, model = "negb", mar1 = rep(1, 3)) [,1] [,2] [1,] 0.7417 1.085 [2,] 0.8391 1.825 [3,] 2.0142 2.280 > pbvevd(c(1, 1.2), dep = .4, asy = c(.4, .6), model = "an", mar1 = rep(1, 3)) [1] 0.173 > tmp.quant <- matrix(c(1,1.2,1,2), ncol = 2, byrow = TRUE) > tmp.mar <- matrix(c(1,1,1,1.2,1.2,1.2), ncol = 3, byrow = TRUE) > pbvevd(tmp.quant, dep = .4, asy = c(.4, .6), model = "an", mar1 = tmp.mar) [1] 0.173 0.175 > dbvevd(c(1, 1.2), alpha = .2, beta = .6, model = "ct", mar1 = rep(1, 3)) [1] 0.1213 > dbvevd(tmp.quant, alpha = 0.2, beta = 0.6, model = "ct", mar1 = tmp.mar) [1] 0.1213 0.0586 \end{verbatim} %The logistic and asymmetric logistic models respectively are simulated using bivariate versions of Algorithms 1.1 and 1.2 in \citet{step02a}. %All other models are simulated using a root finding algorithm to generate random vectors from the conditional distribution function. %The simulation of the the bilogistic or negative bilogistic model is relatively slow (about 2.8 seconds per 1000 random vectors on a 450MHz PIII, 512Mb RAM) because each evaluation of either distribution function requires a root finding algorithm to evaluate $\gamma$. \section{Multivariate Extreme Value Distributions} \setcounter{footnote}{0} \label{mult} Let $z=(z_1,\dots,z_d)$. The $d$-dimensional logistic model \citep{gumb60b} has distribution function \begin{equation*} G(z) = \exp\left\{-\left(\sum\nolimits_{j=1}^d y_j^{1/\alpha}\right)^\alpha\right\} \end{equation*} where $\alpha\in(0,1]$ and $(y_1,\dots,y_d)$ is defined by the transformations \eqref{mtrans}. This distribution can be extended to an asymmetric form. Let $B$ be the set of all non-empty subsets of $\{1,\dots,d\}$, let $B_1=\{b \in B:|b|=1\}$, where $|b|$ denotes the number of elements in the set $b$, and let $B_{(i)}=\{b \in B:i \in b\}$. The multivariate asymmetric logistic model \citep{tawn90} is given by \begin{equation*} G(z)=\exp\left\{-\sum\nolimits_{b \in B} \left[\sum\nolimits_{i \in b}(\theta_{i,b}y_i)^{1/\alpha_b}\right]^{\alpha_b}\right\} \end{equation*} where the dependence parameters $\alpha_b\in(0,1]$ for all $b\in B \setminus B_1$, and the asymmetry parameters $\theta_{i,b}\in[0,1]$ for all $b\in B$ and $i\in b$. The constraints $\sum_{b \in B_{(i)}}\theta_{i,b}=1$ for $i=1,\dots,d$ ensure that the marginal distributions are GEV. There exists further constraints which arise from the possible redundancy of asymmetry parameters in the expansion of the distributional form. Specifically, if $\alpha_b=1$ for some $b\in B \setminus B_1$ then $\theta_{i,b}=0$ for all $i \in b$. Let $b_{-i_0}=\{i \in b:i \neq i_0\}$. If, for some $b \in B \setminus B_1$, $\theta_{i,b}=0$ for all $i \in b_{-i_0}$, then $\theta_{i_0,b}=0$. %The model contains $2^d-d-1$ dependence parameters and $d2^{d-1}$ asymmetry parameters (excluding the constraints). %The logistic model \eqref{multlog} can be obtained by setting $\theta_{i,12 \dots d}=1$ for all $i = 1,\dots,d$ (which implies that $\theta_{i,b}=0$ whenever $|b| rmvevd(3, dep = .6, model = "log", d = 5) [,1] [,2] [,3] [,4] [,5] [1,] 0.1335 0.2878 1.07886 1.55515 1.310 [2,] 1.7100 0.9453 1.02070 -0.02553 1.527 [3,] -0.3376 -0.5814 0.07426 0.10906 2.827 > tmp.mar <- matrix(c(1,1,1,1,1,1.5,1,1,2), ncol = 3, byrow = TRUE) > rmvevd(3, dep = .6, d = 5, mar = tmp.mar) [,1] [,2] [,3] [,4] [,5] [1,] 2.803 4.6415 1.8531 3.5569 8.854 [2,] 0.751 0.9704 2.3328 2.6537 1.233 [3,] 4.641 1.4321 0.5825 0.6041 2.021 > tmp.quant <- matrix(rep(c(1,1.5,2), 5), ncol = 5) > pmvevd(tmp.quant, dep = .6, d = 5, mar = tmp.mar) [1] 0.07233 0.16387 0.21949 > dmvevd(tmp.quant, dep = .6, d = 5, mar = tmp.mar, log = TRUE) [1] -3.564 -6.610 -9.460 \end{verbatim} For the asymmetric logistic model \verb+dep+ should be a vector of length $2^{\verb+d+}-\verb+d+-1$ containing the dependence parameters. For example, when $\verb+d+ = 4$ \begin{equation*} \verb+dep+ = \texttt{c}(\alpha_{12},\alpha_{13},\alpha_{14},\alpha_{23},\alpha_{24},\alpha_{34},\alpha_{123},\alpha_{124},\alpha_{134},\alpha_{234},\alpha_{1234}). \end{equation*} The asymmetry parameters should be passed to \verb+asy+ in a list with $2^{\verb+d+}-1$ elements, where each element is a vector (including vectors of length one) corresponding to a set $b \in B$, containing $\{\theta_{i,b}:i \in b\}$. For example, when $\verb+d+ = 4$ \begin{align*} \texttt{asy} = \texttt{list}&(\theta_{1,1}, \theta_{2,2}, \theta_{3,3}, \theta_{4,4}, \texttt{c}(\theta_{1,12},\theta_{2,12}), \texttt{c}(\theta_{1,13},\theta_{3,13}), \texttt{c}(\theta_{1,14},\theta_{4,14}), \texttt{c}(\theta_{2,23},\theta_{3,23}), \\ &\texttt{c}(\theta_{2,24},\theta_{4,24}), \texttt{c}(\theta_{3,34},\theta_{4,34}), \texttt{c}(\theta_{1,123},\theta_{2,123},\theta_{3,123}), \texttt{c}(\theta_{1,124},\theta_{2,124},\theta_{4,124}), \\ &\texttt{c}(\theta_{1,134},\theta_{3,134},\theta_{4,134}), \texttt{c}(\theta_{2,234},\theta_{3,234},\theta_{4,234}), \texttt{c}(\theta_{1,1234},\theta_{2,1234},\theta_{3,1234},\theta_{4,1234})). \end{align*} All the constraints, including $\sum_{b \in B_{(i)}}\theta_{i,b}=1$ for $i=1,\dots,d$, must be satisfied or an error will occur. Some examples are given below. The dependence parameters used in the following trivariate asymmetric logistic model are $(\alpha_{12},\alpha_{13},\alpha_{23},\alpha_{123})=(.6,.5,.8,.3)$. The asymmetry parameters are $\theta_{1,1}=.4$, $\theta_{2,2}=0$, $\theta_{3,3}=.6$, $(\theta_{1,12},\theta_{2,12})=(.3,.2)$, $(\theta_{1,13},\theta_{3,13})=(.1,.1)$, $(\theta_{2,23},\theta_{3,23})=(.4,.1)$ and finally $(\theta_{1,123},\theta_{2,123},\theta_{3,123})=(.2,.4,.2)$. \begin{verbatim} > asy <- list(.4, 0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2)) > rmvevd(3, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) [,1] [,2] [,3] [1,] 0.52375 -0.8844 1.4898 [2,] 1.16174 -0.4368 -0.7404 [3,] -0.03737 1.5139 -0.5996 > dmvevd(c(2, 2, 2), dep = c(.6,.5,.8,.3), asy = asy, model = "a", d = 3) [1] 0.006636 > tmp.quant <- matrix(rep(c(1,1.5,2), 3), ncol = 3) > pmvevd(tmp.quant, dep = c(.6,.5,.8,.3), asy = asy, model = "a", d = 3) [1] 0.4131 0.5849 0.7223 \end{verbatim} The dependence parameters used in the following four dimensional asymmetric logistic model are $\alpha_b = 1$ for $|b| = 2$\footnote{ The values taken by $\alpha_b$ when $|b| = 2$ are irrelevant here because $\theta_{i,b}=0$ for all $i \in b$ when $|b|=2$.} and $(\alpha_{123},\alpha_{124},\alpha_{134},\alpha_{234},\alpha_{1234})=(.7,.3,.8,.7,.5)$. The asymmetry parameters are $\theta_{i,b}=0$ for all $i \in b$ when $|b|\leq2$, $(\theta_{1,123},\theta_{2,123},\theta_{3,123})=(.2,.1,.2)$, $(\theta_{1,124},\theta_{2,124},\theta_{4,124})=(.1,.1,.2)$, $(\theta_{1,134},\theta_{3,134},\theta_{4,134})=(.3,.4,.1)$, $(\theta_{2,234},\theta_{3,234},\theta_{4,234})=(.2,.2,.2)$ and finally $(\theta_{1,1234},\theta_{2,1234},\theta_{3,1234},\theta_{4,1234})=(.4,.6,.2,.5)$. \begin{verbatim} > asy <- list(0, 0, 0, 0, c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(.2,.1,.2), c(.1,.1,.2), c(.3,.4,.1), c(.2,.2,.2), c(.4,.6,.2,.5)) > rmvevd(3, dep = c(rep(1,6),.7,.3,.8,.7,.5), asy = asy, model = "alog", d = 4) [,1] [,2] [,3] [,4] [1,] -0.5930 -0.1916 1.0211 0.6113 [2,] 4.3522 -1.0050 2.3618 -0.1875 [3,] 0.5805 0.4443 -0.5958 0.9717 \end{verbatim} %I will end this section with some examples that may be helpful in deciphering errors. %\begin{verbatim} %> asy <- list(.4, 0, .5, c(.3,.2), c(.1,.15), c(.4,.075), c(.2,.4,.25)) %> rmvevd(3, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) %Error in mvalog.check(asy, dep, d = d) : % `asy' does not satisfy the appropriate constraints % %# 0.5 + 0.15 + 0.075 + 0.25 does not equal one; the sum constraint on the third %margin is not satisfied. % %> asy <- list(.4, 0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2)) %> rmvevd(3, dep = c(.6,1,.8,.3), asy = asy, model = "alog", d = 3) %Error in mvalog.check(asy, dep, d = d) : % `asy' does not satisfy the appropriate constraints % %# A dependence parameter is equal to one but the corresponding asymmetry %parameters are not zero (the first `further constraint'). %# One possible alternative which preserves dep (and still satisfies the sum %constraints) is % %> asy <- list(.4, 0, .6, c(.3,.2), c(0,0), c(.4,.1), c(.3,.4,.3)) %> rmvevd(3, dep = c(.6,1,.8,.3), asy = asy, model = "alog", d = 3) % [,1] [,2] [,3] %[1,] 4.627 2.9125 0.9414 %[2,] 1.200 0.1556 0.2048 %[3,] -1.159 -0.8749 -1.0340 % %> asy <- list(.5, 0, .6, c(.3,.2), c(0,.1), c(.4,.1), c(.2,.4,.2)) %> rmvevd(3, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) %Error in mvalog.check(asy, dep, d = d) : % `asy' does not satisfy the appropriate constraints % %# The fifth element in asy contains exactly one non-zero asymmetry parameter %(the second `further constraint'). % %> asy <- list(.4, 0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2)) %> rmvevd(3, dep = c(.6,.5,.8,.3), asy = asy, model = "alog") %Error in mvalog.check(asy, dep, d = d) : % `asy' should be a list of length 3 % %# the dimension has been mis-specified (the default dimension is 2). %\end{verbatim} \section{Dependence Functions} \setcounter{footnote}{0} \label{depfun} Let $z=(z_1,\dots,z_d)$ and $\omega=(\omega_1,\dots,\omega_d)$. Any $d$-dimensional extreme value distribution function can be represented in the form \begin{equation*} G(z) = \exp\left\{ - \left\{\sum\nolimits_{j=1}^d y_j \right\} A\left(\frac{y_1}{\sum\nolimits_{j=1}^d y_j}, \dots, \frac{y_d}{\sum\nolimits_{j=1}^d y_j} \right)\right\}, \end{equation*} where $(y_1,\dots,y_d)$ is defined by the transformations \eqref{mtrans}. It follows that $A(\omega)=-\log\{G(y_1^{-1}(\omega_1),\dots,y_d^{-1}(\omega_d))\}$, defined on the simplex $S_d =\{\omega \in \mathbb{R}^d_+: \sum_{j=1}^d \omega_j = 1\}$. $A(\cdot)$ is known as the dependence function. The dependence function characterises the dependence structure of $G$. It can be shown that $A(\omega)=1$ when $\omega$ is one of the $d$ vertices of $S_d$ (i.e.\ when one component of $\omega$ is equal to one, and all remaining components are equal to zero), and that $A$ is a convex function with $\max(\omega_1,\dots,\omega_d) \leq A(\omega) \leq 1$ for all $\omega \in S_d$. The lower and upper bounds are obtained at complete dependence and mutual independence respectively. In particular, $A(1/d,\dots,1/d)$ is equal to $1/d$ at complete dependence, and $1$ at mutual independence. The dependence function of a \emph{bivariate} extreme value distribution is a special case (because the sets $S_2$ and [0,1] are equivalent), and is typically defined as follows. Any bivariate extreme value distribution function can be represented in the form \begin{equation} G(z_1,z_2) = \exp\left\{ - (y_1 + y_2)A\left(\frac{y_1}{y_1+y_2}\right)\right\}, \label{bvdepfn} \end{equation} so that $A(\omega)=-\log\{G(y_1^{-1}(\omega),y_2^{-1}(1-\omega))\}$, defined on $0\leq\omega\leq1$.\footnote{Some authors \citep[e.g.][]{pick81} use $A(\omega)=-\log\{G(y_1^{-1}(1-\omega),y_2^{-1}(\omega))\}$.} It follows that $A(0)=A(1)=1$, and that $A(\cdot)$ is a convex function with $\max(\omega,1-\omega) \leq A(\omega) \leq 1$ for all $0\leq\omega\leq1$. At independence $A(1/2) = 1$. At complete dependence $A(1/2) = 0.5$. Dependence functions for parametric bivariate and trivariate extreme value models can be calculated and plotted, at given parameter values, using the functions \verb+abvevd+ and \verb+amvevd+. Non-parametric estimators of dependence functions can also be calculated and plotted, using the functions \verb+abvnonpar+ and \verb+amvnonpar+. Some examples are given below. The last lines of code produce Figure \ref{depfns}. %Non-parametric estimators of dependence functions of bivariate extreme value models are constructed as follows. %Suppose $(z_{i1},z_{i2})$ for $i=1,\dots,n$ are $n$ bivariate observations that are passed to \verb+abvnonpar+ using the argument \verb+data+. %The marginal parameters are estimated (under the assumption of independence) and the data is transformed using %\begin{align} %y_{i1} &= \{1+\hat{\xi}_1(z_{i1}-\hat{\mu}_1)/\hat{\sigma}_1\}_{+}^{-1/\hat{\xi}_1} \notag \\ %y_{i2} &= \{1+\hat{\xi}_2(z_{i2}-\hat{\mu}_2)/\hat{\sigma}_2\}_{+}^{-1/\hat{\xi}_2} %\label{transtoexp} %\end{align} %for $i=1,\dots,n$, where $(\hat{\mu}_1,\hat{\sigma}_1,\hat{\xi}_1)$ and $(\hat{\mu}_2,\hat{\sigma}_2,\hat{\xi}_2)$ are the maximum likelihood estimates for the location, scale and shape parameters on the first and second margins. %If non-stationary fitting is implemented using the \verb+nsloc1+ or \verb+nsloc2+ arguments (see Sections \ref{unifit} and \ref{bivfit}) the marginal location parameters may depend on $i$. %The estimator is specified using the argument \verb+method+. A number of different estimators are implemented. A short simulation study given in Appendix A compares the properties of these estimators. The default estimator is the estimator of \citet{capefoug97}, which is defined (on $0 \leq \omega \leq 1$) by %which must be either \verb+"pickands"+, \verb+"deheuvels"+, \verb+"cfg"+ (the default), \verb+"tdo"+ or \verb+"hall"+ (or any unique partial match). %These estimators are respectively defined (on $0 \leq \omega \leq 1$) as follows. %\begin{equation*} %\exp\left\{ \{1-p(\omega)\} \int_{0}^{\omega} \frac{H_n(x) - x}{x(1-x)} \, \text{d}x - p(\omega) \int_{\omega}^{1} \frac{H_n(x) - x}{x(1-x)} \, \text{d}x \right\} %\end{equation*} %\citet{pick81} %\begin{equation*} %A_p(\omega) = n\left\{\sum_{i=1}^n \min\left(\frac{y_{i1}}{\omega},\frac{y_{i2}}{1-\omega}\right)\right\}^{-1} %\end{equation*} %\citet{dehe91} %\begin{equation*} %A_d(\omega) = n\left\{\sum_{i=1}^n \min\left(\frac{y_{i1}}{\omega},\frac{y_{i2}}{1-\omega}\right) - \omega\sum_{i=1}^n y_{i1} - (1-\omega)\sum_{i=1}^n y_{i2} + n\right\}^{-1} %\end{equation*} %\citet{capefoug97} %\begin{equation*} %A_c(\omega) = \exp\left\{ \{1-p(\omega)\} \int_{0}^{\omega} \frac{H_n(x) - x}{x(1-x)} \, \text{d}x - p(\omega) \int_{\omega}^{1} \frac{H_n(x) - x}{x(1-x)} \, \text{d}x \right\} %\end{equation*} %\citet{tiag97} %\begin{equation*} %A_t(\omega) = 1 - \frac{1}{1 + \log n} \sum_{i=1}^n \min\left(\frac{\omega}{1+ny_{i1}},\frac{1-\omega}{1+ny_{i2}}\right) %\end{equation*} %\citet{halltajv00} %\begin{equation*} %A_h(\omega) = n\left\{\sum_{i=1}^n \min\left(\frac{y_{i1}}{\bar{y}_1 \omega},\frac{y_{i2}}{\bar{y}_2 (1-\omega)}\right)\right\}^{-1} %\end{equation*} %where $H_n(x)$ is the empirical distribution function of $x_1,\dots,x_n$, with $x_i = y_{i1} / (y_{i1} + y_{i2})$ for $i=1,\dots,n$, and $p(\cdot)$ is any bounded function on $[0,1]$, which can be specified using the argument \verb+wf+. %By default $p(\cdot)$ is the identity function. %In the estimator of \citet{halltajv00}, $\bar{y}_1 = n^{-1}\sum_{i=1}^n y_{i1}$ and $\bar{y}_2 = n^{-1}\sum_{i=1}^n y_{i2}$. %Let $A_n(\cdot)$ be any estimator of $A(\cdot)$. %$A_n(\cdot)$ will not necessarily satisfy $\max(\omega,1-\omega) \leq A_n(\omega) \leq 1$ for all $0\leq\omega\leq1$. %An obvious modification is %\begin{equation*} %A_n^{'}(\omega) = \min(1, \max\{A_n(\omega), \omega, 1-\omega\}). %\end{equation*} %The function \verb+abvnonpar+ always implements this modification. %Another estimator $A_n^{''}(\omega)$ can be derived by taking the convex minorant of $A_n^{'}(\omega)$. %This can be achieved by setting the argument \verb+convex+ to \verb+TRUE+. %Some examples of the functions described in this section are given below. %The last eight lines of code produce Figure \ref{depfns}. \begin{verbatim} > bvlsm <- rmvevd(100, dep = 0.6, model = "log", d = 2) > tvlsm <- rmvevd(100, dep = 0.6, model = "log", d = 3) > abvevd(seq(0,1,0.25), dep = 0.3, asy = c(.7,.9), model = "alog") [1] 1.0000 0.8272 0.7013 0.7842 1.0000 > abvnonpar(seq(0,1,0.25), data = bvlsm) [1] 1.0000 0.8634 0.8158 0.8392 1.0000 > abvnonpar(data = bvlsm, plot = TRUE, blty = 1, lty = 2) > abvevd(dep = .3, asy = c(.5, .9), model = "al", add = TRUE) > abvevd(dep = 1.05, model = "hr", add = TRUE) > amvnonpar(data = tvlsm, plot = TRUE, lower = 0.6) \end{verbatim} \begin{figure} \hspace*{2.5cm} \scalebox{0.25}{\includegraphics{depfns5.ps}} \hspace{0cm} \scalebox{0.3}{\includegraphics[0,100][260,580]{depfns6.ps}} \vspace{-1cm} \caption{Left: Parametric (solid lines) and non-parametric (dashed line) dependence functions for bivariate distributions. The triangular border represents the constraint $\max(\omega,1-\omega) \leq A(\omega) \leq 1$ for all $\omega \in [0,1]$. Right: non-parametric dependence function for a trivariate distribution. Darker colours depict smaller values, and hence stronger dependence.} \label{depfns} \end{figure} \section{Stochastic Processes} \setcounter{footnote}{0} \label{stochproc} The evd package contains four functions that simulate from stochastic processes associated with extreme value theory. The functions \verb+marma+, \verb+mar+ and \verb+mma+ generate max autoregressive moving average processes, and the function \verb+evmc+ generates Markov chains with extreme value dependence structures. The function \verb+clusters+ identifies extreme clusters of a stochastic process, and \verb+exi+ estimates a quantity known as the Extremal Index. A max autoregressive moving average process $\{X_k\}$, denoted by MARMA($p$, $q$), satisfies \begin{equation*} X_k = \max\{\phi_1 X_{k-1}, \dots, \phi_p X_{k-p}, \epsilon_k, \theta_1 \epsilon_{k-1}, \dots, \theta_q \epsilon_{k-q}\} \end{equation*} where $(\phi_1, \dots, \phi_p)$ and $(\theta_1, \ldots, \theta_p)$ are vectors of non-negative parameters, and $\{\epsilon_k\}$ is a series of \emph{iid} random variables with a common distribution defined by the argument \verb+rand.gen+. The standard Fr\'{e}chet distribution is used by default. A max autoregressive process $\{X_k\}$, denoted by MAR($p$), is equivalent to a MARMA($p$, 0) process, so that \begin{equation*} X_k = \max\{\phi_1 X_{k-1}, \dots, \phi_p X_{k-p}, \epsilon_k\}. \end{equation*} A max moving average process $\{X_k\}$, denoted by MMA($q$), is equivalent to a MARMA(0, $q$) process, so that \begin{equation*} X_k = \max\{\epsilon_k, \theta_1 \epsilon_{k-1}, \dots, \theta_q \epsilon_{k-q}\}. \end{equation*} The functions \verb+mar+, \verb+mma+ and \verb+marma+ generate MAR($p$), MMA($q$) and MARMA($p$, $q$) processes respectively. Examples of calls to these functions are given below. The \verb+n.start+ argument denotes the burn-in period, which can be specified so that the output series is not unduly influenced by the $p$ starting values, which are all zero by default. \begin{verbatim} > marma(100, p = 1, q = 1, psi = 0.75, theta = 0.65) > mar(100, psi = 0.85, n.start = 20) > mma(100, q = 2, theta = c(0.75, 0.8)) \end{verbatim} The function \verb+evmc+ generates first order Markov chains. Informally, a first order Markov chain $X_1, \ldots, X_n$ is a stochastic process such that at any given time $t$ the probability distribution of $X_{t+1}$ is independent the past $X_1, \ldots, X_{t-1}$, given the current state $X_t$. The \verb+evmc+ function generates a first order Markov chain such that each pair of consecutive values has the dependence structure of a parametric bivariate extreme value model. The main arguments of \verb+evmc+ are the same as those of \verb+rbvevd+. The function \verb+evmc+ also has the argument \verb+margin+, which denotes the marginal distribution of each value. This must be either \verb+"uniform"+ (the default), \verb+"rweibull"+, \verb+"frechet"+ or \verb+"gumbel"+ (or any unique partial match), for the uniform, standard reversed Weibull, standard Gumbel and standard Fr\'{e}chet distributions respectively. Examples of calls to \verb+evmc+ are given below. \begin{verbatim} > evmc(100, alpha = 0.1, beta = 0.1, model = "bilog") > evmc(100, dep = 10, model = "hr", margins = "gum") \end{verbatim} The function \verb+clusters+ identifies extreme clusters within (stationary) stochastic processes. A simple way of determining clusters is to specify a threshold $u$ and define consecutive exceedances of $u$ to belong to the same cluster. It is more common though to consider a cluster to be active until $r$ consecutive values fall below (or are equal to) $u$, for some given clustering interval length $r$. %If $r > 1$ the clusters may contain any arbitrarily low value. %To avoid this problem a lower threshold $u_l < u$ can be specified so that a cluster is terminated whenever any values fall below (or are equal to) $u_l$. The following code uses \verb+clusters+ to generate the plots depicted in Figure \ref{clust}. These plots identify clusters graphically. If the argument \verb+plot+ is \verb+FALSE+ (the default), then \verb+clusters+ returns a list of extreme clusters. \begin{verbatim} > set.seed(150) > x <- evmc(50, dep = 0.55, model ="log") > clusters(x, 0.8, plot = TRUE) > clusters(x, 0.8, 4, plot = TRUE) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{clust1.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{clust3.ps}} \end{center} \vspace{0cm} \caption{The identification of extreme clusters in a stochastic process. The clustering interval lengths are $r = 1$ (left) and $r=4$ (right). The threshold in each case is $u = 0.8$.} \label{clust} \end{figure} The function \verb+exi+ returns estimates of the Extremal Index of a (stationary) stochastic process. The Extremal Index is defined in Chapter 3 of \citet{leadling83}. A more informal treatment is given in Chapter 5 of \citet{cole01}. The extremal index can be estimated using the inverse of the average size of extreme clusters, where the cluster size is defined as the number of exceedances that it contains. \section{Fitting Univariate Distributions} \setcounter{footnote}{0} \label{unifit} This section presents functions that produce maximum likelihood estimates for some of the distributions introduced in Section \ref{uni}. Peaks over threshold models are discussed in Section \ref{potfit}. Maximum likelihood estimates for bivariate extreme value distributions are discussed in Section \ref{bivfit}. For illustrative purposes Sections \ref{unifit}, \ref{potfit} and \ref{bivfit} use only simulated data. Three practical examples using the data sets \verb+oxford+, \verb+rain+ and \verb+sealevel+ are given in Sections \ref{egoxford}, \ref{egrain} and \ref{egsealevel} respectively. The function \verb+fgev+ produces maximum likelihood estimates for the GEV distribution \eqref{gev}. The first argument should be a numeric vector containing data to be fitted. Missing values are allowed. If the argument \verb+start+ is given it should be a named list containing starting values, the names of which should be the parameters over which the likelihood is to be maximised. If \verb+start+ is omitted the routine attempts to find good starting values for the optimisation using moment estimators. If any of the parameters are to be set to fixed values, they can be given as separate arguments. For example, the Gumbel distribution \eqref{gumbel} can be fitted using \verb+shape = 0+. Arguments of the optimisation function \verb+optim+ can also be specified. This includes the optimisation method, which can be passed using the argument \verb+method+. Two examples of the \verb+fgev+ function are given below. \begin{verbatim} > data1 <- rgev(1000, loc = 0.13, scale = 1.1, shape = 0.2) > m1 <- fgev(data1) > m1 Call: fgev(x = data1) Deviance: 3650 Estimates loc scale shape 0.127 1.125 0.224 Standard Errors loc scale shape 0.0400 0.0321 0.0248 Optimization Information Convergence: successful Function Evaluations: 51 Gradient Evaluations: 12 > m2 <- fgev(data1, loc = 0, scale = 1) > fitted(m2) shape 0.236 \end{verbatim} In the first example the likelihood is maximised over (\verb+loc+, \verb+scale+, \verb+shape+). In the second example the likelihood is maximised over \verb+shape+, with the location and scale parameters fixed at zero and one respectively. The maximum likelihood estimators do not necessarily have the usual asymptotic properties, since the end points of the GEV distribution depend on the model parameters. \citet{smit85} shows that the usual asymptotic properties hold when $\xi > -0.5$. When $-1 < \xi \leq -0.5$ the maximum likelihood estimators do not have the standard asymptotic properties, but generally exist. When $\xi \leq -1$ maximum likelihood estimators do not often exist. This occurs because of the large mass near the upper end point. The likelihood increases without bound as the upper end point is estimated to be closer and closer to the largest observed value. In terms of the reversed Weibull shape parameter $\alpha$, the usual asymptotic properties hold when $\alpha>2$, the asymptotic properties are not standard for $1<\alpha\leq2$, and maximum likelihood estimators do not often exist for $\alpha<1$. When the usual asymptotic properties hold (as here) the standard errors of the maximum likelihood estimates, approximated using the inverse of the observed information matrix, can be extracted from the fitted object using \begin{verbatim} > std.errors(m1) loc scale shape 0.03999 0.03214 0.02479 \end{verbatim} %When the usual asymptotic properties do not hold the \verb+std.errors+ component will still be based on the inverse of the observed information matrix, but these values must be \emph{interpreted with caution} \citep{smit85}. Likelihood ratio tests can be performed using the function \verb+anova+. We can compare the two models \verb+m1+ and \verb+m2+ to test the null hypothesis that the location parameter is zero and the scale parameter is one. \begin{verbatim} > anova(m1, m2) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) m1 3 3650 m2 1 3669 2 18.8 8.2e-05 \end{verbatim} The deviance difference, \verb+deviance(m2)+ minus \verb+deviance(m1)+, is about $18.8$, which yields a p-value of $8.2 \times 10^{-5}$ when compared with a chi-squared distribution on two degrees of freedom. Diagnostic plots and profile traces for fitted models can be constructed using the functions \verb+plot+, \verb+profile+ and \verb+profile2d+ (see Section \ref{egoxford}). By default the maximum likelihood estimates are calculated under the assumption that the data to be fitted are the observed values of independent random variables $Z_1,\dots,Z_n$, where $Z_i \sim \text{GEV}(\mu,\sigma,\xi)$ for each $i=1,\dots,n$. The \verb+nsloc+ argument allows non-stationary models of the form $Z_i \sim \text{GEV}(\mu_i,\sigma,\xi)$, where \begin{equation*} \mu_i = \beta_0 + \beta_1x_{i1} + \dots + \beta_kx_{ik}. \end{equation*} The parameters $(\beta_0,\dots,\beta_k)$ are to be estimated. In matrix notation $\boldsymbol{\mu} = \boldsymbol{\beta_0} + X \boldsymbol{\beta} $, where $ \boldsymbol{\mu}= (\mu_1,\dots,\mu_n)^T$, $\boldsymbol{\beta_0} = (\beta_0,\dots,\beta_0)^T$, $\boldsymbol{\beta} = (\beta_1,\dots,\beta_k)^T$ and $X$ is the $n \times k$ covariate matrix (excluding the intercept) with $ij$th element $x_{ij}$. The \verb+nsloc+ argument must be a data frame containing the matrix $X$, or a numeric vector which is converted into a single column data frame with column name ``trend''. The column names of the data frame are used to derive names for the estimated parameters. This allows any of the $k+3$ parameters $(\beta_0,\dots,\beta_k,\sigma,\xi)$ to be set to fixed values within the optimisation. The covariates must be (at least approximately) \emph{centred and scaled}, not only for numerical reasons, but also because the starting value (if \verb+start+ is not given) for each corresponding coefficient is taken to be zero. When a linear trend is present in the data, the location parameter is often modelled as \begin{equation*} \mu_i = \beta_0 + \beta_1t_i, \end{equation*} where $t_i$ is some centred and scaled version of the time of the $i$th observation. More complex changes in $\mu$ may also be appropriate. For example, a change-point model \begin{equation*} \mu_i = \beta_0 + \beta_1x_i \qquad \text{where} \qquad x_i = \begin{cases} 0 & i \leq i_0 \\ 1 & i > i_0 \end{cases}, \end{equation*} or a quadratic trend \begin{equation*} \mu_i = \beta_0 + \beta_1t_i + \beta_2t_i^2. \end{equation*} See Sections \ref{egoxford} and \ref{egsealevel} for examples of non-stationary modelling. The function \verb+fgev+ also has an argument called \verb+prob+. If $\verb+prob+ = p$ is passed a value in the interval [0,1], \verb+fgev+ again produces maximum likelihood estimates for the GEV distribution, but the model is re-parameterised from $(\mu,\sigma,\xi)$ to $(z_p,\sigma,\xi)$, where $z_p$ is the quantile corresponding to the upper tail probability $p$. This argument can be used to calculate and plot profile log-likelihoods of extreme quantiles (see Section \ref{egoxford}). If \verb+prob+ is zero/one, then $z_p$ is defined as the upper/lower end point $\mu - \sigma/\xi$, and $\xi$ is restricted to the negative/positive axis. Under non-stationarity the model is re-parameterised from $(\beta_0,\beta_1,\dots,\beta_k,\sigma,\xi)$ to $(z_p,\beta_1,\dots,\beta_k,\sigma,\xi)$, so that $z_p$ is the quantile corresponding to the upper tail probability $p$ for the distribution obtained when all covariates are zero. The \verb+fextreme+ function produces maximum likelihood estimates for the distributions \eqref{maxdens} and \eqref{mindens} given an integer $m$ and an arbitrary distribution function $F$. The first argument should be a numeric vector containing the data to be fitted, which should represent maxima (if the argument \verb+largest+ is \verb+TRUE+, the default) or minima (if \verb+largest+ is \verb+FALSE+). The argument \verb+start+ (which cannot be missing) should be a named list containing starting values, the names of which should be the parameters over which the likelihood is to be maximised. If any of the parameters are to be set to fixed values, they can be given as separate arguments. Arguments of the optimisation function \verb+optim+ can also be specified. The example given below produces maximum likelihood estimates for the distribution \eqref{maxdens}, where $m = 365$ and $F$ is the normal distribution. \begin{verbatim} > d2 <- rextreme(100, distn = "norm", mean = 0.56, mlen = 365) # Simulate yearly maxima using normal distribution > sv <- list(mean = 0, sd = 1) > nm <- fextreme(d2, start = sv, distn = "norm", mlen = 365) > fitted(nm) mean sd 0.685 0.959 \end{verbatim} The \verb+forder+ function yields maximum likelihood estimates for the distribution \eqref{orderdens} given integers $m$ and $j \in \{1,\dots,m\}$, and an arbitrary distribution function $F$. An example is given below, where $m = 365$, $j = 2$ and $F$ is the normal distribution. \begin{verbatim} > d3 <- rorder(100, distn = "norm", mean = 0.56, mlen = 365, j = 2) > sv <- list(mean = 0, sd = 1) > nm2 <- forder(d3, sv, distn = "norm", mlen = 365, j = 2) > fitted(nm2) mean sd 0.483 1.042 \end{verbatim} \section{Fitting Peaks Over Threshold Models} \setcounter{footnote}{0} \label{potfit} Suppose $X_1,\dots,X_n$ is a sequence of independent and identically distributed random variables, with $M_n = \{X_1,\dots,X_n\}$. Suppose that $n$ is large, so that (assuming certain regularity conditions) the distribution of $M_n$ is approximately GEV\@. Then for large enough $u$, the exceedances of the threshold $u$ are approximately distributed as generalised Pareto, with location parameter $u$. The function \verb+fpot+ fits this distribution to the exceedances, and hence produces maximum likelihood estimates for the shape and scale parameters. The value of the threshold $u$ must be specified by the user. It is typically chosen to be as small as possible, subject to the limit model providing a reasonable approximation. The functions \verb+mrlplot+ and \verb+tcplot+\footnote{Both of these functions are heavily based on code by Stuart Coles.} produce diagnostic plots that facilitate the specification of $u$. The function \verb+mrlplot+ produces the empirical mean residual life plot, which is a plot of the empirical mean of the excesses of $u$ (i.e.\ the exceedances of $u$ minus $u$), plotted against $u$. If the exceedances of a threshold $u_0$ are generalised Pareto, the empirical mean residual life plot should be approximately linear for all $u > u_0$. The function \verb+tcplot+ calculates maximum likelihood estimates for the shape and modified scale parameters using a number of different thresholds, and plots these estimates against $u$. If the exceedances of a threshold $u_0$ are generalised Pareto, the shape and modified scale parameters should be approximately constant with respect to all thresholds $u > u_0$. Threshold identification plots produced from the example given below are depicted in Figure \ref{threshid}. In this case, the threshold $u = 1$ was chosen. \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{threshid1.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{threshid2.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{threshid3.ps}} \end{center} \caption{The identification of a threshold for the (generalised Pareto) peaks over threshold model. From left to right; the empirical mean residual life plot, modified scale parameter estimates and shape parameter estimates.} \label{threshid} \end{figure} The following code generates $n = 500$ independent standard normal random variables and fits the (generalised Pareto) peaks over threshold model to the exceedances of the threshold $u = 1$. The function \verb+fpot+ performs the fit. Many of the arguments of \verb+fpot+ are similar to those of \verb+fgev+. In particular, either of the \verb+scale+ or \verb+shape+ parameters can be set to fixed values by giving those parameters as arguments. For example, an exponential distribution for the excesses (or equivalently, a shifted exponential distribution for the exceedances) can be fitted using \verb+shape = 0+. \begin{verbatim} > tmp <- rnorm(500) > mrlplot(tmp, tlim = c(-1,1.5)) > tcplot(tmp, tlim = c(-1,1.5)) > pot1 <- fpot(tmp, 1) > pot1 Call: fpot(x = tmp, threshold = 1) Deviance: 40.5 Threshold: 1 Number Above: 76 Proportion Above: 0.152 Estimates scale shape 0.593 -0.211 \end{verbatim} The fitted model \verb+pot1+ gives the estimates for the scale and shape parameters of the generalised Pareto distribution fitted to the exceedances. Also given is the proportion of values above the threshold, or equivalently, the maximum likelihood estimate for the probability of an exceedance. Diagnostic plots and profile traces for fitted models can be constructed using the functions \verb+plot+ and \verb+profile+ (see Section \ref{egrain}). The peaks over thresholds model is typically extended to stationary series via declustering, which corresponds to a filtering of dependent observations to obtain a set of threshold exceedances which are approximately independent. An empirical rule is used to identify clusters of exceedances, and the generalised Pareto model is then fitted to the cluster maxima, assuming those maxima to be independent. The empirical rule, as given in Section \ref{stochproc}, is defined by the function \verb+clusters+. A model of this form can be implemented by setting the logical argument \verb+cmax+ to \verb+TRUE+. The clusters are identified using the threshold of the peaks over threshold model. An illustration of this technique is given below. The argument \verb+r+ is the clustering interval length. \begin{verbatim} > tmp2 <- evmc(500, dep = 0.8, margins = "gum") > pot2 <- fpot(tmp2, 1.5, cmax = TRUE, r = 3) > pot2 Call: fpot(x = tmp, threshold = 1, cmax = TRUE, r = 3) Deviance: 101.1 Threshold: 1.5 Number Above: 92 Proportion Above: 0.184 Clustering Interval: 3 Number of Clusters: 41 Extremal Index: 0.446 Estimates scale shape 1.657 -0.272 \end{verbatim} The Extremal Index is a quantity briefly discussed in Section \ref{stochproc}. The estimate of the Extremal Index is simply the number of clusters divided by the number of exceedances. The function \verb+fpot+ also has an argument called \verb+mper+. If $\verb+mper+ = m$ is passed a positive value, \verb+fpot+ again produces maximum likelihood estimates for the generalised Pareto model, but the model is re-parameterised from $(\sigma,\xi)$ to $(z_m,\xi)$, where $z_m$ is the $m$-period return level, defined as follows. Let $G$ be the fitted generalised Pareto distribution function, with location parameter equal to the specified threshold $u$, so that $1 - G(z)$ is the fitted probability of an exceedance over $z > u$ given an exceedance over $u$. The fitted probability of an exceedance over $z > u$ is therefore $p(1 - G(z))$, where $p$ is the estimated probability of exceeding $u$, which is given by the empirical proportion of exceedances. The $m$-period return level $z_m$ satisfies $p(1 - G(z_m)) = 1/(mN\hat{\theta})$, where $N$ is the number of observations per period, and $\hat{\theta}$ is the estimate of the extremal index if cluster maxima are fitted, with $\hat{\theta} = 1$ otherwise. The value $N$ can be specified using the argument \verb+npp+. For example, if observations are recorded weekly and $\verb+npp+ = 52$, then $z_m$ is the $m$-year return level. If \verb+mper+ is \verb+Inf+, then $z_m$ is defined as the upper end point $u - \sigma/\xi$, and $\xi$ is then restricted to be negative. The argument \verb+mper+ can be used to calculate and plot profile log-likelihoods of return levels (see Section \ref{egrain}). %The peaks over threshold model permits an alternative characterization in terms of point processes. %Suppose again that $X_1,\dots,X_n$ is a sequence of independent and identically distributed random variables, with $M_n = \{X_1,\dots,X_n\}$, and that $n$ is large, so that (assuming certain regularity conditions) the distribution of $M_n$ is approximately \text{GEV}($\mu,\sigma,\xi$), with (possibly infinite) end points\footnote{If $\xi > 0$, $z_- = \mu - \sigma/\xi$ and $z_+ = \infty$. If $\xi < 0$, $z_- = -\infty$ and $z_+ = \mu - \sigma/\xi$. If $\xi = 0$, the expressions given are all defined by continuity, with $z_- = -\infty$ and $z_+ = \infty$.} $z_-$ and $z_+$. Then for large enough $u > z_-$, the sequence $\{X_1,\dots,X_n\}$ viewed on the interval $(u,z_+)$ can be approximated by a non-homogeneous Poisson process \citep{cole01}. %The approximation leads to a likelihood for ($\mu,\sigma,\xi$), and hence maximum likelihood estimates can be obtained. %The likelihood can be easily adjusted so that the maxima of a given (large) number $N \leq n$ of random variables is approximately distributed as \text{GEV}($\mu,\sigma,\xi$), so that e.g.\ if observations are recorded weekly and $N = 52$, then ($\mu,\sigma,\xi$) corresponds to the distribution of annual maxima. %The point process characterization can be fitted using the \verb+fpot+ function with \verb+model = "pp"+. %The value $N$ can by specified using the argument \verb+npp+. %If \verb+npp+ is unspecified the default value $N = n$ is used. %The following code uses the point process characterization to fit a peaks over threshold model to the simulated data \verb+tmp+. %The models \verb+pot3+ and \verb+pot4+ are equivalent; the estimates in \verb+pot3+ correspond to the GEV distribution for the maxima of the data set, whereas those in \verb+pot4+ correspond to the GEV distribution for annual maxima, assuming the observations are recorded daily. %\begin{verbatim} %> pot3 <- fpot(tmp, 1, model = "pp", npp = 500) %> pot4 <- fpot(tmp, 1, model = "pp", npp = 365.25) %> fitted(pot3) % loc scale shape % 2.6839 0.2380 -0.2108 %> fitted(pot4) % loc scale shape % 2.6065 0.2542 -0.2108 % %> fitted(pot1) % scale shape % 0.593 -0.211 %\end{verbatim} %Also given above is the parameter estimates for the model \verb+pot1+, fitted using the generalised Pareto characterization. Let $(\tilde{\sigma}, \tilde{\xi})$ denote the scale and shape parameters of the \text{GPD}. The relationship between the two characterizations is then given by $\tilde{\xi} = \xi$ and $\tilde{\sigma} = \sigma + \xi(u - \mu)$, where $u$ is the threshold. %This relationship can be seen in the above estimates. %Under the generalized Pareto characterization, the parameter $\tilde{\sigma} - \tilde{\xi} u$ is referred to as the modified scale parameter, as plotted in the centre panel of Figure \ref{threshid}. Unlike $\tilde{\sigma} = \tilde{\sigma}(u)$, the modified scale parameter does not depend on the threshold $u$. \section{Fitting Bivariate Extreme Value Distributions} \setcounter{footnote}{0} \label{bivfit} The function \verb+fbvevd+ produces maximum likelihood estimates for nine bivariate extreme value models. The first argument should be a numeric matrix (or a data frame) with two columns containing the data to be fitted. Missing values are allowed. If the argument \verb+start+ is given it should be a named list containing starting values, the names of which should be the parameters over which the likelihood is to be maximised. If \verb+start+ is omitted the routine attempts to find good starting values for the optimisation using maximum likelihood estimators under the assumption of independence. If any of the parameters are to be set to fixed values, they can be given as separate arguments. Common marginal parameters can be fitted using the arguments \verb+cshape+, \verb+cscale+ and \verb+cloc+, and the dependence function can be constrained to symmetry using the argument \verb+sym+ (see the \verb+fbvevd+ help file for details). The \verb+nsloc1+ and \verb+nsloc2+ arguments allow non-stationary modelling of the location parameters on the first and second margins respectively. They should be used in the same manner as the \verb+nsloc+ argument of \verb+fgev+. Examples of bivariate models with non-stationary margins are given in Section \ref{egsealevel}. %For numerical reasons the parameters of each model are subject to the artificial constraints depicted in Table \ref{contab}. The scale parameters on each GEV margin are artificially constrained to be greater than or equal to $0.01$. These constraints only apply to the functions discussed in this section. %\begin{table} %\begin{center} %\begin{tabular}{l|c} %Bivariate Model & Constraints \\ \hline %Logistic & $0.1\leq\alpha\leq1$ \\ %Asymmetric Logistic & $0.1\leq\alpha\leq1$, $0.001\leq\theta_1,\theta_2\leq1$ \\ %H\"{u}sler-Reiss & $0.2\leq\lambda\leq10$ \\ %Negative Logistic & $0.05\leq r \leq5$ \\ %Asymmetric Negative Logistic & $\quad0.05\leq r \leq5$, $0.001\leq\theta_1,\theta_2\leq1\quad$ \\ %Bilogistic & $0.1\leq\alpha,\beta\leq0.999$ \\ %Negative Bilogistic & $0.1\leq\alpha,\beta\leq20$ \\ %Coles-Tawn & $0.001\leq\alpha,\beta\leq30$ \\ \hline %\end{tabular} %\caption{For numerical reasons the parameters of each model are subject to the artificial constraints depicted here.