maxLik/0000755000176200001440000000000014077570432011510 5ustar liggesusersmaxLik/NAMESPACE0000644000176200001440000000453714077553264012744 0ustar liggesusersimportFrom("generics", "glance", "tidy") importFrom("methods", "new", "show", "slot", "slot<-", "slotNames", "validObject") importFrom("miscTools", "nObs", "nParam", "sumKeepAttr") importFrom( "miscTools", "stdEr" ) importFrom("sandwich", "bread", "estfun", "sandwich") importFrom("stats", "coef", "logLik", "optim", "pnorm", "printCoefmat", "vcov", "AIC", "qnorm") importFrom("utils", "head", "str", "tail") export( "activePar" ) export( "compareDerivatives" ) export( "condiNumber" ) export( "fnSubset" ) export( "glance" ) export( "gradient" ) export( "hessian" ) export( "maxBFGS" ) export( "maxBFGSR" ) export( "maxBHHH" ) export( "maxCG", "maxSGA", "maxAdam") export( "maximType" ) export( "maxValue" ) export( "maxLik" ) export( "maxNM" ) export( "maxNR" ) export( "maxSANN" ) export( "nIter" ) export( "numericGradient" ) export( "numericHessian" ) export( "numericNHessian" ) export( "objectiveFn" ) export( "returnCode" ) export( "returnMessage" ) export("storedParameters") export("storedValues") export("sumt") export( "tidy" ) exportClasses("MaxControl") exportMethods("maxControl") exportMethods("show") S3method( "activePar", "default" ) S3method( "AIC", "maxLik" ) S3method( "bread", "maxLik" ) S3method( "coef", "maxim" ) S3method( "coef", "maxLik" ) S3method( "coef", "summary.maxLik" ) S3method( "condiNumber", "default" ) S3method("condiNumber", "maxLik" ) S3method("confint", "maxLik" ) S3method("estfun", "maxLik" ) S3method("glance", "maxLik") S3method( "gradient", "maxim" ) S3method( "hessian", "default" ) S3method( "logLik", "maxLik" ) S3method( "logLik", "summary.maxLik" ) S3method( "maximType", "default" ) S3method( "maximType", "maxim" ) S3method( "maxValue", "maxim" ) S3method( "nIter", "default" ) S3method( "nObs", "maxLik" ) S3method( "nParam", "maxim" ) S3method( "print", "maxLik" ) S3method( "print", "summary.maxim" ) S3method( "print", "summary.maxLik" ) S3method( "objectiveFn", "maxim" ) S3method( "returnCode", "default" ) S3method( "returnCode", "maxim" ) S3method( "returnCode", "maxLik" ) S3method( "returnMessage", "default" ) S3method( "returnMessage", "maxim" ) S3method( "returnMessage", "maxLik" ) S3method( "stdEr", "maxLik" ) S3method( "storedParameters", "maxim" ) S3method( "storedValues", "maxim" ) S3method( "summary", "maxim" ) S3method( "summary", "maxLik" ) S3method("tidy", "maxLik") S3method("vcov", "maxLik") maxLik/man/0000755000176200001440000000000014077525067012267 5ustar liggesusersmaxLik/man/sumt.Rd0000644000176200001440000001231314077525067013546 0ustar liggesusers\name{sumt} \Rdversion{1.1} \alias{sumt} \title{ Equality-constrained optimization } \description{ Sequentially Unconstrained Maximization Technique (SUMT) based optimization for linear equality constraints. This implementation is primarily intended to be called from other maximization routines, such as \code{\link{maxNR}}. } \usage{ sumt(fn, grad=NULL, hess=NULL, start, maxRoutine, constraints, SUMTTol = sqrt(.Machine$double.eps), SUMTPenaltyTol = sqrt(.Machine$double.eps), SUMTQ = 10, SUMTRho0 = NULL, printLevel=print.level, print.level = 0, SUMTMaxIter = 100, ...) } \arguments{ \item{fn}{ function of a (single) vector parameter. The function may have more arguments (passed by \dots), but those are not treated as the parameter. } \item{grad}{ gradient function of \code{fn}. NULL if missing } \item{hess}{ function, Hessian of the \code{fn}. NULL if missing } \item{start}{ numeric, initial value of the parameter } \item{maxRoutine}{ maximization algorithm, such as \code{\link{maxNR}} } \item{constraints}{list, information for constrained maximization. Currently two components are supported: \code{eqA} and \code{eqB} for linear equality constraints: \eqn{A \beta + B = 0}{A \%*\% beta + B = 0}. The user must ensure that the matrices \code{A} and \code{B} are conformable.} \item{SUMTTol}{ stopping condition. If the estimates at successive outer iterations are close enough, i.e. maximum of the absolute value over the component difference is smaller than SUMTTol, the algorithm stops. Note this does not necessarily mean that the constraints are satisfied. If the penalty function is too \dQuote{weak}, SUMT may repeatedly find the same optimum. In that case a warning is issued. The user may set SUMTTol to a lower value, e.g. to zero. } \item{SUMTPenaltyTol}{ stopping condition. If the barrier value (also called penalty) \eqn{(A \beta + B)'(A \beta + B)}{t(A \%*\% beta + B) \%*\% (A \%*\% beta + B)} is less than \code{SUMTTol}, the algorithm stops } \item{SUMTQ}{ a double greater than one, controlling the growth of the \code{rho} as described in Details. Defaults to 10. } \item{SUMTRho0}{ Initial value for \code{rho}. If not specified, a (possibly) suitable value is selected. See Details. One should consider supplying \code{SUMTRho0} in case where the unconstrained problem does not have a maximum, or the maximum is too far from the constrained value. Otherwise the authomatically selected value may not lead to convergence. } \item{printLevel}{ Integer, debugging information. Larger number prints more details. } \item{print.level}{same as \sQuote{printLevel}, for backward compatibility} \item{SUMTMaxIter}{ Maximum SUMT iterations } \item{\dots}{ Other arguments to \code{maxRoutine} and \code{fn}. } } \details{ The Sequential Unconstrained Minimization Technique is a heuristic for constrained optimization. To minimize a function \eqn{f}{f} subject to constraints, it uses a non-negative penalty function \eqn{P}{P}, such that \eqn{P(x)}{P(x)} is zero iff \eqn{x}{x} satisfies the constraints. One iteratively minimizes \eqn{f(x) + \varrho_k P(x)}{f(x) + rho_k P(x)}, where the \eqn{\varrho}{rho} values are increased according to the rule \eqn{\varrho_{k+1} = q \varrho_k}{rho_{k+1} = q rho_k} for some constant \eqn{q > 1}{q > 1}, until convergence is achieved in the sense that the barrier value \eqn{P(x)'P(x)}{P(x)'P(x)} is close to zero. Note that there is no guarantee that the global constrained optimum is found. Standard practice recommends to use the best solution found in \dQuote{sufficiently many} replications. Any of the maximization algorithms in the \pkg{maxLik}, such as \code{\link{maxNR}}, can be used for the unconstrained step. Analytic gradient and hessian are used if provided. } \value{ Object of class 'maxim'. In addition, a component \item{constraints}{A list, describing the constrained optimization. Includes the following components: \describe{ \item{type}{type of constrained optimization} \item{barrier.value}{value of the penalty function at maximum} \item{code}{code for the stopping condition} \item