} %\label{contab} %\end{center} %\end{table} The first example given below produces maximum likelihood estimates for the (symmetric) logistic model. The second example constrains the model at independence (where $\texttt{dep} = 1$). The estimates produced in the second example are the same as those that would be produced if \verb+fgev+ was separately applied to each margin. \begin{verbatim} > bvdata <- rbvevd(100, dep = 0.6, mar1 = c(1.2,1.4,0), mar2 = c(1,1.6,0.1)) > m1 <- fbvevd(bvdata, model = "log") > m1 Call: fbvevd(x = bvdata, model = "log") Deviance: 728.5 AIC: 742.5 Dependence: 0.3526 Estimates loc1 scale1 shape1 loc2 scale2 shape2 dep 1.2121 1.3831 -0.1813 0.8404 1.4005 0.0834 0.7202 Standard Errors loc1 scale1 shape1 loc2 scale2 shape2 dep 0.1540 0.1091 0.0673 0.1537 0.1144 0.0614 0.0624 Optimization Information Convergence: successful Function Evaluations: 47 Gradient Evaluations: 10 > m2 <- fbvevd(bvdata, model = "log", dep = 1) > fitted(m2) loc1 scale1 shape1 loc2 scale2 shape2 1.2231 1.3776 -0.1914 0.8367 1.4083 0.0868 > std.errors(m2) loc1 scale1 shape1 loc2 scale2 shape2 0.1543 0.1089 0.0725 0.1565 0.1163 0.0670 > c(logLik(m2), deviance(m2), AIC(m2)) [1] -376 752 764 \end{verbatim} The discussion in Section \ref{unifit} regarding the properties of maximum likelihood estimators for the GEV distribution also applies to all bivariate models. The usual asymptotic properties will not hold if either of the marginal shape parameters are less than $-0.5$. %When the usual asymptotic properties do not hold the \verb+std.errors+ component will still be based on the inverse of the observed information matrix, but these values must be \emph{interpreted with caution} \citep{smit85}. The value in the output labelled \verb+Dependence+ is the fitted estimate of $\chi = 2\{1-A(1/2)\} \in [0,1]$ \citep{coleheff99}, where $A(\cdot)$ denotes the dependence function \eqref{bvdepfn}. At independence $\chi = 0$, and at complete dependence $\chi = 1$. Diagnostic plots and profile traces for fitted models can be constructed using the functions \verb+plot+, \verb+profile+ and \verb+profile2d+ (see Section \ref{egsealevel}). The function \verb+anova+ performs likelihood ratio tests. The null hypothesis of the test performed below specifies that the margins are Gumbel distributions ($\texttt{shape1} = \texttt{shape2} = 0$). The deviance of the constrained model is compared with the deviance of the unconstrained model, and the p-value is calculated to be $0.78$. The hypothesis would not be rejected at any reasonable significance level. \begin{verbatim} > m3 <- fbvevd(bvdata, model = "log", shape1 = 0, shape2 = 0) > anova(m1, m3) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) m1 7 708 m3 5 708 2 0.5 0.78 \end{verbatim} In the following example I attempt to fit the asymmetric logistic model to the simulated data set used above, which is known to be distributed as symmetric logistic. \begin{verbatim} > m4 <- fbvevd(bvdata, model = "alog") > fitted(m4) loc1 scale1 shape1 loc2 scale2 shape2 asy1 asy2 dep 1.2097 1.3928 -0.1853 0.8421 1.3831 0.0773 0.8331 0.9996 0.6925 \end{verbatim} A boundary of the parameter space has been reached; the maximum likelihood estimate for the second asymmetry parameter is one. This may cause difficulties for the optimiser. There are two solutions to this problem: the second asymmetry parameter can be fixed at one, or the \verb+L-BFGS-B+ method can be used. The \verb+L-BFGS-B+ method allows box-constraints using the arguments \verb+lower+ and \verb+upper+. The following snippet illustrates both approaches. \begin{verbatim} > mb <- fbvevd(bvdata, model = "alog", asy2 = 1) > round(fitted(mb), 3) loc1 scale1 shape1 loc2 scale2 shape2 asy1 dep 1.212 1.385 -0.176 0.834 1.396 0.086 0.867 0.693 > up <- c(rep(Inf, 6), 1, 1, 1) > mb <- fbvevd(bvdata, model = "alog", method = "L-BFGS-B", upper = up) > round(fitted(mb), 3) loc1 scale1 shape1 loc2 scale2 shape2 asy1 asy2 dep 1.212 1.385 -0.176 0.834 1.396 0.086 0.867 1.000 0.693 \end{verbatim} \section{Example: Oxford Temperature Data} \setcounter{footnote}{0} \label{egoxford} The numeric vector \verb+oxford+ contains annual maximum temperatures (in degrees Fahrenheit) at Oxford, England, from 1901 to 1980. It is included in the evd package, and can be made available using \verb+data(oxford)+. The data has previously been analysed by \citet{tabo83}. I begin by plotting the data. The assumptions of stationarity and independence seem sensible, given the plot (not shown) generated using the code below. \begin{verbatim} > data(oxford) ; ox <- oxford > plot(1901:1980, ox, xlab = "year", ylab = "temperature") \end{verbatim} The following code fits two models based on the GEV distribution. The first model assumes stationarity. The second model allows for a trend term in the location parameter (even though the plot appears to show that this is unnecessary). The \verb+nsloc+ argument is centred and scaled so that the intercept \verb+loc+ represents the location parameter in 1950 and the trend \verb+loctrend+ represents the increase in the location parameter (or decrease, if negative) over a period of 100 years. \begin{verbatim} > ox.fit <- fgev(ox) > tt <- (1901:1980 - 1950)/100 > ox.fit.trend <- fgev(ox, nsloc = tt) > fitted(ox.fit.trend) loc loctrend scale shape 83.6617 -1.8812 4.2233 -0.2841 > std.errors(ox.fit.trend) loc loctrend scale shape 0.5557 1.9675 0.3650 0.0707 \end{verbatim} % Moved for graphics placement. \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{graph3.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph4.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph5.ps}} \end{center} \caption{Diagnostic plots for the model \texttt{ox.fit}.} \label{oxdiag} \end{figure} The trend term not statistically significant (at any reasonable level). The stationary model \verb+ox.fit+ is retained for further analysis. \begin{verbatim} > ox.fit Call: fgev(x = oxford) Deviance: 457.8 Estimates loc scale shape 83.839 4.260 -0.287 Standard Errors loc scale shape 0.5231 0.3658 0.0683 \end{verbatim} The fitted shape is negative, so the fitted distribution is Weibull. It is often of interest to test the hypothesis that the shape is zero (the Gumbel distribution). The code \verb+confint(ox.fit)+ returns the 95\% Wald confidence intervals for the model parameters, roughly equal to the fitted estimates plus or minus twice their standard errors. The interval for the shape parameter is given by $(-0.42,-0.15)$. The corresponding Wald test for $\xi = 0$ would be rejected at significance level $0.05$ since the 95\% confidence interval does not contain zero. A likelihood ratio test for $\xi = 0$ is performed in the following snippet. The hypothesis is rejected at any significance level above $0.00053$. \begin{verbatim} > ox.fit.gum <- fgev(ox, shape = 0) > anova(ox.fit, ox.fit.gum) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) ox.fit 3 458 ox.fit.gum 2 470 1 12 0.00053 \end{verbatim} Diagnostic plots can be produced using \verb+plot(ox.fit)+. The plots produced compare parametric distributions, densities and quantiles to their empirical counterparts (see the \verb+plot.uvevd+ help file for details). Selected diagnostics are depicted in Figure \ref{oxdiag}. The small bars on the P-P, Q-Q and return level plots represent simulated (pointwise) 95\% confidence intervals. The model \verb+ox.prof+ is seen to be a good fit. The fitted density is close to the non-parametric estimator, and most points lie within the confidence intervals. Profile log-likelihoods of the parameters can be plotted using \begin{verbatim} > ox.prof <- profile(ox.fit) > plot(ox.prof) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{graph7.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph8.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph9.ps}} \end{center} \caption{Profile log-likelihoods for the model \texttt{ox.fit}.} \label{oxprof} \end{figure} The profile log-likelihoods for the scale and shape parameters are the first two plots of Figure \ref{oxprof}. A horizontal line is (optionally) drawn on each plot so that the intersection of the line with the profile log-likelihood yields a profile confidence interval, with (default) confidence coefficient 0.95. The end points of the intervals can be derived using \verb+confint(ox.prof)+. The profile confidence intervals for the location and shape parameters are approximately the same as the Wald confidence intervals, since the profile log-likelihoods are approximately symmetric. The profile log-likelihood for the scale parameter is asymmetric; both end points of the profile confidence interval $(3.64, 5.12)$ are larger than the corresponding end points of the Wald interval $(3.54, 4.98)$. The joint profile log-likelihood of the scale and shape parameters can be plotted using \begin{verbatim} > ox.prof2d <- profile2d(ox.fit, ox.prof, which = c("scale", "shape")) > plot(ox.prof2d) \end{verbatim} This produces the image plot in the right panel of Figure \ref{oxprof}. The colours of the image plot represent confidence sets with different confidence coefficients. By default, the lightest colour (ignoring the background colour) represents a confidence set with coefficient 0.995; the darkest colour represents a confidence set with coefficient 0.5. Let $G$ be the GEV distribution function, and let $G(z_p) = 1-p$, so that \begin{equation*} z_p = \begin{cases} \mu - \frac{\sigma}{\xi}[1 - \{-\log(1-p)\}^{-\xi}] & \xi \neq 0 \\ \mu - \sigma \log\{-\log(1-p)\} & \xi = 0, \end{cases} \end{equation*} is the quantile corresponding to the upper tail probability $p$. The profile log-likelihood for $z_{0.1}$ can be plotted using the following. The argument $\verb+prob+ = p$ re-parameterises the GEV distribution so that \verb+fgev+ produces maximum likelihood estimates for $(z_p,\sigma,\xi)$. \begin{verbatim} > ox.qfit <- fgev(ox, prob = 0.1) > ox.qprof <- profile(ox.qfit, which = "quantile") > plot(ox.qprof) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{graph10.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph11.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{graph12.ps}} \end{center} \caption{Profile log-likelihoods for $z_{0.1}$, $z_{0.01}$ and $z_{0.001}$.} \label{quantprof} \end{figure} Figure \ref{quantprof} shows profile log-likelihoods for $z_{0.1}$, $z_{0.01}$ and $z_{0.001}$. The extent of the asymmetry in the profile log-likelihood increases for decreasing (small) $p$. This is to be expected, since the data provide increasingly weaker information in the upper tail of the fitted distribution. If $\verb+prob+ = p$ is zero, then $z_p$ is the upper end point of the GEV distribution, given by $\mu-\sigma/\xi$ when $\xi < 0$. The profile log-likelihood for $z_0$ can be plotted using the following code. \begin{verbatim} > ox.qfit <- fgev(ox, prob = 0) > ox.qprof <- profile(ox.qfit, which = "quantile", conf = 0.99) > plot(ox.qprof) > confint(ox.qprof) lower upper quantile 95.78 113.0 \end{verbatim} The argument \verb+conf+ of the function \verb+profile+ controls the range of the profile trace. The profile trace is constructed so that profile confidence intervals with confidence coefficients \verb+conf+ or less can be derived from it. By default, $\verb+conf+ = 0.999$, though a smaller value is often appropriate when the profile log-likelihood exhibits strong asymmetry. The 95\% profile confidence interval for the upper end point $z_0$ is derived as (95.8,113.0). \section{Example: Rainfall Data} \setcounter{footnote}{0} \label{egrain} The numeric vector \verb+rain+ contains 17531 daily rainfall accumulations at a location in south-west England, recorded over the period 1914 to 1962. The data is not included in the evd package, but it is available in the ismev package, which can be downloaded from CRAN. As usual, the package can be loaded using \verb+library(ismev)+, and the data can be made available using \verb+data(rain)+. The plot of the data given in Figure 1.7 of \citet{cole01} shows that an assumption of stationarity is sensible. The example given here follows \citet{cole01}, pages 84--86. \begin{verbatim} > mrlplot(rain, tlim = c(0,85), nt = 100) > par(mfrow = c(2,1)) > tcplot(rain, tlim = c(0,50), nt = 20) > potgp <- fpot(rain, 30, npp = 365.25) > potgp2 <- fpot(rain, 30, npp = 365.25, cmax = TRUE, r = 7) > clusters(rain, 30, r = 7, cmax = TRUE) \end{verbatim} The first three lines of code produce the threshold diagnostic plots given in pages 80 and 85 of \citet{cole01}, who subsequently decides to work with the threshold $u = 30$. The model \verb+potgp+ reports that 152 observations lie above the threshold, giving an exceedance probability estimate of 0.00867. The estimates and standard errors of the parameters of \verb+potgp+ agree with those given page 85 of \citet{cole01}. In \verb+potgp2+ the peaks over threshold model is applied to cluster maxima, where clusters are defined using a clustering interval length of seven. As there is little sign of clustering in the data, this leads to relatively small changes in the parameter estimates, and relatively small increases in the standard errors. The final line of code calls the function \verb+clusters+ (see Section \ref{stochproc}) in order to produce the cluster maxima that were used for the fitting of model \verb+potgp2+. Diagnostic plots can be produced using \verb+plot(potgp)+. The plots compare parametric distributions, densities and quantiles to their empirical counterparts (see the \verb+plot.uvevd+ help file for details). Selected diagnostics are given in Figure \ref{potdiag}. The x-axis of the return level plot gives return periods in units of years, since we specified the number of observations per period as $\texttt{npp} = 365.25$. The small bars on the P-P, Q-Q and return level plots represent simulated (pointwise) 95\% confidence intervals. The model \verb+potgp+ is seen to be a good fit. The fitted density tail is close to the non-parametric estimator, and most points lie within the confidence intervals. \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{potdiag2.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{potdiag3.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{potdiag4.ps}} \end{center} \caption{Diagnostic plots for the peaks over threshold model for daily rain data.} \label{potdiag} \end{figure} Profile log-likelihoods of the shape parameter and the 100-year return level (not shown) can be plotted using the following code. The argument $\verb+mper+ = m$ re-parameterises the model so that \verb+fpot+ produces maximum likelihood estimates for $(z_m,\xi)$, where $z_m$ is the $m$ period return level, as defined in Section \ref{potfit}. Horizontal lines denoting 95\% profile confidence intervals are depicted on each plot. The end points of profile confidence intervals can be derived using \verb+confint(prgp3)+. \begin{verbatim} potgp3 <- fpot(rain, 30, npp = 365.25, mper = 100) prgp3 <- profile(potgp3) plot(prgp3) \end{verbatim} %\begin{figure} %\begin{center} %\scalebox{0.18}{\includegraphics{potprof1.ps}} %\vspace{-1.5cm} %\hspace{0cm} %\scalebox{0.18}{\includegraphics{potprof2.ps}} %\end{center} %\caption{Profile deviances for the shape parameter and 100-year return level in the peaks over threshold model for daily rain data.} %\label{potprof} %\end{figure} \section{Example: Sea Level Data} \setcounter{footnote}{0} \label{egsealevel} The \verb+sealevel+ data frame \citep{coletawn90} has two columns containing annual sea level maxima from 1912 to 1992 at Dover and Harwich, two sites on the coast of Britain. It contains 39 missing maxima in total; nine at Dover and thirty at Harwich. There are three years for which the annual maximum is not available at either site. I begin by plotting the data, using the code below. The plot of the Harwich maxima against the Dover maxima, given in the left panel of Figure \ref{seadata}, depicts a reasonable degree of dependence. The outlier corresponds to the 1953 flood resulting from a storm passing over the South-East coast of Britain on 1st February. The marginal plots (not shown) suggest that the Harwich and Dover maxima both increase with time. The last line of code\footnote{The function \texttt{chiplot} is heavily based on code by Jan Heffernan.} plots estimates of $\chi(u)$ and $\bar{\chi}(u)$ for $0 < u < 1$ \citep{coleheff99}, as depicted in Figure \ref{seadata}. For bivariate extreme value distributions, $\chi(u) = \chi$ is constant for all $0 < u < 1$, and $\lim_{u \rightarrow 1}\bar{\chi}(u) = 1$. The conditions do not seem unreasonable given the wide confidence intervals in each plot. \begin{verbatim} > data(sealevel) ; sl <- sealevel > plot(sl, xlab = "Dover Annual Maxima", ylab = "Harwich Annual Maxima") > plot(1912:1992, sl[,1], xlab = "Year", ylab = "Dover Annual Maxima") > plot(1912:1992, sl[,2], xlab = "Year", ylab = "Harwich Annual Maxima") > chiplot(sl) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{bvgraph1.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{chi.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{chibar.ps}} \end{center} \caption{From left to right; Harwich maxima vs Dover maxima, estimated values of $\chi(u)$ vs $u$, estimated values of $\bar{\chi}(u)$ vs $u$.} \label{seadata} \end{figure} The following three expressions fit (symmetric) logistic models. The first model incorporates linear trend terms on both marginal location parameters. The second model incorporates a linear trend on the Dover margin only. The third model assumes stationarity. The \verb+nsloc1+ and \verb+nsloc2+ arguments are centred and scaled so that the intercepts \verb+loc1+ and \verb+loc2+ represent the marginal location parameters in 1950 and the linear trend parameters \verb+loc1trend+ and \verb+loc2trend+ represent the increase in the marginal location parameters (or decrease, if negative) over a period of 100 years. \begin{verbatim} > tt <- (1912:1992 - 1950)/100 > m1 <- fbvevd(sl, model = "log", nsloc1 = tt, nsloc2 = tt) > m2 <- fbvevd(sl, model = "log", nsloc1 = tt) > m3 <- fbvevd(sl, model = "log") \end{verbatim} %I'll leave you to analyse the models in detail. %In particular, notice how the trend terms affect the parameter estimates. %Marginal Weibull distributions (negative shapes) are estimated when the trends are not included, but marginal Fr\'{e}chet distributions (positive shapes) are estimated upon their inclusion. The maximum likelihood estimates of the parameters can be compared with their standard errors to perform Wald tests. Wald confidence intervals can be derived using e.g.\ \verb+confint(m1)+. Likelihood ratio tests are performed in the following snippet. The p-values confirm the statistical significance of the linear trend terms. \begin{verbatim} > anova(m1, m2, m3) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) m1 9 -36.5 m2 8 -29.2 1 7.26 0.007 m3 7 -9.7 1 19.56 9.7e-06 \end{verbatim} Quadratic trends for the location parameter on either or both margins can be incorporated using the following code. Further testing, using the models generated below, suggests that a quadratic trend may be implemented for the location parameter on the Harwich margin. Despite this, I retain the model \verb+m1+ for further analysis. \begin{verbatim} > tdframe <- data.frame(trend = tt, quad = tt^2) > m4 <- fbvevd(sl, model = "log", nsloc1 = tdframe, nsloc2 = tt) > m5 <- fbvevd(sl, model = "log", nsloc1 = tt, nsloc2 = tdframe) > m6 <- fbvevd(sl, model = "log", nsloc1 = tdframe, nsloc2 = tdframe) \end{verbatim} The code given below compares two logistic models that are nested within \verb+m1+. Model \verb+m7+ assumes independence. The maximum likelihood estimates are the same as those that would be produced if \verb+fgev+ was separately applied to each margin. The asymptotic distribution of the deviance difference between models \verb+m7+ and \verb+m1+ is non-regular because the dependence parameter in the restricted (independence) model is fixed at the edge of the parameter space. \cite{tawn88} discusses non-regular cases, including this case, for which the asymptotic distribution is one-half of a chi-squared random variable on one degree of freedom. In these cases the argument \verb+half+ should be set to \verb+TRUE+. The resulting p-value is less than $10^{-6}$, and clearly the independence model is rejected. Model \verb+m8+ assumes that both marginal shape parameters are zero (or equivalently, that both marginal distributions are Gumbel). A likelihood ratio test of this hypothesis provides a p-value of $0.72$. The hypothesis would not be rejected at any reasonable significance level. \begin{verbatim} > m7 <- fbvevd(sl, model = "log", nsloc1 = tt, nsloc2 = tt, dep = 1) > anova(m1, m7, half = TRUE) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) m1 9 -36.5 m7 8 -22.9 1 27.2 1.9e-07 > m8 <- fbvevd(sl, "log", nsloc1 = tt, nsloc2 = tt, shape1 = 0, shape2 = 0) > anova(m1, m8) Analysis of Deviance Table M.Df Deviance Df Chisq Pr(>chisq) m1 9 -36.5 m8 7 -35.8 2 0.67 0.72 \end{verbatim} Diagnostic plots for the fitted (generalised extreme value) marginal distributions can be produced using \verb+plot+ with \verb+mar = 1+ or \verb+mar = 2+. The plots produced are of the same structure as those given in Section \ref{egoxford}. Diagnostic plots for the fitted dependence structure can be produced using \verb+plot+. There are six plots available (see the \verb+plot.bvevd+ help file for details). Two diagnostic plots are depicted within Figure \ref{seadiag}. \begin{verbatim} > plot(m1, mar = 1) > plot(m1, mar = 2) > plot(m1, which = 1:5) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.25}{\includegraphics{bvgraph7.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.25}{\includegraphics{bvqcurve.ps}} \hspace{0cm} \scalebox{0.25}{\includegraphics{bvgraph8.ps}} \end{center} \caption{From left to right; dependence function diagnostic plot, quantile curves diagnostic plot, profile log-likelihood of the dependence parameter.} \label{seadiag} \end{figure} The model \verb+m1+ fits the data reasonably well. There are some minor deviations within the conditional P-P plots (not shown), but they do not represent a serious departure of the empirical estimates from the fitted model. The profile log-likelihood of the dependence parameter \verb+dep+, as given in the right panel of Figure \ref{seadiag}, can be plotted using the following. The argument \verb+xmax+ denotes the upper bound of the parameter. \begin{verbatim} > m1.prof <- profile(m1, which = "dep", xmax = 1) > plot(m1.prof) > confint(m1.prof) lower upper dep 0.528 0.887 \end{verbatim} A horizontal line is (optionally) drawn so that the intersection of the line with the profile log-likelihood yields a profile confidence interval, with (default) confidence coefficient 0.95. The interval is derived as $(0.53,0.89)$. Further analysis with models other than the (symmetric) logistic yields the following conclusions. The two models in Section \ref{biv} that include three parameters with which to describe the dependence structure (the asymmetric logistic and asymmetric negative logistic) are inappropriate. In both cases, the maximum likelihood estimate for the parameter \verb+dep+ is at an artificial boundary, because the fitted model is close to a distribution (obtained in the limit) which contains a singular component. This is clearly illustrated in the density plots of the fitted models, which both depict a ridge of mass extending towards the 1953 outlier. The logistic and the bilogistic models have the lowest deviance of all one and two parameter models respectively. The dependence structure of the fitted bilogistic model is almost symmetric. At symmetry, the bilogistic model reduces to the logistic model, and so the latter would appear to be preferable. A likelihood ratio test between the two (nested) models gives a p-value of $0.93$. %Models that are not nested can be compared by adding penalty terms to the deviances. %The penalty terms take into account the number of parameters fitted. (If both models have the same number of parameters the deviances can be compared directly.) %Three commonly used penalty terms are $2p$ (Akaike's information criterion, or AIC), $p\log(n)$ (Schwarz's criterion, or SC) and $p\{1+\log(n)\}$ (Bayesian information criterion, or BIC), where $p$ is the number of parameters estimated and $n$ is the number of observations.\footnote{Since \texttt{fbvall} compares models for the dependence structure, $n$ is taken as the number of observations which are complete (i.e.\ not missing on either margin).} %Any bivariate extreme value distribution function can be expressed as \citep{haan84} %\begin{equation*} %G(z_1,z_2) = \exp\left\{ - \int_0^1\max\{y_1f_1(x),y_2f_2(x)\} \, \text{d}x \right\} %\end{equation*} %where $(y_1,y_2)$ are again defined by the transformations \eqref{mtrans}, and where $f_1$ and $f_2$ are density functions with support [0,1]. %In particular, if we take the beta densities $f_1(x)=(1-\alpha)x^{-\alpha}$ and $f_2(x)=(1-\beta)(1-x)^{-\beta}$ we obtain \section*{Appendix A: Additional Bivariate Parametric Models} It can be shown, using a representation of \citet{haan84}, that \begin{equation*} G(z_1,z_2) = \exp\left\{ - \int_0^1\max\{y_1(1-\alpha)x^{-\alpha},y_2(1-\beta)(1-x)^{-\beta}\} \, \text{d}x \right\}, \quad \alpha,\beta < 1. \end{equation*} is a bivariate extreme value distribution function. If we further constrain the parameters to be non-negative we obtain the bivariate bilogistic model proposed by \citet{smit90}, which can also be expressed as \begin{equation*} G(z_1,z_2) = \exp\left\{ - y_1\gamma^{1-\alpha} - y_2(1-\gamma)^{1-\beta} \right\}, \quad 0 < \alpha,\beta <1, \end{equation*} where $\gamma=\gamma(y_1,y_2;\alpha,\beta)$ solves $(1-\alpha)y_1(1-\gamma)^\beta=(1-\beta)y_2\gamma^\alpha$. The logistic model is obtained when $\alpha=\beta$. Independence is obtained as $\alpha = \beta \rightarrow1$, and when one of $\alpha,\beta$ is fixed and the other approaches one. Different limits occur when one of $\alpha,\beta$ is fixed and the other approaches zero. Alternatively, if we constrain both parameters to be non-positive and set $\alpha_0=-\alpha > 0$ and $\beta_0=-\beta > 0$ we obtain the negative bilogistic model \citep{coletawn94}, which has the representation \begin{equation*} G(z_1,z_2) = \exp\left\{-y_1-y_2+y_1\gamma^{1+\alpha_0}+y_2(1-\gamma)^{1+\beta_0} \right\}, \quad \alpha_0,\beta_0 > 0, \end{equation*} where $\gamma=\gamma(y_1,y_2;-\alpha_0,-\beta_0)$. The negative logistic model is obtained when $\alpha_0=\beta_0$ (with $r = 1/\alpha_0 = 1/\beta_0$). Independence is obtained as $\alpha_0 = \beta_0 \rightarrow\infty$, and when one of $\alpha_0,\beta_0$ is fixed and the other tends to $\infty$. Different limits occur when one of $\alpha_0,\beta_0$ is fixed and the other approaches zero. The distribution function of the Coles-Tawn model\footnote{\citet{coletawn91} call this the Dirichelet model.} \citep{coletawn91} is given by \begin{equation*} G(z_1,z_2) = \exp\left\{-y_1[1-\text{Be}(u;\alpha+1,\beta)] - y_2\,\text{Be}(u;\alpha,\beta+1) \right\}, \quad \alpha,\beta > 0, \end{equation*} where $u=\alpha y_2/(\alpha y_2+\beta y_1)$ and Be is the incomplete beta function, given by \begin{equation*} \text{Be}(u;\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \int_0^u x^{\alpha-1}(1-x)^{\beta-1} \, \text{d}x. \end{equation*} Complete dependence is obtained in the limit as $\alpha = \beta \rightarrow\infty$. Independence is obtained as $\alpha = \beta \rightarrow0$ and when one of $\alpha,\beta$ is fixed and the other approaches zero. Different limits occur when one of $\alpha,\beta$ is fixed and the other tends to $\infty$. The asymmetric mixed model \citep{tawn88} is typically defined using the corresponding dependence function \eqref{bvdepfn}, which is modelled as a cubic polynomial. Specifically, for $0 \leq t \leq 1$ the dependence function of the asymmetric mixed model is \begin{equation*} A(t) = 1 - (\alpha + \beta)t + \alpha t^2 + \beta t^3, \end{equation*} where both $\alpha$ and $\alpha + 3\beta$ are non-negative, and where both $\alpha + \beta$ and $\alpha + 2\beta$ are less than or equal to one. These constraints imply that $\beta \in [-0.5,0.5]$ and $\alpha \in [0,1.5]$, though $\alpha$ can only be greater than one if $\beta$ is negative. The (symmetric) mixed model is obtained when $\beta = 0$. Complete dependence cannot be obtained. Independence is obtained when $\alpha = \beta = 0$. The asymmetric mixed model is often referred to in the literature because the dependence function has a simple form, and because the $\beta = 0$ case is historically important. However it cannot capture strong dependence, and hence it is of limited use as a statistical model. The extension to an $m$-degree polynomial can be made, but this is of no statistical interest because the additional parameters add little additional flexibility. \bibliography{bibliog} \end{document} \section*{Appendix A: Simulation Study} In this Appendix we use the tools in the package to perform a simulation study to examine the small sample properties of non-parametric estimators for the dependence function $A(\cdot)$ of the bivariate extreme value distribution. The estimators referred to in this Appendix are defined in the documentation file for the function \verb+abvnonpar+. Simulation studies of this form \citep[e.g.][]{halltajv00} typically use the known marginal parameters $(\mu_1,\sigma_1,\xi_1,\mu_2,\sigma_2,\xi_2)$ within the transformations \eqref{transtoexp}. In practice, these parameters need to be estimated. In this study we seek to replicate the behaviour of the estimators when applied to real data, and we have therefore estimated the marginal parameters by maximum likelihood. Figure \ref{simfig} depicts the behaviour of the estimators of \citet{capefoug97}, \citet{pick81} and \citet{tiag97}, which we subsequently denote by $A_c$, $A_p$ and $A_t$ respectively. The estimators of \citet{dehe91} and \citet{halltajv00} are not considered, as they produce plots that are indistinguishable from those of $A_p$. The first, second and third columns of the figure employ simulations from (symmetric) logistic distributions, with $\alpha$ equal to $0.5$, $0.75$ and $1$ respectively. Standard Gumbel marginal distributions were used in each case. The figure shows that the estimator $A_t$ is abysmal when estimating dependence functions with very strong ($\alpha = 0.5$) or very weak ($\alpha = 1$) levels of dependence. The estimators $A_c$ and $A_p$ give more consistent performances across different levels of dependence. The estimator $A_c$ appears to outperform $A_p$, as the estimates of the former appear to cluster more tightly around the true dependence function for each $\alpha = 0.5,0.75,1$. The plots can easily be generated, using e.g. \begin{verbatim} > dep <- 0.5 ; method <- "cfg" > abvevd(dep = dep, plot = TRUE, lty = 0) > set.seed(44) > for(i in 1:50) { sdt <- rbvevd(100, dep = dep) abvnonpar(data = sdt, add = TRUE, method = method, col = "grey") } > abvevd(dep = dep, add = TRUE, lwd = 3) \end{verbatim} \begin{figure} \begin{center} \scalebox{0.18}{\includegraphics{npsim11.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim12.ps}} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim13.ps}} \\ \scalebox{0.18}{\includegraphics{npsim21.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim22.ps}} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim23.ps}} \\ \scalebox{0.18}{\includegraphics{npsim31.ps}} \vspace{-1.5cm} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim32.ps}} \hspace{0cm} \scalebox{0.18}{\includegraphics{npsim33.ps}} \end{center} \caption{Simulated non-parametric dependence function estimates. The grey lines represent estimates derived using the estimators $A_c$ (top row), $A_p$ (middle row) and $A_t$ (bottom row). The thick black lines represent the true dependence functions, which are (symmetric) logistic models with dependence parameters $0.5$ (first column), $0.75$ (second column) and $1$ (third column).} \label{simfig} \end{figure} which generates the plot in the top left corner. Only the first line of code needs to be changed in order to produce the remaining plots. The second line of code establishes the plotting region. The simulation is performed in the \verb+for+ loop, and the last line adds the true dependence function to the plot. The \verb+set.seed+ function sets the seed of the random generator, which ensures that the simulated data sets used for each plot are comparable. Let $A_n(\cdot)$ be any estimator of $A(\cdot)$. Table \ref{simtab} gives median integrated absolute errors for various non-parametric dependence function estimators. The table was constructed as follows. For $\alpha = 0.5,0.75,1$ we simulated $1000$ datasets containing $n=25,100$ bivariate observations, using standard Gumbel margins. Then for each of the $1000$ datasets we estimated the integrated absolute error $\int_0^1|A_n(x) - A(x)| \, \text{d}x$. The table contains the median of the $1000$ values, for each value of $\alpha$ and $n$. We have extended the number of estimators to include the convex minorants of $A_c$ and $A_p$, which we denote by $A_c^*$ and $A_p^*$. The convex minorant of $A_t$ is identical to $A_t$, because $A_t$ is always convex. The table again shows the poor performance of $A_t$ when $\alpha = 0.5$, and particularly when $\alpha = 1$. $A_t$ is the best estimator when $\alpha = 0.75$, which is not surprising given that the estimator only yields adequate estimates at mid-range levels of dependence. The estimator $A_c$ outperforms $A_p$, confirming the impression given by Figure \ref{simfig}. Taking the convex minorant of $A_c$ or $A_p$ leads to an improvement for $\alpha = 0.5$ and $\alpha = 0.75$, but a considerable worsening for $\alpha = 1$. This worsening is expected, since taking the convex minorant always leads to estimates of stronger dependence. The values in the table can be generated using e.g. \begin{verbatim} > dep <- 0.5 ; n <- 25 ; method <- "cfg" ; cv <- FALSE > nn <- 100 ; x <- (1:nn)/(nn + 1) > a <- abvevd(x, dep = dep) > iae <- numeric(1000) > set.seed(44) > for(i in 1:1000) { sdt <- rbvevd(n, dep = dep) anp <- abvnonpar(x, data = sdt, method = method, convex = cv) iae[i] <- sum(abs(a - anp))/nn } > round(10^4 * median(iae)) \end{verbatim} % FOR ENTIRE TABLE %\begin{verbatim} %method <- rep(c("cfg","cfg","pick","pick","tdo"), 6) %cv <- rep(c(FALSE, TRUE, FALSE, TRUE, FALSE), 6) %dep <- rep(rep(c(0.5, 0.75, 1), each = 5), 2) %n <- rep(c(25, 100), each = 15) %sim.all <- numeric(30) % %for(j in 1:30) { % print(j) % nn <- 100 ; x <- (1:nn)/(nn+1) % a <- abvevd(x, dep = dep[j]) % iae <- numeric(1000) % set.seed(44) % for(i in 1:1000) { % sdt <- rbvevd(n[j], dep = dep[j]) % anp <- abvnonpar(x, data = sdt, method = method[j], convex = cv[j]) % iae[i] <- sum(abs(a - anp))/nn % } % sim.all[j] <- median(iae) %} %round(10^4 * matrix(sim.all, nrow = 5, ncol = 6)) %\end{verbatim} \begin{table} \begin{center} \begin{tabular}{|l|ccc|ccc|} \hline & \multicolumn{3}{c|}{$n=25$} & \multicolumn{3}{c|}{$n=100$} \\ & $\alpha = 0.5$ & $\alpha = 0.75$ & $\alpha = 1$ & $\alpha = 0.5$ & $\alpha = 0.75$ & $\alpha = 1$ \\ \hline $A_c$ & 210 & 415 & 110 & 104 & 198 & 62 \\ $A_c^*$ & 205 & 363 & 340 & 103 & 194 & 168 \\ $A_p$ & 243 & 469 & 211 & 134 & 242 & 113 \\ $A_p^*$ & 218 & 357 & 554 & 126 & 215 & 285 \\ $A_t$ & 393 & 189 & 983 & 334 & 155 & 830 \\ \hline \end{tabular} \caption{Median integrated absolute errors $\times$ $10^4$ for non-parametric estimates of the dependence function of the bivariate extreme value distribution, using datasets containing $n=25,100$ bivariate observations, simulated from the (symmetric) logistic model with dependence parameter $\alpha=0.5,0.75,1$. The estimators $A_c^*$ and $A_p^*$ are the convex minorants of $A_c$ and $A_p$ respectively.} \label{simtab} \end{center} \end{table} which generates the value in the top left corner. Only the first line of code needs to be changed in order to produce the remaining values. The integrated absolute error is estimated by evaluating the absolute difference between true dependence function and the non-parametric estimate at $\verb+nn+ = 100$ equally spaced points in the interval $[0,1]$. The function \verb+numeric+ merely initializes the object \verb+iae+ to be a vector of $1000$ zeros. evd/inst/doc/Multivariate_Extremes.Rnw0000644000175100001440000004631212637167310017616 0ustar hornikusers\documentclass[11pt,a4paper]{article} \usepackage{amsmath,amssymb} \pagestyle{plain} \setlength{\parindent}{0in} \setlength{\parskip}{1.5ex plus 0.5ex minus 0.5ex} \setlength{\oddsidemargin}{0in} \setlength{\topmargin}{-0.5in} \setlength{\textwidth}{6.3in} \setlength{\textheight}{9.8in} %\VignetteIndexEntry{Statistics Of Extremes: Chapter 9} \begin{document} \title{Statistics of Multivariate Extremes} \author{Alec Stephenson} \maketitle \begin{center} \LARGE \textbf{Summary} \\ \end{center} \normalsize \vspace{0.5cm} This vignette uses the \textbf{evd} package to reproduce the figures, tables and analysis in Chapter 9 of Beirlant et al.\ (2001). The chapter was written by Segers and Vandewalle (2004). The code reproduces almost all figures, but for space reasons only some are shown. Deviations from the book are given as footnotes. Differences will inevitably exist due to numerical optimization and random number generation. \normalsize \section{Introduction} \label{Intro} The methods used here are illustrated using the \texttt{lossalae} dataset, which contains observations on $1500$ liability claims. The indemnity payment (loss) and the allocated loss adjustment expense (ALAE) is recorded in USD for each claim. The ALAE is the additional expenses associated with the settlement of the claim (e.g.\ claims investigation expenses and legal fees). The dataset also has an attribute called \texttt{capped}, which gives the row names of the indemnity payments that were capped at their policy limit. We first scale the data so that one unit corresponds to $100\,000$ USD. Putting the data on a sensible scale assists with the numerical optimization involved in maximum likelihood estimation\footnote{The book reports an unsatisfactory fit of the GEV model to the margins. It therefore uses only empirical marginal distributions. This was perhaps due to not scaling the data. In this document we use either fully nonparametric or fully parametric methods.}. The code below plots the raw data using the log scale for both axes (see Figure \ref{rawdata}), and plots the data transformed to uniform $(0,1)$ margins using an empirical transform. <>= options(show.signif.stars=FALSE) library(evd); nn <- nrow(lossalae) loss <- lossalae/1e+05; lts <- c(1e-04, 100) plot(loss, log = "xy", xlim = lts, ylim = lts) @ <<>>= ula <- apply(loss, 2, rank)/(nn + 1) plot(ula) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Scatterplot of ALAE verses Loss: original data (log-scale).} \label{rawdata} \end{figure} \section{Parametric Models} Any bivariate extreme value distribution function can be represented in the form \begin{equation*} G(z_1,z_2) = \exp\left\{ - (y_1 + y_2)A\left(\frac{y_1}{y_1+y_2}\right)\right\}, \label{bvdepfn} \end{equation*} where \begin{equation*} y_j = y_j(z_j) = \{1+\xi_j(z_j-\mu_j)/\sigma_j\}_{+}^{-1/\xi_j} \label{mtrans} \end{equation*} for $\sigma_j > 0$ and $j=1,2$, and where \begin{equation*} A(\omega)=-\log\{G(y_1^{-1}(\omega),y_2^{-1}(1-\omega))\}, \label{dep} \end{equation*} defined on $0\leq\omega\leq1$ is called the dependence function\footnote{The book uses the definition $B(\omega) = A(1-\omega)$.}. The marginal distributions are generalized extreme value (GEV), given by $G_j(z_j) = \exp(-y_j)$. It follows that $A(0)=A(1)=1$, and that $A(\cdot)$ is a convex function with $\max(\omega,1-\omega) \leq A(\omega) \leq 1$ for all $0\leq\omega\leq1$. At independence $A(1/2) = 1$. At complete dependence $A(1/2) = 0.5$. The dependence function represents only the dependence structure of the distribution, and hence only the dependence parameters of parametric models need to be specified in order to produce dependence function plots. The code below plots dependence functions for four different parametric models. The first of these is given in Figure \ref{asylogdfn}. <>= abvevd(dep = 0.5, asy = c(1,1), model = "alog", plot = TRUE) abvevd(dep = 0.5, asy = c(0.6,0.9), model = "alog", add = TRUE, lty = 2) abvevd(dep = 0.5, asy = c(0.8,0.5), model = "alog", add = TRUE, lty = 3) @ <<>>= abvevd(dep = -1/(-2), model = "neglog", plot = TRUE) abvevd(dep = -1/(-1), model = "neglog", add = TRUE, lty = 2) abvevd(dep = -1/(-0.5), model = "neglog", add = TRUE, lty = 3) @ <<>>= abvevd(alpha = 1, beta = -0.2, model = "amix", plot = TRUE) abvevd(alpha = 0.6, beta = 0.1, model = "amix", add = TRUE, lty = 2) abvevd(alpha = 0.2, beta = 0.2, model = "amix", add = TRUE, lty = 3) @ <<>>= abvevd(dep = 1/1.25, model = "hr", plot = TRUE) abvevd(dep = 1/0.83, model = "hr", add = TRUE, lty = 2) abvevd(dep = 1/0.5, model = "hr", add = TRUE, lty = 3) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Dependence functions: asymmetric logistic model.} \label{asylogdfn} \end{figure} \section{Componentwise Maxima} For demonstration purposes we use the data introduced in Section \ref{Intro} to create a dataset of componentwise block maxima by randomly taking $k=50$ groups of size $m=30$, producing $k$ componentwise maxima taken over $m$ observations\footnote{The data may be completely different to the book due to random selection.}. Bivariate extreme value distributions are typically used to model data of this type. The code below creates the componentwise maxima data \texttt{cml} and produces two data plots, the first showing the original data and the componentwise maxima, and the second showing the componentwise maxima data transformed to standard exponential margins. <<>>= set.seed(131); cml <- loss[sample(nn),] xx <- rep(1:50, each = 30); lts <- c(1e-04, 100) cml <- cbind(tapply(cml[,1], xx, max), tapply(cml[,2], xx, max)) colnames(cml) <- colnames(loss) plot(loss, log = "xy", xlim = lts, ylim = lts, col = "grey") points(cml) ecml <- -log(apply(cml,2,rank)/51) plot(ecml) @ The following code estimates and plots the dependence function $A(\cdot)$ from the componentwise maxima data. The first code chunk uses various nonparametric estimates of the dependence function, and also uses empirical (i.e.\ nonparametric) estimation of the margins, as specified by \texttt{epmar = TRUE}. The four different estimates are shown in Figure \ref{nonpardfn}. The second code chunk uses maximum likelihood estimation for parametric models. The call to \texttt{fbvevd} fits the model, and the call to \texttt{plot} plots the parametric dependence function estimates. The argument specification \texttt{asy1 = 1} in the first call to \texttt{fbvevd} constrains the model fit so that the first asymmetry parameter of the model is fixed at the value one. <>= pp <- "pickands"; cc <- "cfg" abvnonpar(data = cml, epmar = TRUE, method = pp, plot = TRUE, lty = 3) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 1, lty = 2) abvnonpar(data = cml, epmar = TRUE, method = pp, add = TRUE, madj = 2, lty = 4) abvnonpar(data = cml, epmar = TRUE, method = cc, add = TRUE, lty = 1) @ <<>>= m1 <- fbvevd(cml, asy1 = 1, model = "alog") m2 <- fbvevd(cml, model = "log") m3 <- fbvevd(cml, model = "bilog") plot(m1, which = 4, nplty = 3) plot(m2, which = 4, nplty = 3, lty = 2, add = TRUE) plot(m3, which = 4, nplty = 3, lty = 4, add = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Nonparametric dependence function estimates by Pickands (dotted line), Deheuvels (dashed line), Hall-Tajvidi (dot-dashed line) and Cap\'{e}r\`{a}a-Foug\`{e}res-Genest (solid line) based on componentwise block maxima data and using empirical marginal estimation.} \label{nonpardfn} \end{figure} The objects produced by \texttt{fbvevd} contain information about the parametric fit of the bivariate extreme value distribution. For example, \texttt{m2} contains information on the fit of a (symmetric) logistic extreme value distribution, which has a single dependence parameter and three parameters on each of the GEV margins. Using \texttt{plot(m2)} produces several diagnostic plots, including quantile curves and spectral densities. Using \texttt{deviance(m2)} produces the deviance, which is equal to twice the negative log-likelihood. The following shows the parameter estimates and their standard errors, and gives an analysis of deviance table for testing \texttt{m2} verses \texttt{m3}, which is possible since the models are nested, with \texttt{m3} having one additional dependence parameter. The call to \texttt{exind.test} produces a score test for independence, following Tawn (1988). Omitting the \texttt{method} argument gives a likelihood ratio test, also from Tawn (1988), which is typically more accurate. <<>>= round(rbind(fitted(m2), std.errors(m2)), 3) anova(m3, m2) evind.test(cml, method = "score") @ The code below uses the function \texttt{qcbvnonpar} to plot quantile curves using nonparametric dependence function estimates. Quantile curves are defined as \begin{equation*} Q(F, p) = \{(z_1,z_2): F(z_1,z_2) = p\}, \end{equation*} where $F$ is a distribution function and $p$ is a probability. We use the default nonparametric estimation method and we again use empirical estimation of the margins\footnote{Using parametric marginal estimates tends to produce more sensible quantile curve plots, but we follow the book here. Unlike the book, the quantile curves in Figure \ref{nonparqc} are not step functions because the empirical marginal transforms include interpolation.}, as specified by \texttt{epmar = TRUE}. For parametric dependence models similar plots can be produced using e.g.\ \texttt{plot(m2, which = 5)}. Note that because we plot curves corresponding to the distribution of the original dataset rather than the componentwise maxima, we pass the argument \texttt{mint = 30}. <>= lts <- c(0.01,100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) points(cml); pp <- c(0.98,0.99,0.995) qcbvnonpar(pp, data = cml, epmar = TRUE, mint = 30, add = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Estimated quantile curves $Q(\hat{F},p)$ for $p=0.98,0.99,0.995$ based on the componentwise block maxima data shown as black circles, using the Cap\'{e}r\`{a}a-Foug\`{e}res-Genest nonparametric estimate of the dependence function and using empirical marginal estimation.} \label{nonparqc} \end{figure} \section{Excesses Over A Threshold} We now consider all the $1500$ observations on liability claims. We assume that the data are distributed according to the distribution function $F$, and we are interested in $F(z)$ where $z=(z_1,z_2)$ is in some sense large. The methods we use assume that $F$ is in the domain of attraction of some bivariate extreme value distribution $G$, and we focus on large data points to estimate features of $G$, and hence of $F(z)$ for large $z$. Typically we focus on points $z$ that lie above a certain threshold. The functions \texttt{tcplot} and \texttt{mrlplot} can be used for producing plots on each margin to help determine thresholds $u_1$ and $u_2$ for methods that focus primarily on points $z$ such that $z_1 > u_1$ and $z_2 > u_2$. Alternatively, the function \texttt{bvtcplot} can be used to help determine a single threshold $u^{*}$ for methods that focus on points $z$ such that $r(z) > u^{*}$, where $r(z) = x_1(z_1) + x_2(z_2)$, and $x_j(z_j) = -1/\log \hat{F}_j(z_j)$ for $j=1,2$ where $F_j$ is estimated empirically. Following Segers and Vandewalle (2004), a sensible choice for threshold $u^{*}$ might be found from Figure \ref{bvtc} by taking the $k$th largest $r(z)$, where $k$ is the largest value for which the y-axis is close to two. Figure \ref{bvtc} is plotted below using \texttt{bvtcplot}. The value of $k$ is returned invisibly. Setting \texttt{spectral = TRUE} uses the $k$th largest points to plot a nonparametric estimate of $H([0,\omega])$ where $H$ is the spectral measure of $G$. <>= k0 <- bvtcplot(loss)$k0 bvtcplot(loss, spectral = TRUE) @ <>= k0 <- bvtcplot(loss)$k0 @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{A plot of $(k/n)r_{(n-k)}$ as a function of $k$, where $r_{(1)} \leq \dots \leq r_{(n)}$ are the ordered values of $r$. The y-axis provides an estimate of $H([0,1]) = 2$ for the spectral measure $H$ of $G$.} \label{bvtc} \end{figure} The parametric approach to the problem can employ models similar to those used for bivariate extreme value distributions. We first consider the margins separately by fitting a univariate generalized Pareto distribution to the excesses over the threshold $u_j$ on each margin $j=1,2$. We choose the thresholds so that the number of exceedances is roughly\footnote{The value is chosen so that the thresholds match exactly with those used in the book.} half of the value \texttt{k0}. <<>>= thresh <- apply(loss, 2, sort, decreasing = TRUE)[(k0+5)/2,] mar1 <- fitted(fpot(loss[,1], thresh[1])) mar2 <- fitted(fpot(loss[,2], thresh[2])) rbind(mar1,mar2) @ Parametric threshold models can be fitted using the function \texttt{fbvpot}, with the parametric model specified using the \texttt{model} argument. The default approach uses censored likelihood methodology, where a bivariate extreme value dependence structure is fitted to the data censored at the marginal thresholds $u_1$ and $u_2$. Alternatively, a Poisson process model can be employed using the \texttt{likelihood} argument, employing the methodology of Coles and Tawn (1991). Some examples of parametric fits are given below. Diagnostic plots for the fitted models can be produced using e.g.\ \texttt{plot(m2)}. <<>>= m1 <- fbvpot(loss, thresh, model = "alog", asy1 = 1) m2 <- fbvpot(loss, thresh, model = "bilog") m3 <- fbvpot(loss, thresh, model = "bilog", likelihood = "poisson") round(rbind(fitted(m2), std.errors(m2)), 3) @ The following code plots parametric and nonparametric estimates for the bivariate extreme value dependence structure fitted to the upper tail of $F$. The parametric estimates use the previously fitted models. The nonparametric estimate can be plotted using the \texttt{"pot"} method and takes the value \texttt{k0} to specify the threshold. <<>>= abvnonpar(data = loss, method = "pot", k = k0, epmar = TRUE, plot = TRUE, lty = 3) plot(m1, which = 2, add = TRUE) plot(m2, which = 2, add = TRUE, lty = 4) plot(m3, which = 2, add = TRUE, lty = 2) @ Figure \ref{qcthresh} uses our fitted asymmetric logistic model \texttt{m1} to plot quantile curves at probabilities $p=0.98,0.99,0.995$. The thresholds used for the censored likelihood model fit are also added to the plot. <>= lts <- c(1e-04, 100) plot(loss, log = "xy", col = "grey", xlim = lts, ylim = lts) plot(m1, which = 3, p = c(0.95,0.975,0.99), tlty = 0, add = TRUE) abline(v=thresh[1], h=thresh[2]) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Quantile curves for probabilities $p=0.98,0.99,0.995$ for an asymmetric logistic model fit using censored likelihood estimation, with censoring at marginal thresholds given by the vertical and horizontal lines.} \label{qcthresh} \end{figure} Models based on bivariate extreme value distributions assume that the margins are either asymptotically dependent or are perfectly independent. They cannot account for situations where the dependence between the margins vanishes at increasingly extreme levels. The remainder of this section illustrates the estimation of dependence measures that can identify such cases. We consider three quantities as defined in Coles \textit{et al.} (1999). The coefficient of extremal dependence $\chi \in [0,1]$ is the tendency for one variable to be large given that the other is large. When $\chi = 0$ the variables are asymptotically independent, and when $\chi > 0$ they are asymptotically independent. The second measure $\bar{\chi}$ identifies the strength of dependence for asymptotically independent variables. When $\bar{\chi} = 1$ the variables are asymptotically dependent, and when $-1 \leq \bar{\chi} < 1$ they are asymptotically independent. The third measure is the coefficient of tail dependence $\eta$, which satisfies $\bar{\chi} = 2\eta - 1$. The following code produces Figure \ref{chiplot} which shows estimates of the functions $\chi(u)$ and $\bar{\chi}(u)$, as defined in Coles \textit{et al.} (1999), for $0 < u < 1$. The functions are defined so that $\chi = \lim_{u \rightarrow 1}\chi(u)$ and $\bar{\chi} = \lim_{u \rightarrow 1}\bar{\chi}(u)$. In this case $\chi(u) > 0 $ for all $u$ but there is little evidence that $\bar{\chi}$ is close to one, so it is difficult to specify the form of dependence on the basis of this plot. <>= old <- par(mfrow = c(2,1)) chiplot(loss, ylim1 = c(-0.25,1), ylim2 = c(-0.25,1), nq = 200, qlim = c(0.02,0.98), which = 1:2, spcases = TRUE) par(old) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{The dependence measures $\chi(u)$ and $\bar{\chi}(u)$. Estimates (solid line), 95\% pointwise confidence intervals (dot-dashed lines). The dashed lines represent the theoretical limits of the functions and the exact independence case at zero.} \label{chiplot} \end{figure} We now consider the coefficient of tail dependence $\eta$. We can estimate $\eta$ using univariate theory because of its relationship with $T = \min\{x_1(z_1),x_2(z_2)\}$. If we fit a generalized Pareto distribution to the data points in $T$ that exceed a large fixed threshold, then the estimated shape parameter of the fitted distribution provides an estimate of $\eta$. The call to \texttt{tcplot} plots estimates of $\eta$ at different thresholds in order to assist with threshold choice. The plot seems roughly linear after $u=0.8$, so we take the 80th percentile of $T$ as our threshold. Finally, we use \texttt{anova} to perform a likelihood ratio test for asymptotic dependence, with the null hypothesis $\eta = 1$ versus the alternative $\eta < 1$. <>= fla <- apply(-1/log(ula), 1, min) thresh <- quantile(fla, probs = c(0.025, 0.975)) tcplot(fla, thresh, nt = 100, pscale = TRUE, which = 2, vci = FALSE, cilty = 2, type = "l", ylim = c(-0.2,1.2), ylab = "Tail Dependence") abline(h = c(0,1)) @ <<>>= thresh <- quantile(fla, probs = 0.8) m1 <- fpot(fla, thresh = thresh) cat("Tail Dependence:", fitted(m1)["shape"], "\n") @ <<>>= m2 <- fpot(fla, thresh = thresh, shape = 1) anova(m1, m2, half = TRUE) @ \begin{figure}[ht] \begin{center} <>= <> @ \end{center} \vspace{-1cm} \caption{Maximum likelihood estimates (solid line) and 95\% pointwise confidence intervals (dot-dashed lines) for $\eta$ at different threshold probabilities.} \label{etaplot} \end{figure} \section*{Bibliography} Beirlant, J., Goegebeur, Y., Segers, J and Teugels, J. (2004) \textit{Statistics of Extremes: Theory and Applications}. Wiley, U.K. Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analysis. \textit{Extremes}, \textbf{2}, 339--365. Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. \textit{J.\ R.\ Statist.\ Soc.\ B}, \textbf{53}, 377--392. Segers, J. and Vandewalle, B. (2004). Statistics of Multivariate Extremes. In Beirlant et al. (eds.), \textit{Statistics of Extremes: Theory and Applications}. Wiley, U.K. Tawn, J. A. (1988). Bivariate extreme value theory: Models and estimation. \textit{Biometrika}, \textbf{75}, 397--415. \end{document} evd/inst/doc/guide22.pdf0000644000175100001440000157645212637167310014556 0ustar hornikusers%PDF-1.3 %Çì¢ 6 0 obj <> stream xœ½[‰{\UWlE‚_ÙªH‰KIœòîÛ_EÆP¨Ã¤“e:(BLi“ØIiÚH¥S*(ˆˆˆ"¨¸/­„7(ŠIT\iiÓÄ´)\Šâöxï=w}Ëd& ||šyw9÷,¿ó;ç^Œ(eØÿöl¬Û\wnÁNõ ÕmN!ú‰ÿOÏÆÔźs›R(Uì­ã3Òäÿ Ó[a¦<×Çÿ.n¬[þ²sŠêOF×Õ-ù+Ž[¼èøWÿªŽ;±ãª_½¤cËÉ'²èÄS—žöšS¿vÉ¢Óñºe'½áõg™zãÙoª?3õæúEo}KÇò·uœƒç5·´cKãié·Ÿ»æŠz‹®nØnÊtlºí:¾·28ïç¿3ã5½kÕ«¼¦ W_”}÷’±ž›B.¢#/^Ë6ç×ä[ÚZ 홵ùâ{.!##H¥aÄ{Ï»4ó>üÏeä÷4>™—BC?]þþŽº·‰G›ô§æ¦Î®–¶b[7ùÝôƒüÜÑqåLJ>²XŸëêY¿._lìì*¶µöõâY­toÛOÁ1¨(ø¯4üÙ[ßI×°,¶ê ý>@7²¤¥æÒ@¡•+œo¿±ÓUt$òùÈ–Á\v3ws©µ€gŠæš¡-pH;ðSÈòè°’Ÿ|“Ïast}Ïç‚?]eÌÀWÓÝ ¤0øˆÖÂVº¬#¤Gõ¤l½Ø Å­¹+iÛº<=wˆ]­½ÜÙÕ¼4—m¡{º¶¨#4ÎG¬‡,—þÄVö½¾v{f{Çòg¯ëºž|²,¹E¿]6üæ(®/å†óoë åw_ÛÙ…ÕîtV]"$¶Íu{Iöc`uSøA[±¢òšKkÁ+QÄûÄþ½ƒÔ„˾òãt„-´µ¶!—Å~ r‚„ zDÓN9qnšËv3E{6V4Œ¹”™”týKøK±m톛nÆC·Á¦QÝßâë¡ ƒ ¡†¥iÛNÊÂáF>|‚Jgú 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108 0 0 0 0 0 d1 endstream endobj 71 0 obj <> stream 0 0 0 13 43 104 d1 43 0 0 91 0 13 cm BI /IM true /W 43 /H 91 /BPC 1 /F /CCF /DP <> ID & \Ð'‚3 ð·¤ý7ÓëáÃÿýoAÿÿÿÿÿÿÿÿÿÿ Ü×ÿÿÿÿÿÿÿÿÿÿÿÿÿüI‹É«àød€o÷ßßÿÿ¿ÿÿïÿÿÿÿð EI endstream endobj 72 0 obj <> stream 71 0 0 0 0 0 d1 endstream endobj 73 0 obj <> stream 54 0 0 0 0 0 d1 endstream endobj 74 0 obj <> stream 0 0 0 39 66 130 d1 66 0 0 91 0 39 cm BI /IM true /W 66 /H 91 /BPC 1 /F /CCF /DP <> ID &¨˜ _ÿùOÃÿÿÿÿÿÿÿÿÿä:?ò(Ïð¡øX~Aƒôú~)¾›ø}þøO÷Óýÿì?ýþÿÿÿü7ÿÿÿÿÿÿÿþ¼/ÿÿõëÿþ‚ÿºô¸¯a/­§ VÒÊO¾ØA{ x<†!dCD EI endstream endobj 75 0 obj <> stream 62 0 0 0 0 0 d1 endstream endobj 76 0 obj <> stream 0 0 0 2 66 102 d1 66 0 0 100 0 2 cm BI /IM true /W 66 /H 100 /BPC 1 /F /CCF /DP <> ID &¨˜ÿÿò.„i†0_ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿýÿÿÃÿÿ¿×š'_¿î—ßðkí¥àÌÆ½à¼1^AÑÿÿÿÿÿÿÿÿÿÿÿÿÿÿë…ÊOÿÿùð EI endstream endobj 77 0 obj <> stream 75 0 0 0 0 0 d1 endstream endobj 78 0 obj <> stream 0 0 0 39 66 102 d1 66 0 0 63 0 39 cm BI /IM true /W 66 /H 63 /BPC 1 /F /CCF /DP <> ID &¨˜ÿÿò.„i†0_ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿýÿÿÃÿÿ¿×š'_¿«¥…¿)<ûix31¯x/ VD9Gð EI endstream endobj 79 0 obj <> stream 0 0 0 38 44 104 d1 44 0 0 66 0 38 cm BI /IM true /W 44 /H 66 /BPC 1 /F /CCF /DP <> ID &¨À! 1þÁ6ø@ýo´çÄ}?O῾¿øO÷¯ÿþ¾¿á¥õëŠà¹ä3„ë äðAap‚Â\h— F ÊÉÑ…¤C*úÿáõÿáÿmßᯮÚì0¸ Ì!õÃAÿÈ`7€€ EI endstream endobj 80 0 obj <> stream 76 0 0 0 0 0 d1 endstream endobj 81 0 obj <> stream 0 0 0 38 61 104 d1 61 0 0 66 0 38 cm BI /IM true /W 61 /H 66 /BPC 1 /F /CCF /DP <> ID & \†þˆ  P‰Mo `ðú}&ôƒzÃÒo&ÿÒoýoý&ÿëôÿã¿ÿÿÿÿ&¾¿ÿÿöÂ_ÿ¼/÷®×zíà -ëm¶ívÒÃÃÁa†,…ø2 EI endstream endobj 82 0 obj <> stream 53 0 0 0 0 0 d1 endstream endobj 83 0 obj <> stream 0 0 0 0 66 80 d1 66 0 0 80 0 0 cm BI /IM true /W 66 /H 80 /BPC 1 /F /CCF /DP <> ID & ÔC ’eÿC ·„H àð‚ ðO„ƒzMëá&ô˜|'è>·¯Ózý?øKû¯ß ýÿƒÖ¿ÿÿÿÿÿÿû`¿ÿïõÃÿðÚÿköÂßÚá­¾¶¶×`°Á£ë 0‚]ÂðbŸð`¹ Ü@ EI endstream endobj 84 0 obj <> stream 0 0 0 28 47 79 d1 47 0 0 51 0 28 cm BI /IM true /W 47 /H 51 /BPC 1 /F /CCF /DP <> ID & A@ð@ðFc…‡ ƒz ôøI‡­ÿ¤Þ·Óÿ­ÿÂÿñÃÿÿÿüšø_ÿÿ .õþõÚÿ¶–õþAm¥Á‚á†[ƒƒ EI endstream endobj 85 0 obj <> stream 0 0 0 29 54 99 d1 54 0 0 70 0 29 cm BI /IM true /W 54 /H 70 /BPC 1 /F /CCF /DP <> ID *Ã7òj2€Ô†ÿÿÿÿÿÿÿ‘)ÿ‚5üñ‡ëèà |S}?A‡íÿÓ|+÷ÿÛÿÿÿðÃÿÿÿÿÿÿ ÿ ÿý~¿úᯯuÚ_ Ñ0”l\0¸1 š‰ @ EI endstream endobj 86 0 obj <> stream 0 0 0 30 52 100 d1 52 0 0 70 0 30 cm BI /IM true /W 52 /H 70 /BPC 1 /F /CCF /DP <> ID &¤fЃÐxGôÃn-ûÛýäÕ(xh?øÞÿ·ýþÿßè?ÿÓý?ç×ü$ßúß¾”?úþ·ÓëðŸü-ôúßý?úÃè>·ï×­ÿ¼-ÿ Á‰¨ü@ EI endstream endobj 87 0 obj <> stream 57 0 0 0 0 0 d1 endstream endobj 88 0 obj <> stream 0 0 0 29 37 77 d1 37 0 0 48 0 29 cm BI /IM true /W 37 /H 48 /BPC 1 /F /CCF /DP <> ID $ÂþMFPá…ÿÿÿÿÿÿÿÿÿÿÿÿü?ÿþÈXç” §ï…¿¿WÇ~±X0¤Ôp€€ EI endstream endobj 89 0 obj <> stream 0 0 0 4 24 77 d1 24 0 0 73 0 4 cm BI /IM true /W 24 /H 73 /BPC 1 /F /CCF /DP <> ID òj2‚†ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþõÊ×ÿò'ãÿÿÿäKÁ§ÿÿÿµƒ  EI endstream endobj 90 0 obj <> stream 42 0 0 0 0 0 d1 endstream endobj 91 0 obj <> stream 0 0 0 28 50 101 d1 50 0 0 73 0 28 cm BI /IM true /W 50 /H 73 /BPC 1 /F /CCF /DP <> ID & ÁÔêðƒÁƒÂ' áðƒé7 ƒxX}> stream 30 0 0 0 0 0 d1 endstream endobj 93 0 obj <> stream 0 0 0 1 55 77 d1 55 0 0 76 0 1 cm BI /IM true /W 55 /H 76 /BPC 1 /F /CCF /DP <> ID *Ïÿ&£( å†aÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü?ÿûÿïð¼úuÿí¥ì|0¾ W†ƒãÿÿÿÿÿÿÿÿÿø]cÿ“Q0 EI endstream endobj 94 0 obj <> stream 0 0 0 10 33 79 d1 33 0 0 69 0 10 cm BI /IM true /W 33 /H 69 /BPC 1 /F /CCF /DP <> ID & A˜þx#6·¤Ò~ŸÿºøOÿÿÿÿÿþÿÿÿÿÿÿÿÿÿÿÿÄ„^M_ œ¾÷ýÿÿýÿÿÿà EI endstream endobj 95 0 obj <> stream 0 0 0 24 42 75 d1 42 0 0 51 0 24 cm BI /IM true /W 42 /H 51 /BPC 1 /F /CCF /DP <> ID & A@'ÁÁŠ zƒö“ðƒá+Òaëøn—ß]AÿŽÿÿüšþùùºí„ýûýû‡Ãl-†°¸a‘)±    EI endstream endobj 96 0 obj <> stream 112 0 0 0 0 0 d1 endstream endobj 97 0 obj <> stream 0 0 0 0 97 102 d1 97 0 0 102 0 0 cm BI /IM true /W 97 /H 102 /BPC 1 /F /CCF /DP <> ID & ØS†¼†¢LƒPž<|êÂ$pð@ƒÖðAoI½oI½ Ãá>o§Öô›ÿ ú ¿õ‡Óáo§ÿ§Öÿÿ úßÿðŸÿõ¿ÿOÿÿÿÿÿÿÿkÿÞ¿ÿ†¿ÿ½vÿöÒÿü6—ý®ð¿Øivý´¶Òÿ ¥°Ð]…ÛKm-ë Ka‚ †  ¶ Áà°Î á`Ä/ÃÔIlO EI endstream endobj 98 0 obj <> stream 0 0 0 5 44 78 d1 44 0 0 73 0 5 cm BI /IM true /W 44 /H 73 /BPC 1 /F /CCF /DP <> ID &¨/ÿÛß}ÿ àeû}ý¿aýÝ·†ûöû öûÃíì>öÞ‡·ÞÞÿ·†·û÷ÿ—mþ>Ÿÿÿÿ×ý-˜ Åÿ~éwö» %½v‚ÃisA8b¶  @ EI endstream endobj 99 0 obj <> stream 0 0 0 5 44 81 d1 44 0 0 76 0 5 cm BI /IM true /W 44 /H 76 /BPC 1 /F /CCF /DP <> ID & Á@/ôxo`õ½&ôo§ÿ 0ÿÖÿéõ¿ÿÿ[ÿ§ÿÿõ¿ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÞ¿ÿÿÿÛKÿÿÿzÿm/û\<-ëµØa.×a„°b k  EI endstream endobj 100 0 obj <> stream 0 0 0 5 44 81 d1 44 0 0 76 0 5 cm BI /IM true /W 44 /H 76 /BPC 1 /F /CCF /DP <> ID & Á@/áÂ3ÖôƒzMôA0ð•ÿ«õûzÿý××ÿÿõÿÿÿÿþëÿ_û¯þ»Kþ °¼Ÿ×¶‚öÒø÷`¼^=ïÿû÷þCgm§éáý‡ÿß·í¾ï†Á­†ƒ ±Xa`ÂÁ… EI endstream endobj 101 0 obj <> stream 0 0 0 0 73 75 d1 73 0 0 75 0 0 cm BI /IM true /W 73 /H 75 /BPC 1 /F /CCF /DP <> ID !A”IC;“P„0Πa~ ?¿oØ~þÛû÷û÷ûÿÿaÿÿ߇ÿÿÿûÿÿÿÿÿÿÿÿ_ÿÿþëÿÿø^¿^¿ú__ú^Ð^—×¥à—…Ðd3ë  EI endstream endobj 102 0 obj <> stream 0 0 0 26 42 77 d1 42 0 0 51 0 26 cm BI /IM true /W 42 /H 51 /BPC 1 /F /CCF /DP <> ID & A@'ÁÁŠ zƒæ“|'ÂMé0ÿÔ7Kï® ÿÿ ?ÿ’°ß4Mü?þ·ývÿûAoïXl%¶— vKb ,8`@ EI endstream endobj 103 0 obj <> stream 82 0 0 0 0 0 d1 endstream endobj 104 0 obj <> stream 0 0 0 28 49 79 d1 49 0 0 51 0 28 cm BI /IM true /W 49 /H 51 /BPC 1 /F /CCF /DP <> ID & „àç‹ù°àŠzÚz Ìî“i7§„ýþ”þÿþÿÿÞûû~ÃðÃðþ@œ üx>AF¹ŸÿüŸzú„¿ÿתÚK‚Kƒ]ƒ Bà 耀 EI endstream endobj 105 0 obj <> stream 52 0 0 0 0 0 d1 endstream endobj 106 0 obj <> stream 0 0 0 29 85 77 d1 85 0 0 48 0 29 cm BI /IM true /W 85 /H 48 /BPC 1 /F /CCF /DP <> ID *Ì«?ÿòj2€ÎPÊ  /ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþ?ÿÿ»ÿû¿Â…Â4Ngºÿý[I´£`郮XapbŠÃ0¤ÔH0@Á@@ EI endstream endobj 107 0 obj <> stream 0 0 0 27 55 75 d1 55 0 0 48 0 27 cm BI /IM true /W 55 /H 48 /BPC 1 /F /CCF /DP <> ID *Ïÿ&£( å†aÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü?ÿûÿïð¸F‰×þ­¥\0¸1XaI¨`  EI endstream endobj 108 0 obj <> stream 47 0 0 0 0 0 d1 endstream endobj 109 0 obj <> stream 0 0 0 0 38 77 d1 38 0 0 77 0 0 cm BI /IM true /W 38 /H 77 /BPC 1 /F /CCF /DP <> ID &¨¤ ÿò*0Âÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿø‘‚~MB Ÿÿÿÿÿÿÿÿød$÷ý;û÷ß¿ µ°a…dž² ) EI endstream endobj 110 0 obj <> stream 0 0 0 0 48 80 d1 48 0 0 80 0 0 cm BI /IM true /W 48 /H 80 /BPC 1 /F /CCF /DP <> ID &¨À0H l…ÿ„aà·Ú4ß›àƒôÃéþúøMõûõýÿÿ×þ¿ÿâ¾´°¿¥…ÈeÈlQ]'à– k„‚ár „K‚Šòœ2èê 8\.µ ÓõÿÒÿÿ…ÿýþ×ÿ]þ×uì.]ƒ ¼õØdJ{ƒÿƒàÁ<@ EI endstream endobj 111 0 obj <> stream 74 0 0 0 0 0 d1 endstream endobj 112 0 obj <> stream 0 0 0 28 36 79 d1 36 0 0 51 0 28 cm BI /IM true /W 36 /H 51 /BPC 1 /F /CCF /DP <> ID 0œ?†N»GÂàƒuì:ûºúýxKéAár × úÁ, °º @¹¬Ê@…Ï]_ëòkµÿýµìf‚pÄâá§Á‚x€ EI endstream endobj 113 0 obj <> stream 31 0 0 0 0 0 d1 endstream endobj 114 0 obj <> stream 0 0 0 -1 54 77 d1 54 0 0 78 0 -1 cm BI /IM true /W 54 /H 78 /BPC 1 /F /CCF /DP <> ID & A@‡Äþ9¨7‚3>AH<œ°ƒÖˆZO¯Óëø~½ÿˆ_ÿÿÿÿÿüšÿ‡ÿÿûûûaív×ì4mpÚ_ àÅx`¼†ƒøÿÿÿÿÿÿÿÿÿÿ…×!›WÿþC[à EI endstream endobj 115 0 obj <> stream 0 0 0 -3 74 75 d1 74 0 0 78 0 -3 cm BI /IM true /W 74 /H 78 /BPC 1 /F /CCF /DP <> ID !¿þMLè F¡Á„[ Þû¯Öÿøa¥ÿ¯~õÿ…ïÞ»_÷…ÇÿµÌŽiÿk½ávëÿ^ýáÚÿ½p×ü=Úï_ÿ¶ÿ׿x_úí×ü†¿ÿkÿkÿaÀ@ EI endstream endobj 116 0 obj <> stream 98 0 0 0 0 0 d1 endstream endobj 117 0 obj <> stream 60 0 0 0 0 0 d1 endstream endobj 118 0 obj <> stream 0 0 0 1 25 77 d1 25 0 0 76 0 1 cm BI /IM true /W 25 /H 76 /BPC 1 /F /CCF /DP <> ID òj2à ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ¬òj&  EI endstream endobj 119 0 obj <> stream 48 0 0 0 0 0 d1 endstream endobj 120 0 obj <> stream 0 0 0 0 63 75 d1 63 0 0 75 0 0 cm BI /IM true /W 63 /H 75 /BPC 1 /F /CCF /DP <> ID #`Óù5CË ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ‘P_ð|œ4ø~@º=¿á‡÷÷ï÷ïÿßÿÿÿÿÿýéëõõá}át4ˆ¨(RZð EI endstream endobj 121 0 obj <> stream 97 0 0 0 0 0 d1 endstream endobj 122 0 obj <> stream 43 0 0 0 0 0 d1 endstream endobj 123 0 obj <> stream 0 0 0 1 54 79 d1 54 0 0 78 0 1 cm BI /IM true /W 54 /H 78 /BPC 1 /F /CCF /DP <> ID & `Ð4|‰@¾Á~°}£€áö“Aþ˜|mÿÕð›ÿßïÿÿðÿÿÿÿÿÿÿÿ…áÿÒÿë‡úö—i|.}0—°i|0¾ BðÁx`¼ÿÿÿÿÿÿÿÿÿ ¬òj&  EI endstream endobj 124 0 obj <> stream 0 0 0 66 13 99 d1 13 0 0 33 0 66 cm BI /IM true /W 13 /H 33 /BPC 1 /F /CCF /DP <> ID &¹wÞû{÷ûðýþýûÿÿ—|!¯“]a‚€€ EI endstream endobj 125 0 obj <> stream 55 0 0 0 0 0 d1 endstream endobj 126 0 obj <> stream 0 0 0 0 74 75 d1 74 0 0 75 0 0 cm BI /IM true /W 74 /H 75 /BPC 1 /F /CCF /DP <> ID *ÃLÀZþ²jD0ÖàÂö´¾ÿ_ý}õð¿ÿ辿Âý¯Òÿü/é_ÿ ¿Kÿõü%ý~¿×ëú ÿúý/Âÿþ—é 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&¡Ê¿È£4ƒÂOù8¯Atw~þ~ïiþöŸß÷ïò?ù×Â>›ÓÿCÊaz>ž“ÿôƒÿ_ÿÿ ÿÿð½ÿ[¯ýÖÚö¼4¶Âáëƒ ± ö°Öà ÿ EI endstream endobj 207 0 obj <> stream 129 0 0 0 0 0 d1 endstream endobj 208 0 obj <> stream 0 0 0 2 67 77 d1 67 0 0 75 0 2 cm BI /IM true /W 67 /H 75 /BPC 1 /F /CCF /DP <> ID *@A$rj†ì ý¿¿aÿíÿÿ†ÿÿÿÿÿýáô¿ø^—¯¥á/‚ð—_üœ3‡ä#ÃÃïÃ~ßþÿûÿÿÿÿÿÿázÿéõéxKÂè0‚ˆ¨PR0 EI endstream endobj 209 0 obj <> stream 32 0 0 0 0 0 d1 endstream endobj 210 0 obj <> stream 0 0 0 29 37 79 d1 37 0 0 50 0 29 cm BI /IM true /W 37 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ „G3·„ˆ'êÃè>­ôûÕ¾ýûðÿÿûÿÿä¸c$ÂX>h(? ”ì7¸oï°û¾áÝ·ý¾ß~K ?†¸b°kf0 EI endstream endobj 211 0 obj <> stream 0 0 0 8 31 79 d1 31 0 0 71 0 8 cm BI /IM true /W 31 /H 71 /BPC 1 /F /CCF /DP <> ID &¢þ0õ¿aêþþûÿÛï÷ûÝÿïÿÇïÿïÿü?ýûÿßûÿÞ$brkä<ÿ¿ü?ðÿ÷ïÿïÿ}…ð EI endstream endobj 212 0 obj <> stream 0 0 0 29 46 79 d1 46 0 0 50 0 29 cm BI /IM true /W 46 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ ÁÏ ü Á‚>ÃG§¬5½ oo¦?Ýõ„÷ÿ¨wþÛ÷ÿáÿ¿ä!¯ûþ÷ýÿïûßðÿíÿý·ÿì;ÿ¿mvï …Øk†&Í`Ðá… EI endstream endobj 213 0 obj <> stream 0 0 0 1 25 79 d1 25 0 0 78 0 1 cm BI /IM true /W 25 /H 78 /BPC 1 /F /CCF /DP <> ID &¤xᣓÂßñ°÷þÿ÷&ªÿ÷ÿã‡þþÿßþÿßþýÿïýÿáÿ‡ÿ¿÷ÿ¿÷ÿ¿ûÿøáþG_ÿ~Az€ EI endstream endobj 214 0 obj <> stream 0 0 0 66 12 77 d1 12 0 0 11 0 66 cm BI /IM true /W 12 /H 11 /BPC 1 /F /CCF /DP <> ID &°-=>ÿ÷÷ƒ  EI endstream endobj 215 0 obj <> stream 0 0 0 27 55 97 d1 55 0 0 70 0 27 cm BI /IM true /W 55 /H 70 /BPC 1 /F /CCF /DP <> ID & ÑÿÈkkÿÿÿÿÿÿÿ B?ð@ü˜¯ÖÂA¿A ü$Ç«ÿ¥ê¯ßKÿøÿ…ÿÿÿÿü>Mÿÿ÷î¿íû…ïÛF׆¾Ÿ ¥í¤ü0_+Ãä4<@ EI endstream endobj 216 0 obj <> stream 104 0 0 0 0 0 d1 endstream endobj 217 0 obj <> stream 0 0 0 0 50 69 d1 50 0 0 69 0 0 cm BI /IM true /W 50 /H 69 /BPC 1 /F /CCF /DP <> ID & ¹8ø@Éi‚ 8O§ÂÈcZ3 µø'Ò~¿¤ÿð¸ýëÿÿÿÿ¿þMÃý{÷_öþ^]†cÃï²Úðax0^?ÿÿÿÿÿä2õ×ÿÿj  EI endstream endobj 218 0 obj <> stream 0 0 0 19 51 94 d1 51 0 0 75 0 19 cm BI /IM true /W 51 /H 75 /BPC 1 /F /CCF /DP <> ID & ÁÔ3Áa€ZžŸ:…è 6ÂÖúx‚ ý?ÿë÷Rk×µá®Ã!£O…±[_µÁr æypk¤?×ÿ“§‚5ÌCïaþåÂÞ‚øOÓôúßýÿÿÿÿ°¿ï]¯kÃR;Ø`šxb/ÞþÉt°aÁš  EI endstream endobj 219 0 obj <> stream 0 0 0 30 51 77 d1 51 0 0 47 0 30 cm BI /IM true /W 51 /H 47 /BPC 1 /F /CCF /DP <> ID & ÜÌ6Sü'ÿú?¿ÿÖúÿÖ ÿÿI¿ÿÂßOÿúA¿ÿÖúÿÖü'ÿ[éÿÊBŒ‹Oÿù5I¨€ EI endstream endobj 220 0 obj <> stream 0 0 0 19 50 91 d1 50 0 0 72 0 19 cm BI /IM true /W 50 /H 72 /BPC 1 /F /CCF /DP <> ID &¨„ ¡òj–Dÿÿÿÿÿÿÿÿ#?þ¨‚ÓâÍDø oï§è0úÿMÿÿ·…ÿ¿ÿÿÿÿÿ_úÿ†¾½×ý„kØ\¥”Eu5%ø`¤Õ ab(€ EI endstream endobj 221 0 obj <> stream 0 0 0 20 51 69 d1 51 0 0 49 0 20 cm BI /IM true /W 51 /H 49 /BPC 1 /F /CCF /DP <> ID & Á@(ø@Éé‚ :¯ÉÅZ2¿á?‡÷ÿßÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü­H}c_ÿÿ“Tš€€ EI endstream endobj 222 0 obj <> stream 0 0 0 19 57 67 d1 57 0 0 48 0 19 cm BI /IM true /W 57 /H 48 /BPC 1 /F /CCF /DP <> ID &¨ÔΦ¢áÝBÿ÷MR…Y. ž ùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿáßÿÿÿÝý¦½ß•Z b"¾@°¾HI©5I¦±#™ 8€ EI endstream endobj 223 0 obj <> stream 0 0 0 0 50 69 d1 50 0 0 69 0 0 cm BI /IM true /W 50 /H 69 /BPC 1 /F /CCF /DP <> ID &  œù )‚ÓâÍDø oï§è0úÿMÿÿ·…ÿ¿ÿÿÿÿÿ_úÿ†¾½×ý„kØ^ â¿®}/ü0^ / ÿÿÿÿÿåküš¥ EI endstream endobj 224 0 obj <> stream 0 0 0 10 45 77 d1 45 0 0 67 0 10 cm BI /IM true /W 45 /H 67 /BPC 1 /F /CCF /DP <> ID &¸_ ÿ°¿!žÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÕþ¿ûð  EI endstream endobj 225 0 obj <> stream 0 0 0 19 50 67 d1 50 0 0 48 0 19 cm BI /IM true /W 50 /H 48 /BPC 1 /F /CCF /DP <> ID &¨Šî?“Uë “ÿÿÿÿÿÿÿÿþÿÿø}ÿ}ÿd¸iðÓ”)äBq½¯“Uè5‰<@ EI endstream endobj 226 0 obj <> stream 0 0 0 20 53 67 d1 53 0 0 47 0 20 cm BI /IM true /W 53 /H 47 /BPC 1 /F /CCF /DP <> ID & †`‡)„û_ýáú<³ˆ?ÿí_ÿÿúÿßõÖþÿÿþ¿ÿþ? ·Ðkÿÿö¿ÿkÔo ÿÿÿúßý?ÿå!Dv¿Ç“U®ž  EI endstream endobj 227 0 obj <> stream 0 0 0 8 40 77 d1 40 0 0 69 0 8 cm BI /IM true /W 40 /H 69 /BPC 1 /F /CCF /DP <> ID &©Gz“UÈ1?ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÌ ÿÿøkÿÿ ÅáÿÂÿ†¿ À@ EI endstream endobj 228 0 obj <> stream 0 0 0 -2 44 67 d1 44 0 0 69 0 -2 cm BI /IM true /W 44 /H 69 /BPC 1 /F /CCF /DP <> ID &±?ÿý¬‚óÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿüåüšë ¼ÿÿÿÿû ‘Ø öÓýƒáûX|Ãð EI endstream endobj 229 0 obj <> stream 0 0 0 28 42 79 d1 42 0 0 51 0 28 cm BI /IM true /W 42 /H 51 /BPC 1 /F /CCF /DP <> ID & Á@?Â5„‚žýGÐ Þ¿ :^Ÿ\ ˜_øõñ×ÿýòj¿ßá†Cf¶í>ý¿Bwö<5ûXk°Î¡@ EI endstream endobj 230 0 obj <> stream 0 0 0 10 51 77 d1 51 0 0 67 0 10 cm BI /IM true /W 51 /H 67 /BPC 1 /F /CCF /DP <> ID &¨„%Ÿÿ“TšÈžC<ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþÿýÿö½ý¯ãá>—þðkÃãÿÿÿÿÿùZÇÿ&©@@ EI endstream endobj 231 0 obj <> stream 0 0 0 17 45 79 d1 45 0 0 62 0 17 cm BI /IM true /W 45 /H 62 /BPC 1 /F /CCF /DP <> ID &  œ?‚Aè?ý ð›Âü&ûéÿÿÿü5ãÿÿÿÿÿÿÿÿÿÿ”UÿɪYŸÿÿÿÿÿᨀ EI endstream endobj 232 0 obj <> stream 0 0 0 30 51 77 d1 51 0 0 47 0 30 cm BI /IM true /W 51 /H 47 /BPC 1 /F /CCF /DP <> ID &¨ˆv±ßäÕk§ÈI^}®ð¶Â_[k‡K¿a¥è-ýºØ¯ë oúá>ôÿFJþ‚o ýzMûzù“„FPšÿÿÿᦠ EI endstream endobj 233 0 obj <> stream 0 0 0 0 54 72 d1 54 0 0 72 0 0 cm BI /IM true /W 54 /H 72 /BPC 1 /F /CCF /DP <> ID &¡˜¿Oû_ÔŸÿÿÿÿÿÿþAƒÿàø ýô=†/A×á>‚ÓûéûÄ/÷¯ÿÿÿïù5ÿ¯ÿµÛ_öýµàÁxav<>û*×ü0¼^A¨€ EI endstream endobj 234 0 obj <> stream 99 0 0 0 0 0 d1 endstream endobj 235 0 obj <> stream 0 0 0 18 51 69 d1 51 0 0 51 0 18 cm BI /IM true /W 51 /H 51 /BPC 1 /F /CCF /DP <> ID &¡É—ðD`@°ƒ.º-~H-t ý k§ë÷ÿÿûÿÛøì²øcÃþA`ù ƒ2 Sÿü–/øN´×Âú Á~¸­¯Ã[ ad8p EI endstream endobj 236 0 obj <> stream 51 0 0 0 0 0 d1 endstream 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0 0 120 0 -12 cm BI /IM true /W 52 /H 120 /BPC 1 /F /CCF /DP <> ID &¨À2 ‡òj½ÿ{þ¿ï߇ýïÞÿ‡ïûß÷áÿ{þýÿxß¿áïû÷ýïø~ÿ½ÿaÿíÿíÿá¿ý¿ý‡ÿ·ÿ·ÿ†ÿöÿöþßþß÷‡ýûþÿ¿Þÿ‡ïûÚ€€ EI endstream endobj 266 0 obj <> stream 0 0 0 24 67 80 d1 67 0 0 56 0 24 cm BI /IM true /W 67 /H 56 /BPC 1 /F /CCF /DP <> ID &¡°è /ò a¼ øAðŸD~?PÄê=5¯ÿ‡ÿßÿýÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÕrpÿÿÿÿ¤.D@ EI endstream endobj 267 0 obj <> stream 128 0 0 0 0 0 d1 endstream endobj 268 0 obj <> stream 120 0 0 0 0 0 d1 endstream endobj 269 0 obj <> stream 0 0 0 20 14 67 d1 14 0 0 47 0 20 cm BI /IM true /W 14 /H 47 /BPC 1 /F /CCF /DP <> ID &¬žœ_ä×[[Xÿÿÿÿÿ—ú§òk¯ÚÚ€€ EI endstream endobj 270 0 obj <> stream 0 0 0 -9 45 77 d1 45 0 0 86 0 -9 cm BI /IM true /W 45 /H 86 /BPC 1 /F /CCF /DP <> ID &¹ÀÖ¯ÿ¿Ûýûß¿Ã÷ï÷ï÷ïöÿÃß¿Ûýÿ¿Ûü?öÿ~÷ï÷þðü?Ûýûý¿ß¼?ðÿ~÷ïöÿ~ÿ ?a@@ EI endstream endobj 272 0 obj <> stream 0 0 0 39 37 47 d1 37 0 0 8 0 39 cm BI /IM true /W 37 /H 8 /BPC 1 /F /CCF /DP <> ID &º‹ù5Ô@ EI endstream endobj 273 0 obj <> stream 0 0 0 8 35 103 d1 35 0 0 95 0 8 cm BI 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&¢š‚‚Fc|'ˆl;¿Ûÿ“TŸjÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþ¾²¯ü†zHfÿÿÿò3=?ÿþÖP EI endstream endobj 279 0 obj <> stream 0 0 0 4 44 80 d1 44 0 0 76 0 4 cm BI /IM true /W 44 /H 76 /BPC 1 /F /CCF /DP <> ID & ¹ ‚ø xF‚½aჽÃàŸH7¥ýó@ÙÓ¿ûý¿ÿÿµÚÿ ë_þ¿K¬.‚Òë,ƒ×ýdŸÃß{{ðýÿävßø ý?ÿÿÿþ¿]ªáÞ»Aa†—'†+a`Á`Á@@ EI endstream endobj 280 0 obj <> stream 0 0 0 4 44 80 d1 44 0 0 76 0 4 cm BI /IM true /W 44 /H 76 /BPC 1 /F /CCF /DP <> ID & Á@/òF xF€ÃÒ >ôÞ¡½~}êý_ÿÿõþ×÷ ý¥¸^í‚ú[øl%¶àÂí °Å`ÖÂÐZà‚Óýàz ¼-ôÐ&ú~…zü&úü?_ÿð¿ÿÜ/½vþÒÃûh-°Kš Ø­…ƒƒ EI endstream endobj 281 0 obj <> stream 0 0 0 0 60 78 d1 60 0 0 78 0 0 cm BI /IM true /W 60 /H 78 /BPC 1 /F /CCF /DP <> ID &¡³ÿþCK_ÿÿÿÿÿÿ‰&þfg@¤×~þÃöþÿöü?·ì?¿¿ ûÛÿßÛðßß°ÿ÷öý¿¿ ÿïì{þøßßÃïûïð EI endstream endobj 282 0 obj <> stream 0 0 0 0 76 78 d1 76 0 0 78 0 0 cm BI /IM true /W 76 /H 78 /BPC 1 /F /CCF /DP <> ID &¡¨l5?è?ý?ÿÓü'ÿúT ¡ÿÐõoþ½o§ÿ ôúß¿ ÖýÖýúÿÖú|-ÿÓÿ­ûè(ôýo߯ý&ÿÂßý?ú·Âõ‡ÿAÿÖú éÞIpË%Ãÿ EI endstream endobj 283 0 obj <> stream 0 0 0 -4 42 106 d1 42 0 0 110 0 -4 cm BI /IM true /W 42 /H 110 /BPC 1 /F /CCF /DP <> ID &¨Àk~ÿïöÿáþûßûýÿ¿Ûýÿ‡ø~ÿïöÿï÷þûýÿ¿Ûýÿ¿ß‡øï÷þ÷ï÷þ÷þáÿ¿Ûýÿ¿ß¿ßø‡þ÷ï÷þïø€ EI endstream endobj 284 0 obj <> stream 0 0 0 0 54 69 d1 54 0 0 69 0 0 cm BI /IM true /W 54 /H 69 /BPC 1 /F /CCF /DP <> ID & Ú|¾P€ƒðœZéüš¥üÏG5þ¿ÿþÂüÿÿÿÿÿÿÿ×áx__úâºÿïûæaßÃðoÃûûýÿïÿÿÿëÿׯ¯…ô¼áe õAIª ””  EI endstream endobj 285 0 obj <> stream 0 0 0 0 55 80 d1 55 0 0 80 0 0 cm BI /IM true /W 55 /H 80 /BPC 1 /F /CCF /DP <> ID &¡œè þCð‰€x ðƒÐ| @Þ&õ‡Â é?ü|?Nÿíÿûÿÿ®ý® øÿýëÖE¿N¸kßÃKápa. %ÇÏ ¼0^ /C ÿÿÿÿÿ€c)f‡ü>¾ÿ¾ÿf~l4ð EI endstream endobj 286 0 obj <> stream 0 0 0 79 45 88 d1 45 0 0 9 0 79 cm BI /IM true /W 45 /H 9 /BPC 1 /F /CCF /DP <> ID &¸_ ÿöü@ EI endstream endobj 287 0 obj <> stream 0 0 0 8 47 79 d1 47 0 0 71 0 8 cm BI /IM true /W 47 /H 71 /BPC 1 /F /CCF /DP <> ID &  ààá= ÿ†+ ëÐ z…öºÇ„xH?OTúúõF†ú§Õoáð“…ÿOÿªßý§ÿÿÿÿÿÿÿö»ÿ÷ÿµÃþ_ -Þ»µÝ¿¿wöP¼0Â^ãØdžÁ®þà aŠýö°Á`Ö /€€ EI endstream endobj 288 0 obj <> stream 0 0 0 0 55 80 d1 55 0 0 80 0 0 cm BI /IM true /W 55 /H 80 /BPC 1 /F /CCF /DP <> ID & Ü†>— A„ á>t¤ 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d1 endstream endobj 295 0 obj <> stream 0 0 0 45 53 48 d1 53 0 0 3 0 45 cm BI /IM true /W 53 /H 3 /BPC 1 /F /CCF /DP <> ID à EI endstream endobj 296 0 obj <> stream 0 0 0 0 25 26 d1 25 0 0 26 0 0 cm BI /IM true /W 25 /H 26 /BPC 1 /F /CCF /DP <> ID &¡Ìßõùß´ôðð¸<ýaˆ…ƒÿ‚øFy„`ßü,=­¬qþÿ¨€ EI endstream endobj 297 0 obj <> stream 0 0 0 1 62 66 d1 62 0 0 65 0 1 cm BI /IM true /W 62 /H 65 /BPC 1 /F /CCF /DP <> ID #?þMY@f”Eæ¶¿_÷®Âï ÿû]ë×ü0ëþ?Úþl3÷¯û ¼/ÿÛKÿ^‡¯û_÷®×ýëþÂï ÿöÒÿ× íþ×ì/ÿ€€ EI endstream endobj 298 0 obj <> stream 0 0 0 3 20 66 d1 20 0 0 63 0 3 cm BI /IM true /W 20 /H 63 /BPC 1 /F /CCF /DP <> ID òjdçkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿXÿ“Rü@ EI endstream endobj 299 0 obj <> stream 25 0 0 0 0 0 d1 endstream endobj 300 0 obj <> stream 0 0 0 25 40 68 d1 40 0 0 43 0 25 cm BI /IM true /W 40 /H 43 /BPC 1 /F /CCF /DP <> ID & ¹ cä=XFf‡ A½ ?ô›á>·þúÿŽÿÿÿù5ZÿÿðÚ_ï_ð×m-áoXa‚[ãÃ^ EI endstream endobj 301 0 obj <> stream 0 0 0 2 32 66 d1 32 0 0 64 0 2 cm BI /IM true /W 32 /H 64 /BPC 1 /F /CCF /DP <> ID &¨ƒÿäµÿÿÿÿÿÿÿÿÿÿÿÿÿÿœäÔf`ÇÿÿÿÿÿþÈ—þáþø~ÃîöÚÁˆ_†@ EI endstream endobj 302 0 obj <> stream 0 0 0 11 29 68 d1 29 0 0 57 0 11 cm BI /IM true /W 29 /H 57 /BPC 1 /F /CCF /DP <> ID & ¹òø xFmëxA?Oþ•÷ÿÿÿÿÿÿÿÿÿÿÿÿÿþ$ºòjÿ†h}ÿÿ¿þÿÿÿ EI endstream endobj 303 0 obj <> stream 0 0 0 3 46 66 d1 46 0 0 63 0 3 cm BI /IM true /W 46 /H 63 /BPC 1 /F /CCF /DP <> ID ïüš‘899Zkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþÿÿáÿö½×3O÷¯`×Áˆ/ü0¼ÿÿÿÿÿÿÿXÿ“Qþ  EI endstream endobj 304 0 obj <> stream 0 0 0 25 36 68 d1 36 0 0 43 0 25 cm BI /IM true /W 36 /H 43 /BPC 1 /F /CCF /DP <> ID & ¹ _õazÃÁôƒ}?O¬:øAá~õÿ"¡¾hüš¯ýÿ‡…ë°ÿû m®õ·X6ãØ,†) EI endstream endobj 305 0 obj <> stream 0 0 0 27 46 66 d1 46 0 0 39 0 27 cm BI /IM true /W 46 /H 39 /BPC 1 /F /CCF /DP <> ID 5‰ÿ“SAAáaëuïÛ-ÒÞ°Ã^ÒïØ­…¿]k®Ÿóÿ 7¬=+Ô7ÓôÞ‚¼,=Å£Zÿÿ EI endstream endobj 306 0 obj <> stream 0 0 0 25 42 68 d1 42 0 0 43 0 25 cm BI /IM true /W 42 /H 43 /BPC 1 /F /CCF /DP <> ID &¡ òU‚ÐÚ[ô´g1LB{á?ÿÃÿø~M~û~þÃð`þÃAÆd×!•ÿÿ—ÿú„¿õôº­‚­Ÿ.<X2pØ  EI endstream endobj 307 0 obj <> stream 0 0 0 26 73 66 d1 73 0 0 40 0 26 cm BI /IM true /W 73 /H 40 /BPC 1 /F /CCF /DP <> ID âÿÿ&¤N( • Âkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¸ÿÿÿþÂk§3NMÛ÷§­ƒIƒ `ÅüšŽ Á@@ EI endstream endobj 308 0 obj <> stream 0 0 0 26 45 84 d1 45 0 0 58 0 26 cm BI /IM true /W 45 /H 58 /BPC 1 /F /CCF /DP <> ID ÁGäÔ‰Ã6×ÿÿÿÿÿ’§þ£cÇ}0øA¿øA‡Óýÿíÿÿßûÿÿÿÿÿÿ¥ÿþ¿_]¯Ó(´§U­†‚Ã_&£ƒ  EI endstream endobj 309 0 obj <> stream 0 0 0 25 31 68 d1 31 0 0 43 0 25 cm BI /IM true /W 31 /H 43 /BPC 1 /F /CCF /DP <> ID 0)8Oa‰î7L:éûᥭAÈc_ÈDù"úá.¸,#X ”|âõþ·Éª¯ømv/«ä!Þ  EI endstream endobj 310 0 obj <> stream 0 0 0 26 32 66 d1 32 0 0 40 0 26 cm BI /IM true /W 32 /H 40 /BPC 1 /F /CCF /DP <> ID !üš‘8/kÿÿÿÿÿÿÿÿÿÿ¿ÿìý?Ðx¼éûmlBù50  EI endstream endobj 311 0 obj <> stream 0 0 0 26 46 66 d1 46 0 0 40 0 26 cm BI /IM true /W 46 /H 40 /BPC 1 /F /CCF /DP <> ID ïüš‘899ZkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþÿÿáÿöºuDýë`× ¾MG (€ EI endstream endobj 312 0 obj <> stream 0 0 0 3 47 68 d1 47 0 0 65 0 3 cm BI /IM true /W 47 /H 65 /BPC 1 /F /CCF /DP <> ID &¡ƒAO‹ü9Ô7„|ƒáað‚:AËL-'ÿ_H?ÿð¸ÿÿÿÿÿÿ“_ðÿ÷ÿÛü/k¶»kƒÍ®Á‚^‚ÿÈ/Oÿÿÿÿÿÿÿð¹úÿþA«þ  EI endstream endobj 313 0 obj <> stream 0 0 0 5 20 66 d1 20 0 0 61 0 5 cm BI /IM true /W 20 /H 61 /BPC 1 /F /CCF /DP <> ID òjdçkÿÿÿÿÿÿÿÿÿÿÿÿÿõþMKÿÿä»Oôÿÿö¶  EI endstream endobj 314 0 obj <> stream 0 0 0 25 42 87 d1 42 0 0 62 0 25 cm BI /IM true /W 42 /H 62 /BPC 1 /F /CCF /DP <> ID & ‡@Ä.àŒÂ„0zÞð“}ŽéÿÿþMS_ ÚXl.õ†=­t …È@z< Zÿëÿrtì¨ÅìÐôúMôúßú ¿ÿÿÿÿÿzá¯ûi¾Öëƒh5±97ñÁš  Ô@ EI endstream endobj 315 0 obj <> stream 0 0 0 26 46 68 d1 46 0 0 42 0 26 cm BI /IM true /W 46 /H 42 /BPC 1 /F /CCF /DP <> ID & ¹9™ø çPÞ™Ö:²Ai¬/ïÿÿûÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿªÆD×ÿþMG ¼à EI endstream endobj 316 0 obj <> stream 26 0 0 0 0 0 d1 endstream endobj 317 0 obj <> stream 0 0 0 25 36 68 d1 36 0 0 43 0 25 cm BI /IM true /W 36 /H 43 /BPC 1 /F /CCF /DP <> ID & ¹ _õa<ô‚f“}>¯Ô:ü< ¾Åÿÿ&«ßù;ðÂ{ýûû~áðØk`Èÿ ,‚瀀 EI endstream endobj 318 0 obj <> stream 0 0 0 27 63 68 d1 63 0 0 41 0 27 cm BI /IM true /W 63 /H 41 /BPC 1 /F /CCF /DP <> ID &¡œÀg0ÿ¦Ÿÿè0Ÿÿÿ¦Ÿý®3ó÷ÿÕµ¿ÿ®ëk}>ÿ×ÿë -ôŠÿôMÿáx~ÿTÿIoáz­÷ ât‚5ÿÿþ  EI endstream endobj 319 0 obj <> stream 24 0 0 0 0 0 d1 endstream endobj 320 0 obj <> stream 0 0 0 27 37 66 d1 37 0 0 39 0 27 cm BI /IM true /W 37 /H 39 /BPC 1 /F /CCF /DP <> ID & äàAáÓÓü"‚z ‡­ðõ¾Ÿ§ÿ[þÿÿÿÿû /û_öÒÿm-ƒÚÚà à EI endstream endobj 321 0 obj <> stream 0 0 0 27 44 87 d1 44 0 0 60 0 27 cm BI /IM true /W 44 /H 60 /BPC 1 /F /CCF /DP <> ID &¨ˆÿäÕ,ŽÿÿÿÿÿÿäéàAq¾ó©¾7÷Â~›éü?ÿ_ÿÿÿ_ø_öþÒÿƒ pÒÊŠ…S¬š¤/à EI endstream endobj 322 0 obj <> stream 0 0 0 17 38 66 d1 38 0 0 49 0 17 cm BI /IM true /W 38 /H 49 /BPC 1 /F /CCF /DP <> ID &¡œ XAá<' ÿšè>·¯¿ÿÿðaxÿÿÿÿÿÿÿþP”CþMP_³ÿÿÿÿÚ€€ EI endstream endobj 323 0 obj <> stream 0 0 0 9 36 66 d1 36 0 0 57 0 9 cm BI /IM true /W 36 /H 57 /BPC 1 /F /CCF /DP <> ID &º×¿­òŸÿÿÿÿÿÿÿÿÿÿÿÿæ¯ÿí~?ÿÇ~žŸÿþÖ×À@ EI endstream endobj 324 0 obj <> stream 0 0 0 27 46 66 d1 46 0 0 39 0 27 cm BI /IM true /W 46 /H 39 /BPC 1 /F /CCF /DP <> ID &¨ˆ5Ÿüš¤ÖG2 Ïÿÿÿÿÿÿÿÿÿÿÿÿÿûÿïïÿ†¼YBÑ©ÖRj0¿€€ EI endstream endobj 325 0 obj <> stream 0 0 0 27 35 66 d1 35 0 0 39 0 27 cm BI /IM true /W 35 /H 39 /BPC 1 /F /CCF /DP <> ID &¹²!„¦PŽ!÷ÃþtòpÇïOß_ì%ë#!d@xKÐZÍ@ÆN (c8H?ÿáû Çûïƒàθ_À@ EI endstream endobj 326 0 obj <> stream 0 0 0 3 24 74 d1 24 0 0 71 0 3 cm BI /IM true /W 24 /H 71 /BPC 1 /F /CCF /DP <> ID & Bý8ZÂÒáh-/ÒÂðºÒýzý/ýx_ÿÿÿÜšÿÿáïÿ·ýïû}ððû·°ø{p÷‡µð EI endstream endobj 327 0 obj <> stream 0 0 0 11 44 66 d1 44 0 0 55 0 11 cm BI /IM true /W 44 /H 55 /BPC 1 /F /CCF /DP <> ID & …°ƒN|!×Z'ÔtƒëðŸ ÿôÿÇï_ÿÿßòj¿ý¯ÃkþØ]†}ðøf¯Áÿñÿÿÿÿÿ £Mÿj  EI endstream endobj 328 0 obj <> stream 0 0 0 26 43 88 d1 43 0 0 62 0 26 cm BI /IM true /W 43 /H 62 /BPC 1 /F /CCF /DP <> ID &¡‚@/!àƒý„Ÿ!‰Â( ø@ñÃúÿµù5L\.¸ö¿ap\ŒÁ§ üÔðÿÄ=¾z('ÂÿO­ðŸÿÿÿþí¥ÿåöà 'ˆkc¼<3¦°Áø†  EI endstream endobj 329 0 obj <> stream 0 0 0 27 39 50 d1 39 0 0 23 0 27 cm BI /IM true /W 39 /H 23 /BPC 1 /F /CCF /DP <> ID &°_üš†¿ÿù( üšÁ|@ EI endstream endobj 330 0 obj <> stream 95 0 0 0 0 0 d1 endstream endobj 331 0 obj <> stream 0 0 0 9 42 66 d1 42 0 0 57 0 9 cm BI /IM true /W 42 /H 57 /BPC 1 /F /CCF /DP <> ID & @-?ÿÚÈÿÿÿÿü¢ˆ‡áMsá ªø¿‡ï÷ïíÿ÷ïá¿ý‡ÿß·ïÿ·áþþ¿þÇ¿þøÚø€ EI endstream endobj 332 0 obj <> stream 0 0 0 3 23 74 d1 23 0 0 71 0 3 cm BI /IM true /W 23 /H 71 /BPC 1 /F /CCF /DP <> ID &¨Øâ;¹5Aïƒ{·Þø}‡ýïûýïÿýÃÿÿÿÿü.¿ÿ^¿ô¿^¿AauëKô°KX…U MP_ EI endstream endobj 333 0 obj <> stream 0 0 0 56 11 84 d1 11 0 0 28 0 56 cm BI /IM true /W 11 /H 28 /BPC 1 /F /CCF /DP <> ID &¨À÷ÃÛÃïÞýþÿÝÿþzTzäÕ/Ú€€ EI endstream endobj 334 0 obj <> stream 0 0 0 11 46 66 d1 46 0 0 55 0 11 cm BI /IM true /W 46 /H 55 /BPC 1 /F /CCF /DP <> ID &¨ˆ5Ÿüš¤ÖG2 Ïÿÿÿÿÿÿÿÿÿÿÿÿÿûÿïïÿ†¼\Âë™ËÁ…ÿÿÿÿÿåiòj”@ EI endstream endobj 335 0 obj <> stream 0 0 0 27 44 66 d1 44 0 0 39 0 27 cm BI /IM true /W 44 /H 39 /BPC 1 /F /CCF /DP <> ID &¡O|-ŸkÿÿáoŸM_ÿÿëßÿÿõøÿÇû_ÿÿþ½ôÓû_ÿÿëðŸÿò“’Oµéɪ×ð EI endstream endobj 336 0 obj <> stream 0 0 0 54 13 66 d1 13 0 0 12 0 54 cm BI /IM true /W 13 /H 12 /BPC 1 /F /CCF /DP <> ID &¬§ÿþ×á… EI endstream endobj 337 0 obj <> stream 0 0 0 11 38 66 d1 38 0 0 55 0 11 cm BI /IM true /W 38 /H 55 /BPC 1 /F /CCF /DP <> ID &±(OÿðÖA'ÿÿÿÿÿÿÿÿÿÿÿÿùBQù5A~A'ÿÿý‘Ó‡ð}ÞÖÖAB  EI endstream endobj 338 0 obj <> stream 0 0 0 27 43 66 d1 43 0 0 39 0 27 cm BI /IM true /W 43 /H 39 /BPC 1 /F /CCF /DP <> ID &¡I,0Nu®´P)Ð\ xøAéÿþø|šàÿöu±þø>\fC.‘G\?ái…ðKúØ[ƒ à EI endstream endobj 339 0 obj <> stream 0 0 0 27 42 66 d1 42 0 0 39 0 27 cm BI /IM true /W 42 /H 39 /BPC 1 /F /CCF /DP <> ID &¨…üš¥³ÿÿÿÿÿÿÿïÿ¿ï†C¿aÿh<>QPÈÏaa©5H_À@ EI endstream endobj 340 0 obj <> stream 0 0 0 11 42 66 d1 42 0 0 55 0 11 cm BI /IM true /W 42 /H 55 /BPC 1 /F /CCF /DP <> ID &¨ˆ ‚ÿ“T²Sÿÿÿÿÿÿÿ!{Óÿÿøÿÿÿ™„ÿÿþ×ÿù ™ÿ ÿÿÿÿ”T>MPP EI endstream endobj 341 0 obj <> stream 0 0 0 9 42 66 d1 42 0 0 57 0 9 cm BI /IM true /W 42 /H 57 /BPC 1 /F /CCF /DP <> ID &¨ÖAG ¯NMVN„›×ÿÿþÚ þ?ÿÃ_üøŸÿØiÿÿÿ¶—ÿÿþõÿkÿÿï ÿÿÿü=qÿÿí~×À@ EI endstream endobj 342 0 obj <> stream 0 0 0 11 41 66 d1 41 0 0 55 0 11 cm BI /IM true /W 41 /H 55 /BPC 1 /F /CCF /DP <> ID &©v95_#™ ÿÿÿÿÿÿÚñÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÊ/Ü|š¯P EI endstream endobj 343 0 obj <> stream 0 0 0 9 37 68 d1 37 0 0 59 0 9 cm BI /IM true /W 37 /H 59 /BPC 1 /F /CCF /DP <> ID &¹±I¦aœ_‡üê?(}7ÿO÷„ÿÿÿk]k®½d4kÀÌ„Q„— ŠpÅyÐ4Q@Âë_ Gp“ÿÿýxûkþØ^/ûáðÍW &¿€€ EI endstream endobj 344 0 obj <> stream 0 0 0 11 43 66 d1 43 0 0 55 0 11 cm BI /IM true /W 43 /H 55 /BPC 1 /F /CCF /DP <> ID &¨(‡ÉªÉL†Éÿÿÿÿÿöÿù ÞŸÿÿÇÿÿüÌ'ÿÿö¼ÿÈlÏý?ÿÿÿÿÊ)ù5J  EI endstream endobj 345 0 obj <> stream 0 0 0 56 10 66 d1 10 0 0 10 0 56 cm BI /IM true /W 10 /H 10 /BPC 1 /F /CCF /DP <> ID &¸Zq~MRý¨€ EI endstream endobj 347 0 obj <> stream 0 0 0 0 64 94 d1 64 0 0 94 0 0 cm BI /IM true /W 64 /H 94 /BPC 1 /F /CCF /DP <> ID ýòj»þ}÷ßðÞû>}·Ûø~oß¿·íý݃ ÷íöÈfšáì<7¾ÞÁ÷·†~öûÛØ†öûïaþýþßþ#»‚(œ'Óÿñÿÿ |%“Uú®ìÌ—¯h,0aw­´¶Kv °kaa‘°"@‚Q¬@ EI endstream endobj 351 0 obj <> stream 0 0 0 -5 112 96 d1 112 0 0 101 0 -5 cm BI /IM true /W 112 /H 101 /BPC 1 /F /CCF /DP <> ID & ÊF§Èmä5Åòe¸@ðƒä?Œ"¨Öð‚ú}Þ¯÷¤z¿O÷„Ÿýoÿ~¿½ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿø yAD3ÿÿÿ EI endstream endobj 352 0 obj <> stream 0 0 0 30 75 96 d1 75 0 0 66 0 30 cm BI /IM true /W 75 /H 66 /BPC 1 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endstream endobj 357 0 obj <> stream 0 0 0 -6 79 96 d1 79 0 0 102 0 -6 cm BI /IM true /W 79 /H 102 /BPC 1 /F /CCF /DP <> ID &¡®t ¯ž@€}‚ @FÁÚ°‰ðD«†A¾!7ðú ôß þú¿ý‡ÿÿøÿûÿÿÿÿÿÿÿÿÿþ¿ÿÂÿõÿÿ ×þÒá×ý¥Ú\0¼Apa.jð¼1àÂÿà È>ž?ÿÿÿÿÿÿÿÿÿÿÿÿ…ÿäÔ>  EI endstream endobj 358 0 obj <> stream 0 0 0 7 42 79 d1 42 0 0 72 0 7 cm BI /IM true /W 42 /H 72 /BPC 1 /F /CCF /DP <> ID & A@'ÁÁŠ zƒæ“|'ÂMé0ÿÔ7Kï® ÿÿ ?ÿ’°ß4Mü?þ·ývÿûAoïXl%¶— vKb ,8b?¯ü>aá‡¾ÃØ|=¼>ûïûP EI endstream endobj 359 0 obj <> stream 0 0 0 2 106 80 d1 106 0 0 78 0 2 cm BI /IM true /W 106 /H 78 /BPC 1 /F /CCF /DP <> ID & Ò`5ŒMoþÿ¬>×ÿõ¾×ÿôÂÿþO”/üŸÿú ·áëÿõµ‡×‡÷¯ÿÖÕ¿ÿ§¯_¿÷é°—ÿá~þ×éá?ÿ¯ß0ÒÿÿÖÖOAÿÿÿ[[é÷þ¿ÿJ·ï_ÿ…µ¾½ýáþ±·ÿ zðÿÓ_þ½úýSÿëß…O „ôA„$!0 ƒÿÿð EI endstream endobj 360 0 obj <> stream 0 0 0 0 77 80 d1 77 0 0 80 0 0 cm BI /IM true /W 77 /H 80 /BPC 1 /F /CCF /DP <> ID & \ ²Áþ20ˆa®áÁ†AIôƒzøWéø^·ï¯×ðÿÒÿûÿÃÿ÷á|ƒHÿøÿ÷ûÿðÿï÷ ¥}øý¿ï¿‡ï}øoÛ þ÷·ð×oÃÛ]†¼5Á†ýìO‡§Á ðÅûÕ.òD` EI endstream endobj 361 0 obj <> stream 0 0 0 -5 25 104 d1 25 0 0 109 0 -5 cm BI /IM true /W 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/F /CCF /DP <> ID &¨à ¿ïäÕ{ÿÃÿÿïü™Èc÷ÿAñ80Ñè?µ¾`è>Óøwá?ïû»ýß~~ÿ‡ÿòÖïÿwÿû¿ýßÿîÿ÷ÿ»÷ÿ‡ÿÿîÿÿþá…°†  EI endstream endobj 407 0 obj <> stream 0 0 0 -5 42 105 d1 42 0 0 110 0 -5 cm BI /IM true /W 42 /H 110 /BPC 1 /F /CCF /DP <> ID &¹€Õ§ßþü?ýûÿÛÿÃ÷ÿ·ÿ¿ÿ~ÿöÿðýÿ{þü?ý¿ýûÿÃ÷ÿ·ÿ¿ÿoÿ~ÿðýÿ{þü?ý¿ýûÿÃû÷ÿ¿ÿoÿ~ÿðß÷ö  EI endstream endobj 408 0 obj <> stream 0 0 0 30 59 79 d1 59 0 0 49 0 30 cm BI /IM true /W 59 /H 49 /BPC 1 /F /CCF /DP <> ID &¡ÊSáÂ<0x,= >ƒ„Vô›ûûŒ0÷ý¾ÿÛûû“_øøûûýûýýû .Û­µívClJ¾ 8ƒ  EI endstream endobj 409 0 obj <> stream 0 0 0 -5 17 105 d1 17 0 0 110 0 -5 cm BI /IM true /W 17 /H 110 /BPC 1 /F /CCF /DP <> ID ù5ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü EI endstream endobj 410 0 obj <> stream 0 0 0 -19 28 34 d1 28 0 0 53 0 -19 cm BI /IM true /W 28 /H 53 /BPC 1 /F /CCF /DP <> ID &«ü„¦Âÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ™Øù5!{÷ÃÀ@ EI endstream endobj 411 0 obj <> stream 0 0 0 -26 32 55 d1 32 0 0 81 0 -26 cm BI /IM true /W 32 /H 81 /BPC 1 /F /CCF /DP <> ID &¨Øeÿïï÷þÿ~ïýþýÿ¿ßûßø¿xï÷þÿ~ÿáþßáÿ¿ßûßûýø{ÿ‡þÿ~ÿïÞà EI 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/CCF /DP <> ID & A@/ðAáÁ§ ÿ” ô>·ÓxIøOÓz‡ÿ¨uûëÿÿ_ÿÿ_ì ¿ÿ´»^ x`—×_ÂæšÜ0¼0^AÇïñýûß¿ £0à ÿ¶Ÿ~ß°~ï† %±þÖø5† \ÿ EI endstream endobj 425 0 obj <> stream 0 0 0 64 19 93 d1 19 0 0 29 0 64 cm BI /IM true /W 19 /H 29 /BPC 1 /F /CCF /DP <> ID &¹ÀôCáÝɯÁ¼>ßþ~òAÖ°¿áõ‡ÚØP EI endstream endobj 426 0 obj <> stream 0 0 0 31 49 57 d1 49 0 0 26 0 31 cm BI /IM true /W 49 /H 26 /BPC 1 /F /CCF /DP <> ID &°Zq~Muÿÿœ/“]~Ô@ EI endstream endobj 427 0 obj <> stream 0 0 0 8 47 79 d1 47 0 0 71 0 8 cm BI /IM true /W 47 /H 71 /BPC 1 /F /CCF /DP <> ID &¡œœø ðéá=?Ñ@Þ‚ ?ôƒxI¾ÿ¤ßÿé7ÿAÿÖý?ÿëÂÿÿÿÿÿÿÿÿì4¿ÿþ×x_ÿØ]ëµÿzíxk†ÒÛ °Â b¿k ma‚Á…ð EI endstream endobj 428 0 obj <> stream 0 0 0 8 37 77 d1 37 0 0 69 0 8 cm BI /IM true /W 37 /H 69 /BPC 1 /F /CCF /DP <> ID &ºþÿÖù Oÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ<ý÷ñÿï‘uýðøß÷ýÿkà EI endstream endobj 429 0 obj <> stream 0 0 0 1 29 87 d1 29 0 0 86 0 1 cm BI /IM true /W 29 /H 86 /BPC 1 /F /CCF /DP <> ID &¹Àn.¹5ß{xaï†öÿooÃöÿ~÷þÿïðÿü;ÿÿÿÿÿõø_ø_ëõþµþµý,/ ôµÒ׬% °ºQUP\šê  EI endstream endobj 430 0 obj <> stream 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&¡°-?ÿý¬ƒGÿÿÿÿÿþpX¿\Ð6Mpýþýþÿöýþü?ÃÿÛ÷û÷ûû÷ðÿßïïßßïßïáøý÷ü5ð EI endstream endobj 436 0 obj <> stream 0 0 0 8 47 79 d1 47 0 0 71 0 8 cm BI /IM true /W 47 /H 71 /BPC 1 /F /CCF /DP <> ID & Á ø" < ôŸázAt ›Òaôýéõ¿á?ÿÿÿ zíÛ.×zØ0–`—Ö,[X ðƒÐx#¨Ãàˆ:8 úMë}>aÿÓÿÿÿÿÚí„ü0» /ö ‰¦¯ÚØXa`ÂÁÁ EI endstream endobj 437 0 obj <> stream 0 0 0 10 49 77 d1 49 0 0 67 0 10 cm BI /IM true /W 49 /H 67 /BPC 1 /F /CCF /DP <> ID &¹ÿÉ®²/ÿÿÿÿÿÿÿÿù w§ÿÿÿÇÿÿÿσÿÿÿý¯ÿÿÿ!—ÞŸÿÿÿÿüáCòk¨€ EI endstream endobj 438 0 obj <> stream 0 0 0 8 51 77 d1 51 0 0 69 0 8 cm BI /IM true /W 51 /H 69 /BPC 1 /F /CCF /DP <> ID &¨Œ ¿‹Oòj“_äs!²ÿÚÿ‡¯ÿÿ†»×ÿþÂþfÿ°»×ÿÿÚÿ½ÿý®¿ÿý놿ÿÿûÂíÿýëÿÿÿýŠÿÿÃ_µð EI endstream endobj 439 0 obj <> stream 0 0 0 10 49 77 d1 49 0 0 67 0 10 cm BI /IM true /W 49 /H 67 /BPC 1 /F /CCF /DP <> ID &ºìy5þD™óÿÿÿÿÿÿû^?ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÂwɯê  EI endstream endobj 440 0 obj <> stream 0 0 0 8 45 79 d1 45 0 0 71 0 8 cm BI /IM true /W 45 /H 71 /BPC 1 /F /CCF /DP <> ID &¹°ä€ÇôÈÄxAñ|>ÿ”â>ÿ‚ ôý}øÓÿÿÿì%…ø]t°½d kÐ3!… °K… Z"Šáࢎ™“ƒV´¿H†VxO¯ÿÿõÿþ×µívÁÙ ^ÿ}öT\/ _À@ EI endstream endobj 441 0 obj <> stream 0 0 0 10 50 77 d1 50 0 0 67 0 10 cm BI /IM true /W 50 /H 67 /BPC 1 /F /CCF /DP <> ID &º‹òkä_!—?ÿÿÿÿÿÿ†¼ÿÿÈ?zÿÿüÿÿÿÏ‚ÿÿkÿÿÿ!•ÞŸÿÿÿÿÿþpXÿä×P EI endstream endobj 442 0 obj <> stream 114 0 0 0 0 0 d1 endstream endobj 444 0 obj <> stream 0 0 0 0 49 67 d1 49 0 0 67 0 0 cm BI /IM true /W 49 /H 67 /BPC 1 /F /CCF /DP <> ID &¢8 ÓAÿÿÿÿÿÿÿÿ÷ÿÌ[úq|šë!&BÏþïýßÿÿÿÿÿÿþïýÃÿœ/“]mdÈyÿÿÿðïwÿÿÿÿÿÿÿý„Ô@ EI endstream endobj 448 0 obj <> stream 0 0 0 0 49 67 d1 49 0 0 67 0 0 cm BI /IM true /W 49 /H 67 /BPC 1 /F /CCF /DP <> ID &¨Š;ƒ¾MSÈæ@‚>ü?†‡ÿßïýÿÿÿý}ëÿ„¯×„¸¯ë‚ápûáó0ÃðoÃûýÿíÿáÿÿÿü/ÿ^¿ø_Kü¢¸®B“TFÁp EI endstream endobj 449 0 obj <> stream 117 0 0 0 0 0 d1 endstream endobj 450 0 obj <> stream 0 0 0 10 51 77 d1 51 0 0 67 0 10 cm BI /IM true /W 51 /H 67 /BPC 1 /F /CCF /DP <> ID & …X%?ÿý¬ƒyÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿåvC„ÿÿÿÿÿÄ&©@@ EI endstream endobj 451 0 obj <> stream 0 0 0 0 53 69 d1 53 0 0 69 0 0 cm BI /IM true /W 53 /H 69 /BPC 1 /F /CCF /DP <> ID & Üœ6|x@ôðžŸè oAúA¾Ÿü ›éÿÖÿÿ§ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ(A "-?ÿ&©5 EI endstream endobj 452 0 obj <> stream 0 0 0 0 39 67 d1 39 0 0 67 0 0 cm BI /IM true /W 39 /H 67 /BPC 1 /F /CCF /DP <> ID &©Eù5K ¼ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿò…¢üš¥ EI endstream endobj 453 0 obj <> stream 0 0 0 2 78 77 d1 78 0 0 75 0 2 cm BI /IM true /W 78 /H 75 /BPC 1 /F /CCF /DP <> ID "¡µù5 C‹]ÿáûýÿûýÿûýÈh¿ÿáÿßþü=ÿëÚÿ±ÿÊÞáü?¿·¿ïÿ —ðï÷¿ÿðûÿßÿþÿ÷ý>»^@Æ’%ÿø€ EI endstream endobj 455 0 obj <> stream 0 0 0 2 90 77 d1 90 0 0 75 0 2 cm BI /IM true /W 90 /H 75 /BPC 1 /F /CCF /DP <> ID )9ÿ&ªùd3*ÈT°ÓàÚ_ì5Û…¿‡öÒïÛ]‡[ýøl%ß¶»·öü6‚ÿmvëaý¥ß†=­÷ëõáè>ôùB~þÞaýêßAýý½ßÃÕ¾ƒûzMü<+}7áú½`û¼ Â5”†q D˜oÿÃÿ EI endstream endobj 456 0 obj <> stream 0 0 0 -11 13 1 d1 13 0 0 12 0 -11 cm BI /IM true /W 13 /H 12 /BPC 1 /F /CCF /DP <> ID &¬ã‚ñ&©~ @ EI endstream endobj 457 0 obj <> stream 0 0 0 58 69 96 d1 69 0 0 38 0 58 cm BI /IM true /W 69 /H 38 /BPC 1 /F /CCF /DP <> ID &¦l4çÇÿM0Ÿ9ÿÿPowïýû½ÿ»ÿÝØo»ÿ¿ýîöþíÿ¿ü8wÿîøwþÿ¾MVïá·ÿÏ/‡œß{˜:õ†°ÂìPc>ƒ …†`‚€€ EI endstream endobj 458 0 obj <> stream 0 0 0 0 76 78 d1 76 0 0 78 0 0 cm BI /IM true /W 76 /H 78 /BPC 1 /F /CCF /DP <> ID & ¹@2ÈqŸ„œ@õƒÐD1M ß@øA7Ó[öã ÓíïÃï|?{ßßï÷òj¯ÿÿîýÿýáÃÿß÷þïÿßû¿ýÿîÿðÿ‡þïßÿÞ?ýßÿîþŸwØ# å \Ÿýÿ EI endstream endobj 459 0 obj <> stream 0 0 0 -5 39 105 d1 39 0 0 110 0 -5 cm BI /IM true /W 39 /H 110 /BPC 1 /F /CCF /DP <> ID & ÓÊ4(AaÂõ¥×è/ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü/õ®½a.°@²ƒ“TGL<7ß{ïÞÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ‡¿í÷Þ<>xd0ßÈ4Ì@ EI endstream endobj 460 0 obj <> stream 0 0 0 -5 39 105 d1 39 0 0 110 0 -5 cm BI /IM true /W 39 /H 110 /BPC 1 /F /CCF /DP <> ID 3’e&¥Ã}½ÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿðÿ~ûßaC[ð—4-®½u¯ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ×ék®.Q ™ƒH€ EI endstream endobj 461 0 obj <> stream 0 0 0 0 67 75 d1 67 0 0 75 0 0 cm BI /IM true /W 67 /H 75 /BPC 1 /F /CCF /DP <> ID +Aù5 H ïr 4ì?ƒýáß·÷Ü7ïÿwï¿ïðûý÷¿ü?û¿ÚñïÿßïÿßïÿßïÿÃýÿûÃÿ÷ûÿ÷ûÿ÷ÃÈ+ÿÿ EI endstream endobj 462 0 obj <> stream 0 0 0 -27 25 48 d1 25 0 0 75 0 -27 cm BI /IM true /W 25 /H 75 /BPC 1 /F /CCF /DP <> ID òj)AZÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ×ð¹êÿü‰Lÿÿü‰˜ úÿû_ƒð EI endstream endobj 463 0 obj <> stream 0 0 0 0 56 48 d1 56 0 0 48 0 0 cm BI /IM true /W 56 /H 48 /BPC 1 /F /CCF /DP <> ID $ßþME( å´×ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿßÿýÿ÷úèÑ_øPÜ(Øa. pÅ`Ôš‰ œ EI endstream endobj 464 0 obj <> stream 0 0 0 0 65 75 d1 65 0 0 75 0 0 cm BI /IM true /W 65 /H 75 /BPC 1 /F /CCF /DP <> ID #`×ù5ê °Áÿÿÿÿÿÿÿÿÿÿÿÿÿò, ~“†€È#ÃßÃ~ßßÿ~ÿü?ÿÿÿÿÿø_ú_ÿé}xKÐ^2kP ¤4` EI endstream endobj 465 0 obj <> stream 0 0 0 27 38 75 d1 38 0 0 48 0 27 cm BI /IM true /W 38 /H 48 /BPC 1 /F /CCF /DP <> ID &~MFH ð×ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÙ ú„ôýð¿ß­ãaø|1XjMD†  EI endstream endobj 466 0 obj <> stream 0 0 0 29 55 79 d1 55 0 0 50 0 29 cm BI /IM true /W 55 /H 50 /BPC 1 /F /CCF /DP <> ID &¤h3?„ z80Â>`ð¶·ÚA»z ï¼8…MUÛÿü7†aû[ãÞïÿýþ;ÿ÷ûÿ÷ûþ^_ÿî¸d-ïÿ„a÷»ÿÃwß{s›í…l> aö)ŠÁ‚A‚€€ EI endstream endobj 467 0 obj <> stream 0 0 0 29 90 79 d1 90 0 0 50 0 29 cm BI /IM true /W 90 /H 50 /BPC 1 /F /CCF /DP <> ID &¢ £a˜fþ ‚aÝ„~¬=ú 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/BPC 1 /F /CCF /DP <> ID &¬ã‚ÿäÕ/ü(€ EI endstream endobj 475 0 obj <> stream 0 0 0 2 44 98 d1 44 0 0 96 0 2 cm BI /IM true /W 44 /H 96 /BPC 1 /F /CCF /DP <> ID &¦h è= ~ü}·÷íøaÿ»á‚¿ñ~ÿïÿïÿü?ýûÿßûÿßûÿßûÿÃþA2ÿí¿ûýÞïÃû÷ðõ·íÂà×cÁ…†?ÿÿÿÕ¼ßþþ (€ EI endstream endobj 476 0 obj <> stream 0 0 0 46 35 117 d1 35 0 0 71 0 46 cm BI /IM true /W 35 /H 71 /BPC 1 /F /CCF /DP <> ID &¬ÌB:z|ôƒöß°þÿÃMãïýÿáÿ¿ýÿ‡ÿ¿ûÿûÿyÿ‡ïßáßømÿÛ­¿ƒ]ad6~?ÿþC2ÿ@ÿ÷µ EI endstream endobj 477 0 obj <> stream 0 0 0 0 145 153 d1 145 0 0 153 0 0 cm BI /IM true /W 145 /H 153 /BPC 1 /F /CCF /DP <> ID &¨Ê7ûïÃßðû{4Ãïlà ka<2ÕùÁ¤¾ᦟl?~Øolpøa°ü?~Û{o»í·†~ÿooÝöý¶ðølàÚkÛÛïaá½ûß½½‡‡ÛÛþö÷Ão½¿ßo >öö÷þôŸ§…ýtº×­p‚Òþ´µÒþÒýtº×……ÒÒþ´µÂ úÒÒýt¸X^µÒÒ xjßô–‚ð«¥ú]zA-%õÕ, —¯P‚è.¡t‚ZA~¸D јHR‚‚zD¼ã+ê´>ÿõ EI endstream endobj 478 0 obj <> stream 0 0 0 0 38 57 d1 38 0 0 57 0 0 cm BI /IM true /W 38 /H 57 /BPC 1 /F /CCF /DP <> ID &¹° øÿ´rzÃ÷“_†þ¾íÿ¿ý÷áÿÿï¾>µðKëb ’gï™§»û}¶BßÃ>ßûûoíƒø>ìx`׆¼wþÿ÷þÿ÷þÿ"‘ÿ~€€ EI endstream endobj 479 0 obj <> stream 0 0 0 24 56 47 d1 56 0 0 23 0 24 cm BI /IM true /W 56 /H 23 /BPC 1 /F /CCF /DP <> ID &«ÿ(5ÿÿÿùKð EI endstream endobj 480 0 obj <> stream 0 0 0 2 35 57 d1 35 0 0 55 0 2 cm BI /IM true /W 35 /H 55 /BPC 1 /F /CCF /DP <> ID &¡Êü ð˜xXz7ÓôúÞ‚ ÿþ·ÿOÿñÃÿÿÿÿÿÿÿ“_ ÿÿÿµÞ¿Þ¿á…ÛKý´°Ú ,0  EI endstream endobj 481 0 obj <> stream 0 0 0 150 50 228 d1 50 0 0 78 0 150 cm BI /IM true /W 50 /H 78 /BPC 1 /F /CCF /DP <> ID &±°0x?ðšÃ~ÿÒo}‡ÿÚ¿ÿ÷¾»ÿýï}ÿûÿÿÇuúí|/„¼Äq3Á¢ÃækÙïÛûݾØd1~ÿ†Á»wïÛaýÞá¿°Á÷ðÇÁ…‡ xðÿ÷þÿ÷ïÿïÿïò¯þä3Ò  EI endstream endobj 482 0 obj <> stream 0 0 0 15 54 80 d1 54 0 0 65 0 15 cm BI /IM true /W 54 /H 65 /BPC 1 /F /CCF /DP <> ID &¡”² Æä®°‰—@°]†•.‚Â_¥¥ýh/ׯëãúä‘=O&þM÷ýïû¿ÛÛü7°ø{{!ƒD`ðȪÝäÈe$€€ EI endstream endobj 487 0 obj <> stream 0 0 0 0 67 96 d1 67 0 0 96 0 0 cm BI /IM true /W 67 /H 96 /BPC 1 /F /CCF /DP <> ID & \Ö_!œÆCx"Tƒæ N‰±èaáoIôß»íôÿ÷ÿ‡ÿïÿ°¿Úÿ ÿ¬/õúõ¥ÂëK\%‚ÂÈlä˜hýpò 5àÁàßoaïÞþò oþD"ôßOøzÿÿÿÿÿ×¼/ú]‚®¶–Ýr/Ã:„XbÂÁ…ƒ$ÃGÈeU EI endstream endobj 488 0 obj <> stream 0 0 0 -5 97 94 d1 97 0 0 99 0 -5 cm BI /IM true /W 97 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>!7áøL>¿¿¿ýþÿÿÿÿÿÿÿúõÿÿ¯K¿†½„is5¯a ¼1ÿƒ Çÿÿÿÿÿÿÿõù5à EI endstream endobj 599 0 obj <> stream 0 0 0 27 45 86 d1 45 0 0 59 0 27 cm BI /IM true /W 45 /H 59 /BPC 1 /F /CCF /DP <> ID &¦f _Aàû‹aï‡ÛþäÕ'Æ÷þÿ~ðŸéÿþŸéÿ3]aðƒÿ…¾Ÿýo§ÖÿéÿÒoýaÿÂþoïTõ åêÿÿø€ EI endstream endobj 600 0 obj <> stream 0 0 0 26 52 68 d1 52 0 0 42 0 26 cm BI /IM true /W 52 /H 42 /BPC 1 /F /CCF /DP <> ID &¡IÃ9âþ¬(<ñCGç¬zDjúß õûá ÿûÿü}ÝɪÿýüÍßþÃÿïÃï}û_~û…ï¾ÿ†ÂW¿°Õì:ï@àÃK A~Ce  EI endstream endobj 601 0 obj <> stream 0 0 0 -1 6 89 d1 6 0 0 90 0 -1 cm BI /IM true /W 6 /H 90 /BPC 1 /F /CCF /DP <> ID &º‹ÿÿÿÿÿÿÿÿÿÿþMu EI endstream endobj 602 0 obj <> stream 0 0 0 3 36 68 d1 36 0 0 65 0 3 cm BI /IM true /W 36 /H 65 /BPC 1 /F /CCF /DP <> ID &¡ À’ÓG=`ôƒ~ßAð·ì?ão½ÿoïîMWðÿ÷ü?ÿðÿþÿ…ßßß¶—úæˆ0–áŠÿ°Ï„ãûÿøáÿïýÿåëßþCÏà EI endstream endobj 603 0 obj <> stream 0 0 0 6 36 66 d1 36 0 0 60 0 6 cm BI /IM true /W 36 /H 60 /BPC 1 /F /CCF /DP <> ID &¨/þ÷ß~lx?mðß¿¶Ý‡ï·Ã`þÞÃïooÞÃïooöÿ‡†ÿîlè>ŸßÿëýxU°_kÿa-µÞ -†— ,0X0  EI endstream endobj 604 0 obj <> stream 0 0 0 2 38 68 d1 38 0 0 66 0 2 cm BI /IM true /W 38 /H 66 /BPC 1 /F /CCF /DP <> ID &¢ŸÃïVMûé7á¿ã½‡¿í÷ýïö¿áï÷÷&¿ãþ÷ÿæ€Çýï¿÷ßýý‡þø÷ì?ûí÷¯ðߨ]‡[k½mø1øj  EI endstream endobj 605 0 obj <> stream 0 0 0 65 9 84 d1 9 0 0 19 0 65 cm BI /IM true /W 9 /H 19 /BPC 1 /F /CCF /DP <> ID &¨¹o½‡†ÿîÿʘõɪP EI endstream endobj 606 0 obj <> stream 0 0 0 6 38 69 d1 38 0 0 63 0 6 cm BI /IM true /W 38 /H 63 /BPC 1 /F /CCF /DP <> ID &¡ƒ@y Wèù‡„0ð·Ð}o§Òaÿ­ÿÿú ¿ÿÿÿã¿ÿÿÿÿÿÿÿä××ÿÿÿû /ÿÞ¿íw…½v¸l%Á…öC€€ EI endstream endobj 607 0 obj <> stream 0 0 0 6 25 68 d1 25 0 0 62 0 6 cm BI /IM true /W 25 /H 62 /BPC 1 /F /CCF /DP <> ID &¡ø z==oÛöýõ}¿áÿØ÷½µÿŽßáÿ‡û”ü[÷&«ðßü?í÷÷[~þÝlWàÁcÿÿù/@ÿ^ýõ†@ EI endstream endobj 608 0 obj <> stream 0 0 0 15 46 71 d1 46 0 0 56 0 15 cm BI /IM true /W 46 /H 56 /BPC 1 /F /CCF /DP <> ID & Ýd1 ÖŠD X]Ð]ipµÂ_Ö—ÿQü)OS€ÚɪÿÞßáöø}íï°ðÁðxdI[ƒÃÈ7 ð EI endstream endobj 609 0 obj <> stream 0 0 0 2 42 69 d1 42 0 0 67 0 2 cm BI /IM true /W 42 /H 67 /BPC 1 /F /CCF /DP <> ID &¨À! 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Zÿëÿrtì¨ÅìÐôúMôúßú ¿ÿÿÿÿÿzá¯ûi¾Öëƒh5±97ñÁš  Ô@ EI endstream endobj 621 0 obj <> stream 0 0 0 11 41 15 d1 41 0 0 4 0 11 cm BI /IM true /W 41 /H 4 /BPC 1 /F /CCF /DP <> ID ð EI endstream endobj 622 0 obj <> stream 0 0 0 6 31 66 d1 31 0 0 60 0 6 cm BI /IM true /W 31 /H 60 /BPC 1 /F /CCF /DP <> ID &«ÿ§ÚÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿŸ<¡cäÔ¹ OÃ€ EI endstream endobj 631 0 obj <> stream 0 0 0 0 67 96 d1 67 0 0 96 0 0 cm BI /IM true /W 67 /H 96 /BPC 1 /F /CCF /DP <> ID &¡–uB‰ XAá< |Ô'AƺÞô›ö^}oïKÿß ÿ­ÿ ÿëÿÿÿÿ_ÿþÿÿ÷_ú÷_ýv\§Oý~^ÃKÞ¼0Áö+ýØ^/ øÿì?ýü†lý?ÿÝÿ~ÿ~þþ~·íƒ wÛ aµà× Kb°k ä5€€ EI endstream endobj 632 0 obj <> stream 0 0 0 -6 70 96 d1 70 0 0 102 0 -6 cm BI /IM true /W 70 /H 102 /BPC 1 /F /CCF /DP <> ID 6F /á!%‚`ƒ°ƒÁå8>Â:ƒw€Í}Á=0àž›¯º þŸþ¯þºÿáI¨ëZë\ m1Ôüƒ@¶@Á^C`. aø%„µÒÂ\–‚ýPÑD° ä 32$PˆkþU‚¦Kƒ&Umÿ…×ô¿ô¿òj¿…ÿµõÃÿí¯¯`» wö\6 ÍAû©ö¼0Ÿ ða?ùú<@ EI endstream endobj 633 0 obj <> stream 0 0 0 -6 80 94 d1 80 0 0 100 0 -6 cm BI /IM true /W 80 /H 100 /BPC 1 /F /CCF /DP <> ID &¨Š 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0 0 58 43 110 d1 43 0 0 52 0 58 cm BI /IM true /W 43 /H 52 /BPC 1 /F /CCF /DP <> ID )ÀÇ“S3¿÷ÿ‡þÿÉþÿ„‰ƒï¦i¾ŸßÃ{ý°ÿÿvÿïûü;ÿ|›þÿ»ÿßø}ïÃmXyÄþö뵆»¶kh0  EI endstream endobj 639 0 obj <> stream 0 0 0 -46 37 3 d1 37 0 0 49 0 -46 cm BI /IM true /W 37 /H 49 /BPC 1 /F /CCF /DP <> ID &¡‚pÿ!SázÁèž~õ/×ú×ÿÿÿûÿþ@‡‡ûòp1öÿ½ø{áöûÛØ<`ðÈ…¸<‚€à EI endstream endobj 640 0 obj <> stream 0 0 0 58 35 110 d1 35 0 0 52 0 58 cm BI /IM true /W 35 /H 52 /BPC 1 /F /CCF /DP <> ID &¡Âÿ^¿÷ûÿ÷ùÃÿ‚èûw¬z úáþSý¿¿ýßɯÿáßû‡ÿoûïßÛðþØAØs µìožAÀ@ EI endstream endobj 645 0 obj <> stream 0 0 0 1 50 79 d1 50 0 0 78 0 1 cm BI /IM true /W 50 /H 78 /BPC 1 /F /CCF /DP <> ID &¡ ÁÏþ`ƒÁa£ÓÖÞ7·Óîú¿Â{ÿÔ;ýÃ÷ÿÿ‡ÿãïÿwþÿ¿ýßûþý¸ÿ¶ÿxaÿ÷í§·ïá°»í<17ðkà ñûÿÃÿÿ~ÿ÷þÿ÷þÿ ×Wÿ~A¶€ EI endstream endobj 646 0 obj <> stream 0 0 0 -2 56 75 d1 56 0 0 77 0 -2 cm BI /IM true /W 56 /H 77 /BPC 1 /F /CCF /DP <> ID &¨«?ÿù3£ Âÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþ#þMBÿÿÿÿÿÿÿÿÿÿÿ‡ø]þ׿oÛðþ7ØABŒ  EI endstream endobj 647 0 obj <> stream 0 0 0 58 45 96 d1 45 0 0 38 0 58 cm BI /IM true 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/F /CCF /DP <> ID &¨ˆŸäÕ&¿ÈæCdÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþ?ÿÿÿæa³ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ” ˆ#ø´ÿ&©5ü@ EI endstream endobj 825 0 obj <> stream 0 0 0 10 51 77 d1 51 0 0 67 0 10 cm BI /IM true /W 51 /H 67 /BPC 1 /F /CCF /DP <> ID & ¹úû_o?ÿÿÿÿÿÿÿÿÿÓÿü'úÏýoýaôúßü'Âßý>·ÿO­ÿ¤ôúß ð·þ·Óå'kC¶ôš­@@ EI endstream endobj 826 0 obj <> stream 0 0 0 27 36 66 d1 36 0 0 39 0 27 cm BI /IM true /W 36 /H 39 /BPC 1 /F /CCF /DP <> ID &¡‚@@ð@ôôÿEð‚ ×Âz^ƒ]ëýÿÿ¿÷ÙÛý†žûí`Ö À@ EI endstream endobj 827 0 obj <> stream 0 0 0 11 37 66 d1 37 0 0 55 0 11 cm BI /IM true /W 37 /H 55 /BPC 1 /F /CCF /DP <> ID &ºÓÿÚÈbÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùËëÿûP EI endstream endobj 828 0 obj <> stream 0 0 0 5 39 69 d1 39 0 0 64 0 5 cm BI /IM true /W 39 /H 64 /BPC 1 /F /CCF /DP <> ID &¡ Ã< ‚> stream 0 0 0 5 64 66 d1 64 0 0 61 0 5 cm BI /IM true /W 64 /H 61 /BPC 1 /F /CCF /DP <> ID &ÿɨ…` [Mÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿñÿÿ4 ¿ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿúiÄŒ˜ ÿà EI endstream endobj 830 0 obj <> stream 0 0 0 2 57 66 d1 57 0 0 64 0 2 cm BI /IM true /W 57 /H 64 /BPC 1 /F /CCF /DP <> ID %Ì„ÿÉ©† 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stream 0 0 0 0 69 78 d1 69 0 0 78 0 0 cm BI /IM true /W 69 /H 78 /BPC 1 /F /CCF /DP <> ID &¨— ÿüHÃMnøý¿ýý¿þï÷X¿¿¿ýÿÎÎþßÿk¿þîÿðßþþßÿ~ÿßïÃýûý®ÿûuÿÛÿÃûü{þÿûáÿý…ûÀ@ EI endstream endobj 880 0 obj <> stream 0 0 0 30 54 100 d1 54 0 0 70 0 30 cm BI /IM true /W 54 /H 70 /BPC 1 /F /CCF /DP <> ID !§Éªätÿß¿ýÿ¿ü?ðÿÉ~ÿ„´lÁñßL=„~ý7íÿ½ûø}áÿÛÿ‡oþ÷þïÿßîÿ‘zÿÿîÿ°ÿ÷Ö¿á…îÿëyõûØio·ö°×Šb¶‚Á‚ (€ EI endstream endobj 881 0 obj <> stream 0 0 0 -1 74 80 d1 74 0 0 81 0 -1 cm BI /IM true /W 74 /H 81 /BPC 1 /F /CCF /DP <> ID &¡‚pË'!S‚þ0Aáˆ'Âà H Þ‚ ú ?M^¡§þô‚_ Ÿÿ¤è1þý&ü$ÿþ‚ozaúþ¡½º÷§õ½é¾Âøt{½>ÚPöí|7Mì~Óï{ ¿‡÷ÒêÕ쀘}?4_NAT×ß·÷«ôÃÿÃ~ÿûûÿÿ áÿïÿëß½¶¸zá®Ç†°`  EI endstream endobj 882 0 obj <> stream 0 0 0 7 49 79 d1 49 0 0 72 0 7 cm BI /IM true /W 49 /H 72 /BPC 1 /F /CCF /DP <> ID & „àç‹ù°àŠzÚz Ìî“i7§„ýþ”þÿþÿÿÞûû~ÃðÃðþ@œ üx>AF¹ŸÿüŸzú„¿ÿתÚK‚Kƒ]ƒ Bà èÇò¯ø]p\%¥Ö,%®—]pºý¨€ EI endstream endobj 883 0 obj <> stream 0 0 0 3 72 78 d1 72 0 0 75 0 3 cm BI /IM true /W 72 /H 75 /BPC 1 /F /CCF /DP <> ID &¨–†^BÁFaä"l°Áø7÷»}°ý÷ü7÷Ûÿ»ýÿßíïÿÿÿ_ÿÚ[þ¾—a/…º\Bÿäà¸{Ð ø`û¿ îßþÃá‡û÷ÿý¾÷ÿïï¯÷úßõúí|Èg ë…ÉP EI endstream endobj 884 0 obj <> stream 0 0 0 1 46 79 d1 46 0 0 78 0 1 cm BI /IM true /W 46 /H 78 /BPC 1 /F /CCF /DP <> ID &¹ÀØx?âÁ´rz߇“UÛÿí_ûÿß~áÿþððûÿðÿÿñßôa}}-ëˆ\ð[|Í{*ý¿½Ø}·áý²0îÿaßw¸oíƒ]ÃýÃ[ƒãßþÿßþýÿáÿ‡ÿ¿÷ù Wÿ" EI endstream endobj 885 0 obj <> stream 0 0 0 5 53 79 d1 53 0 0 74 0 5 cm BI /IM true /W 53 /H 74 /BPC 1 /F /CCF /DP <> ID &¡œêþAÔ`á„Öò@léô ¿õ õ§×ëûÿ_ÿ…ëµáoð]®ÂÜÛ ¼,0×lá…Ø¬> pX%®‚Òáh*Ðx[è0øA¾ ÐMôÿðWÂ~¾¾ÿû_þ¸zí}vKzØh-”\x0°Ád4i EI endstream endobj 886 0 obj <> stream 0 0 0 5 55 77 d1 55 0 0 72 0 5 cm BI /IM true /W 55 /H 72 /BPC 1 /F /CCF /DP <> ID & Gÿä4ÿÿÿÿÿÿ‰Vùð0H M·÷ì?¿ ýûûûÛûûðþþß°þÿöü?·÷ÿ°ý¿¿ ¾ûþÿ‡ß÷ß÷Ãü@ EI endstream endobj 887 0 obj <> stream 0 0 0 2 48 77 d1 48 0 0 75 0 2 cm BI /IM true /W 48 /H 75 /BPC 1 /F /CCF /DP <> ID &¨ÿ!Ã]ÿïßïÿßïÿßïÿÃü?ÿ{ÿ÷ûÿ÷ûÿ÷ûÿðÿÿÞÿýþÿýþÿýðòÌ€€ EI endstream endobj 888 0 obj <> stream 0 0 0 29 50 79 d1 50 0 0 50 0 29 cm BI /IM true /W 50 /H 50 /BPC 1 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HøvýŒÿÑ´px§V–çÁô ¨aƒªaqËÌI…Ù’Yç0¸¸A™ŒAènÀ0ÅýœEùåz0?ð‚¢ªHüt&ç$£{”+%±$Q/­¨än%9­ìevd/man/0000755000175100001440000000000014260535126011625 5ustar hornikusersevd/man/evind.test.Rd0000644000175100001440000000254212637167310014204 0ustar hornikusers\name{evind.test} \alias{evind.test} \title{Perform Hypothesis Test Of Independence} \description{ Perform score and likelihood ratio tests of independence for bivariate data, assuming a logistic dependence model as the alternative. } \usage{ evind.test(x, method = c("ratio", "score"), verbose = FALSE) } \arguments{ \item{x}{A matrix or data frame, ordinarily with two columns, which may contain missing values.} \item{method}{The test methodology; either \code{"ratio"} for the likelihood ratio test or \code{"score"} for the score test.} \item{verbose}{If \code{TRUE}, shows estimates of the marginal parameters in addition to the dependence parameter.} } \details{ This simple function fits a stationary bivariate logistic model to the data and performs a hypothesis test of \eqn{\code{dep} = 1} versus \eqn{\code{dep} < 1} using the methodology in Tawn (1988). The null distributions for the printed test statistics are chi-squared on one df for the likelihood ratio test, and standard normal for the score test. } \value{ An object of class \code{"htest"}. } \references{ Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. \emph{Biometrika}, \bold{75}, 397--415. } \seealso{\code{\link{fbvevd}}, \code{\link{t.test}}} \examples{ evind.test(sealevel) evind.test(sealevel, method = "score") } \keyword{htest} evd/man/sealevel2.Rd0000644000175100001440000000246612637167310014010 0ustar hornikusers\name{sealevel2} \alias{sealevel2} \title{Annual Sea Level Maxima at Dover and Harwich with Indicator} \usage{sealevel2} \description{ The \code{sealevel2} data frame has 81 rows and 3 columns. The first two columns contain annual sea level maxima from 1912 to 1992 at Dover and Harwich respectively, two sites on the coast of Britain. The third column is a logical vector denoting whether or not the maxima in a given year are assumed to have derived from the same storm event; this assumption is made if the times of obsevation of the maxima are at most 48 hours apart. The row names give the years of observation. There are 39 missing data values. There are only nine non-missing logical values. } \format{ This data frame contains the following columns: \describe{ \item{dover}{A numeric vector containing annual sea level maxima at Dover, including 9 missing values.} \item{harwich}{A numeric vector containing sea annual level maxima at Harwich, including 30 missing values.} \item{case}{A logical vector denoting whether or not the maxima are assumed to have derived from the same storm event.} } } \source{ Coles, S. G. and Tawn, J. A. (1990) Statistics of coastal flood prevention. \emph{Phil. Trans. R. Soc. Lond., A} \bold{332}, 457--476. } \keyword{datasets} evd/man/mvevd.Rd0000644000175100001440000002162312637167310013243 0ustar hornikusers\name{mvevd} \alias{pmvevd} \alias{rmvevd} \alias{dmvevd} \title{Parametric Multivariate Extreme Value Distributions} \description{ Density function, distribution function and random generation for the multivariate logistic and multivariate asymmetric logistic models. } \usage{ pmvevd(q, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0), lower.tail = TRUE) rmvevd(n, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0)) dmvevd(x, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0), log = FALSE) } \arguments{ \item{x, q}{A vector of length \code{d} or a matrix with \code{d} columns, in which case the density/distribution is evaluated across the rows.} \item{n}{Number of observations.} \item{dep}{The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length \eqn{2^d-d-1}, or a single value, in which case the value is used for each of the \eqn{2^d-d-1} parameters (see \bold{Details}).} \item{asy}{The asymmetry parameters for the asymmetric logistic model. Should be a list with \eqn{2^d-1} vector elements containing the asymmetry parameters for each separate component (see \bold{Details}).} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default) or \code{"alog"} (or any unique partial match), for the logistic and asymmetric logistic models respectively.} \item{d}{The dimension.} \item{mar}{A vector of length three containing marginal parameters for every univariate margin, or a matrix with three columns where each column represents a vector of values to be passed to the corresponding marginal parameter. It can also be a list with \code{d} elements, such that each element is either a vector of length three or a matrix with three columns, in which case the \eqn{i}th element represents the marginal parameters on the \eqn{i}th margin.} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), the distribution function is returned; the survivor function is returned otherwise.} } \details{ Define \deqn{y_i = y_i(z_i) = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}}{ yi = yi(zi) = {1+si(zi-ai)/bi}^(-1/si)} for \eqn{1+s_i(z_i-a_i)/b_i > 0}{1+si(zi-ai)/bi > 0} and \eqn{i = 1,\ldots,d}{i = 1,\ldots,d}, where the marginal parameters are given by \eqn{(a_i,b_i,s_i)}{(ai,bi,si)}, \eqn{b_i > 0}{bi > 0}. If \eqn{s_i = 0}{si = 0} then \eqn{y_i}{yi} is defined by continuity. Let \eqn{z = (z_1,z_2,\ldots,z_d)}{z = (z1,z2,\ldots,zd)}. In each of the multivariate distributions functions \eqn{G(z)} given below, the univariate margins are generalized extreme value, so that \eqn{G(z_i) = \exp(-y_i)}{G(zi) = \exp(-yi)} for \eqn{i = 1,\ldots,d}{i = 1,\ldots,d}. If \eqn{1+s_i(z_i-a_i)/b_i \leq 0}{1+si(zi-ai)/bi <= 0} for some \eqn{i = 1,\ldots,d}{i = 1,\ldots,d}, the value \eqn{z_i}{zi} is either greater than the upper end point (if \eqn{s_i < 0}{si < 0}), or less than the lower end point (if \eqn{s_i > 0}{si > 0}), of the \eqn{i}th univariate marginal distribution. \code{model = "log"} (Gumbel, 1960) The \code{d} dimensional multivariate logistic distribution function with parameter \eqn{\code{dep} = r} is \deqn{G(z) = \exp\left\{-\left(\sum\nolimits_{i = 1}^{d} y_i^{1/r}\right)^r\right\}}{ G(z) = exp{-[sum_{i=1}^d yi^(1/r)]^r}} where \eqn{0 < r \leq 1}{0 < r <= 1}. This is a special case of the multivariate asymmetric logistic model. \code{model = "alog"} (Tawn, 1990) Let \eqn{B} be the set of all non-empty subsets of \eqn{\{1,\ldots,d\}}{{1,\ldots,d}}, let \eqn{B_1=\{b \in B:|b|=1\}}{B1={b in B:|b|=1}}, where \eqn{|b|} denotes the number of elements in the set \eqn{b}, and let \eqn{B_{(i)}=\{b \in B:i \in b\}}{B(i)={b in B:i in b}}. The \code{d} dimensional multivariate asymmetric logistic distribution function is \deqn{G(z)=\exp\left\{-\sum\nolimits_{b \in B} \left[\sum\nolimits_ {i\in b}(t_{i,b}y_i)^{1/r_b}\right]^{r_b}\right\},}{G(z) = exp{-sum{b in B} [sum{i in b}(t{i,b}yi)^(1/r{b})]^r{b}},} where the dependence parameters \eqn{r_b\in(0,1]}{r{b} in (0,1]} for all \eqn{b\in B \setminus B_1}{b in B\B1}, and the asymmetry parameters \eqn{t_{i,b}\in[0,1]}{t{i,b} in [0,1]} for all \eqn{b\in B}{b in B} and \eqn{i\in b}{i in b}. The constraints \eqn{\sum_{b \in B_{(i)}}t_{i,b}=1}{sum{b in B(i)} t{i,b}=1} for \eqn{i = 1,\ldots,d} ensure that the marginal distributions are generalized extreme value. Further constraints arise from the possible redundancy of asymmetry parameters in the expansion of the distribution form. Let \eqn{b_{-i_0} = \{i \in b:i \neq i_0\}}{ b_{-i0} = {i in b:i is not i_0}}. If \eqn{r_b = 1}{r{b} = 1} for some \eqn{b\in B \setminus B_1}{b in B\B1} then \eqn{t_{i,b} = 0}{t{i,b} = 0} for all \eqn{i\in b}{i in b}. Furthermore, if for some \eqn{b\in B \setminus B_1}{b in B\B1}, \eqn{t_{i,b} = 0}{t{i,b} = 0} for all \eqn{i\in b_{-i_0}}{i in b_{-i0}}, then \eqn{t_{i_0,b} = 0}{t{i0,b} = 0}. \code{dep} should be a vector of length \eqn{2^d-d-1} which contains \eqn{\{r_b:b\in B \setminus B_1\}}{{r{b}:b in B\B1}}, with the order defined by the natural set ordering on the index. For example, for the trivariate model, \eqn{\code{dep} = (r_{12},r_{13},r_{23},r_{123})}{ \code{dep} = (r{12},r{13},r{23},r{123})}. \code{asy} should be a list with \eqn{2^d-1} elements. Each element is a vector which corresponds to a set \eqn{b\in B}{b in B}, containing \eqn{t_{i,b}}{t{i,b}} for every integer \eqn{i\in b}{i in b}. The elements should be given using the natural set ordering on the \eqn{b\in B}{b in B}, so that the first \eqn{d} elements are vectors of length one corresponding to the sets \eqn{\{1\},\ldots,\{d\}}{{1},\ldots,{d}}, and the last element is a a vector of length \eqn{d}, corresponding to the set \eqn{\{1,\ldots,d\}}{{1,\ldots,d}}. \code{asy} must be constructed to ensure that all constraints are satisfied or an error will occur. } \value{ \code{pmvevd} gives the distribution function, \code{dmvevd} gives the density function and \code{rmvevd} generates random deviates, for the multivariate logistic or multivariate asymmetric logistic model. } \note{ Multivariate extensions of other bivariate models are more complex. A multivariate extension of the Husler-Reiss model exists, involving a multidimensional integral and one parameter for each bivariate margin. Multivariate extensions for the negative logistic model can be derived but are considerably more complex and appear to be less flexible. The ``multivariate negative logistic model'' often presented in the literature (e.g. Kotz \emph{et al}, 2000) is not a valid distribution function and should not be used. The logistic and asymmetric logistic models respectively are simulated using Algorithms 2.1 and 2.2 in Stephenson(2003b). The density function of the logistic model is evaluated using the representation of Shi(1995). The density function of the asymmetric logistic model is evaluated using the representation given in Stephenson(2003a). } \references{ Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. \emph{Publ. Inst. Statist. Univ. Paris}, \bold{9}, 171--173. Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000) \emph{Continuous Multivariate Distributions}, vol. 1. New York: John Wiley & Sons, 2nd edn. Shi, D. (1995) Fisher information for a multivariate extreme value distribution. \emph{Biometrika}, \bold{82}(3), 644--649. Stephenson, A. G. (2003a) \emph{Extreme Value Distributions and their Application}. Ph.D. Thesis, Lancaster University, Lancaster, UK. Stephenson, A. G. (2003b) Simulating multivariate extreme value distributions of logistic type. \emph{Extremes}, \bold{6}(1), 49--60. Tawn, J. A. (1990) Modelling multivariate extreme value distributions. \emph{Biometrika}, \bold{77}, 245--253. } \seealso{\code{\link{rbvevd}}, \code{\link{rgev}}} \examples{ pmvevd(matrix(rep(0:4,5), ncol=5), dep = .7, model = "log", d = 5) pmvevd(rep(4,5), dep = .7, model = "log", d = 5) rmvevd(10, dep = .7, model = "log", d = 5) dmvevd(rep(-1,20), dep = .7, model = "log", d = 20, log = TRUE) asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2)) pmvevd(rep(2,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) asy <- list(.4, .0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2)) rmvevd(10, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) dmvevd(rep(0,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3) asy <- list(0, 0, 0, 0, c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(.2,.1,.2), c(.1,.1,.2), c(.3,.4,.1), c(.2,.2,.2), c(.4,.6,.2,.5)) rmvevd(10, dep = .7, asy = asy, model = "alog", d = 4) rmvevd(10, dep = c(rep(1,6), rep(.7,5)), asy = asy, model = "alog", d = 4) } \keyword{distribution} evd/man/order.Rd0000644000175100001440000000456112637167310013237 0ustar hornikusers\name{order} \alias{dorder} \alias{porder} \alias{rorder} \title{Distributions of Order Statistics} \description{ Density function, distribution function and random generation for a selected order statistic of a given number of independent variables from a specified distribution. } \usage{ dorder(x, densfun, distnfun, \dots, distn, mlen = 1, j = 1, largest = TRUE, log = FALSE) porder(q, distnfun, \dots, distn, mlen = 1, j = 1, largest = TRUE, lower.tail = TRUE) rorder(n, quantfun, \dots, distn, mlen = 1, j = 1, largest = TRUE) } \arguments{ \item{x, q}{Vector of quantiles.} \item{n}{Number of observations.} \item{densfun, distnfun, quantfun}{Density, distribution and quantile function of the specified distribution. The density function must have a \code{log} argument (a simple wrapper can always be constructed to achieve this).} \item{\dots}{Parameters of the specified distribution.} \item{distn}{A character string, optionally specified as an alternative to \code{densfun}, \code{distnfun} and \code{quantfun} such that the density, distribution and quantile functions are formed upon the addition of the prefixes \code{d}, \code{p} and \code{q} respectively.} \item{mlen}{The number of independent variables.} \item{j}{The order statistic, taken as the \code{j}th largest (default) or smallest of \code{mlen}, according to the value of \code{largest}.} \item{largest}{Logical; if \code{TRUE} (default) use the \code{j}th largest order statistic, otherwise use the \code{j}th smallest.} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default) probabilities are P[X <= x], otherwise P[X > x].} } \value{ \code{dorder} gives the density function, \code{porder} gives the distribution function and \code{qorder} gives the quantile function of a selected order statistic from a sample of size \code{mlen}, from a specified distibution. \code{rorder} generates random deviates. } \seealso{\code{\link{rextreme}}, \code{\link{rgev}}} \examples{ dorder(2:4, dnorm, pnorm, mean = 0.5, sd = 1.2, mlen = 5, j = 2) dorder(2:4, distn = "norm", mean = 0.5, sd = 1.2, mlen = 5, j = 2) dorder(2:4, distn = "exp", mlen = 2, j = 2) porder(2:4, distn = "exp", rate = 1.2, mlen = 2, j = 2) rorder(5, qgamma, shape = 1, mlen = 10, j = 2) } \keyword{distribution} evd/man/gumbel.Rd0000644000175100001440000000311412637167310013370 0ustar hornikusers\name{gumbel} \alias{dgumbel} \alias{pgumbel} \alias{qgumbel} \alias{rgumbel} \title{The Gumbel Distribution} \description{ Density function, distribution function, quantile function and random generation for the Gumbel distribution with location and scale parameters. } \usage{ dgumbel(x, loc=0, scale=1, log = FALSE) pgumbel(q, loc=0, scale=1, lower.tail = TRUE) qgumbel(p, loc=0, scale=1, lower.tail = TRUE) rgumbel(n, loc=0, scale=1) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{loc, scale}{Location and scale parameters (can be given as vectors).} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} } \details{ The Gumbel distribution function with parameters \eqn{\code{loc} = a} and \eqn{\code{scale} = b} is \deqn{G(z) = \exp\left\{-\exp\left[-\left(\frac{z-a}{b}\right) \right]\right\}}{G(x) = exp{-exp[-(z-a)/b]}} for all real \eqn{z}, where \eqn{b > 0}. } \value{ \code{dgumbel} gives the density function, \code{pgumbel} gives the distribution function, \code{qgumbel} gives the quantile function, and \code{rgumbel} generates random deviates. } \seealso{\code{\link{rfrechet}}, \code{\link{rgev}}, \code{\link{rrweibull}}} \examples{ dgumbel(-1:2, -1, 0.5) pgumbel(-1:2, -1, 0.5) qgumbel(seq(0.9, 0.6, -0.1), 2, 0.5) rgumbel(6, -1, 0.5) p <- (1:9)/10 pgumbel(qgumbel(p, -1, 2), -1, 2) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/fox.Rd0000644000175100001440000000163012637167310012712 0ustar hornikusers\name{fox} \alias{fox} \title{Maximum Annual Flood Discharges of the Fox River} \usage{fox} \description{ The \code{fox} data frame has 33 rows and 2 columns. The columns contain maximum annual flood discharges, in units of 1000 cubed feet per second, from the Fox River in Wisconsin, USA at Berlin (upstream) and Wrightstown (downstream), for the years 1918 to 1950. The row names give the years of observation. } \format{ This data frame contains the following columns: \describe{ \item{berlin}{A numeric vector containing maximum annual flood discharges at Berlin (upstream).} \item{wright}{A numeric vector containing maximum annual flood discharges at Wrightstown (downstream).} } } \source{ Gumbel, E. J. and Mustafi, C. K. (1967) Some analytical properties of bivariate extremal distributions. \emph{J. Amer. Statist. Assoc.}, \bold{62}, 569--588. } \keyword{datasets} evd/man/abvnonpar.Rd0000644000175100001440000001656312637167310014117 0ustar hornikusers\name{abvnonpar} \alias{abvnonpar} \title{Non-parametric Estimates for Dependence Functions of the Bivariate Extreme Value Distribution} \description{ Calculate or plot non-parametric estimates for the dependence function \eqn{A} of the bivariate extreme value distribution. } \usage{ abvnonpar(x = 0.5, data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL, method = c("cfg", "pickands", "tdo", "pot"), k = nrow(data)/4, convex = FALSE, rev = FALSE, madj = 0, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0, 1), ylim = c(0.5, 1), xlab = "t", ylab = "A(t)", \dots) } \arguments{ \item{x}{A vector of values at which the dependence function is evaluated (ignored if plot or add is \code{TRUE}). \eqn{A(1/2)} is returned by default since it is often a useful summary of dependence.} \item{data}{A matrix or data frame with two columns, which may contain missing values.} \item{epmar}{If \code{TRUE}, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the \code{nsloc} arguments are ignored.} \item{nsloc1, nsloc2}{A data frame with the same number of rows as \code{data}, for linear modelling of the location parameter on the first/second margin. The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.} \item{method}{The estimation method (see \bold{Details}). Typically either \code{"cfg"} (the default) or \code{"pickands"}. The method \code{"tdo"} performs poorly and is not recommended. The method \code{"pot"} is for peaks over threshold modelling where only large data values are used for estimation.} \item{k}{An integer parameter for the \code{"pot"} method. Only the largest \code{k} values are used, as described in \code{\link{bvtcplot}}.} \item{convex}{Logical; take the convex minorant?} \item{rev}{Logical; reverse the dependence function? This is equivalent to evaluating the function at \code{1-x}.} \item{madj}{Performs marginal adjustments for the \code{"pickands"} method (see \bold{Details}).} \item{kmar}{In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates.} \item{plot}{Logical; if \code{TRUE} the function is plotted. The x and y values used to create the plot are returned invisibly. If \code{plot} and \code{add} are \code{FALSE} (the default), the arguments following \code{add} are ignored.} \item{add}{Logical; add to an existing plot? The existing plot should have been created using either \code{abvnonpar} or \code{\link{abvevd}}, the latter of which plots (or calculates) the dependence function for a number of parametric models.} \item{lty, blty}{Function and border line types. Set \code{blty} to zero to omit the border.} \item{lwd, blwd}{Function and border line widths.} \item{col}{Line colour.} \item{xlim, ylim}{x and y-axis limits.} \item{xlab, ylab}{x and y-axis labels.} \item{\dots}{Other high-level graphics parameters to be passed to \code{plot}.} } \details{ The dependence function \eqn{A(\cdot)}{A()} of the bivariate extreme value distribution is defined in \code{\link{abvevd}}. Non-parametric estimates are constructed as follows. Suppose \eqn{(z_{i1},z_{i2})} for \eqn{i=1,\ldots,n} are \eqn{n} bivariate observations that are passed using the \code{data} argument. If \code{epmar} is \code{FALSE} (the default), then the marginal parameters of the GEV margins are estimated (under the assumption of independence) and the data is transformed using \deqn{y_{i1} = \{1+\hat{s}_1(z_{i1}-\hat{a}_1)/ \hat{b}_1\}_{+}^{-1/\hat{s}_1}}{ y_{i1} = {1 + s'_1(z_{i1}-a'_1)/b'_1}^(-1/s'_1)} and \deqn{y_{i2} = \{1+\hat{s}_2(z_{i2}-\hat{a}_2)/ \hat{b}_2\}_{+}^{-1/\hat{s}_2}}{ y_{i2} = {1 + s'_2(z_{i2}-a'_2)/b'_2}^(-1/s'_2)} for \eqn{i = 1,\ldots,n}, where \eqn{(\hat{a}_1,\hat{b}_1,\hat{s}_1)}{(a'_1,b'_1,s'_1)} and \eqn{(\hat{a}_2,\hat{b}_2,\hat{s}_2)}{(a'_2,b'_2,s'_2)} are the maximum likelihood estimates for the location, scale and shape parameters on the first and second margins. If \code{nsloc1} or \code{nsloc2} are given, the location parameters may depend on \eqn{i} (see \code{\link{fgev}}). Two different estimators of the dependence function can be implemented. They are defined (on \eqn{0 \leq w \leq 1}{0 <= w <= 1}) as follows. \code{method = "cfg"} (Caperaa, Fougeres and Genest, 1997) \deqn{\log(A_c(w)) = \frac{1}{n} \left\{ \sum_{i=1}^n \log(\max[(1-w)y_{i1}, wy_{i1}]) - (1-w)\sum_{i=1}^n y_{i1} - w \sum_{i=1}^n y_{i2} \right\}}{log(A_c(w)) = 1/n { sum_{i=1}^n log (max[(1-w)y_{i1}, wy_{i1}]) - (1-w)sum_{i=1}^n y_{i1} - w sum_{i=1}^n y_{i2} }} \code{method = "pickands"} (Pickands, 1981) \deqn{A_p(w) = n\left\{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right)\right\}^{-1}}{ A_p(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}/(1-w)]}} Two variations on the estimator \eqn{A_p(\cdot)}{A_p()} are also implemented. If the argument \code{madj = 1}, an adjustment given in Deheuvels (1991) is applied. If the argument \code{madj = 2}, an adjustment given in Hall and Tajvidi (2000) is applied. These are marginal adjustments; they are only useful when empirical marginal estimation is used. Let \eqn{A_n(\cdot)}{A_n()} be any estimator of \eqn{A(\cdot)}{A()}. None of the estimators satisfy \eqn{\max(w,1-w) \leq A_n(w) \leq 1}{max(w,1-w) <= A_n(w) <= 1} for all \eqn{0\leq w \leq1}{0 <= w <= 1}. An obvious modification is \deqn{A_n^{'}(w) = \min(1, \max\{A_n(w), w, 1-w\}).}{ A'_n(w) = min(1, max{A_n(w), w, 1-w}).} This modification is always implemented. Convex estimators can be derived by taking the convex minorant, which can be achieved by setting \code{convex} to \code{TRUE}. } \note{ I have been asked to point out that Hall and Tajvidi (2000) suggest putting a constrained smoothing spline on their modified Pickands estimator, but this is not done here. } \value{ \code{abvnonpar} calculates or plots a non-parametric estimate of the dependence function of the bivariate extreme value distribution. } \references{ Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric estimation procedure for bivariate extreme value copulas. \emph{Biometrika}, \bold{84}, 567--577. Pickands, J. (1981) Multivariate extreme value distributions. \emph{Proc. 43rd Sess. Int. Statist. Inst.}, \bold{49}, 859--878. Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. \emph{Statist. Probab. Letters}, \bold{12}, 429--439. Hall, P. and Tajvidi, N. (2000) Distribution and dependence-function estimation for bivariate extreme-value distributions. \emph{Bernoulli}, \bold{6}, 835--844. } \seealso{\code{\link{abvevd}}, \code{\link{amvnonpar}}, \code{\link{bvtcplot}}, \code{\link{fgev}}} \examples{ bvdata <- rbvevd(100, dep = 0.7, model = "log") abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE) abvnonpar(data = bvdata, method = "pick", plot = TRUE) M1 <- fitted(fbvevd(bvdata, model = "log")) abvevd(dep = M1["dep"], model = "log", plot = TRUE) abvnonpar(data = bvdata, add = TRUE, lty = 2) } \keyword{nonparametric} evd/man/plot.profile.evd.Rd0000644000175100001440000000451414224765754015326 0ustar hornikusers\name{plot.profile.evd} \alias{plot.profile.evd} \title{Plot Profile Log-likelihoods} \description{ Displays profile log-likelihoods from a model profiled with \code{\link{profile.evd}}. } \usage{ \method{plot}{profile.evd}(x, which = names(x), main = NULL, ask = nb.fig < length(which) && dev.interactive(), ci = 0.95, clty = 2, \dots) } \arguments{ \item{x}{An object of class \code{"profile.evd"}.} \item{which}{A character vector giving the parameters for which the profile deviance is plotted, and for which profile confidence intervals are calculated. By default all profiled parameters in \code{x} are used.} \item{main}{Title of each plot; a character vector, the same length as \code{which}.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{ci}{A numeric vector. For each parameter in \code{which} profile confidence intervals are calculated, for each confidence coefficient in \code{ci} (but see \bold{Warning}). The intervals are returned invisibly as a list of vectors/matrices. Each plot then (by default) includes horizonal lines that represent each interval.} \item{clty}{The line type of the horizontal lines that represent the profile confidence intervals. To omit the lines set \code{clty} to zero.} \item{\dots}{Other graphics parameters.} } \value{ Profile devainces are plotted for each parameter in \code{which}. For calculation of profile confidence intervals, use the \code{\link{confint.profile.evd}} function. } \section{Warning}{ The profile confidence intervals may not have confidence coefficient \code{ci}, because the usual asymptotic properties of maximum likelihood estimators may not hold. For the GEV model, the usual asymptotic properties hold when the shape parameter is greater than \eqn{-0.5} (Smith, 1985). } \references{ Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. } \seealso{\code{\link{confint.profile.evd}}, \code{\link{plot.profile2d.evd}}, \code{\link{profile.evd}}, \code{\link{profile2d.evd}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata) \dontrun{M1P <- profile(M1)} \dontrun{par(mfrow = c(2,2))} \dontrun{cint <- plot(M1P, ci = c(0.95, 0.99))} \dontrun{cint} } \keyword{hplot} evd/man/exi.Rd0000644000175100001440000000411312637167310012702 0ustar hornikusers\name{exi} \alias{exi} \title{Estimates of the Extremal Index} \description{ Estimates of the extremal index. } \usage{ exi(data, u, r = 1, ulow = -Inf, rlow = 1) } \arguments{ \item{data}{A numeric vector, which may contain missing values.} \item{u}{A single value giving the threshold, unless a time varying threshold is used, in which case \code{u} should be a vector of thresholds, typically with the same length as \code{data} (or else the usual recycling rules are applied).} \item{r}{Either a postive integer denoting the clustering interval length, or zero, in which case the intervals estimator of Ferro and Segers (2003) is used and following arguments are ignored. By default the interval length is one.} \item{ulow}{A single value giving the lower threshold, unless a time varying lower threshold is used, in which case \code{ulow} should be a vector of lower thresholds, typically with the same length as \code{data} (or else the usual recycling rules are applied). By default there is no lower threshold (or equivalently, the lower threshold is \code{-Inf}).} \item{rlow}{A postive integer denoting the lower clustering interval length. By default the interval length is one.} } \details{ If \code{r} is a positive integer the extremal index is estimated using the inverse of the average cluster size, using the clusters of exceedences derived from \code{\link{clusters}}. If \code{r} is zero, an estimate based on inter-exceedance times is used (Ferro and Segers, 2003). If there are no exceedances of the threshold, the estimate is \code{NaN}. If there is only one exceedance, the estimate is one. } \value{ A single value estimating the extremal index. } \references{ Ferro, C. A. T. and Segers, J. (2003) Inference for clusters of extreme values. \emph{JRSS B}, \bold{65}, 545--556. } \seealso{\code{\link{clusters}}, \code{\link{exiplot}}} \examples{ exi(portpirie, 4.2, r = 3, ulow = 3.8) tvu <- c(rep(4.2, 20), rep(4.1, 25), rep(4.2, 20)) exi(portpirie, tvu, r = 1) exi(portpirie, tvu, r = 0) } \keyword{manip} evd/man/anova.evd.Rd0000644000175100001440000000426512637167310014006 0ustar hornikusers\name{anova.evd} \alias{anova.evd} \title{Compare Nested EVD Objects} \description{ Compute an analysis of deviance table for two or more nested evd objects. } \usage{ \method{anova}{evd}(object, object2, \dots, half = FALSE) } \arguments{ \item{object}{An object of class \code{"evd"}.} \item{object2}{An object of class \code{"evd"} that represents a model nested within \code{object}.} \item{\dots}{Further successively nested objects.} \item{half}{For some non-regular tesing problems the deviance difference is known to be one half of a chi-squared random variable. Set \code{half} to \code{TRUE} in these cases.} } \value{ An object of class \code{c("anova", "data.frame")}, with one row for each model, and the following five columns \item{M.Df}{The number of parameters.} \item{Deviance}{The deviance.} \item{Df}{The number of parameters of the model in the previous row minus the number of parameters.} \item{Chisq}{The deviance minus the deviance of the model in the previous row (or twice this if \code{half} is \code{TRUE}).} \item{Pr(>chisq)}{The p-value calculated by comparing the quantile \code{Chisq} with a chi-squared distribution on \code{Df} degrees of freedom.} } \section{Warning}{ Circumstances may arise such that the asymptotic distribution of the test statistic is not chi-squared. In particular, this occurs when the smaller model is constrained at the edge of the parameter space. It is up to the user recognize this, and to interpret the output correctly. In some cases the asymptotic distribution is known to be one half of a chi-squared; you can set \code{half = TRUE} in these cases. } \seealso{\code{\link{fbvevd}}, \code{\link{fextreme}}, \code{\link{fgev}}, \code{\link{forder}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) trend <- (-49:50)/100 M1 <- fgev(uvdata, nsloc = trend) M2 <- fgev(uvdata) M3 <- fgev(uvdata, shape = 0) anova(M1, M2, M3) bvdata <- rbvevd(100, dep = 0.75, model = "log") M1 <- fbvevd(bvdata, model = "log") M2 <- fbvevd(bvdata, model = "log", dep = 0.75) M3 <- fbvevd(bvdata, model = "log", dep = 1) anova(M1, M2) anova(M1, M3, half = TRUE) } \keyword{models} evd/man/portpirie.Rd0000644000175100001440000000110712637167310014132 0ustar hornikusers\name{portpirie} \alias{portpirie} \title{Annual Maximum Sea Levels at Port Pirie} \usage{portpirie} \description{ A numeric vector containing annual maximum sea levels, in metres, from 1923 to 1987 at Port Pirie, South Australia. } \format{A vector containing 65 observations.} \source{ Tawn, J. A. (1993) Extreme sea-levels, in \emph{Statistics in the Environment}, 243--263, eds. V. Barnett and F. Turkman, Wiley. } \references{ Coles, S. G. (2001) \emph{An Introduction to Statistical Modeling of Extreme Values}. London: Springer-Verlag. } \keyword{datasets} evd/man/plot.profile2d.evd.Rd0000644000175100001440000000424614260534766015553 0ustar hornikusers\name{plot.profile2d.evd} \alias{plot.profile2d.evd} \title{Plot Joint Profile Log-likelihoods} \description{ Displays an image plot of the joint profile log-likelihood from a model profiled with \code{\link{profile.evd}} and \code{\link{profile2d.evd}}. } \usage{ \method{plot}{profile2d.evd}(x, main = NULL, ci = c(0.5, 0.8, 0.9, 0.95, 0.975, 0.99, 0.995), col = heat.colors(8), intpts = 75, xaxs = "r", yaxs = "r", \dots) } \arguments{ \item{x}{An object of class \code{"profile2d.evd"}.} \item{main}{Title of plot; a character string.} \item{ci}{A numeric vector whose length is one less than the length of \code{col}. The colours of the image plot, excluding the background colour, represent confidence sets with confidence coefficients \code{ci} (but see \bold{Warning}).} \item{col}{A list of colors such as that generated by \code{rainbow}, \code{heat.colors}, \code{topo.colors}, \code{terrain.colors} or similar functions.} \item{intpts}{If the package \CRANpkg{interp} is available, interpolation is performed using \code{intpts} points for each parameter. The function is interpolated at \code{intpts^2} points in total.} \item{xaxs,yaxs}{Graphics parameters (see \code{\link{par}}). The default, \code{"r"}, overrides the default set by \code{image}.} \item{\dots}{Other parameters to be passed to \code{image}.} } \section{Warning}{ The sets represented by different colours may not be confidence sets with confidence coefficients \code{ci}, because the usual asymptotic properties of maximum likelihood estimators may not hold. For the GEV model, the usual asymptotic properties hold when the shape parameter is greater than \eqn{-0.5} (Smith, 1985). } \references{ Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. } \seealso{\code{\link{plot.profile.evd}}, \code{\link{profile.evd}}, \code{\link{profile2d.evd}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata) \dontrun{M1P <- profile(M1)} \dontrun{M1JP <- profile2d(M1, M1P, which = c("scale", "shape"))} \dontrun{plot(M1JP)} } \keyword{hplot} evd/man/oxford.Rd0000644000175100001440000000065312637167310013423 0ustar hornikusers\name{oxford} \alias{oxford} \title{Annual Maximum Temperatures at Oxford} \usage{oxford} \description{ A numeric vector containing annual maximum temperatures, in degrees Fahrenheit, from 1901 to 1980 at Oxford, England. } \format{A vector containing 80 observations.} \source{ Tabony, R. C. (1983) Extreme value analysis in meteorology. \emph{The Meteorological Magazine} \bold{112}, 77--98. } \keyword{datasets} evd/man/failure.Rd0000644000175100001440000000057412637167310013553 0ustar hornikusers\name{failure} \alias{failure} \title{Failure Times} \usage{failure} \description{ Failure times. } \format{A vector containing 24 observations.} \source{ van Montfort, M. A. J. and Otten, A. (1978) On testing a shape parameter in the presence of a scale parameter. \emph{Math. Operations Forsch. Statist., Ser. Statistics}, \bold{9}, 91--104. } \keyword{datasets} evd/man/exiplot.Rd0000644000175100001440000000260212637167310013602 0ustar hornikusers\name{exiplot} \alias{exiplot} \title{Plot Estimates of the Extremal Index} \description{ Plots estimates of the extremal index. } \usage{ exiplot(data, tlim, r = 1, ulow = -Inf, rlow = 1, add = FALSE, nt = 100, lty = 1, xlab = "Threshold", ylab = "Ext. Index", ylim = c(0,1), \dots) } \arguments{ \item{data}{A numeric vector, which may contain missing values.} \item{tlim}{A numeric vector of length two, giving the limits for the (time invariant) thresholds at which the estimates are evaluated.} \item{r, ulow, rlow}{The estimation method. See \code{\link{exi}}.} \item{add}{Add to an existing plot?} \item{nt}{The number of thresholds at which the estimates are evaluated.} \item{lty}{Line type.} \item{xlab, ylab}{x and y axis labels.} \item{ylim}{y axis limits.} \item{\dots}{Other arguments passed to \code{plot} or \code{lines}.} } \details{ The estimates are calculated using the function \code{\link{exi}}. } \value{ A list with components \code{x} and \code{y} is invisibly returned. The first component contains the thresholds, the second contains the estimates. } \seealso{\code{\link{clusters}}, \code{\link{exi}}} \examples{ sdat <- mar(100, psi = 0.5) tlim <- quantile(sdat, probs = c(0.4,0.9)) exiplot(sdat, tlim) exiplot(sdat, tlim, r = 4, add = TRUE, lty = 2) exiplot(sdat, tlim, r = 0, add = TRUE, lty = 4) } \keyword{hplot} evd/man/gev.Rd0000644000175100001440000000457412637167310012711 0ustar hornikusers\name{gev} \alias{dgev} \alias{pgev} \alias{qgev} \alias{rgev} \title{The Generalized Extreme Value Distribution} \description{ Density function, distribution function, quantile function and random generation for the generalized extreme value (GEV) distribution with location, scale and shape parameters. } \usage{ dgev(x, loc=0, scale=1, shape=0, log = FALSE) pgev(q, loc=0, scale=1, shape=0, lower.tail = TRUE) qgev(p, loc=0, scale=1, shape=0, lower.tail = TRUE) rgev(n, loc=0, scale=1, shape=0) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{loc, scale, shape}{Location, scale and shape parameters; the \code{shape} argument cannot be a vector (must have length one).} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} } \details{ The GEV distribution function with parameters \eqn{\code{loc} = a}, \eqn{\code{scale} = b} and \eqn{\code{shape} = s} is \deqn{G(z) = \exp\left[-\{1+s(z-a)/b\}^{-1/s}\right]}{ G(x) = exp[-{1+s(z-a)/b}^(-1/s)]} for \eqn{1+s(z-a)/b > 0}, where \eqn{b > 0}. If \eqn{s = 0} the distribution is defined by continuity. If \eqn{1+s(z-a)/b \leq 0}{1+s(z-a)/b <= 0}, the value \eqn{z} is either greater than the upper end point (if \eqn{s < 0}), or less than the lower end point (if \eqn{s > 0}). The parametric form of the GEV encompasses that of the Gumbel, Frechet and reverse Weibull distributions, which are obtained for \eqn{s = 0}, \eqn{s > 0} and \eqn{s < 0} respectively. It was first introduced by Jenkinson (1955). } \value{ \code{dgev} gives the density function, \code{pgev} gives the distribution function, \code{qgev} gives the quantile function, and \code{rgev} generates random deviates. } \references{ Jenkinson, A. F. (1955) The frequency distribution of the annual maximum (or minimum) of meteorological elements. \emph{Quart. J. R. Met. Soc.}, \bold{81}, 158--171. } \seealso{\code{\link{fgev}}, \code{\link{rfrechet}}, \code{\link{rgumbel}}, \code{\link{rrweibull}}} \examples{ dgev(2:4, 1, 0.5, 0.8) pgev(2:4, 1, 0.5, 0.8) qgev(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8) rgev(6, 1, 0.5, 0.8) p <- (1:9)/10 pgev(qgev(p, 1, 2, 0.8), 1, 2, 0.8) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/frechet.Rd0000644000175100001440000000333512637167310013542 0ustar hornikusers\name{frechet} \alias{dfrechet} \alias{pfrechet} \alias{qfrechet} \alias{rfrechet} \title{The Frechet Distribution} \description{ Density function, distribution function, quantile function and random generation for the Frechet distribution with location, scale and shape parameters. } \usage{ dfrechet(x, loc=0, scale=1, shape=1, log = FALSE) pfrechet(q, loc=0, scale=1, shape=1, lower.tail = TRUE) qfrechet(p, loc=0, scale=1, shape=1, lower.tail = TRUE) rfrechet(n, loc=0, scale=1, shape=1) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{loc, scale, shape}{Location, scale and shape parameters (can be given as vectors).} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} } \details{ The Frechet distribution function with parameters \eqn{\code{loc} = a}, \eqn{\code{scale} = b} and \eqn{\code{shape} = s} is \deqn{G(z) = \exp\left\{-\left(\frac{z-a}{b}\right)^{-s} \right\}}{G(x) = exp{-[(z-a)/b]^(-s)}} for \eqn{z > a} and zero otherwise, where \eqn{b > 0} and \eqn{s > 0}. } \value{ \code{dfrechet} gives the density function, \code{pfrechet} gives the distribution function, \code{qfrechet} gives the quantile function, and \code{rfrechet} generates random deviates. } \seealso{\code{\link{rgev}}, \code{\link{rgumbel}}, \code{\link{rrweibull}}} \examples{ dfrechet(2:4, 1, 0.5, 0.8) pfrechet(2:4, 1, 0.5, 0.8) qfrechet(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8) rfrechet(6, 1, 0.5, 0.8) p <- (1:9)/10 pfrechet(qfrechet(p, 1, 2, 0.8), 1, 2, 0.8) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/lossalae.Rd0000644000175100001440000000247612637167310013732 0ustar hornikusers\name{lossalae} \alias{lossalae} \title{General Liability Claims} \usage{lossalae} \description{ The \code{lossalae} data frame has 1500 rows and 2 columns. The columns contain the indemnity payment (loss), and the allocated loss adjustment expense (alae), both in USD. The latter is the additional expenses associated with the settlement of the claim (e.g. claims investigation expenses and legal fees). The dataset also has an attribute called \code{capped}, which gives the row names of the indemnity payments that were capped at their policy limit. } \format{ This data frame contains the following columns: \describe{ \item{Loss}{A numeric vector containing the indemnity payments.} \item{ALAE}{A numeric vector containing the allocated loss adjustment expenses.} } } \source{ Frees, E. W. and Valdez, E. A. (1998) Understanding relationships using copulas. \emph{North American Actuarial Journal}, \bold{2}, 1--15. } \references{ Klugman, S. A. and Parsa, R. (1999) Fitting bivariate loss distributions with copulas. \emph{Insurance: Mathematics and Economics}, \bold{24}, 139--148. Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L. (2004) \emph{Statistics of Extremes: Theory and Applications.}, Chichester, England: John Wiley and Sons. } \keyword{datasets} evd/man/plot.bvevd.Rd0000644000175100001440000001074613270066762014214 0ustar hornikusers\name{plot.bvevd} \alias{plot.bvevd} \title{Plot Diagnostics for a Bivariate EVD Object} \description{ Six plots (selectable by \code{which}) are currently provided: two conditional P-P plots (1,2), conditioning on each margin, a density plot (3), a dependence function plot (4), a quantile curves plot (5) and a spectral density plot (6). Plot diagnostics for the generalized extreme value margins (selectable by \code{mar} and \code{which}) are also available. } \usage{ \method{plot}{bvevd}(x, mar = 0, which = 1:6, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, grid = 50, legend = TRUE, nplty = 2, blty = 3, method = "cfg", convex = FALSE, rev = FALSE, p = seq(0.75, 0.95, 0.05), mint = 1, half = FALSE, \dots) } \arguments{ \item{x}{An object of class \code{"bvevd"}.} \item{mar}{If \code{mar = 1} or \code{mar = 2} diagnostics are given for the first or second genereralized extreme value margin respectively.} \item{which}{A subset of the numbers \code{1:6} selecting the plots to be shown. By default all are plotted.} \item{main}{Title of each plot. If given, should be a character vector with the same length as \code{which}.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{ci}{Logical; if \code{TRUE} (the default), plot simulated 95\% confidence intervals for the conditional P-P plots.} \item{cilwd}{Line width for confidence interval lines.} \item{a}{Passed through to \code{ppoints} for empirical estimation. Larger values give less probability for extreme events.} \item{grid}{Argument for the density plot. The (possibly transformed) data is plotted with a contour plot of the bivariate density of the fitted model. The density is evaluated at \code{grid^2} points.} \item{legend}{If \code{legend} is \code{TRUE} and if the fitted data contained a third column of mode \code{logical}, then a legend is included in the density and quantile curve plots.} \item{method, convex, rev}{Arguments to the dependence function plot. The dependence function for the fitted model is plotted and (optionally) compared to a non-parameteric estimate. See \code{\link{abvnonpar}} for a description of the arguments.} \item{nplty, blty}{Line types for the dependence function plot. \code{nplty} is the line type of the non-parametric estimate. To omit the non-parametric estimate set \code{nplty} to zero. \code{blty} is the line type of the triangular border. To omit the border estimate set \code{blty} to zero.} \item{p, mint}{Arguments to the quantile curves plot. See \code{\link{qcbvnonpar}} for a description of the plot and the arguments.} \item{half}{Argument to the spectral density plot. See \code{\link{hbvevd}}.} \item{\dots}{Other arguments to be passed through to plotting functions.} } \details{ In all plots we assume that the fitted model is stationary. For non-stationary models the data are transformed to stationarity. The plot then corresponds to the distribution obtained when all covariates are zero. In particular, the density and quanitle curves plots will not plot the original data for non-stationary models. A conditional P-P plot is a P-P plot for the condition distribution function of a bivariate evd object. Let \eqn{G(.|.)} be the conditional distribution of the first margin given the second, under the fitted model. Let \eqn{z_1,\ldots,z_m} be the data used in the fitted model, where \eqn{z_j = (z_{1j}, z_{2j})} for \eqn{j = 1,\ldots,m}. The plot that (by default) is labelled Conditional Plot Two, conditioning on the second margin, consists of the points \deqn{\{(p_i, c_i), i = 1,\ldots,m\}}{ {(p_i, c_i), i = 1,\ldots,m}} where \eqn{p_1,\ldots,p_m} are plotting points defined by \code{\link{ppoints}} and \eqn{c_i} is the \eqn{i}th largest value from the sample \eqn{\{G(z_{j1}|z_{j2}), j = 1,\ldots,m\}.}{ {G(z_{j1}|z_{j2}), j = 1,\ldots,m}.} The margins are reversed for Conditional Plot One, so that \eqn{G(.|.)} is the conditional distribution of the second margin given the first. } \seealso{\code{\link{plot.uvevd}}, \code{\link{contour}}, \code{\link{jitter}}, \code{\link{abvnonpar}}, \code{\link{qcbvnonpar}}} \examples{ bvdata <- rbvevd(100, dep = 0.6, model = "log") M1 <- fbvevd(bvdata, model = "log") \dontrun{par(mfrow = c(2,2))} \dontrun{plot(M1, which = 1:5)} \dontrun{plot(M1, mar = 1)} \dontrun{plot(M1, mar = 2)} } \keyword{hplot} evd/man/mtransform.Rd0000644000175100001440000000320312637167310014304 0ustar hornikusers\name{mtransform} \alias{mtransform} \title{GEV Transformations} \description{ Transforms to exponential margins under the GEV model. } \usage{ mtransform(x, p, inv = FALSE, drp = FALSE) } \arguments{ \item{x}{A matrix with n rows and d columns, or a vector. In the latter case, if \code{p} is a list with the same length as the vector, it is treated as a matrix with one row. If \code{p} is not a list, it is treated as a matrix with one column.} \item{p}{A vector of length three or a matrix with n rows and three columns. It can also be a list of length d, in which case each element must be a vector of length three or a matrix with n rows and three columns.} \item{inv}{Logical; use the inverse transformation?} \item{drp}{Logical; return a vector rather than a single row matrix?. Note that a single column matrix is always returned as a vector.} } \details{ Let \eqn{x_i} denote a vector of observations for \eqn{i = 1,\ldots,n}. This function implements the transformation \deqn{y_{i} = \{1+s_i(x_{i}-a_i)/b_i\}_{+}^{-1/s_i}} to each column of the matrix \code{x}. The values \eqn{(a_i,b_i,s_i)} are contained in the ith row of the n by 3 matrix \code{p}. If \code{p} is a vector of length three, the parameters are the same for every \eqn{i = 1,\ldots,n}. Alternatively, \code{p} can be a list with d elements, in which case the jth element is used to transform the jth column of \code{x}. This function is mainly for internal use. It is used by bivariate and multivariate routines to calculate marginal transformations. } \value{ A numeric matrix or vector. } \keyword{manip} evd/man/plot.uvevd.Rd0000644000175100001440000001455114225014401014215 0ustar hornikusers\name{plot.uvevd} \alias{plot.uvevd} \alias{plot.gumbelx} \title{Plot Diagnostics for a Univariate EVD Object} \description{ Four plots (selectable by \code{which}) are currently provided: a P-P plot, a Q-Q plot, a density plot and a return level plot. } \usage{ \method{plot}{uvevd}(x, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, adjust = 1, jitter = FALSE, nplty = 2, \dots) \method{plot}{gumbelx}(x, interval, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), ci = TRUE, cilwd = 1, a = 0, adjust = 1, jitter = FALSE, nplty = 2, \dots) } \arguments{ \item{x}{An object that inherits from class \code{"uvevd"}.} \item{which}{If a subset of the plots is required, specify a subset of the numbers \code{1:4}.} \item{main}{Title of each plot. If given, must be a character vector with the same length as \code{which}.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{ci}{Logical; if \code{TRUE} (the default), plot simulated 95\% confidence intervals for the P-P, Q-Q and return level plots.} \item{cilwd}{Line width for confidence interval lines.} \item{a}{Passed through to \code{ppoints} for empirical estimation. Larger values give less probability for extreme events.} \item{adjust, jitter, nplty}{Arguments to the density plot. The density of the fitted model is plotted with a rug plot and (optionally) a non-parameteric estimate. The argument \code{adjust} controls the smoothing bandwidth for the non-parametric estimate (see \code{\link{density}}). \code{jitter} is logical; if \code{TRUE}, the (possibly transformed) data are jittered to produce the rug plot. This need only be used if the data contains repeated values. \code{nplty} is the line type of the non-parametric estimate. To omit the non-parametric estimate set \code{nplty} to zero.} \item{interval}{A vector of length two, for the gumbelx (maximum of two Gumbels) model. This is passed to the uniroot function to calculate quantiles for the Q-Q and return level plots. The interval should be large enough to contain all plotted quantiles or an error from uniroot will occur.} \item{\dots}{Other parameters to be passed through to plotting functions.} } \details{ The following discussion assumes that the fitted model is stationary. For non-stationary generalized extreme value models the data are transformed to stationarity. The plot then corresponds to the distribution obtained when all covariates are zero. The P-P plot consists of the points \deqn{\{(G_n(z_i), G(z_i)), i = 1,\ldots,m\}}{ {(G_n(z_i), G(z_i)), i = 1,\ldots,m}} where \eqn{G_n} is the empirical distribution function (defined using \code{\link{ppoints}}), G is the model based estimate of the distribution (generalized extreme value or generalized Pareto), and \eqn{z_1,\ldots,z_m} are the data used in the fitted model, sorted into ascending order. The Q-Q plot consists of the points \deqn{\{(G^{-1}(p_i), z_i), i = 1,\ldots,m\}}{ {(G^{-1}(p_i), z_i), i = 1,\ldots,m}} where \eqn{G^{-1}} is the model based estimate of the quantile function (generalized extreme value or generalized Pareto), \eqn{p_1,\ldots,p_m} are plotting points defined by \code{\link{ppoints}}, and \eqn{z_1,\ldots,z_m} are the data used in the fitted model, sorted into ascending order. The return level plot for generalized extreme value models is defined as follows. Let \eqn{G} be the generalized extreme value distribution function, with location, scale and shape parameters \eqn{a}, \eqn{b} and \eqn{s} respectively. Let \eqn{z_t} be defined by \eqn{G(z_t) = 1 - 1/t}. In common terminology, \eqn{z_t} is the return level associated with the return period \eqn{t}. Let \eqn{y_t = -1/\log(1 - 1/t)}{y_t = -1/log(1 - 1/t)}. It follows that \deqn{z_t = a + b(y_t^s - 1)/s.}{ z_t = a + b((y_t)^s - 1)/s.} When \eqn{s = 0}, \eqn{z_t} is defined by continuity, so that \deqn{z_t = a + b\log(y_t).}{ z_t = a + b log(y_t).} The curve within the return level plot is \eqn{z_t} plotted against \eqn{y_t} on a logarithmic scale, using maximum likelihood estimates of \eqn{(a,b,s)}. If the estimate of \eqn{s} is zero, the curve will be linear. For large values of \eqn{t}, \eqn{y_t} is approximately equal to the return period \eqn{t}. It is usual practice to label the x-axis as the return period. The points on the plot are \deqn{\{(-1/\log(p_i), z_i), i = 1,\ldots,m\}}{ {(-1/log(p_i), z_i), i = 1,\ldots,m}} where \eqn{p_1,\ldots,p_m} are plotting points defined by \code{\link{ppoints}}, and \eqn{z_1,\ldots,z_m} are the data used in the fitted model, sorted into ascending order. For a good fit the points should lie ``close'' to the curve. The return level plot for peaks over threshold models is defined as follows. Let \eqn{G} be the generalized Pareto distribution function, with location, scale and shape parameters \eqn{u}, \eqn{b} and \eqn{s} respectively, where \eqn{u} is the model threshold. Let \eqn{z_m} denote the \eqn{m} period return level (see \code{\link{fpot}} and the notation therein). It follows that \deqn{z_m = u + b((pmN)^s - 1)/s.}{ z_m = u + b((pmN)^s - 1)/s.} When \eqn{s = 0}, \eqn{z_m} is defined by continuity, so that \deqn{z_m = u + b\log(pmN).}{ z_m = u + b log(pmN).} The curve within the return level plot is \eqn{z_m} plotted against \eqn{m} on a logarithmic scale, using maximum likelihood estimates of \eqn{(b,s,p)}. If the estimate of \eqn{s} is zero, the curve will be linear. The points on the plot are \deqn{\{(1/(pN(1-p_i)), z_i), i = 1,\ldots,m\}}{ {(1/(pN(1-p_i)), z_i), i = 1,\ldots,m}} where \eqn{p_1,\ldots,p_m} are plotting points defined by \code{\link{ppoints}}, and \eqn{z_1,\ldots,z_m} are the data used in the fitted model, sorted into ascending order. For a good fit the points should lie ``close'' to the curve. } \seealso{\code{\link{plot.bvevd}}, \code{\link{density}}, \code{\link{jitter}}, \code{\link{rug}}, \code{\link{ppoints}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata) \dontrun{par(mfrow = c(2,2))} \dontrun{plot(M1)} uvdata <- rgpd(100, loc = 0, scale = 1.1, shape = 0.2) M1 <- fpot(uvdata, 1) \dontrun{par(mfrow = c(2,2))} \dontrun{plot(M1)} } \keyword{hplot} evd/man/fbvevd.Rd0000644000175100001440000002774012637167310013404 0ustar hornikusers\name{fbvevd} \alias{fbvevd} \alias{print.bvevd} \title{Maximum-likelihood Fitting of Bivariate Extreme Value Distributions} \description{ Fit models for one of nine parametric bivariate extreme value distributions, including linear modelling of the marginal location parameters, and allowing any of the parameters to be held fixed if desired. } \usage{ fbvevd(x, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), start, \dots, sym = FALSE, nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) } \arguments{ \item{x}{A matrix or data frame, ordinarily with two columns, which may contain missing values. A data frame may also contain a third column of mode \code{logical}, which itself may contain missing values (see \bold{More Details}).} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in \code{\link{rbvevd}}.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized. If \code{start} is omitted the routine attempts to find good starting values using marginal maximum likelihood estimators.} \item{\dots}{Additional parameters, either for the bivariate extreme value model or for the optimization function \code{optim}. If parameters of the model are included they will be held fixed at the values given (see \bold{Examples}).} \item{sym}{Logical; if \code{TRUE}, the dependence structure of the models \code{"alog"}, \code{"aneglog"} or \code{"ct"} are constrained to be symmetric (see \bold{Details}). For all other models, the argument is ignored (and a warning is given).} \item{nsloc1, nsloc2}{A data frame with the same number of rows as \code{x}, for linear modelling of the location parameter on the first/second margin (see \bold{Details}). The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.} \item{cshape}{Logical; if \code{TRUE}, a common shape parameter is fitted to each margin.} \item{cscale}{Logical; if \code{TRUE}, a common scale parameter is fitted to each margin, and the default value of \code{cshape} is then \code{TRUE}, so that under this default common scale and shape parameters are fitted.} \item{cloc}{Logical; if \code{TRUE}, a common location parameter is fitted to each margin, and the default values of \code{cshape} and \code{cscale} are then \code{TRUE}, so that under these defaults common marginal parameters are fitted.} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} \item{warn.inf}{Logical; if \code{TRUE} (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.} } \details{ The dependence parameter names are one or more of \code{dep}, \code{asy1}, \code{asy2}, \code{alpha} and \code{beta}, depending on the model selected (see \code{\link{rbvevd}}). The marginal parameter names are \code{loc1}, \code{scale1} and \code{shape1} for the first margin, and \code{loc2}, \code{scale2} and \code{shape2} for the second margin. If \code{nsloc1} is not \code{NULL}, so that a linear model is implemented for the first marginal location parameter, the parameter names for the first margin are \code{loc1}, \code{loc1}\emph{x1}, \dots, \code{loc1}\emph{xn}, \code{scale} and \code{shape}, where \emph{x1}, \dots, \emph{xn} are the column names of \code{nsloc1}, so that \code{loc1} is the intercept of the linear model, and \code{loc1}\emph{x1}, \dots, \code{loc1}\emph{xn} are the \code{ncol(nsloc1)} coefficients. When \code{nsloc2} is not \code{NULL}, the parameter names for the second margin are constructed similarly. It is recommended that the covariates within the linear models for the location parameters are (at least approximately) centered and scaled (i.e. that the columns of \code{nsloc1} and \code{nsloc2} are centered and scaled), particularly if automatic starting values are used, since the starting values for the associated parameters are then zero. If \code{cloc} is \code{TRUE}, both \code{nsloc1} and \code{nsloc2} must be identical, since a common linear model is then implemented on both margins. If \code{cshape} is true, the models are constrained so that \code{shape2 = shape1}. The parameter \code{shape2} is then taken to be specified, so that e.g. the common shape parameter can only be fixed at zero using \code{shape1 = 0}, since using \code{shape2 = 0} gives an error. Similar comments apply for \code{cscale} and \code{cloc}. If \code{sym} is \code{TRUE}, the asymmetric logistic and asymmetric negative logistic models are constrained so that \code{asy2 = asy1}, and the Coles-Tawn model is constrained so that \code{beta = alpha}. The parameter \code{asy2} or \code{beta} is then taken to be specified, so that e.g. the parameters \code{asy1} and \code{asy2} can only be fixed at \code{0.8} using \code{asy1 = 0.8}, since using \code{asy2 = 0.8} gives an error. Bilogistic and negative bilogistic models constrained to symmetry are logistic and negative logistic models respectively. The (symmetric) mixed model (e.g. Tawn, 1998) can be obtained as a special case of the asymmetric logistic or asymmetric mixed models (see \bold{Examples}). The value \code{Dependence} given in the printed output is \eqn{2(1-A(1/2))}, where \eqn{A} is the estimated dependence function (see \code{\link{abvevd}}). It measures the strength of dependence, and lies in the interval [0,1]; at independence and complete dependence it is zero and one respectively (Coles, Heffernan and Tawn, 1999). See \code{\link{chiplot}} for further information. } \value{ Returns an object of class \code{c("bvevd","evd")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The functions \code{profile} and \code{profile2d} can be used to obtain deviance profiles. The function \code{anova} compares nested models, and the function \code{AIC} compares non-nested models. The function \code{plot} produces diagnostic plots. An object of class \code{c("bvevd","evd")} is a list containing the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{fixed}{A vector containing the parameters that have been fixed at specific values within the optimization.} \item{fixed2}{A vector containing the parameters that have been set to be equal to other model parameters.} \item{param}{A vector containing all parameters (those optimized, those fixed to specific values, and those set to be equal to other model parameters).} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{dep.summary}{The estimate of \eqn{2(1-A(1/2))}.} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{data}{The data passed to the argument \code{x}.} \item{tdata}{The data, transformed to stationarity (for non-stationary models).} \item{nsloc1, nsloc2}{The arguments \code{nsloc1} and \code{nsloc2}.} \item{n}{The number of rows in \code{x}.} \item{sym}{The argument \code{sym}.} \item{cmar}{The vector \code{c(cloc, cscale, cshape)}.} \item{model}{The argument \code{model}.} \item{call}{The call of the current function.} } \section{More Details}{ If \code{x} is a data frame with a third column of mode \code{logical}, then the model is fitted using the likelihood derived by Stephenson and Tawn (2004). This is appropriate when each bivariate data point comprises componentwise maxima from some underlying bivariate process, and where the corresponding logical value denotes whether or not the maxima were caused by the same event within that process. Under this scheme the diagnostic plots that are produced using \code{plot} are somewhat different to those described in \code{\link{plot.bvevd}}: the density, dependence function and quantile curves plots contain fitted functions for observations where the logical case is unknown, and the conditional P-P plots condition on both the logical case and the given margin (which requires numerical integration at each data point). } \section{Artificial Constraints}{ For numerical reasons parameters are subject to artificial constraints. Specifically, these constraints are: marginal scale parameters not less than 0.01; \code{dep} not less than [0.1] [0.2] [0.05] in [logistic] [Husler-Reiss] [negative logistic] models; \code{dep} not greater than [10] [5] in [Husler-Reiss] [negative logistic] models; \code{asy1} and \code{asy2} not less than 0.001; \code{alpha} and \code{beta} not less than [0.1] [0.1] [0.001] in [bilogistic] [negative bilogistic] [Coles-Tawn] models; \code{alpha} and \code{beta} not greater than [0.999] [20] [30] in [bilogistic] [negative bilogistic] [Coles-Tawn] models. } \section{Warning}{ The standard errors and the correlation matrix in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution when either of the marginal shape parameters are less than \eqn{-0.5}, because the usual asymptotic properties of maximum likelihood estimators do not then hold (Smith, 1985). } \references{ Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. \emph{Extremes}, \bold{2}, 339--365. Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. \emph{Biometrika} \bold{92}(1), 213--217. Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. \emph{Biometrika}, \bold{75}, 397--415. } \seealso{\code{\link{anova.evd}}, \code{\link{optim}}, \code{\link{plot.bvevd}}, \code{\link{profile.evd}}, \code{\link{profile2d.evd}}, \code{\link{rbvevd}}} \examples{ bvdata <- rbvevd(100, dep = 0.6, model = "log", mar1 = c(1.2,1.4,0.4)) M1 <- fbvevd(bvdata, model = "log") M2 <- fbvevd(bvdata, model = "log", dep = 0.75) anova(M1, M2) par(mfrow = c(2,2)) plot(M1) plot(M1, mar = 1) plot(M1, mar = 2) \dontrun{par(mfrow = c(1,1))} \dontrun{M1P <- profile(M1, which = "dep")} \dontrun{plot(M1P)} trend <- (-49:50)/100 rnd <- runif(100, min = -.5, max = .5) fbvevd(bvdata, model = "log", nsloc1 = trend) fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend = trend, random = rnd)) fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend = trend, random = rnd), loc2random = 0) bvdata <- rbvevd(100, dep = 1, asy = c(0.5,0.5), model = "anegl") anlog <- fbvevd(bvdata, model = "anegl") mixed <- fbvevd(bvdata, model = "anegl", dep = 1, sym = TRUE) anova(anlog, mixed) amixed <- fbvevd(bvdata, model = "amix") mixed <- fbvevd(bvdata, model = "amix", beta = 0) anova(amixed, mixed) } \keyword{models} evd/man/extreme.Rd0000644000175100001440000000474512637167310013601 0ustar hornikusers\name{extreme} \alias{dextreme} \alias{pextreme} \alias{qextreme} \alias{rextreme} \title{Distributions of Maxima and Minima} \description{ Density function, distribution function, quantile function and random generation for the maximum/minimum of a given number of independent variables from a specified distribution. } \usage{ dextreme(x, densfun, distnfun, \dots, distn, mlen = 1, largest = TRUE, log = FALSE) pextreme(q, distnfun, \dots, distn, mlen = 1, largest = TRUE, lower.tail = TRUE) qextreme(p, quantfun, \dots, distn, mlen = 1, largest = TRUE, lower.tail = TRUE) rextreme(n, quantfun, \dots, distn, mlen = 1, largest = TRUE) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{densfun, distnfun, quantfun}{Density, distribution and quantile function of the specified distribution. The density function must have a \code{log} argument (a simple wrapper can always be constructed to achieve this).} \item{\dots}{Parameters of the specified distribution.} \item{distn}{A character string, optionally given as an alternative to \code{densfun}, \code{distnfun} and \code{quantfun} such that the density, distribution and quantile functions are formed upon the addition of the prefixes \code{d}, \code{p} and \code{q} respectively.} \item{mlen}{The number of independent variables.} \item{largest}{Logical; if \code{TRUE} (default) use maxima, otherwise minima.} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default) probabilities are P[X <= x], otherwise P[X > x].} } \value{ \code{dextreme} gives the density function, \code{pextreme} gives the distribution function and \code{qextreme} gives the quantile function of the maximum/minimum of \code{mlen} independent variables from a specified distibution. \code{rextreme} generates random deviates. } \seealso{\code{\link{rgev}}, \code{\link{rorder}}} \examples{ dextreme(2:4, dnorm, pnorm, mean = 0.5, sd = 1.2, mlen = 5) dextreme(2:4, distn = "norm", mean = 0.5, sd = 1.2, mlen = 5) dextreme(2:4, distn = "exp", mlen = 2, largest = FALSE) pextreme(2:4, distn = "exp", rate = 1.2, mlen = 2) qextreme(seq(0.9, 0.6, -0.1), distn = "exp", rate = 1.2, mlen = 2) rextreme(5, qgamma, shape = 1, mlen = 10) p <- (1:9)/10 pexp(qextreme(p, distn = "exp", rate = 1.2, mlen = 1), rate = 1.2) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/fgev.Rd0000644000175100001440000001732714212006502013041 0ustar hornikusers\name{fgev} \alias{fgev} \alias{fgumbel} \alias{fitted.evd} \alias{std.errors} \alias{std.errors.evd} \alias{vcov.evd} \alias{print.evd} \alias{logLik.evd} \title{Maximum-likelihood Fitting of the Generalized Extreme Value Distribution} \description{ Maximum-likelihood fitting for the generalized extreme value distribution, including linear modelling of the location parameter, and allowing any of the parameters to be held fixed if desired. } \usage{ fgev(x, start, \dots, nsloc = NULL, prob = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) } \arguments{ \item{x}{A numeric vector, which may contain missing values.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized. If \code{start} is omitted the routine attempts to find good starting values using moment estimators.} \item{\dots}{Additional parameters, either for the GEV model or for the optimization function \code{optim}. If parameters of the model are included they will be held fixed at the values given (see \bold{Examples}).} \item{nsloc}{A data frame with the same number of rows as the length of \code{x}, for linear modelling of the location parameter. The data frame is treated as a covariate matrix (excluding the intercept). A numeric vector can be given as an alternative to a single column data frame.} \item{prob}{Controls the parameterization of the model (see \bold{Details}). Should be either \code{NULL} (the default), or a probability in the closed interval [0,1].} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} \item{warn.inf}{Logical; if \code{TRUE} (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.} } \details{ If \code{prob} is \code{NULL} (the default): For stationary models the parameter names are \code{loc}, \code{scale} and \code{shape}, for the location, scale and shape parameters respectively. For non-stationary models, the parameter names are \code{loc}, \code{loc}\emph{x1}, \dots, \code{loc}\emph{xn}, \code{scale} and \code{shape}, where \emph{x1}, \dots, \emph{xn} are the column names of \code{nsloc}, so that \code{loc} is the intercept of the linear model, and \code{loc}\emph{x1}, \dots, \code{loc}\emph{xn} are the \code{ncol(nsloc)} coefficients. If \code{nsloc} is a vector it is converted into a single column data frame with column name \code{trend}, and hence the associated trend parameter is named \code{loctrend}. If \eqn{\code{prob} = p} is a probability: The fit is performed using a different parameterization. Let \eqn{a}, \eqn{b} and \eqn{s} denote the location, scale and shape parameters of the GEV distribution. For stationary models, the distribution is parameterized using \eqn{(z_p,b,s)}, where \deqn{z_p = a - b/s (1 - (-\log(1 - p))^s)}{ z_p = a - b/s (1 - (-log(1 - p))^s)} is such that \eqn{G(z_p) = 1 - p}, where \eqn{G} is the GEV distribution function. \eqn{\code{prob} = p} is therefore the probability in the upper tail corresponding to the quantile \eqn{z_p}. If \code{prob} is zero, then \eqn{z_p} is the upper end point \eqn{a - b/s}, and \eqn{s} is restricted to the negative (Weibull) axis. If \code{prob} is one, then \eqn{z_p} is the lower end point \eqn{a - b/s}, and \eqn{s} is restricted to the positive (Frechet) axis. The parameter names are \code{quantile}, \code{scale} and \code{shape}, for \eqn{z_p}, \eqn{b} and \eqn{s} respectively. For non-stationary models the parameter \eqn{z_p} is again given by the equation above, but \eqn{a} becomes the intercept of the linear model for the location parameter, so that \code{quantile} replaces (the intercept) \code{loc}, and hence the parameter names are \code{quantile}, \code{loc}\emph{x1}, \dots, \code{loc}\emph{xn}, \code{scale} and \code{shape}, where \emph{x1}, \dots, \emph{xn} are the column names of \code{nsloc}. In either case: For non-stationary fitting it is recommended that the covariates within the linear model for the location parameter are (at least approximately) centered and scaled (i.e.\ that the columns of \code{nsloc} are centered and scaled), particularly if automatic starting values are used, since the starting values for the associated parameters are then zero. } \value{ Returns an object of class \code{c("gev","uvevd","evd")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The functions \code{profile} and \code{profile2d} are used to obtain deviance profiles for the model parameters. In particular, profiles of the quantile \eqn{z_p} can be calculated and plotted when \eqn{\code{prob} = p}. The function \code{anova} compares nested models. The function \code{plot} produces diagnostic plots. An object of class \code{c("gev","uvevd","evd")} is a list containing at most the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{fixed}{A vector containing the parameters of the model that have been held fixed.} \item{param}{A vector containing all parameters (optimized and fixed).} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{data}{The data passed to the argument \code{x}.} \item{tdata}{The data, transformed to stationarity (for non-stationary models).} \item{nsloc}{The argument \code{nsloc}.} \item{n}{The length of \code{x}.} \item{prob}{The argument \code{prob}.} \item{loc}{The location parameter. If \code{prob} is \code{NULL} (the default), this will also be an element of \code{param}.} \item{call}{The call of the current function.} } \section{Warning}{ The standard errors and the correlation matrix in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution when the shape parameter is less than \eqn{-0.5}, because the usual asymptotic properties of maximum likelihood estimators do not then hold (Smith, 1985). } \references{ Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. } \seealso{\code{\link{anova.evd}}, \code{\link{optim}}, \code{\link{plot.uvevd}}, \code{\link{profile.evd}}, \code{\link{profile2d.evd}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) trend <- (-49:50)/100 M1 <- fgev(uvdata, nsloc = trend, control = list(trace = 1)) M2 <- fgev(uvdata) M3 <- fgev(uvdata, shape = 0) M4 <- fgev(uvdata, scale = 1, shape = 0) anova(M1, M2, M3, M4) par(mfrow = c(2,2)) plot(M2) \dontrun{M2P <- profile(M2)} \dontrun{plot(M2P)} rnd <- runif(100, min = -.5, max = .5) fgev(uvdata, nsloc = data.frame(trend = trend, random = rnd)) fgev(uvdata, nsloc = data.frame(trend = trend, random = rnd), locrandom = 0) uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata, prob = 0.1) M2 <- fgev(uvdata, prob = 0.01) \dontrun{M1P <- profile(M1, which = "quantile")} \dontrun{M2P <- profile(M2, which = "quantile")} \dontrun{plot(M1P)} \dontrun{plot(M2P)} } \keyword{models} evd/man/sask.Rd0000644000175100001440000000113612637167310013060 0ustar hornikusers\name{sask} \alias{sask} \title{Maximum Annual Flood Discharges of the North Saskachevan River} \usage{sask} \description{ A numeric vector containing maximum annual flood discharges, in units of 1000 cubic feet per second, of the North Saskachevan River at Edmonton, over a period of 47 years. Unfortunately, the data are ordered from largest to smallest. } \format{A vector containing 47 observations.} \source{ van Montfort, M. A. J. (1970) On testing that the distribution is of type I when type II is the alternative. \emph{J. Hydrology}, \bold{11}, 421--427. } \keyword{datasets} evd/man/oldage.Rd0000644000175100001440000000146112637167310013353 0ustar hornikusers\name{oldage} \alias{oldage} \title{Oldest Ages for Swedish Males and Females} \usage{oldage} \description{ The \code{oldage} data frame has 66 rows and 2 columns. The columns contain the oldest ages at death for men and women in Sweden, for the period 1905--1970. The row names give the years of observation. } \format{ This data frame contains the following columns: \describe{ \item{men}{A numeric vector containing the oldest ages at death for men.} \item{women}{A numeric vector containing the oldest ages at death for women.} } } \source{ Fransen, A. and Tiago de Oliveira, J. (1984) Statistical choice of univariate extreme models, part II, in \emph{Statistical Extremes and Applications}, J. Tiago de Oliveira ed., 373--394, D. Reidel, Dordrect. } \keyword{datasets} evd/man/tcplot.Rd0000644000175100001440000001035112637167310013423 0ustar hornikusers\name{tcplot} \alias{tcplot} \title{Threshold Choice Plot} \description{ Plots of parameter estimates at various thresholds for peaks over threshold modelling, using the Generalized Pareto or Point Process representation. } \usage{ tcplot(data, tlim, model = c("gpd","pp"), pscale = FALSE, cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, nt = 25, which = 1:npar, conf = 0.95, lty = 1, lwd = 1, type = "b", cilty = 1, vci = TRUE, xlab, xlim, ylabs, ylims, ask = nb.fig < length(which) && dev.interactive(), \dots) } \arguments{ \item{data}{A numeric vector.} \item{tlim}{A numeric vector of length two, giving the limits for the thresholds at which the model is fitted.} \item{model}{The model; either \code{"gpd"} (the default) or \code{"pp"}, for the Generalized Pareto or Point Process representations respectively.} \item{pscale}{If \code{TRUE}, then the x-axis gives the threshold exceedance probability rather than the threshold itself.} \item{cmax}{Logical; if \code{FALSE} (the default), the models are fitted using all exceedences over the thresholds. If \code{TRUE}, the models are fitted using cluster maxima, using clusters of exceedences derived from \code{clusters}.} \item{r, ulow, rlow}{Arguments used for the identification of clusters of exceedences (see \code{\link{clusters}}). Ignored if \code{cmax} is \code{FALSE} (the default).} \item{nt}{The number of thresholds at which the model is fitted.} \item{which}{If a subset of the plots is required, specify a subset of the numbers \code{1:npar}, where \code{npar} is the number of parameters, so that \code{npar = 2} when \code{model = "gpd"} (the default) and \code{npar = 3} when \code{model = "pp"}.} \item{conf}{The (pointwise) confidence coefficient for the plotted confidence intervals. Use zero to suppress.} \item{lty, lwd}{The line type and width of the line connecting the parameter estimates.} \item{type}{The form taken by the line connecting the parameter estimates and the points denoting these estimates. Possible values include \code{"b"} (the default) for points joined by lines, \code{"o"} for overplotted points and lines, and \code{"l"} for an unbroken line with no points.} \item{cilty}{The line type of the lines depicting the confidence intervals.} \item{vci}{If \code{TRUE} (the default), confidence intervals are plotted using vertical lines.} \item{xlab, xlim}{Label and limits for the x-axis; if given, these arguments apply to every plot.} \item{ylabs, ylims}{A vector of y-axis labels and a matrix of y-axis limits. If given, \code{ylabs} should have the same length as \code{which}, and \code{ylims} should have two columns and \code{length(which)} rows. If the length of \code{which} is one, then \code{ylims} can be a vector of length two.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{\dots}{Other arguments to be passed to the model fit function \code{fpot}.} } \details{ For each of the \code{nt} thresholds a peaks over threshold model is fitted using the function \code{fpot}. When \code{model = "gpd"} (the default), the maximum likelihood estimates for the shape and the modified scale parameter (modified by subtracting the shape multiplied by the threshold) are plotted against the thresholds. When \code{model = "pp"} the maximum likelihood estimates for the location, scale and shape parameters are plotted against the thresholds. (The modified scale parameter in the \code{"gpd"} case is equivalent to the scale parameter in the \code{"pp"} case.) If the threshold \code{u} is a valid threshold to be used for peaks over threshold modelling, the parameter estimates depicted should be approximately constant above \code{u}. } \value{ A list is invisibly returned. Each component is a matrix with three columns giving parameter estimates and confidence limits. } \author{Stuart Coles and Alec Stephenson} \seealso{\code{\link{fpot}}, \code{\link{mrlplot}}, \code{\link{clusters}}} \examples{ tlim <- c(3.6, 4.2) \dontrun{tcplot(portpirie, tlim)} \dontrun{tcplot(portpirie, tlim, nt = 100, lwd = 3, type = "l")} \dontrun{tcplot(portpirie, tlim, model = "pp")} } \keyword{hplot} evd/man/ocmulgee.Rd0000644000175100001440000000165012637167310013720 0ustar hornikusers\name{ocmulgee} \alias{ocmulgee} \title{Maximum Annual Flood Discharges of the Ocmulgee River} \usage{ocmulgee} \description{ The \code{ocmulgee} data frame has 40 rows and 2 columns. The columns contain maximum annual flood discharges, in units of 1000 cubed feet per second, from the Ocmulgee River in Georgia, USA at Hawkinsville (upstream) and Macon (downstream), for the years 1910 to 1949. The row names give the years of observation. } \format{ This data frame contains the following columns: \describe{ \item{hawk}{A numeric vector containing maximum annual flood discharges at Hawkinsville (upstream).} \item{macon}{A numeric vector containing maximum annual flood discharges at Macon (downstream).} } } \source{ Gumbel, E. J. and Goldstein, N. (1964) Analysis of empirical bivariate extremal distributions. \emph{J. Amer. Statist. Assoc.}, \bold{59}, 794--816. } \keyword{datasets} evd/man/bvtcplot.Rd0000644000175100001440000000336012637167310013755 0ustar hornikusers\name{bvtcplot} \alias{bvtcplot} \title{Bivariate Threshold Choice Plot} \description{ Produces a diagnostic plot to assist with threshold choice for bivariate data. } \usage{ bvtcplot(x, spectral = FALSE, xlab, ylab, \dots) } \arguments{ \item{x}{A matrix or data frame, ordinarily with two columns, which may contain missing values.} \item{spectral}{If \code{TRUE}, an estimate of the spectral measure is plotted instead of the diagnostic plot.} \item{ylab, xlab}{Graphics parameters.} \item{\dots}{Other arguments to be passed to the plotting function.} } \details{ If \code{spectral} is \code{FALSE} (the default), produces a threshold choice plot as illustrated in Beirlant et al. (2004). With \eqn{n} non-missing bivariate observations, the integers \eqn{k = 1,\dots,n-1}{k = 1,...,n-1} are plotted against the values \eqn{(k/n)r_{(n-k)}}{(k/n)r_(n-k)}, where \eqn{r_{(n-k)}}{r_(n-k)} is the \eqn{(n-k)}th order statistic of the sum of the margins following empirical transformation to standard Frechet. A vertical line is drawn at \code{k0}, where \code{k0} is the largest integer for which the y-axis is above the value two. If \code{spectral} is \code{FALSE}, the largest \code{k0} data points are used to plot an estimate of the spectal measure \eqn{H([0, w])} versus \eqn{w}. } \value{ A list is invisibly returned giving \code{k0} and the values used to produce the plot. } \references{ Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L. (2004) \emph{Statistics of Extremes: Theory and Applications.}, Chichester, England: John Wiley and Sons. } \seealso{\code{\link{fbvpot}}, \code{\link{tcplot}}} \examples{ \dontrun{bvtcplot(lossalae)} \dontrun{bvtcplot(lossalae, spectral = TRUE)} } \keyword{hplot} evd/man/marma.Rd0000644000175100001440000000614112637167310013215 0ustar hornikusers\name{marma} \alias{marma} \alias{mar} \alias{mma} \title{Simulate MARMA(p,q) Processes} \description{ Simulation of MARMA(p,q) processes. } \usage{ marma(n, p = 0, q = 0, psi, theta, init = rep(0, p), n.start = p, rand.gen = rfrechet, \dots) mar(n, p = 1, psi, init = rep(0, p), n.start = p, rand.gen = rfrechet, \dots) mma(n, q = 1, theta, rand.gen = rfrechet, \dots) } \arguments{ \item{n}{The number of observations.} \item{p}{The AR order of the MARMA process.} \item{q}{The MA order of the MARMA process.} \item{psi}{A vector of non-negative parameters, of length \code{p}. Can be omitted if \code{p} is zero.} \item{theta}{A vector of non-negative parameters, of length \code{q}. Can be omitted if \code{q} is zero.} \item{init}{A vector of non-negative starting values, of length \code{p}.} \item{n.start}{A non-negative value denoting the length of the burn-in period. If \code{n.start} is less than \code{p}, then \code{p} minus \code{n.start} starting values will be included in the output series.} \item{rand.gen}{A simulation function to generate the innovations.} \item{\dots}{Additional arguments for \code{rand.gen}. Most usefully, the scale and shape parameters of the innovations generated by \code{rfrechet} can be specified by \code{scale} and \code{shape} respectively.} } \details{ A max autoregressive moving average process \eqn{\{X_k\}}{{X_k}}, denoted by MARMA(p,q), is defined in Davis and Resnick (1989) as satisfying \deqn{X_k = \max\{\phi_1 X_{k-1}, \ldots, \phi_p X_{k-p}, \epsilon_k, \theta_1 \epsilon_{k-1}, \ldots, \theta_q \epsilon_{k-q}\}}{ X_k = max[phi_1 X_{k-1}, \ldots, phi_p X_{k-p}, epsilon_k, theta_1 epsilon_{k-1}, \ldots, theta_q epsilon_{k-q}]} where \eqn{\code{phi} = (\phi_1, \ldots, \phi_p)}{ \code{phi} = (phi_1, \ldots, phi_p)} and \eqn{\code{theta} = (\theta_1, \ldots, \theta_q)}{ \code{theta} = (theta_1, \ldots, theta_q)} are non-negative vectors of parameters, and where \eqn{\{\epsilon_k\}}{{epsilon_k}} is a series of \emph{iid} random variables with a common distribution defined by \code{rand.gen}. The functions \code{mar} and \code{mma} generate MAR(p) and MMA(q) processes respectively. A MAR(p) process \eqn{\{X_k\}}{{X_k}} is equivalent to a MARMA(p, 0) process, so that \deqn{X_k = \max\{\phi_1 X_{k-1}, \ldots, \phi_p X_{k-p}, \epsilon_k\}.}{X_k = max[phi_1 X_{k-1}, \ldots, phi_p X_{k-p}, epsilon_k].} A MMA(q) process \eqn{\{X_k\}}{{X_k}} is equivalent to a MARMA(0, q) process, so that \deqn{X_k = \max\{\epsilon_k, \theta_1 \epsilon_{k-1}, \ldots, \theta_q \epsilon_{k-q}\}.}{X_k = max[epsilon_k, theta_1 epsilon_{k-1}, \ldots, theta_q epsilon_{k-q}].} } \value{ A numeric vector of length \code{n}. } \references{ Davis, R. A. and Resnick, S. I. (1989) Basic properties and prediction of max-arma processes. \emph{Adv. Appl. Prob.}, \bold{21}, 781--803. } \seealso{\code{\link{evmc}}} \examples{ marma(100, p = 1, q = 1, psi = 0.75, theta = 0.65) mar(100, psi = 0.85, n.start = 20) mma(100, q = 2, theta = c(0.75, 0.8)) } \keyword{distribution} evd/man/qcbvnonpar.Rd0000644000175100001440000001104512637167310014270 0ustar hornikusers\name{qcbvnonpar} \alias{qcbvnonpar} \title{Non-parametric Estimates for Bivariate Quantile Curves} \description{ Calculate or plot non-parametric estimates for quantile curves of bivariate extreme value distributions. } \usage{ qcbvnonpar(p = seq(0.75, 0.95, 0.05), data, epmar = FALSE, nsloc1 = NULL, nsloc2 = NULL, mint = 1, method = c("cfg", "pickands", "tdo"), convex = FALSE, madj = 0, kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, xlim = range(data[,1], na.rm = TRUE), ylim = range(data[,2], na.rm = TRUE), xlab = colnames(data)[1], ylab = colnames(data)[2], \dots) } \arguments{ \item{p}{A vector of lower tail probabilities. One quantile curve is calculated or plotted for each probability.} \item{data}{A matrix or data frame with two columns, which may contain missing values.} \item{epmar}{If \code{TRUE}, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the \code{nsloc} arguments are ignored.} \item{nsloc1, nsloc2}{A data frame with the same number of rows as \code{data}, for linear modelling of the location parameter on the first/second margin. The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.} \item{mint}{An integer \eqn{m}. Quantile curves are plotted or calculated using the lower tail probabilities \eqn{p^m}.} \item{method, kmar}{Arguments for the non-parametric estimate of the dependence function. See \code{\link{abvnonpar}}.} \item{convex, madj}{Other arguments for the non-parametric estimate of the dependence function.} \item{plot}{Logical; if \code{TRUE} the data is plotted along with the quantile curves. If \code{plot} and \code{add} are \code{FALSE} (the default), the arguments following \code{add} are ignored.} \item{add}{Logical; add quantile curves to an existing data plot? The existing plot should have been created using either \code{qcbvnonpar} or \code{\link{plot.bvevd}}, the latter of which can plot quantile curves for parametric fits.} \item{lty, lwd}{Line types and widths.} \item{col}{Line colour.} \item{xlim, ylim}{x and y-axis limits.} \item{xlab, ylab}{x and y-axis labels.} \item{\dots}{Other high-level graphics parameters to be passed to \code{plot}.} } \details{ Let G be a fitted bivariate distribution function with margins \eqn{G_1} and \eqn{G_2}. A quantile curve for a fitted distribution function G at lower tail probability p is defined by \deqn{Q(G, p) = \{(y_1,y_1):G(y_1,y_2) = p\}.}{ Q(G, p) = {(y_1,y_1):G(y_1,y_2) = p}.} For bivariate extreme value distributions, it consists of the points \deqn{\left\{G_1^{-1}(p_1),G_2^{-1}(p_2))\right\}}{ {G_1^{-1}(p_1),G_2^{-1}(p_2))}} where \eqn{p_1 = p^{t/A(t)}} and \eqn{p_2 = p^{(1-t)/A(t)}}, with \eqn{A} being the estimated dependence function defined in \code{\link{abvevd}}, and where \eqn{t} lies in the interval \eqn{[0,1]}. By default the margins \eqn{G_1} and \eqn{G_2} are modelled using estimated generalized extreme value distributions. For non-stationary generalized extreme value margins the plotted data are transformed to stationarity, and the plot corresponds to the distribution obtained when all covariates are zero. If \code{epmar} is \code{TRUE}, empirical transformations are used in preference to generalized extreme value models. Note that the marginal empirical quantile functions are evaluated using \code{\link{quantile}}, which linearly interpolates between data points, hence the curve will not be a step function. The idea behind the argument \eqn{\code{mint} = m} is that if G is fitted to a dataset of componentwise maxima, and the underlying observations are \emph{iid} distributed according to F, then if \eqn{m} is the size of the blocks over which the maxima were taken, approximately \eqn{F^m = G}, leading to \eqn{Q(F, p) = Q(G, p^m)}. } \value{ \code{qcbvnonpar} calculates or plots non-parametric quantile curve estimates for bivariate extreme value distributions. If \code{p} has length one it returns a two column matrix giving points on the curve, else it returns a list of such matrices. } \seealso{\code{\link{abvevd}}, \code{\link{abvnonpar}}, \code{\link{plot.bvevd}}} \examples{ bvdata <- rbvevd(100, dep = 0.7, model = "log") qcbvnonpar(c(0.9,0.95), data = bvdata, plot = TRUE) qcbvnonpar(c(0.9,0.95), data = bvdata, epmar = TRUE, plot = TRUE) } \keyword{nonparametric} evd/man/mrlplot.Rd0000644000175100001440000000435512637167310013616 0ustar hornikusers\name{mrlplot} \alias{mrlplot} \title{Empirical Mean Residual Life Plot} \description{ The empirical mean residual life plot. } \usage{ mrlplot(data, tlim, pscale = FALSE, nt = max(100, length(data)), lty = c(2,1,2), col = 1, conf = 0.95, main = "Mean Residual Life Plot", xlab = "Threshold", ylab = "Mean Excess", \dots) } \arguments{ \item{data}{A numeric vector.} \item{tlim}{A numeric vector of length two, giving the limits for the thresholds at which the mean residual life plot is evaluated. If \code{tlim} is not given, sensible defaults are used.} \item{pscale}{If \code{TRUE}, then the x-axis gives the threshold exceedance probability rather than the threshold itself.} \item{nt}{The number of thresholds at which the mean residual life plot is evaluated.} \item{lty, col}{Arguments passed to \code{matplot}. The first and last elements of \code{lty} correspond to the lower and upper confidence limits respectively. Use zero to supress.} \item{conf}{The (pointwise) confidence coefficient for the plotted confidence intervals.} \item{main}{Plot title.} \item{xlab, ylab}{x and y axis labels.} \item{\dots}{Other arguments to be passed to \code{matplot}.} } \details{ The empirical mean residual life plot is the locus of points \deqn{\left(u,\frac{1}{n_u} \sum\nolimits_{i=1}^{n_u} (x_{(i)} - u) \right)}{{u,1/n_u \sum_{i=1}^{n_u} (x(i) - u)}} where \eqn{x_{(1)}, \dots, x_{(n_u)}}{x(1), \dots, x(n_u)} are the \eqn{n_u} observations that exceed the threshold \eqn{u}. If the exceedances of a threshold \eqn{u_0}{u0} are generalized Pareto, the empirical mean residual life plot should be approximately linear for \eqn{u > u_0}{u > u0}. The confidence intervals within the plot are symmetric intervals based on the approximate normality of sample means. } \value{ A list with components \code{x} and \code{y} is invisibly returned. The components contain those objects that were passed to the formal arguments \code{x} and \code{y} of \code{matplot} in order to create the mean residual life plot. } \author{Stuart Coles and Alec Stephenson} \seealso{\code{\link{fpot}}, \code{\link{matplot}}, \code{\link{tcplot}}} \examples{ mrlplot(portpirie) } \keyword{hplot} evd/man/fpot.Rd0000644000175100001440000002177212637167310013077 0ustar hornikusers\name{fpot} \alias{fpot} \alias{print.pot} \title{Peaks Over Threshold Modelling using the Generalized Pareto or Point Process Representation} \description{ Maximum-likelihood fitting for peaks over threshold modelling, using the Generalized Pareto or Point Process representation, allowing any of the parameters to be held fixed if desired. } \usage{ fpot(x, threshold, model = c("gpd", "pp"), start, npp = length(x), cmax = FALSE, r = 1, ulow = -Inf, rlow = 1, mper = NULL, \dots, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) } \arguments{ \item{x}{A numeric vector. If this contains missing values, those values are treated as if they fell below the threshold.} \item{threshold}{The threshold.} \item{model}{The model; either \code{"gpd"} (the default) or \code{"pp"}, for the Generalized Pareto or Point Process representations respectively.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized. If \code{start} is omitted the routine attempts to find good starting values using moment estimators.} \item{npp}{The data should contain \code{npp} observations per ``period'', where the return level plot produced by \code{plot.pot} will represent return periods in units of ``periods''. By default \code{npp = length(x)}, so that the ``period'' is the period of time over which the entire data set is collected. It may often be useful to change this default so that more sensible units are used. For example, if yearly periodic units are required, use \code{npp = 365.25} for daily data and \code{npp = 52.18} for weekly data. The argument only makes a difference to the actual fit if \code{mper} is not \code{NULL} or if \code{model = "pp"} (see \bold{Details}).} \item{cmax}{Logical; if \code{FALSE} (the default), the model is fitted using all exceedences over the threshold. If \code{TRUE}, the model is fitted using cluster maxima, using clusters of exceedences derived from \code{clusters}.} \item{r, ulow, rlow}{Arguments used for the identification of clusters of exceedences (see \code{\link{clusters}}). Ignored if \code{cmax} is \code{FALSE} (the default).} \item{mper}{Controls the parameterization of the generalized Pareto model. Should be either \code{NULL} (the default), or a positive number (see \bold{Details}). If \code{mper} is not \code{NULL} and \code{model = "pp"}, an error is returned.} \item{\dots}{Additional parameters, either for the model or for the optimization function \code{optim}. If parameters of the model are included they will be held fixed at the values given (see \bold{Examples}).} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} \item{warn.inf}{Logical; if \code{TRUE} (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.} } \details{ The exeedances over the threshold \code{threshold} (if \code{cmax} is \code{FALSE}) or the maxima of the clusters of exeedances (if \code{cmax} is \code{TRUE}) are (if \code{model = "gpd"}) fitted to a generalized Pareto distribution (GPD) with location \code{threshold}. If \code{model = "pp"} the exceedances are fitted to a non-homogeneous Poisson process (Coles, 2001). If \code{mper} is \code{NULL} (the default), the parameters of the model (if \code{model = "gpd"}) are \code{scale} and \code{shape}, for the scale and shape parameters of the GPD. If \code{model = "pp"} the parameters are \code{loc}, \code{scale} and \code{shape}. Under \code{model = "pp"} the parameters can be interpreted as parameters of the Generalized Extreme Value distribution, fitted to the maxima of \code{npp} random variables. In this case, the value of \code{npp} should be reasonably large. For both characterizations, the shape parameters are equivalent. The scale parameter under the generalized Pareto characterization is equal to \eqn{b + s(u - a)}, where \eqn{a}, \eqn{b} and \eqn{s} are the location, scale and shape parameters under the Point Process characterization, and where \eqn{u} is the threshold. If \eqn{\code{mper} = m} is a positive value, then the generalized Pareto model is reparameterized so that the parameters are \code{rlevel} and \code{shape}, where \code{rlevel} is the \eqn{m} ``period'' return level, where ``period'' is defined via the argument \code{npp}. The \eqn{m} ``period'' return level is defined as follows. Let \eqn{G} be the fitted generalized Pareto distribution function, with location \eqn{\code{threshold} = u}, so that \eqn{1 - G(z)} is the fitted probability of an exceedance over \eqn{z > u} given an exceedance over \eqn{u}. The fitted probability of an exceedance over \eqn{z > u} is therefore \eqn{p(1 - G(z))}, where \eqn{p} is the estimated probabilty of exceeding \eqn{u}, which is given by the empirical proportion of exceedances. The \eqn{m} ``period'' return level \eqn{z_m} satisfies \eqn{p(1 - G(z_m)) = 1/(mN)}, where \eqn{N} is the number of points per period (multiplied by the estimate of the extremal index, if cluster maxima are fitted). In other words, \eqn{z_m} is the quantile of the fitted model that corresponds to the upper tail probability \eqn{1/(mN)}. If \code{mper} is infinite, then \eqn{z_m} is the upper end point, given by \code{threshold} minus \eqn{\code{scale}/\code{shape}}, and the shape parameter is then restricted to be negative. } \value{ Returns an object of class \code{c("pot","uvevd","pot")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The function \code{profile} can be used to obtain deviance profiles for the model parameters. In particular, profiles of the \eqn{m} \code{period} return level \eqn{z_m} can be calculated and plotted when \eqn{\code{mper} = m}. The function \code{anova} compares nested models. The function \code{plot} produces diagnostic plots. An object of class \code{c("pot","uvevd","evd")} is a list containing the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{fixed}{A vector containing the parameters of the model that have been held fixed.} \item{param}{A vector containing all parameters (optimized and fixed).} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{threshold, r, ulow, rlow, npp}{The arguments of the same name.} \item{nhigh}{The number of exceedences (if \code{cmax} is \code{FALSE}) or the number of clusters of exceedences (if \code{cmax} is \code{TRUE}).} \item{nat, pat}{The number and proportion of exceedences.} \item{extind}{The estimate of the extremal index (i.e. \code{nhigh} divided by \code{nat}). If \code{cmax} is \code{FALSE}, this is \code{NULL}.} \item{data}{The data passed to the argument \code{x}.} \item{exceedances}{The exceedences, or the maxima of the clusters of exceedences.} \item{mper}{The argument \code{mper}.} \item{scale}{The scale parameter for the fitted generalized Pareto distribution. If \code{mper} is \code{NULL} and \code{model = "gpd"} (the defaults), this will also be an element of \code{param}.} \item{call}{The call of the current function.} } \section{Warning}{ The standard errors and the correlation matrix in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution when the shape parameter is less than \eqn{-0.5}, because the usual asymptotic properties of maximum likelihood estimators do not then hold (Smith, 1985). } \references{ Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. } \seealso{\code{\link{anova.evd}}, \code{\link{optim}}, \code{\link{plot.uvevd}}, \code{\link{profile.evd}}, \code{\link{profile2d.evd}}, \code{\link{mrlplot}}, \code{\link{tcplot}}} \examples{ uvdata <- rgpd(100, loc = 0, scale = 1.1, shape = 0.2) M1 <- fpot(uvdata, 1) M2 <- fpot(uvdata, 1, shape = 0) anova(M1, M2) par(mfrow = c(2,2)) plot(M1) \dontrun{M1P <- profile(M1)} \dontrun{plot(M1P)} M1 <- fpot(uvdata, 1, mper = 10) M2 <- fpot(uvdata, 1, mper = 100) \dontrun{M1P <- profile(M1, which = "rlevel", conf=0.975, mesh=0.1)} \dontrun{M2P <- profile(M2, which = "rlevel", conf=0.975, mesh=0.1)} \dontrun{plot(M1P)} \dontrun{plot(M2P)} } \keyword{models} evd/man/evd-internal.Rd0000644000175100001440000000750614225012305014502 0ustar hornikusers\name{evd-internal} \alias{abvalog} \alias{abvaneglog} \alias{abvhr} \alias{abvlog} \alias{abvneglog} \alias{abvbilog} \alias{abvnegbilog} \alias{abvct} \alias{abvamix} \alias{hbvalog} \alias{hbvaneglog} \alias{hbvhr} \alias{hbvlog} \alias{hbvneglog} \alias{hbvbilog} \alias{hbvnegbilog} \alias{hbvct} \alias{hbvamix} \alias{rbvlog} \alias{rbvalog} \alias{rbvhr} \alias{rbvneglog} \alias{rbvaneglog} \alias{rbvbilog} \alias{rbvnegbilog} \alias{rbvct} \alias{rbvamix} \alias{dbvlog} \alias{dbvalog} \alias{dbvhr} \alias{dbvneglog} \alias{dbvaneglog} \alias{dbvbilog} \alias{dbvnegbilog} \alias{dbvct} \alias{dbvamix} \alias{pbvlog} \alias{pbvalog} \alias{pbvhr} \alias{pbvneglog} \alias{pbvaneglog} \alias{pbvbilog} \alias{pbvnegbilog} \alias{pbvct} \alias{pbvamix} \alias{amvalog} \alias{amvlog} \alias{rmvlog} \alias{rmvalog} \alias{pmvlog} \alias{pmvalog} \alias{dmvlog} \alias{dmvalog} \alias{tvdepfn} \alias{mvalog.check} \alias{subsets} \alias{fgev.quantile} \alias{fgev.norm} \alias{fpot.quantile} \alias{fpot.norm} \alias{fbvlog} \alias{fbvalog} \alias{fbvhr} \alias{fbvneglog} \alias{fbvaneglog} \alias{fbvbilog} \alias{fbvnegbilog} \alias{fbvct} \alias{fbvamix} \alias{fbvcpot} \alias{fbvclog} \alias{fbvcalog} \alias{fbvcaneglog} \alias{fbvcbilog} \alias{fbvcct} \alias{fbvcnegbilog} \alias{fbvcneglog} \alias{fbvchr} \alias{fbvcamix} \alias{fbvppot} \alias{fbvplog} \alias{fbvpbilog} \alias{fbvpct} \alias{fbvpnegbilog} \alias{fbvpneglog} \alias{fbvphr} \alias{bvpost.optim} \alias{bvstart.vals} \alias{sep.bvdata} \alias{dens} \alias{pp} \alias{qq} \alias{rl} \alias{dens.gev} \alias{pp.gev} \alias{qq.gev} \alias{rl.gev} \alias{dens.pot} \alias{pp.pot} \alias{qq.pot} \alias{rl.pot} \alias{dens.gumbelx} \alias{pp.gumbelx} \alias{qq.gumbelx} \alias{rl.gumbelx} \alias{bvcpp} \alias{bvdens} \alias{bvdp} \alias{bvqc} \alias{bvh} \alias{bvcpp.bvevd} \alias{bvdens.bvevd} \alias{bvdp.bvevd} \alias{bvqc.bvevd} \alias{bvh.bvevd} \alias{bvdens.bvpot} \alias{bvdp.bvpot} \alias{bvqc.bvpot} \alias{bvh.bvpot} \title{Internal Functions} \description{ The evd package contains many internal functions that are not designed to be called by the user. Plotting: The generic functions \code{dens}, \code{pp}, \code{qq} and \code{rl} create the diagnostic plots generated by \code{plot.uvevd}. Similarly, \code{bvdens}, \code{bvcpp}, \code{bvdp}, \code{bvqc} and \code{bvh} create the diagnostic plots generated by \code{plot.bvevd} and \code{plot.bvpot}. Distribution: There are internal, simulation, distribution, density, dependence and spectral density functions for separate bivariate and multivariate parametric models, which are called from functions such as \code{abvevd} and \code{pmvevd}. Additionally, the three functions \code{mvalog.check} (checks and transforms asymmetry parameters), \code{subsets} (generates all subsets of a set) and \code{tvdepfn} (plots trivariate dependence functions) are called from functions associated with multivariate distributions. Univariate Fitting: The fitting function \code{fgev} calls the internal functions \code{fgev.quantile} and \code{fgev.norm} for fits under different parameterizations. The fitting function \code{fpot} calls the internal functions \code{fpot.norm} and \code{fpot.quantile}. Bivariate Fitting: For fitting bivariate distributions, internal functions exist for each model. For fitting bivariate threshold models, internal functions exist for the censored and (undocumented) point process likelihoods, and each of these calls a further internal function corresponding to the specified model. The functions \code{bvpost.optim} (post-optimisation processing), \code{bvstart.vals} (starting values) and \code{sep.bvdata} (separation of data) are additionally used in the fitting of bivariate distributions and bivariate threshold models. } \keyword{internal} evd/man/venice2.Rd0000644000175100001440000000166112637167310013455 0ustar hornikusers\name{venice2} \alias{venice2} \title{Largest Sea Levels in Venice} \usage{venice2} \description{ The \code{venice2} data frame has 125 rows and 10 columns. The data was kindly provided by Anthony Davison. The jth column contains the jth largest sea levels in Venice, for the years 1887--2011. This is a larger version of the dataset \code{venice}. Only the largest six measurements are available for the year 1935, and only the largest is available for 1922; the corresponding rows contain missing values. The years for each set of measurements are given as row names. } \format{A data frame with 125 rows and 10 columns.} \source{ Smith, R. L. (1986) Extreme value theory based on the \eqn{r} largest annual events. \emph{Journal of Hydrology}, \bold{86}, 27--43. } \references{ Coles, S. G. (2001) \emph{An Introduction to Statistical Modeling of Extreme Values}. London: Springer-Verlag. } \keyword{datasets} evd/man/fextreme.Rd0000644000175100001440000000731312637167310013741 0ustar hornikusers\name{fextreme} \alias{fextreme} \title{Maximum-likelihood Fitting of Maxima and Minima} \description{ Maximum-likelihood fitting for the distribution of the maximum/minimum of a given number of independent variables from a specified distribution. } \usage{ fextreme(x, start, densfun, distnfun, \dots, distn, mlen = 1, largest = TRUE, std.err = TRUE, corr = FALSE, method = "Nelder-Mead") } \arguments{ \item{x}{A numeric vector.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized.} \item{densfun, distnfun}{Density and distribution function of the specified distribution.} \item{\dots}{Additional parameters, either for the specified distribution or for the optimization function \code{optim}. If parameters of the distribution are included they will be held fixed at the values given (see \bold{Examples}). If parameters of the distribution are not included either here or as a named component in \code{start} they will be held fixed at the default values specified in the corresponding density and distribution functions (assuming they exist; an error will be generated otherwise).} \item{distn}{A character string, optionally specified as an alternative to \code{densfun} and \code{distnfun} such that the density and distribution functions are formed upon the addition of the prefixes \code{d} and \code{p} respectively.} \item{mlen}{The number of independent variables.} \item{largest}{Logical; if \code{TRUE} (default) use maxima, otherwise minima.} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} } \details{ Maximization of the log-likelihood is performed. The estimated standard errors are taken from the observed information, calculated by a numerical approximation. If the density and distribution functions are user defined, the order of the arguments must mimic those in R base (i.e. data first, parameters second). Density functions must have \code{log} arguments. } \value{ Returns an object of class \code{c("extreme","evd")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The function \code{anova} compares nested models. An object of class \code{c("extreme","evd")} is a list containing at most the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{call}{The call of the current function.} \item{data}{The data passed to the argument \code{x}.} \item{n}{The length of \code{x}.} } \seealso{\code{\link{anova.evd}}, \code{\link{forder}}, \code{\link{optim}}} \examples{ uvdata <- rextreme(100, qnorm, mean = 0.56, mlen = 365) fextreme(uvdata, list(mean = 0, sd = 1), distn = "norm", mlen = 365) fextreme(uvdata, list(rate = 1), distn = "exp", mlen = 365, method = "Brent", lower=0.01, upper=10) fextreme(uvdata, list(scale = 1), shape = 1, distn = "gamma", mlen = 365, method = "Brent", lower=0.01, upper=10) fextreme(uvdata, list(shape = 1, scale = 1), distn = "gamma", mlen = 365) } \keyword{models} evd/man/evmc.Rd0000644000175100001440000000431312637167310013051 0ustar hornikusers\name{evmc} \alias{evmc} \title{Simulate Markov Chains With Extreme Value Dependence Structures} \description{ Simulation of first order Markov chains, such that each pair of consecutive values has the dependence structure of one of nine parametric bivariate extreme value distributions. } \usage{ evmc(n, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), margins = c("uniform","rweibull","frechet","gumbel")) } \arguments{ \item{n}{Number of observations.} \item{dep}{Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.} \item{asy}{A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.} \item{alpha, beta}{Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in \code{\link{rbvevd}}. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.} \item{margins}{The marginal distribution of each value; a character string. Must be either \code{"uniform"} (the default), \code{"rweibull"}, \code{"frechet"} or \code{"gumbel"} (or any unique partial match), for the uniform, standard reverse Weibull, standard Gumbel and standard Frechet distributions respectively.} } \value{ A numeric vector of length \code{n}. } \seealso{\code{\link{marma}}, \code{\link{rbvevd}}} \examples{ evmc(100, alpha = 0.1, beta = 0.1, model = "bilog") evmc(100, dep = 10, model = "hr", margins = "gum") } \keyword{distribution} evd/man/plot.bvpot.Rd0000644000175100001440000000504412637167310014230 0ustar hornikusers\name{plot.bvpot} \alias{plot.bvpot} \title{Plot Diagnostics for a Bivariate POT EVD Object} \description{ Four plots (selectable by \code{which}) are currently provided: a density plot (1), a dependence function plot (2), a quantile curves plot (3) and a spectral density plot (4). Plot diagnostics for the generalized Pareto peaks-over-threshold margins (selectable by \code{mar} and \code{which}) are also available. } \usage{ \method{plot}{bvpot}(x, mar = 0, which = 1:4, main, ask = nb.fig < length(which) && dev.interactive(), grid = 50, above = FALSE, levels = NULL, tlty = 1, blty = 3, rev = FALSE, p = seq(0.75, 0.95, 0.05), half = FALSE, \dots) } \arguments{ \item{x}{An object of class \code{"bvpot"}.} \item{mar}{If \code{mar = 1} or \code{mar = 2} diagnostics are given for the first or second generalized Pareto margin respectively.} \item{which}{A subset of the numbers \code{1:4} selecting the plots to be shown. By default all are plotted.} \item{main}{Title of each plot. If given, should be a character vector with the same length as \code{which}.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{grid, levels}{Arguments for the density plot. The data is plotted with a contour plot of the bivariate density of the fitted model in the tail region. The density is evaluated at \code{grid^2} points, and contours are plotted at the values given in the numeric vector \code{levels}. If \code{levels} is \code{NULL} (the default), the routine attempts to find sensible values.} \item{above}{Logical; if \code{TRUE}, only data points above both marginal thresholds are plotted.} \item{tlty}{Line type for the lines identifying the thresholds.} \item{rev, blty}{Arguments to the dependence function plot. See \code{\link{abvevd}}.} \item{p}{Lower tail probabilities for the quantile curves plot. The plot is of the same type as given by the function \code{\link{qcbvnonpar}}, but applied to the parametric bivariate threshold model.} \item{half}{Argument to the spectral density plot. See \code{\link{hbvevd}}.} \item{\dots}{Other arguments to be passed through to plotting functions.} } \seealso{\code{\link{plot.bvevd}}, \code{\link{contour}}, \code{\link{abvnonpar}}, \code{\link{qcbvnonpar}}, \code{\link{hbvevd}}} \examples{ bvdata <- rbvevd(500, dep = 0.6, model = "log") M1 <- fbvpot(bvdata, threshold = c(0,0), model = "log") \dontrun{plot(M1)} \dontrun{plot(M1, mar = 1)} \dontrun{plot(M1, mar = 2)} } \keyword{hplot} evd/man/gpd.Rd0000644000175100001440000000360612637167310012675 0ustar hornikusers\name{gpd} \alias{dgpd} \alias{pgpd} \alias{qgpd} \alias{rgpd} \title{The Generalized Pareto Distribution} \description{ Density function, distribution function, quantile function and random generation for the generalized Pareto distribution (GPD) with location, scale and shape parameters. } \usage{ dgpd(x, loc=0, scale=1, shape=0, log = FALSE) pgpd(q, loc=0, scale=1, shape=0, lower.tail = TRUE) qgpd(p, loc=0, scale=1, shape=0, lower.tail = TRUE) rgpd(n, loc=0, scale=1, shape=0) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{loc, scale, shape}{Location, scale and shape parameters; the \code{shape} argument cannot be a vector (must have length one).} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} } \details{ The generalized Pareto distribution function (Pickands, 1975) with parameters \eqn{\code{loc} = a}, \eqn{\code{scale} = b} and \eqn{\code{shape} = s} is \deqn{G(z) = 1 - \{1+s(z-a)/b\}^{-1/s}}{ G(z) = 1 - {1+s(z-a)/b}^(-1/s)} for \eqn{1+s(z-a)/b > 0} and \eqn{z > a}, where \eqn{b > 0}. If \eqn{s = 0} the distribution is defined by continuity. } \value{ \code{dgpd} gives the density function, \code{pgpd} gives the distribution function, \code{qgpd} gives the quantile function, and \code{rgpd} generates random deviates. } \references{ Pickands, J. (1975) Statistical inference using extreme order statistics. \emph{Annals of Statistics}, \bold{3}, 119--131. } \seealso{\code{\link{fpot}}, \code{\link{rgev}}} \examples{ dgpd(2:4, 1, 0.5, 0.8) pgpd(2:4, 1, 0.5, 0.8) qgpd(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8) rgpd(6, 1, 0.5, 0.8) p <- (1:9)/10 pgpd(qgpd(p, 1, 2, 0.8), 1, 2, 0.8) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/bvevd.Rd0000644000175100001440000003552712637167310013240 0ustar hornikusers\name{bvevd} \alias{dbvevd} \alias{pbvevd} \alias{rbvevd} \title{Parametric Bivariate Extreme Value Distributions} \description{ Density function, distribution function and random generation for nine parametric bivariate extreme value models. } \usage{ dbvevd(x, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) pbvevd(q, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0, 1, 0), mar2 = mar1, lower.tail = TRUE) rbvevd(n, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), mar1 = c(0, 1, 0), mar2 = mar1) } \arguments{ \item{x, q}{A vector of length two or a matrix with two columns, in which case the density/distribution is evaluated across the rows.} \item{n}{Number of observations.} \item{dep}{Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.} \item{asy}{A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.} \item{alpha, beta}{Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.} \item{mar1, mar2}{Vectors of length three containing marginal parameters, or matrices with three columns where each column represents a vector of values to be passed to the corresponding marginal parameter.} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), the distribution function is returned; the survivor function is returned otherwise.} } \details{ Define \deqn{y_i = y_i(z_i) = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}}{ yi = yi(zi) = {1+si(zi-ai)/bi}^(-1/si)} for \eqn{1+s_i(z_i-a_i)/b_i > 0}{1+si(zi-ai)/bi > 0} and \eqn{i = 1,2}, where the marginal parameters are given by \eqn{\code{mari} = (a_i,b_i,s_i)}{\code{mari} = (ai,bi,si)}, \eqn{b_i > 0}{bi > 0}. If \eqn{s_i = 0}{si = 0} then \eqn{y_i}{yi} is defined by continuity. In each of the bivariate distributions functions \eqn{G(z_1,z_2)}{G(z1,z2)} given below, the univariate margins are generalized extreme value, so that \eqn{G(z_i) = \exp(-y_i)}{G(zi) = \exp(-yi)} for \eqn{i = 1,2}. If \eqn{1+s_i(z_i-a_i)/b_i \leq 0}{1+si(zi-ai)/bi <= 0} for some \eqn{i = 1,2}, the value \eqn{z_i}{zi} is either greater than the upper end point (if \eqn{s_i < 0}{si < 0}), or less than the lower end point (if \eqn{s_i > 0}{si > 0}), of the \eqn{i}th univariate marginal distribution. \code{model = "log"} (Gumbel, 1960) The bivariate logistic distribution function with parameter \eqn{\code{dep} = r} is \deqn{G(z_1,z_2) = \exp\left[-(y_1^{1/r}+y_2^{1/r})^r\right]}{ G(z1,z2) = exp{-[y1^(1/r)+y2^(1/r)]^r}} where \eqn{0 < r \leq 1}{0 < r <= 1}. This is a special case of the bivariate asymmetric logistic model. Complete dependence is obtained in the limit as \eqn{r} approaches zero. Independence is obtained when \eqn{r = 1}. \code{model = "alog"} (Tawn, 1988) The bivariate asymmetric logistic distribution function with parameters \eqn{\code{dep} = r} and \eqn{\code{asy} = (t_1,t_2)}{\code{asy} = (t1,t2)} is \deqn{G(z_1,z_2) = \exp\left\{-(1-t_1)y_1-(1-t_2)y_2- [(t_1y_1)^{1/r}+(t_2y_2)^{1/r}]^r\right\}}{ G(z1,z2) = exp{-(1-t1)y1-(1-t2)y2-[(t1y1)^(1/r)+(t2y2)^(1/r)]^r}} where \eqn{0 < r \leq 1}{0 < r <= 1} and \eqn{0 \leq t_1,t_2 \leq 1}{0 <= t1,t2 <= 1}. When \eqn{t_1 = t_2 = 1}{t1 = t2 = 1} the asymmetric logistic model is equivalent to the logistic model. Independence is obtained when either \eqn{r = 1}, \eqn{t_1 = 0}{t1 = 0} or \eqn{t_2 = 0}{t2 = 0}. Complete dependence is obtained in the limit when \eqn{t_1 = t_2 = 1}{t1 = t2 = 1} and \eqn{r} approaches zero. Different limits occur when \eqn{t_1}{t1} and \eqn{t_2}{t2} are fixed and \eqn{r} approaches zero. \code{model = "hr"} (Husler and Reiss, 1989) The Husler-Reiss distribution function with parameter \eqn{\code{dep} = r} is \deqn{G(z_1,z_2) = \exp\left(-y_1\Phi\{r^{-1}+{\textstyle\frac{1}{2}} r[\log(y_1/y_2)]\} - y_2\Phi\{r^{-1}+{\textstyle\frac{1}{2}}r [\log(y_2/y_1)]\}\right)}{ G(z1,z2) = exp(-y1 Phi{r^{-1}+r[log(y1/y2)]/2} - y2 Phi{r^{-1}+r[log(y2/y1)]/2}} where \eqn{\Phi(\cdot)}{Phi()} is the standard normal distribution function and \eqn{r > 0}. Independence is obtained in the limit as \eqn{r} approaches zero. Complete dependence is obtained as \eqn{r} tends to infinity. \code{model = "neglog"} (Galambos, 1975) The bivariate negative logistic distribution function with parameter \eqn{\code{dep} = r} is \deqn{G(z_1,z_2) = \exp\left\{-y_1-y_2+ [y_1^{-r}+y_2^{-r}]^{-1/r}\right\}}{ G(z1,z2) = exp{-y1-y2+[y1^(-r)+y2^(-r)]^(-1/r)}} where \eqn{r > 0}. This is a special case of the bivariate asymmetric negative logistic model. Independence is obtained in the limit as \eqn{r} approaches zero. Complete dependence is obtained as \eqn{r} tends to infinity. The earliest reference to this model appears to be Galambos (1975, Section 4). \code{model = "aneglog"} (Joe, 1990) The bivariate asymmetric negative logistic distribution function with parameters parameters \eqn{\code{dep} = r} and \eqn{\code{asy} = (t_1,t_2)}{\code{asy} = (t1,t2)} is \deqn{G(z_1,z_2) = \exp\left\{-y_1-y_2+ [(t_1y_1)^{-r}+(t_2y_2)^{-r}]^{-1/r}\right\}}{ G(z1,z2) = exp{-y1-y2+[(t1y1)^(-r)+(t2y2)^(-r)]^(-1/r)}} where \eqn{r > 0} and \eqn{0 < t_1,t_2 \leq 1}{0 < t1,t2 <= 1}. When \eqn{t_1 = t_2 = 1}{t1 = t2 = 1} the asymmetric negative logistic model is equivalent to the negative logistic model. Independence is obtained in the limit as either \eqn{r}, \eqn{t_1}{t1} or \eqn{t_2}{t2} approaches zero. Complete dependence is obtained in the limit when \eqn{t_1 = t_2 = 1}{t1 = t2 = 1} and \eqn{r} tends to infinity. Different limits occur when \eqn{t_1}{t1} and \eqn{t_2}{t2} are fixed and \eqn{r} tends to infinity. The earliest reference to this model appears to be Joe (1990), who introduces a multivariate extreme value distribution which reduces to \eqn{G(z_1,z_2)}{G(z1,z2)} in the bivariate case. \code{model = "bilog"} (Smith, 1990) The bilogistic distribution function with parameters \eqn{\code{alpha} = \alpha}{\code{alpha} = alpha} and \eqn{\code{beta} = \beta}{\code{beta} = beta} is \deqn{G(z_1,z_2) = \exp\left\{-y_1 q^{1-\alpha} - y_2 (1-q)^{1-\beta}\right\}}{ G(z1,z2) = exp{- y1 q^(1-alpha) - y2 (1-q)^(1-beta)}} where \eqn{q = q(y_1,y_2;\alpha,\beta)}{q = q(y1,y2;alpha,beta)} is the root of the equation \deqn{(1-\alpha) y_1 (1-q)^\beta - (1-\beta) y_2 q^\alpha = 0,}{ (1-alpha) y1 (1-q)^beta - (1-beta) y2 q^alpha = 0,} \eqn{0 < \alpha,\beta < 1}{0 < alpha,beta < 1}. When \eqn{\alpha = \beta}{alpha = beta} the bilogistic model is equivalent to the logistic model with dependence parameter \eqn{\code{dep} = \alpha = \beta}{\code{dep} = alpha = beta}. Complete dependence is obtained in the limit as \eqn{\alpha = \beta}{alpha = beta} approaches zero. Independence is obtained as \eqn{\alpha = \beta}{alpha = beta} approaches one, and when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other approaches one. Different limits occur when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other approaches zero. A bilogistic model is fitted in Smith (1990), where it appears to have been first introduced. \code{model = "negbilog"} (Coles and Tawn, 1994) The negative bilogistic distribution function with parameters \eqn{\code{alpha} = \alpha}{\code{alpha} = alpha} and \eqn{\code{beta} = \beta}{\code{beta} = beta} is \deqn{G(z_1,z_2) = \exp\left\{- y_1 - y_2 + y_1 q^{1+\alpha} + y_2 (1-q)^{1+\beta}\right\}}{ G(z1,z2) = exp{- y1 - y2 + y1 q^(1+alpha) + y2 (1-q)^(1+beta)}} where \eqn{q = q(y_1,y_2;\alpha,\beta)}{q = q(y1,y2;alpha,beta)} is the root of the equation \deqn{(1+\alpha) y_1 q^\alpha - (1+\beta) y_2 (1-q)^\beta = 0,}{ (1+alpha) y1 q^alpha - (1+beta) y2 (1-q)^beta = 0,} \eqn{\alpha > 0}{alpha > 0} and \eqn{\beta > 0}{beta > 0}. When \eqn{\alpha = \beta}{alpha = beta} the negative bilogistic model is equivalent to the negative logistic model with dependence parameter \eqn{\code{dep} = 1/\alpha = 1/\beta}{ \code{dep} = 1/alpha = 1/beta}. Complete dependence is obtained in the limit as \eqn{\alpha = \beta}{alpha = beta} approaches zero. Independence is obtained as \eqn{\alpha = \beta}{alpha = beta} tends to infinity, and when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other tends to infinity. Different limits occur when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other approaches zero. \code{model = "ct"} (Coles and Tawn, 1991) The Coles-Tawn distribution function with parameters \eqn{\code{alpha} = \alpha > 0}{\code{alpha} = alpha > 0} and \eqn{\code{beta} = \beta > 0}{\code{beta} = beta > 0} is \deqn{G(z_1,z_2) = \exp\left\{-y_1 [1 - \mbox{Be}(q;\alpha+1,\beta)] - y_2 \mbox{Be}(q;\alpha,\beta+1) \right\}}{ G(z1,z2) = exp{- y1 [1 - Be(q;alpha+1,beta)] - y2 Be(q;alpha,beta+1)}} where \eqn{q = \alpha y_2 / (\alpha y_2 + \beta y_1)}{ q = alpha y2 / (alpha y2 + beta y1)} and \eqn{\mbox{Be}(q;\alpha,\beta)}{Be(q;alpha,beta)} is the beta distribution function evaluated at \eqn{q} with \eqn{\code{shape1} = \alpha}{\code{shape1} = alpha} and \eqn{\code{shape2} = \beta}{\code{shape2} = beta}. Complete dependence is obtained in the limit as \eqn{\alpha = \beta}{alpha = beta} tends to infinity. Independence is obtained as \eqn{\alpha = \beta}{alpha = beta} approaches zero, and when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other approaches zero. Different limits occur when one of \eqn{\alpha,\beta}{alpha,beta} is fixed and the other tends to infinity. \code{model = "amix"} (Tawn, 1988) The asymmetric mixed distribution function with parameters \eqn{\code{alpha} = \alpha}{\code{alpha} = alpha} and \eqn{\code{beta} = \beta}{\code{beta} = beta} has a dependence function with the following cubic polynomial form. \deqn{A(t) = 1 - (\alpha +\beta)t + \alpha t^2 + \beta t^3}{ A(t) = 1 - (\alpha +\beta)t + \alpha t^2 + \beta t^3} where \eqn{\alpha}{alpha} and \eqn{\alpha + 3\beta}{alpha + 3beta} are non-negative, and where \eqn{\alpha + \beta}{alpha + beta} and \eqn{\alpha + 2\beta}{alpha + 2beta} are less than or equal to one. These constraints imply that beta lies in the interval [-0.5,0.5] and that alpha lies in the interval [0,1.5], though alpha can only be greater than one if beta is negative. The strength of dependence increases for increasing alpha (for fixed beta). Complete dependence cannot be obtained. Independence is obtained when both parameters are zero. For the definition of a dependence function, see \code{\link{abvevd}}. } \value{ \code{dbvevd} gives the density function, \code{pbvevd} gives the distribution function and \code{rbvevd} generates random deviates, for one of nine parametric bivariate extreme value models. } \note{ The logistic and asymmetric logistic models respectively are simulated using bivariate versions of Algorithms 1.1 and 1.2 in Stephenson(2003). All other models are simulated using a root finding algorithm to simulate from the conditional distributions. The simulation of the bilogistic and negative bilogistic models requires a root finding algorithm to evaluate \eqn{q} within the root finding algorithm used to simulate from the conditional distributions. The generation of bilogistic and negative bilogistic random deviates is therefore relatively slow (about 2.8 seconds per 1000 random vectors on a 450MHz PIII, 512Mb RAM). The bilogistic and negative bilogistic models can be represented under a single model, using the integral of the maximum of two beta distributions (Joe, 1997). The Coles-Tawn model is called the Dirichelet model in Coles and Tawn (1991). } \references{ Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. \emph{J. Roy. Statist. Soc., B}, \bold{53}, 377--392. Coles, S. G. and Tawn, J. A. (1994) Statistical methods for multivariate extremes: an application to structural design (with discussion). \emph{Appl. Statist.}, \bold{43}, 1--48. Galambos, J. (1975) Order statistics of samples from multivariate distributions. \emph{J. Amer. Statist. Assoc.}, \bold{70}, 674--680. Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. \emph{Publ. Inst. Statist. Univ. Paris}, \bold{9}, 171--173. Husler, J. and Reiss, R.-D. (1989) Maxima of normal random vectors: between independence and complete dependence. \emph{Statist. Probab. Letters}, \bold{7}, 283--286. Joe, H. (1990) Families of min-stable multivariate exponential and multivariate extreme value distributions. \emph{Statist. Probab. Letters}, \bold{9}, 75--81. Joe, H. (1997) \emph{Multivariate Models and Dependence Concepts}, London: Chapman & Hall. Smith, R. L. (1990) Extreme value theory. In \emph{Handbook of Applicable Mathematics} (ed. W. Ledermann), vol. 7. Chichester: John Wiley, pp. 437--471. Stephenson, A. G. (2003) Simulating multivariate extreme value distributions of logistic type. \emph{Extremes}, \bold{6}(1), 49--60. Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. \emph{Biometrika}, \bold{75}, 397--415. } \seealso{\code{\link{abvevd}}, \code{\link{rgev}}, \code{\link{rmvevd}}} \examples{ pbvevd(matrix(rep(0:4,2), ncol=2), dep = 0.7, model = "log") pbvevd(c(2,2), dep = 0.7, asy = c(0.6,0.8), model = "alog") pbvevd(c(1,1), dep = 1.7, model = "hr") margins <- cbind(0, 1, seq(-0.5,0.5,0.1)) rbvevd(11, dep = 1.7, model = "hr", mar1 = margins) rbvevd(10, dep = 1.2, model = "neglog", mar1 = c(10, 1, 1)) rbvevd(10, alpha = 0.7, beta = 0.52, model = "bilog") dbvevd(c(0,0), dep = 1.2, asy = c(0.5,0.9), model = "aneglog") dbvevd(c(0,0), alpha = 0.75, beta = 0.5, model = "ct", log = TRUE) dbvevd(c(0,0), alpha = 0.7, beta = 1.52, model = "negbilog") } \keyword{distribution} evd/man/profile.evd.Rd0000644000175100001440000000575114224766050014343 0ustar hornikusers\name{profile.evd} \alias{profile.evd} \title{Method for Profiling EVD Objects} \description{ Calculate profile traces for fitted models. } \usage{ \method{profile}{evd}(fitted, which = names(fitted$estimate), conf = 0.999, mesh = fitted$std.err[which]/4, xmin = rep(-Inf, length(which)), xmax = rep(Inf, length(which)), convergence = FALSE, method = "BFGS", control = list(maxit = 500), \dots) } \arguments{ \item{fitted}{An object of class \code{"evd"}.} \item{which}{A character vector giving the model parameters that are to be profiled. By default, all parameters are profiled.} \item{conf}{Controls the range over which the parameters are profiled. The profile trace is constructed so that (assuming the usual asymptotic properties hold) profile confidence intervals with confidence coefficients \code{conf} or less can be derived from it.} \item{mesh}{A numeric vector containing one value for each parameter in \code{which}. The values represent the distance between the points profiled. By default \code{mesh} is one quarter of the standard errors. If the fitted object does not contain standard errors the argument must be specified. The argument should also be specified when an estimator is on or close to a parameter boundary, since the approximated ``standard error'' will then be close to zero.} \item{xmin, xmax}{Numeric vectors containing one value for each parameter in \code{which}. Each value represents the theoretical lower/upper bound of the corresponding parameter. The arguments are needed only when a parameter has a lower/upper bound at which the likelihood is non-zero. Do not use these arguments to specify plotting ranges in a subsequent plot (as they are used in the calculation of profile confidence intervals); to do this use \code{xlim} in the call to \code{plot}.} \item{convergence}{Logical; print convergence code after each optimization? (A warning is given for each non-zero convergence code, irrespective of the value of \code{convergence}.)} \item{method}{The optimization method.} \item{control}{Passed to \code{optim}. See \code{\link{optim}} for details.} \item{\dots}{Ignored.} } \value{ An object of class \code{"profile.evd"}, which is a list with an element for each parameter being profiled. The elements are matrices. The first column contains the values of the profiled parameter. The second column contains profile deviances. The remaining columns contain the constrained maximum likelihood estimates for the remaining model parameters. For calculation of profile confidence intervals, use the \code{\link{confint.profile.evd}} function. } \seealso{\code{\link{confint.profile.evd}}, \code{\link{profile2d.evd}}, \code{\link{plot.profile.evd}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata) \dontrun{M1P <- profile(M1)} \dontrun{par(mfrow = c(2,2))} \dontrun{cint <- plot(M1P)} \dontrun{cint} } \keyword{models} evd/man/amvevd.Rd0000644000175100001440000001403112637167310013377 0ustar hornikusers\name{amvevd} \alias{amvevd} \title{Parametric Dependence Functions of Multivariate Extreme Value Models} \description{ Calculate the dependence function \eqn{A} for the multivariate logistic and multivariate asymmetric logistic models; plot the estimated function in the trivariate case. } \usage{ amvevd(x = rep(1/d,d), dep, asy, model = c("log", "alog"), d = 3, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) } \arguments{ \item{x}{A vector of length \code{d} or a matrix with \code{d} columns, in which case the dependence function is evaluated across the rows (ignored if plot is \code{TRUE}). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. \eqn{A(1/d,\dots,1/d)} is returned by default since it is often a useful summary of dependence.} \item{dep}{The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length \eqn{2^d-d-1}, or a single value, in which case the value is used for each of the \eqn{2^d-d-1} parameters (see \code{\link{rmvevd}}).} \item{asy}{The asymmetry parameters for the asymmetric logistic model. Should be a list with \eqn{2^d-1} vector elements containing the asymmetry parameters for each separate component (see \code{\link{rmvevd}} and \bold{Examples}).} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default) or \code{"alog"} (or any unique partial match), for the logistic and asymmetric logistic models respectively. The definition of each model is given in \code{\link{rmvevd}}.} \item{d}{The dimension; an integer greater than or equal to two. The trivariate case \code{d = 3} is the default.} \item{plot}{Logical; if \code{TRUE}, and the dimension \code{d} is three (the default dimension), the dependence function of a trivariate model is plotted. For plotting in the bivariate case, use \code{\link{abvevd}}. If \code{FALSE} (the default), the following arguments are ignored.} \item{col}{A list of colours (see \code{\link{image}}). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between \code{lower} and one.} \item{blty}{The border line type, for the border that surrounds the triangular image. By default \code{blty} is zero, so no border is plotted. Plotting a border leads to (by default) an increase in \code{grid} (and hence computation time), to ensure that the image fits within it.} \item{grid}{For plotting, the function is evaluated at \code{grid^2} points.} \item{lower}{The minimum value for which colours are plotted. By defualt \eqn{\code{lower} = 1/3} as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.} \item{ord}{A vector of length three, which should be a permutation of the set \eqn{\{1,2,3\}}{{1,2,3}}. The points \eqn{(1,0,0)}, \eqn{(0,1,0)} and \eqn{(0,0,1)} (the vertices of the simplex) are depicted clockwise from the top in the order defined by \code{ord}.The argument alters the way in which the function is plotted; it does not change the function definition.} \item{lab}{A character vector of length three, in which case the \code{i}th margin is labelled using the \code{i}th component, or \code{NULL}, in which case no labels are given. The actual location of the margins, and hence the labels, is defined by \code{ord}.} \item{lcex}{A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if \code{lab} is \code{NULL}.} } \details{ Let \eqn{z = (z_1,\dots,z_d)}{z = (z1,\dots,zd)} and \eqn{w = (w_1,\dots,w_d)}{w = (w1,\dots,wd)}. Any multivariate extreme value distribution can be written as \deqn{G(z) = \exp\left\{- \left\{\sum\nolimits_{j=1}^{d} y_j \right\} A\left(\frac{y_1}{\sum\nolimits_{j=1}^{d} y_j}, \dots, \frac{y_d}{\sum\nolimits_{j=1}^{d} y_j}\right)\right\}}{ G(z) = exp{-(y1+\dots+yd) A[y1/(y1+\dots+yd), \dots, yd/(y1+\dots+yd)]}} for some function \eqn{A} defined on the simplex \eqn{S_d = \{w \in R^d_+ : \sum\nolimits_{j=1}^{d} w_j = 1\}}{S_d = {w: w1 + \dots + wd = 1}}, where \deqn{y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}}{ yi = {1+si(zi-ai)/bi}^(-1/si)} for \eqn{1+s_i(z_i-a_i)/b_i > 0}{1+si(zi-ai)/bi > 0} and \eqn{i = 1,\dots,d}, and where the (generalized extreme value) marginal parameters are given by \eqn{(a_i,b_i,s_i)}{(ai,bi,si)}, \eqn{b_i > 0}{bi > 0}. If \eqn{s_i = 0}{si = 0} then \eqn{y_i}{yi} is defined by continuity. \eqn{A} is called (by some authors) the dependence function. It follows that \eqn{A(w) = 1} when \eqn{w} is one of the \eqn{d} vertices of \eqn{S_d}, and that \eqn{A} is a convex function with \eqn{\max(w_1,\dots,w_d) \leq A(w)\leq 1}{ max(w1,\dots,wd) <= A(w) <= 1} for all \eqn{w} in \eqn{S_d}. The lower and upper limits of \eqn{A} are obtained under complete dependence and mutual independence respectively. \eqn{A} does not depend on the marginal parameters. } \value{ A numeric vector of values. If plotting, the smallest evaluated function value is returned invisibly. } \seealso{\code{\link{amvnonpar}}, \code{\link{abvevd}}, \code{\link{rmvevd}}, \code{\link{image}}} \examples{ amvevd(dep = 0.5, model = "log") s3pts <- matrix(rexp(30), nrow = 10, ncol = 3) s3pts <- s3pts/rowSums(s3pts) amvevd(s3pts, dep = 0.5, model = "log") \dontrun{amvevd(dep = 0.05, model = "log", plot = TRUE, blty = 1)} amvevd(dep = 0.95, model = "log", plot = TRUE, lower = 0.94) asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2)) amvevd(s3pts, dep = 0.15, asy = asy, model = "alog") amvevd(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7) } \keyword{distribution} evd/man/amvnonpar.Rd0000644000175100001440000001265612637167310014131 0ustar hornikusers\name{amvnonpar} \alias{amvnonpar} \title{Non-parametric Estimates for Dependence Functions of the Multivariate Extreme Value Distribution} \description{ Calculate non-parametric estimates for the dependence function \eqn{A} of the multivariate extreme value distribution and plot the estimated function in the trivariate case. } \usage{ amvnonpar(x = rep(1/d,d), data, d = 3, epmar = FALSE, nsloc = NULL, madj = 0, kmar = NULL, plot = FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1) } \arguments{ \item{x}{A vector of length \code{d} or a matrix with \code{d} columns, in which case the dependence function is evaluated across the rows (ignored if plot is \code{TRUE}). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. \eqn{A(1/d,\dots,1/d)} is returned by default since it is often a useful summary of dependence.} \item{data}{A matrix or data frame with \code{d} columns, which may contain missing values.} \item{d}{The dimension; an integer greater than or equal to two. The trivariate case \code{d = 3} is the default.} \item{epmar}{If \code{TRUE}, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the \code{nsloc} argument is ignored.} \item{nsloc}{A data frame with the same number of rows as \code{data}, or a list containing \code{d} elements of this type, for linear modelling of the marginal location parameters. In the former case, the argument is applied to all margins. The data frames are treated as covariate matrices, excluding the intercept. Numeric vectors can be given as alternatives to single column data frames. A list can contain \code{NULL} elements for stationary modelling of selected margins.} \item{madj}{Performs marginal adjustments. See \code{\link{abvnonpar}}.} \item{kmar}{In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates.} \item{plot}{Logical; if \code{TRUE}, and the dimension \code{d} is three (the default dimension), the dependence function of a trivariate extreme value distribution is plotted. For plotting in the bivariate case, use \code{\link{abvnonpar}}. If \code{FALSE} (the default), the following arguments are ignored.} \item{col}{A list of colours (see \code{\link{image}}). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between \code{lower} and one.} \item{blty}{The border line type, for the border that surrounds the triangular image. By default \code{blty} is zero, so no border is plotted. Plotting a border leads to (by default) an increase in \code{grid} (and hence computation time), to ensure that the image fits within it.} \item{grid}{For plotting, the function is evaluated at \code{grid^2} points.} \item{lower}{The minimum value for which colours are plotted. By default \eqn{\code{lower} = 1/3} as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.} \item{ord}{A vector of length three, which should be a permutation of the set \eqn{\{1,2,3\}}{{1,2,3}}. The points \eqn{(1,0,0)}, \eqn{(0,1,0)} and \eqn{(0,0,1)} (the vertices of the simplex) are depicted clockwise from the top in the order defined by \code{ord}. The argument alters the way in which the function is plotted; it does not change the function definition.} \item{lab}{A character vector of length three, in which case the \code{i}th margin is labelled using the \code{i}th component, or \code{NULL}, in which case no labels are given. By default, \code{lab} is \code{as.character(1:3)}. The actual location of the margins, and hence the labels, is defined by \code{ord}.} \item{lcex}{A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if \code{lab} is \code{NULL}.} } \note{ The rows of \code{data} that contain missing values are not used in the estimation of the dependence structure, but every non-missing value is used in estimating the margins. The dependence function of the multivariate extreme value distribution is defined in \code{\link{amvevd}}. The function \code{\link{amvevd}} calculates and plots dependence functions of multivariate logistic and multivariate asymmetric logistic models. The estimator plotted or calculated is a multivariate extension of the bivariate Pickands estimator defined in \code{\link{abvnonpar}}. } \value{ A numeric vector of estimates. If plotting, the smallest evaluated estimate is returned invisibly. } \seealso{\code{\link{amvevd}}, \code{\link{abvnonpar}}, \code{\link{fgev}}} \examples{ s5pts <- matrix(rexp(50), nrow = 10, ncol = 5) s5pts <- s5pts/rowSums(s5pts) sdat <- rmvevd(100, dep = 0.6, model = "log", d = 5) amvnonpar(s5pts, sdat, d = 5) \dontrun{amvnonpar(data = sdat, plot = TRUE)} \dontrun{amvnonpar(data = sdat, plot = TRUE, ord = c(2,3,1), lab = LETTERS[1:3])} \dontrun{amvevd(dep = 0.6, model = "log", plot = TRUE)} \dontrun{amvevd(dep = 0.6, model = "log", plot = TRUE, blty = 1)} } \keyword{nonparametric} evd/man/fgumbelx.Rd0000644000175100001440000001043214225006372013722 0ustar hornikusers\name{fgumbelx} \alias{fgumbelx} \title{Maximum-likelihood Fitting of the Maximum of Two Gumbel Distributions} \description{ Maximum-likelihood fitting for the maximum of two gumbel distributions, allowing any of the parameters to be held fixed if desired. } \usage{ fgumbelx(x, start, \dots, nsloc1 = NULL, nsloc2 = NULL, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) } \arguments{ \item{x}{A numeric vector, which may contain missing values.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized. If \code{start} is omitted the routine attempts to find good starting values using moment estimators.} \item{\dots}{Additional parameters, either for the fitted model or for the optimization function \code{optim}. If parameters of the model are included they will be held fixed at the values given (see \bold{Examples}).} \item{nsloc1}{A data frame with the same number of rows as the length of \code{x}, for linear modelling of the location parameter of the first Gumbel distribution. This is not recommended as the model is already complex.} \item{nsloc2}{A data frame with the same number of rows as the length of \code{x}, for linear modelling of the location parameter of the second Gumbel distribution. This is not recommended as the model is already complex.} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} \item{warn.inf}{Logical; if \code{TRUE} (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.} } \details{ For stationary models the parameter names are \code{loc1}, \code{scale1}, \code{loc2} and \code{scale2} for the location and scale parameters of two Gumbel distributions, where \code{loc2} must be greater or equal to \code{loc1}. The likelihood may have multiple local optima and therefore may be difficult to fit properly; the default starting values use a moment based approach, however it is recommended that the user specify multiple different starting values and experiment with different optimization methods. Using non-stationary models with nsloc1 and nsloc2 is not recommended due to the model complexity; the data also cannot be transformed back to stationarity so diagnostic plots will be misleading in this case. } \value{ Returns an object of class \code{c("gumbelx","evd")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The functions \code{profile} and \code{profile2d} are used to obtain deviance profiles for the model parameters. The function \code{anova} compares nested models. The function \code{plot} produces diagnostic plots. An object of class \code{c("gumbelx","evd")} is a list containing at most the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{fixed}{A vector containing the parameters of the model that have been held fixed.} \item{param}{A vector containing all parameters (optimized and fixed).} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{data}{The data passed to the argument \code{x}.} \item{nsloc1}{The argument \code{nsloc1}.} \item{nsloc2}{The argument \code{nsloc2}.} \item{n}{The length of \code{x}.} \item{call}{The call of the current function.} } \section{Warning}{ This function is experimental and involves optimizing over a potentially complex surface. } \seealso{\code{\link{fgev}}, \code{\link{optim}}, \code{\link{rgumbelx}}} \examples{ uvdata <- rgumbelx(100, loc1 = 0, scale1 = 1, loc2 = 1, scale2 = 1) fgumbelx(uvdata, loc1 = 0, scale1 = 1) } \keyword{models} evd/man/confint.evd.Rd0000644000175100001440000000242214224765374014343 0ustar hornikusers\name{confint.evd} \alias{confint.evd} \alias{confint.profile.evd} \title{Calculate Confidence Intervals} \description{ Calculate profile and Wald confidence intervals of parameters in fitted models. } \usage{ \method{confint}{evd}(object, parm, level = 0.95, \dots) \method{confint}{profile.evd}(object, parm, level = 0.95, \dots) } \arguments{ \item{object}{Either a fitted model object (of class \code{evd}) for Wald confidence intervals, or a profile trace (of class \code{profile.evd}) for profile likelihood confidence intervals.} \item{parm}{A character vector of parameters; a confidence interval is calculated for each parameter. If missing, then intervals are returned for all parameters in the fitted model or profile trace.} \item{level}{A single number giving the confidence level.} \item{\dots}{Not used.} } \value{ A matrix with two columns giving lower and upper confidence limits. For profile confidence intervals, this function assumes that the profile trace is unimodal. If the profile trace is not unimodal then the function will give spurious results. } \seealso{\code{\link{profile.evd}}} \examples{ m1 <- fgev(portpirie) confint(m1) \dontrun{pm1 <- profile(m1)} \dontrun{plot(pm1)} \dontrun{confint(pm1)} } \keyword{manip} evd/man/lisbon.Rd0000644000175100001440000000060412637167310013404 0ustar hornikusers\name{lisbon} \alias{lisbon} \title{Annual Maximum Wind Speeds at Lisbon} \usage{lisbon} \description{ A numeric vector containing annual maximum wind speeds, in kilometers per hour, from 1941 to 1970 at Lisbon, Portugal. } \format{A vector containing 30 observations.} \source{ Tiago de Oliveira, J. (1997) \emph{Statistical Analysis of Extremes.} Pendor. } \keyword{datasets} evd/man/sealevel.Rd0000644000175100001440000000154712637167310013725 0ustar hornikusers\name{sealevel} \alias{sealevel} \title{Annual Sea Level Maxima at Dover and Harwich} \usage{sealevel} \description{ The \code{sealevel} data frame has 81 rows and 2 columns. The columns contain annual sea level maxima from 1912 to 1992 at Dover and Harwich respectively, two sites on the coast of Britain. The row names give the years of observation. There are 39 missing values. } \format{ This data frame contains the following columns: \describe{ \item{dover}{A numeric vector containing annual sea level maxima at Dover, including 9 missing values.} \item{harwich}{A numeric vector containing sea annual level maxima at Harwich, including 30 missing values.} } } \source{ Coles, S. G. and Tawn, J. A. (1990) Statistics of coastal flood prevention. \emph{Phil. Trans. R. Soc. Lond., A} \bold{332}, 457--476. } \keyword{datasets} evd/man/hbvevd.Rd0000644000175100001440000001017712637167310013402 0ustar hornikusers\name{hbvevd} \alias{hbvevd} \title{Parametric Spectral Density Functions of Bivariate Extreme Value Models} \description{ Calculate or plot the density \eqn{h} of the spectral measure \eqn{H} on the interval \eqn{(0,1)}, for nine parametric bivariate extreme value models. } \usage{ hbvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), half = FALSE, plot = FALSE, add = FALSE, lty = 1, \dots) } \arguments{ \item{x}{A vector of values at which the function is evaluated (ignored if plot or add is \code{TRUE}). \eqn{h(1/2)} is returned by default.} \item{dep}{Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.} \item{asy}{A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.} \item{alpha, beta}{Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in \code{\link{rbvevd}}. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.} \item{half}{Logical; if \code{TRUE} the function is divided by two, corresponding to a spectral measure with total mass one rather than two.} \item{plot}{Logical; if \code{TRUE} the function is plotted. The x and y values used to create the plot are returned invisibly.} \item{add}{Logical; add to an existing plot?} \item{lty}{Line type.} \item{\dots}{Other high-level graphics parameters to be passed to \code{plot}.} } \details{ Any bivariate extreme value distribution can be written as \deqn{G(z_1,z_2) = \exp\left[-\int_0^1 \max\{wy_1, (1-w)y_2\} H(dw)\right]}{G(z1,z2) = exp{-\int_0^1 max(w y1, (1-w) y2) H(dw)}} for some function \eqn{H(\cdot)}{H()} defined on \eqn{[0,1]}, satisfying \deqn{\int_0^1 w H(dw) = \int_0^1 (1-w) H(dw) = 1}{ \int_0^1 w H(dw) = int_0^1 (1-w) H(dw) = 1.} In particular, the total mass of H is two. The functions \eqn{y_1}{y1} and \eqn{y_2}{y2} are as defined in \code{\link{abvevd}}. H is called the spectral measure, with density \eqn{h} on the interval \eqn{(0,1)}. } \section{Point Masses}{ For differentiable models H may have up to two point masses: at zero and one. Assuming that the model parameters are in the interior of the parameter space, we have the following. For the asymmetric logistic and asymmetric negative logistic models the point masses are of size \code{1-asy1} and \code{1-asy2} respectively. For the asymmetric mixed model they are of size \code{1-alpha-beta} and \code{1-alpha-2*beta} respectively. For all other models the point masses are zero. At independence, H has point masses of size one at both zero and one. At complete dependence [a non-differentiable model] H has a single point mass of size two at \eqn{1/2}. In either case, \eqn{h} is zero everywhere. } \value{ \code{hbvevd} calculates or plots the spectral density function \eqn{h} for one of nine parametric bivariate extreme value models, at specified parameter values. } \seealso{\code{\link{abvevd}}, \code{\link{fbvevd}}, \code{\link{rbvevd}}, \code{\link{plot.bvevd}}} \examples{ hbvevd(dep = 2.7, model = "hr") hbvevd(seq(0.25,0.5,0.75), dep = 0.3, asy = c(.7,.9), model = "alog") hbvevd(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE) bvdata <- rbvevd(100, dep = 0.7, model = "log") M1 <- fitted(fbvevd(bvdata, model = "log")) hbvevd(dep = M1["dep"], model = "log", plot = TRUE) } \keyword{distribution} evd/man/venice.Rd0000644000175100001440000000154512637167310013374 0ustar hornikusers\name{venice} \alias{venice} \title{Largest Sea Levels in Venice} \usage{venice} \description{ The \code{venice} data frame has 51 rows and 10 columns. The jth column contains the jth largest sea levels in Venice, for the years 1931--1981. Only the largest six measurements are available for the year 1935; the corresponding row contains four missing values. The years for each set of measurements are given as row names. A larger version of this data is available in the dataset \code{venice2}. } \format{A data frame with 51 rows and 10 columns.} \source{ Smith, R. L. (1986) Extreme value theory based on the \eqn{r} largest annual events. \emph{Journal of Hydrology}, \bold{86}, 27--43. } \references{ Coles, S. G. (2001) \emph{An Introduction to Statistical Modeling of Extreme Values}. London: Springer-Verlag. } \keyword{datasets} evd/man/uccle.Rd0000644000175100001440000000150614611655364013220 0ustar hornikusers\name{uccle} \alias{uccle} \title{Rainfall Maxima at Uccle, Belgium} \usage{uccle} \description{ The \code{uccle} data frame has 35 rows and 4 columns. The columns contain annual rainfall maxima (in millimetres) from 1938 to 1972 at Uccle, Belgium, over the durations of one day, one hour, ten minutes and one minute. The row names give the years of observation. } \format{ This data frame contains the following columns: \describe{ \item{day}{Annual daily rainfall maxima.} \item{hour}{Annual hourly rainfall maxima.} \item{tmin}{Annual rainfall maxima over ten minute durations.} \item{min}{Annual rainfall maxima over one minute durations.} } } \source{ Sneyers, R. (1977) L'intensite maximale des precipitations en Belgique. \emph{Inst. Royal Meteor. Belgique, B} \bold{86}. } \keyword{datasets} evd/man/forder.Rd0000644000175100001440000000653712637167310013412 0ustar hornikusers\name{forder} \alias{forder} \title{Maximum-likelihood Fitting of Order Statistics} \description{ Maximum-likelihood fitting for the distribution of a selected order statistic of a given number of independent variables from a specified distribution. } \usage{ forder(x, start, densfun, distnfun, \dots, distn, mlen = 1, j = 1, largest = TRUE, std.err = TRUE, corr = FALSE, method = "Nelder-Mead") } \arguments{ \item{x}{A numeric vector.} \item{start}{A named list giving the initial values for the parameters over which the likelihood is to be maximized.} \item{densfun, distnfun}{Density and distribution function of the specified distribution.} \item{\dots}{Additional parameters, either for the specified distribution or for the optimization function \code{optim}. If parameters of the distribution are included they will be held fixed at the values given (see \bold{Examples}). If parameters of the distribution are not included either here or as a named component in \code{start} they will be held fixed at the default values specified in the corresponding density and distribution functions (assuming they exist; an error will be generated otherwise).} \item{distn}{A character string, optionally specified as an alternative to \code{densfun} and \code{distnfun} such that the density and distribution and functions are formed upon the addition of the prefixes \code{d} and \code{p} respectively.} \item{mlen}{The number of independent variables.} \item{j}{The order statistic, taken as the \code{j}th largest (default) or smallest of \code{mlen}, according to the value of \code{largest}.} \item{largest}{Logical; if \code{TRUE} (default) use the \code{j}th largest order statistic, otherwise use the \code{j}th smallest.} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} } \details{ Maximization of the log-likelihood is performed. The estimated standard errors are taken from the observed information, calculated by a numerical approximation. If the density and distribution functions are user defined, the order of the arguments must mimic those in R base (i.e. data first, parameters second). Density functions must have \code{log} arguments. } \value{ Returns an object of class \code{c("extreme","evd")}. This class is defined in \code{\link{fextreme}}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The function \code{anova} compares nested models. } \seealso{\code{\link{anova.evd}}, \code{\link{fextreme}}, \code{\link{optim}}} \examples{ uvd <- rorder(100, qnorm, mean = 0.56, mlen = 365, j = 2) forder(uvd, list(mean = 0, sd = 1), distn = "norm", mlen = 365, j = 2) forder(uvd, list(rate = 1), distn = "exp", mlen = 365, j = 2, method = "Brent", lower=0.01, upper=10) forder(uvd, list(scale = 1), shape = 1, distn = "gamma", mlen = 365, j = 2, method = "Brent", lower=0.01, upper=10) forder(uvd, list(shape = 1, scale = 1), distn = "gamma", mlen = 365, j = 2) } \keyword{models} evd/man/clusters.Rd0000644000175100001440000001077612637167310013775 0ustar hornikusers\name{clusters} \alias{clusters} \title{Identify Clusters of Exceedences} \description{ Identify clusters of exceedences. } \usage{ clusters(data, u, r = 1, ulow = -Inf, rlow = 1, cmax = FALSE, keep.names = TRUE, plot = FALSE, xdata = seq(along = data), lvals = TRUE, lty = 1, lwd = 1, pch = par("pch"), col = if(n > 250) NULL else "grey", xlab = "Index", ylab = "Data", ...) } \arguments{ \item{data}{A numeric vector, which may contain missing values.} \item{u}{A single value giving the threshold, unless a time varying threshold is used, in which case \code{u} should be a vector of thresholds, typically with the same length as \code{data} (or else the usual recycling rules are applied).} \item{r}{A postive integer denoting the clustering interval length. By default the interval length is one.} \item{ulow}{A single value giving the lower threshold, unless a time varying lower threshold is used, in which case \code{ulow} should be a vector of lower thresholds, typically with the same length as \code{data} (or else the usual recycling rules are applied). By default there is no lower threshold (or equivalently, the lower threshold is \code{-Inf}).} \item{rlow}{A postive integer denoting the lower clustering interval length. By default the interval length is one.} \item{cmax}{Logical; if \code{FALSE} (the default), a list containing the clusters of exceedences is returned. If \code{TRUE} a numeric vector containing the cluster maxima is returned.} \item{keep.names}{Logical; if \code{FALSE}, the function makes no attempt to retain the names/indices of the observations within the returned object. If \code{data} contains a large number of observations, this can make the function run much faster. The argument is mainly designed for internal use.} \item{plot}{Logical; if \code{TRUE} a plot is given that depicts the identified clusters, and the clusters (if \code{cmax} is \code{FALSE}) or cluster maxima (if \code{cmax} is \code{TRUE}) are returned invisibly. If \code{FALSE} (the default), the following arguments are ignored.} \item{xdata}{A numeric vector with the same length as \code{data}, giving the values to be plotted on the x-axis.} \item{lvals}{Logical; should the values below the threshold and the line depicting the lower threshold be plotted?} \item{lty, lwd}{Line type and width for the lines depicting the threshold and the lower threshold.} \item{pch}{Plotting character.} \item{col}{Strips of colour \code{col} are used to identify the clusters. An observation is contained in the cluster if the centre of the corresponding plotting character is contained in the coloured strip. If \code{col} is \code{NULL} the strips are omitted. By default the strips are coloured \code{"grey"}, but are omitted whenever \code{data} contains more than 250 observations.} \item{xlab, ylab}{Labels for the x and y axis.} \item{\dots}{Other graphics parameters.} } \details{ The clusters of exceedences are identified as follows. The first exceedence of the threshold initiates the first cluster. The first cluster then remains active until either \code{r} consecutive values fall below (or are equal to) the threshold, or until \code{rlow} consecutive values fall below (or are equal to) the lower threshold. The next exceedence of the threshold (if it exists) then initiates the second cluster, and so on. Missing values are allowed, in which case they are treated as falling below (or equal to) the threshold, but falling above the lower threshold. } \value{ If \code{cmax} is \code{FALSE} (the default), a list with one component for each identified cluster. If \code{cmax} is \code{TRUE}, a numeric vector containing the cluster maxima. In any case, the returned object has an attribute \code{acs}, giving the average cluster size (where the cluster size is defined as the number of exceedences within a cluster), which will be \code{NaN} if there are no values above the threshold (and hence no clusters). If \code{plot} is \code{TRUE}, the list of clusters, or vector of cluster maxima, is returned invisibly. } \seealso{\code{\link{exi}}, \code{\link{exiplot}}} \examples{ clusters(portpirie, 4.2, 3) clusters(portpirie, 4.2, 3, cmax = TRUE) clusters(portpirie, 4.2, 3, 3.8, plot = TRUE) clusters(portpirie, 4.2, 3, 3.8, plot = TRUE, lvals = FALSE) tvu <- c(rep(4.2, 20), rep(4.1, 25), rep(4.2, 20)) clusters(portpirie, tvu, 3, plot = TRUE) } \keyword{manip} evd/man/rweibull.Rd0000644000175100001440000000500712637167310013745 0ustar hornikusers\name{rweibull} \alias{drweibull} \alias{prweibull} \alias{qrweibull} \alias{rrweibull} \alias{dnweibull} \alias{pnweibull} \alias{qnweibull} \alias{rnweibull} \title{The Reverse Weibull Distribution} \description{ Density function, distribution function, quantile function and random generation for the reverse (or negative) Weibull distribution with location, scale and shape parameters. } \usage{ drweibull(x, loc=0, scale=1, shape=1, log = FALSE) prweibull(q, loc=0, scale=1, shape=1, lower.tail = TRUE) qrweibull(p, loc=0, scale=1, shape=1, lower.tail = TRUE) rrweibull(n, loc=0, scale=1, shape=1) dnweibull(x, loc=0, scale=1, shape=1, log = FALSE) pnweibull(q, loc=0, scale=1, shape=1, lower.tail = TRUE) qnweibull(p, loc=0, scale=1, shape=1, lower.tail = TRUE) rnweibull(n, loc=0, scale=1, shape=1) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{loc, scale, shape}{Location, scale and shape parameters (can be given as vectors).} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} } \details{ The reverse (or negative) Weibull distribution function with parameters \eqn{\code{loc} = a}, \eqn{\code{scale} = b} and \eqn{\code{shape} = s} is \deqn{G(z) = \exp\left\{-\left[-\left(\frac{z-a}{b}\right) \right]^s\right\}}{G(x) = exp{-[-(z-a)/b]^s}} for \eqn{z < a} and one otherwise, where \eqn{b > 0} and \eqn{s > 0}. } \note{ Within extreme value theory the reverse Weibull distibution (also known as the negative Weibull distribution) is often referred to as the Weibull distribution. We make a distinction to avoid confusion with the three-parameter distribution used in survival analysis, which is related by a change of sign to the distribution given above. } \value{ \code{drweibull} and \code{dnweibull} give the density function, \code{prweibull} and \code{pnweibull} give the distribution function, \code{qrweibull} and \code{qnweibull} give the quantile function, \code{rrweibull} and \code{rnweibull} generate random deviates. } \seealso{\code{\link{rfrechet}}, \code{\link{rgev}}, \code{\link{rgumbel}}} \examples{ drweibull(-5:-3, -1, 0.5, 0.8) prweibull(-5:-3, -1, 0.5, 0.8) qrweibull(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8) rrweibull(6, -1, 0.5, 0.8) p <- (1:9)/10 prweibull(qrweibull(p, -1, 2, 0.8), -1, 2, 0.8) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/chiplot.Rd0000644000175100001440000001654312637167310013571 0ustar hornikusers\name{chiplot} \alias{chiplot} \title{Dependence Measure Plots} \description{ Plots of estimates of the dependence measures chi and chi-bar for bivariate data. } \usage{ chiplot(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, trunc = TRUE, spcases = FALSE, lty = 1, cilty = 2, col = 1, cicol = 1, xlim = c(0,1), ylim1 = c(-1,1), ylim2 = c(-1,1), main1 = "Chi Plot", main2 = "Chi Bar Plot", xlab = "Quantile", ylab1 = "Chi", ylab2 = "Chi Bar", ask = nb.fig < length(which) && dev.interactive(), \dots) } \arguments{ \item{data}{A matrix or data frame with two columns. Rows (observations) with missing values are stripped from the data before any computations are performed.} \item{nq}{The number of quantiles at which the measures are evaluated.} \item{qlim}{The limits of the quantiles at which the measures are evaluated (see \bold{Details}).} \item{which}{If only one plot is required, specify \code{1} for chi and \code{2} for chi-bar.} \item{conf}{The confidence coefficient of the plotted confidence intervals.} \item{trunc}{Logical; truncate the estimates at their theoretical upper and lower bounds?} \item{spcases}{If \code{TRUE}, plots greyed lines corresponding to the special cases of perfect positive/negative dependence and exact independence.} \item{lty, cilty}{Line types for the estimates of the measures and for the confidence intervals respectively. Use zero to supress.} \item{col, cicol}{Colour types for the estimates of the measures and for the confidence intervals respectively.} \item{xlim, xlab}{Limits and labels for the x-axis; they apply to both plots.} \item{ylim1}{Limits for the y-axis of the chi plot. If this is \code{NULL} (the default) the upper limit is one, and the lower limit is the minimum of zero and the smallest plotted value.} \item{ylim2}{Limits for the y-axis of the chi-bar plot.} \item{main1, main2}{The plot titles for the chi and chi-bar plots respectively.} \item{ylab1, ylab2}{The y-axis labels for the chi and chi-bar plots respectively.} \item{ask}{Logical; if \code{TRUE}, the user is asked before each plot.} \item{\dots}{Other arguments to be passed to \code{matplot}.} } \details{ These measures are explained in full detail in Coles, Heffernan and Tawn (1999). A brief treatment is also given in Section 8.4 of Coles(2001). A short summary is given as follows. We assume that the data are \emph{iid} random vectors with common bivariate distribution function \eqn{G}, and we define the random vector \eqn{(X,Y)} to be distributed according to \eqn{G}. The chi plot is a plot of \eqn{q} against empirical estimates of \deqn{\chi(q) = 2 - \log(\Pr(F_X(X) < q, F_Y(Y) < q)) / \log(q)}{ chi(q) = 2 - log(Pr(F_X(X) < q, F_Y(Y) < q)) / log(q)} where \eqn{F_X} and \eqn{F_Y} are the marginal distribution functions, and where \eqn{q} is in the interval (0,1). The quantity \eqn{\chi(q)}{chi(q)} is bounded by \deqn{2 - \log(2u - 1)/\log(u) \leq \chi(q) \leq 1}{ 2 - log(2u - 1)/log(u) <= chi(q) <= 1} where the lower bound is interpreted as \code{-Inf} for \eqn{q \leq 1/2}{q <= 1/2} and zero for \eqn{q = 1}. These bounds are reflected in the corresponding estimates. The chi bar plot is a plot of \eqn{q} against empirical estimates of \deqn{\bar{\chi}(q) = 2\log(1-q)/\log(\Pr(F_X(X) > q, F_Y(Y) > q)) - 1}{ chibar(q) = 2log(1-q)/log(Pr(F_X(X) > q, F_Y(Y) > q)) - 1} where \eqn{F_X} and \eqn{F_Y} are the marginal distribution functions, and where \eqn{q} is in the interval (0,1). The quantity \eqn{\bar{\chi}(q)}{chibar(q)} is bounded by \eqn{-1 \leq \bar{\chi}(q) \leq 1}{-1 <= chibar(q) <= 1} and these bounds are reflected in the corresponding estimates. Note that the empirical estimators for \eqn{\chi(q)}{chi(q)} and \eqn{\bar{\chi}(q)}{chibar(q)} are undefined near \eqn{q=0} and \eqn{q=1}. By default the function takes the limits of \eqn{q} so that the plots depicts all values at which the estimators are defined. This can be overridden by the argument \code{qlim}, which must represent a subset of the default values (and these can be determined using the component \code{quantile} of the invisibly returned list; see \bold{Value}). The confidence intervals within the plot assume that observations are independent, and that the marginal distributions are estimated exactly. The intervals are constructed using the delta method; this may lead to poor interval estimates near \eqn{q=0} and \eqn{q=1}. The function \eqn{\chi(q)}{chi(q)} can be interpreted as a quantile dependent measure of dependence. In particular, the sign of \eqn{\chi(q)}{chi(q)} determines whether the variables are positively or negatively associated at quantile level \eqn{q}. By definition, variables are said to be asymptotically independent when \eqn{\chi(1)}{chi(1)} (defined in the limit) is zero. For independent variables, \eqn{\chi(q) = 0}{chi(q) = 0} for all \eqn{q} in (0,1). For perfectly dependent variables, \eqn{\chi(q) = 1}{chi(q) = 1} for all \eqn{q} in (0,1). For bivariate extreme value distributions, \eqn{\chi(q) = 2(1-A(1/2))}{chi(q) = 2(1-A(1/2))} for all \eqn{q} in (0,1), where \eqn{A} is the dependence function, as defined in \code{\link{abvevd}}. If a bivariate threshold model is to be fitted (using \code{\link{fbvpot}}), this plot can therefore act as a threshold identification plot, since e.g. the use of 95\% marginal quantiles as threshold values implies that \eqn{\chi(q)}{chi(q)} should be approximately constant above \eqn{q = 0.95}. The function \eqn{\bar{\chi}(q)}{chibar(q)} can again be interpreted as a quantile dependent measure of dependence; it is most useful within the class of asymptotically independent variables. For asymptotically dependent variables (i.e. those for which \eqn{\chi(1) < 1}{chi(1) < 1}), we have \eqn{\bar{\chi}(1) = 1}{ chibar(1) = 1}, where \eqn{\bar{\chi}(1)}{chibar(1)} is again defined in the limit. For asymptotically independent variables, \eqn{\bar{\chi}(1)}{ chibar(1)} provides a measure that increases with dependence strength. For independent variables \eqn{\bar{\chi}(q) = 0}{chibar(q) = 0} for all \eqn{q} in (0,1), and hence \eqn{\bar{\chi}(1) = 0}{chibar(1) = 0}. } \value{ A list with components \code{quantile}, \code{chi} (if \code{1} is in \code{which}) and \code{chibar} (if \code{2} is in \code{which}) is invisibly returned. The components \code{quantile} and \code{chi} contain those objects that were passed to the formal arguments \code{x} and \code{y} of \code{matplot} in order to create the chi plot. The components \code{quantile} and \code{chibar} contain those objects that were passed to the formal arguments \code{x} and \code{y} of \code{matplot} in order to create the chi-bar plot. } \references{ Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. \emph{Extremes}, \bold{2}, 339--365. Coles, S. G. (2001) \emph{An Introduction to Statistical Modelling of Extreme Values}, London: Springer--Verlag. } \author{Jan Heffernan and Alec Stephenson} \seealso{\code{\link{fbvevd}}, \code{\link{fbvpot}}, \code{\link{matplot}}} \examples{ par(mfrow = c(1,2)) smdat1 <- rbvevd(1000, dep = 0.6, model = "log") smdat2 <- rbvevd(1000, dep = 1, model = "log") chiplot(smdat1) chiplot(smdat2) } \keyword{hplot} evd/man/ccbvevd.Rd0000644000175100001440000000620512637167310013535 0ustar hornikusers\name{ccbvevd} \alias{ccbvevd} \title{Calculate Conditional Copulas for Parametric Bivariate Extreme Value Distributions} \description{ Conditional copula functions, conditioning on either margin, for nine parametric bivariate extreme value models. } \usage{ ccbvevd(x, mar = 2, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), lower.tail = TRUE) } \arguments{ \item{x}{A matrix or data frame, ordinarily with two columns, which may contain missing values. A data frame may also contain a third column of mode \code{logical}, which itself may contain missing values (see \bold{Details}).} \item{mar}{One or two; conditions on this margin.} \item{dep}{Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.} \item{asy}{A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.} \item{alpha, beta}{Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.} \item{lower.tail}{Logical; if \code{TRUE} (default), the conditional distribution function is returned; the conditional survivor function is returned otherwise.} } \details{ The function calculates \eqn{P(U_1 < x_1|U_2 = x_2)}{ P(U1 < x1|U2 = x2)}, where \eqn{(U_1,U_2)}{(U1,U2)} is a random vector with Uniform(0,1) margins and with a dependence structure given by the specified parametric model. By default, the values of \eqn{x_1}{x1} and \eqn{x_1}{x2} are given by the first and second columns of the argument \code{x}. If \code{mar = 1} then this is reversed. If \code{x} has a third column \eqn{x_3}{x3} of mode logical, then the function returns \eqn{P(U_1 < x_1|U_2 = x_2,I = x_3)}{ P(U1 < x1|U2 = x2,I = x3)}, according to inference proceedures derived by Stephenson and Tawn (2004). See \code{\link{fbvevd}}. This requires numerical integration, and hence will be slower. This function is mainly for internal use. It is used by \code{\link{plot.bvevd}} to calculate the conditional P-P plotting diagnostics. } \value{ A numeric vector of probabilities. } \references{ Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. \emph{Biometrika} \bold{92}(1), 213--217. } \seealso{\code{\link{rbvevd}}, \code{\link{fbvevd}}, \code{\link{plot.bvevd}}} \keyword{distribution} evd/man/fbvpot.Rd0000644000175100001440000002136412637167310013424 0ustar hornikusers\name{fbvpot} \alias{fbvpot} \alias{print.bvpot} \title{Maximum-likelihood Fitting of Bivariate Extreme Value Distributions to Threshold Exceedances} \description{ Fit models for one of nine parametric bivariate extreme-value distributions using threshold exceedances, allowing any of the parameters to be held fixed if desired. } \usage{ fbvpot(x, threshold, model = c("log", "bilog", "alog", "neglog", "negbilog", "aneglog", "ct", "hr", "amix"), likelihood = c("censored", "poisson"), start, \dots, sym = FALSE, cshape = cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS", warn.inf = TRUE) } \arguments{ \item{x}{A matrix or data frame with two columns. If this contains missing values, those values are treated as if they fell below the corresponding marginal threshold.} \item{threshold}{A vector of two thresholds.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in \code{\link{rbvevd}}.} \item{likelihood}{The likelihood model; either \code{"censored"} (the default) or \code{"poisson"}. The \code{"poisson"} method is not recommended. See \bold{Details}.} \item{start}{A named list giving the initial values for all of the parameters in the model. If \code{start} is omitted the routine attempts to find good starting values using marginal maximum likelihood estimators.} \item{\dots}{Additional parameters, either for the bivariate extreme value model or for the optimization function \code{optim}. If parameters of the model are included they will be held fixed at the values given (see \bold{Examples}).} \item{sym}{Logical; if \code{TRUE}, the dependence structure of the models \code{"alog"}, \code{"aneglog"} or \code{"ct"} are constrained to be symmetric (see \bold{Details}). For all other models, the argument is ignored (and a warning is given).} \item{cshape}{Logical; if \code{TRUE}, a common shape parameter is fitted to each margin.} \item{cscale}{Logical; if \code{TRUE}, a common scale parameter is fitted to each margin, and the default value of \code{cshape} is then \code{TRUE}, so that under this default common marginal parameters are fitted.} \item{std.err}{Logical; if \code{TRUE} (the default), the standard errors are returned.} \item{corr}{Logical; if \code{TRUE}, the correlation matrix is returned.} \item{method}{The optimization method (see \code{\link{optim}} for details).} \item{warn.inf}{Logical; if \code{TRUE} (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.} } \details{ For the \code{"censored"} method bivariate peaks over threshold models are fitted by maximizing the censored likelihood as given in e.g. Section 8.3.1 of Coles(2001). For the \code{"poisson"} method models are fitted using Equation 5.4 of Coles and Tawn (1991), see also Joe, Smith and Weissman (1992). This method is only available for models whose spectral measure does not contain point masses (see {\link{hbvevd}}). It is not recommended as in practice it can produce poor estimates. For either likelihood the margins are modelled using a generalized Pareto distribution for points above the threshold and an empirical model for those below. For the \code{"poisson"} method data lying below both thresholds is not used. For the \code{"censored"} method the number of points lying below both thresholds is used, but the locations of the those points are not. The dependence parameter names are one or more of \code{dep}, \code{asy1}, \code{asy2}, \code{alpha} and \code{beta}, depending on the model selected (see \code{\link{rbvevd}}). The marginal parameter names are \code{scale1} and \code{shape1} for the first margin, and \code{scale2} and \code{shape2} for the second margin. If \code{cshape} is true, the models are constrained so that \code{shape2 = shape1}. The parameter \code{shape2} is then taken to be specified, so that e.g. the common shape parameter can only be fixed at zero using \code{shape1 = 0}, since using \code{shape2 = 0} gives an error. Similar comments apply for \code{cscale}. If \code{sym} is \code{TRUE}, the asymmetric logistic and asymmetric negative logistic models are constrained so that \code{asy2 = asy1}, and the Coles-Tawn model is constrained so that \code{beta = alpha}. The parameter \code{asy2} or \code{beta} is then taken to be specified, so that e.g. the parameters \code{asy1} and \code{asy2} can only be fixed at \code{0.8} using \code{asy1 = 0.8}, since using \code{asy2 = 0.8} gives an error. Bilogistic and negative bilogistic models constrained to symmetry are logistic and negative logistic models respectively. The (symmetric) mixed model (e.g. Tawn, 1998) can be obtained as a special case of the asymmetric logistic or asymmetric mixed models (see \bold{fbvevd}). For numerical reasons the parameters of each model are subject the artificial constraints given in \code{\link{fbvevd}}. } \value{ Returns an object of class \code{c("bvpot","evd")}. The generic accessor functions \code{\link{fitted}} (or \code{\link{fitted.values}}), \code{\link{std.errors}}, \code{\link{deviance}}, \code{\link{logLik}} and \code{\link{AIC}} extract various features of the returned object. The functions \code{profile} and \code{profile2d} can be used to obtain deviance profiles. The function \code{anova} compares nested models, and the function \code{AIC} compares non-nested models. There is currently no plot method available. An object of class \code{c("bvpot","evd")} is a list containing the following components \item{estimate}{A vector containing the maximum likelihood estimates.} \item{std.err}{A vector containing the standard errors.} \item{fixed}{A vector containing the parameters that have been fixed at specific values within the optimization.} \item{fixed2}{A vector containing the parameters that have been set to be equal to other model parameters.} \item{param}{A vector containing all parameters (those optimized, those fixed to specific values, and those set to be equal to other model parameters).} \item{deviance}{The deviance at the maximum likelihood estimates.} \item{dep.summary}{A value summarizing the strength of dependence in the fitted model (see \bold{fbvevd}).} \item{corr}{The correlation matrix.} \item{var.cov}{The variance covariance matrix.} \item{convergence, counts, message}{Components taken from the list returned by \code{\link{optim}}.} \item{data}{The data passed to the argument \code{x}.} \item{threshold}{The argument \code{threshold}.} \item{n}{The number of rows in \code{x}.} \item{nat}{The vector of length three containing the number of exceedances on the first, second and both margins respectively.} \item{likelihood}{The argument \code{likelihood}.} \item{sym}{The argument \code{sym}.} \item{cmar}{The vector \code{c(cscale, cshape)}.} \item{model}{The argument \code{model}.} \item{call}{The call of the current function.} } \section{Warning}{ The standard errors and the correlation matrix in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution when either of the marginal shape parameters are less than \eqn{-0.5}, because the usual asymptotic properties of maximum likelihood estimators do not then hold (Smith, 1985). } \references{ Coles, S. G. (2001) \emph{An Introduction to Statistical Modelling of Extreme Values}, London: Springer--Verlag. Coles, S. G. and Tawn, J. A. (1991) Modelling multivariate extreme events. \emph{J. R. Statist. Soc. B}, \bold{53}, 377--392. Joe, H., Smith, R. L. and Weissman, I. (1992) Bivariate threshold methods for extremes. \emph{J. R. Statist. Soc. B}, \bold{54}, 171--183. Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. \emph{Biometrika}, \bold{72}, 67--90. } \author{Chris Ferro and Alec Stephenson} \seealso{\code{\link{abvevd}}, \code{\link{anova.evd}}, \code{\link{fbvevd}}, \code{\link{optim}}, \code{\link{rbvevd}}} \examples{ bvdata <- rbvevd(1000, dep = 0.5, model = "log") u <- apply(bvdata, 2, quantile, probs = 0.9) M1 <- fbvpot(bvdata, u, model = "log") M2 <- fbvpot(bvdata, u, "log", dep = 0.5) anova(M1, M2) } \keyword{models} evd/man/profile2d.evd.Rd0000644000175100001440000000353512637167310014567 0ustar hornikusers\name{profile2d.evd} \alias{profile2d} \alias{profile2d.evd} \title{Method for Profiling EVD Objects} \description{ Calculate joint profile traces for fitted models. } \usage{ \method{profile2d}{evd}(fitted, prof, which, pts = 20, convergence = FALSE, method = "Nelder-Mead", control = list(maxit = 5000), \dots) } \arguments{ \item{fitted}{An object of class \code{"evd"}.} \item{prof}{An object of class \code{"profile.evd"}, created using \code{\link{profile.evd}} with argument \code{fitted}. The object must contain the (marginal) profile traces for the two parameters specified in \code{which}.} \item{which}{A character vector of length two containing the original model parameters that are to be jointly profiled.} \item{pts}{The number of distinct values used for each profiled parameter in \code{which}. There are \code{pts^2} optimizations performed in total.} \item{convergence}{Logical; print convergence code after each optimization? (A warning is given for each non-zero convergence code, irrespective of the value of \code{convergence}.)} \item{method}{The optimization method.} \item{control}{Passed to \code{optim}. See \code{\link{optim}} for details.} \item{\dots}{Ignored.} } \value{ An object of class \code{"profile2d.evd"}, which is a list with three elements. The first element, a matrix named \code{trace}, has the same structure as the elements of an object of class \code{"profile.evd"}. The last two elements give the distinct values used for each profiled parameter in \code{which}. } \seealso{\code{\link{profile.evd}}, \code{\link{plot.profile2d.evd}}} \examples{ uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2) M1 <- fgev(uvdata) \dontrun{M1P <- profile(M1)} \dontrun{M1JP <- profile2d(M1, M1P, which = c("scale", "shape"))} \dontrun{plot(M1JP)} } \keyword{models} evd/man/gumbelx.Rd0000644000175100001440000000363614225012534013561 0ustar hornikusers\name{gumbelx} \alias{dgumbelx} \alias{pgumbelx} \alias{qgumbelx} \alias{rgumbelx} \title{Maxima of Two Gumbel Distributions} \description{ Density function, distribution function, quantile function and random generation for the maxima of two Gumbel distributions, each with different location and scale parameters. } \usage{ dgumbelx(x, loc1=0, scale1=1, loc2=0, scale2=1, log = FALSE) pgumbelx(q, loc1=0, scale1=1, loc2=0, scale2=1, lower.tail = TRUE) qgumbelx(p, interval, loc1=0, scale1=1, loc2=0, scale2=1, lower.tail = TRUE, \dots) rgumbelx(n, loc1=0, scale1=1, loc2=0, scale2=1) } \arguments{ \item{x, q}{Vector of quantiles.} \item{p}{Vector of probabilities.} \item{n}{Number of observations.} \item{interval}{A length two vector containing the end-points of the interval to be searched for the quantiles, passed to the uniroot function.} \item{loc1, scale1, loc2, scale2}{Location and scale parameters of the two Gumbel distributions. The second location parameter must be greater than or equal to the first location parameter.} \item{log}{Logical; if \code{TRUE}, the log density is returned.} \item{lower.tail}{Logical; if \code{TRUE} (default), probabilities are P[X <= x], otherwise, P[X > x]} \item{\dots}{Other arguments passed to uniroot.} } \value{ \code{dgumbelx} gives the density function, \code{pgumbelx} gives the distribution function, \code{qgumbelx} gives the quantile function, and \code{rgumbelx} generates random deviates. } \seealso{\code{\link{fgev}}, \code{\link{rfrechet}}, \code{\link{rgumbel}}, \code{\link{rrweibull}}, \code{\link{uniroot}}} \examples{ dgumbelx(2:4, 0, 1.1, 1, 0.5) pgumbelx(2:4, 0, 1.1, 1, 0.5) qgumbelx(seq(0.9, 0.6, -0.1), interval = c(0,10), 0, 1.2, 2, 0.5) rgumbelx(6, 0, 1.1, 1, 0.5) p <- (1:9)/10 pgumbelx(qgumbelx(p, interval = c(0,10), 0, 0.5, 1, 2), 0, 0.5, 1, 2) ## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 } \keyword{distribution} evd/man/abvevd.Rd0000644000175100001440000001122512637167310013366 0ustar hornikusers\name{abvevd} \alias{abvevd} \title{Parametric Dependence Functions of Bivariate Extreme Value Models} \description{ Calculate or plot the dependence function \eqn{A} for nine parametric bivariate extreme value models. } \usage{ abvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"), rev = FALSE, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "t", ylab = "A(t)", \dots) } \arguments{ \item{x}{A vector of values at which the dependence function is evaluated (ignored if plot or add is \code{TRUE}). \eqn{A(1/2)} is returned by default since it is often a useful summary of dependence.} \item{dep}{Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.} \item{asy}{A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.} \item{alpha, beta}{Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.} \item{model}{The specified model; a character string. Must be either \code{"log"} (the default), \code{"alog"}, \code{"hr"}, \code{"neglog"}, \code{"aneglog"}, \code{"bilog"}, \code{"negbilog"}, \code{"ct"} or \code{"amix"} (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in \code{\link{rbvevd}}. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.} \item{rev}{Logical; reverse the dependence function? This is equivalent to evaluating the function at \code{1-x}.} \item{plot}{Logical; if \code{TRUE} the function is plotted. The x and y values used to create the plot are returned invisibly. If \code{plot} and \code{add} are \code{FALSE} (the default), the arguments following \code{add} are ignored.} \item{add}{Logical; add to an existing plot? The existing plot should have been created using either \code{abvevd} or \code{\link{abvnonpar}}, the latter of which plots (or calculates) a non-parametric estimate of the dependence function.} \item{lty, blty}{Function and border line types. Set \code{blty} to zero to omit the border.} \item{lwd, blwd}{Function an border line widths.} \item{col}{Line colour.} \item{xlim, ylim}{x and y-axis limits.} \item{xlab, ylab}{x and y-axis labels.} \item{\dots}{Other high-level graphics parameters to be passed to \code{plot}.} } \details{ Any bivariate extreme value distribution can be written as \deqn{G(z_1,z_2) = \exp\left[-(y_1+y_2)A\left( \frac{y_1}{y_1+y_2}\right)\right]}{ G(z1,z2) = exp{-(y1+y2)A[y1/(y1+y2)]}} for some function \eqn{A(\cdot)}{A()} defined on \eqn{[0,1]}, where \deqn{y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}}{ yi = {1+si(zi-ai)/bi}^(-1/si)} for \eqn{1+s_i(z_i-a_i)/b_i > 0}{1+si(zi-ai)/bi > 0} and \eqn{i = 1,2}, with the (generalized extreme value) marginal parameters given by \eqn{(a_i,b_i,s_i)}{(ai,bi,si)}, \eqn{b_i > 0}{bi > 0}. If \eqn{s_i = 0}{si = 0} then \eqn{y_i}{yi} is defined by continuity. \eqn{A(\cdot)}{A()} is called (by some authors) the dependence function. It follows that \eqn{A(0)=A(1)=1}, and that \eqn{A(\cdot)}{A()} is a convex function with \eqn{\max(x,1-x) \leq A(x)\leq 1}{max(x,1-x) <= A(x) <= 1} for all \eqn{0\leq x\leq1}{0 <= x <= 1}. The lower and upper limits of \eqn{A} are obtained under complete dependence and independence respectively. \eqn{A(\cdot)}{A()} does not depend on the marginal parameters. Some authors take B(x) = A(1-x) as the dependence function. If the argument \code{rev = TRUE}, then B(x) is plotted/evaluated. } \value{ \code{abvevd} calculates or plots the dependence function for one of nine parametric bivariate extreme value models, at specified parameter values. } \seealso{\code{\link{abvnonpar}}, \code{\link{fbvevd}}, \code{\link{rbvevd}}, \code{\link{amvevd}}} \examples{ abvevd(dep = 2.7, model = "hr") abvevd(seq(0,1,0.25), dep = 0.3, asy = c(.7,.9), model = "alog") abvevd(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE) bvdata <- rbvevd(100, dep = 0.7, model = "log") M1 <- fitted(fbvevd(bvdata, model = "log")) abvevd(dep = M1["dep"], model = "log", plot = TRUE) abvnonpar(data = bvdata, add = TRUE, lty = 2) } \keyword{distribution} evd/DESCRIPTION0000644000175100001440000000143514673503722012570 0ustar hornikusersPackage: evd Version: 2.3-7.1 Date: 2024-04-23 Title: Functions for Extreme Value Distributions Author: Alec Stephenson. Function fbvpot by Chris Ferro. Maintainer: Alec Stephenson Imports: stats, grDevices, graphics Suggests: interp Description: Extends simulation, distribution, quantile and density functions to univariate and multivariate parametric extreme value distributions, and provides fitting functions which calculate maximum likelihood estimates for univariate and bivariate maxima models, and for univariate and bivariate threshold models. LazyData: yes License: GPL-3 NeedsCompilation: yes Packaged: 2024-09-21 08:14:54 UTC; hornik Repository: CRAN Date/Publication: 2024-09-21 08:46:10 UTC Depends: R (>= 2.10)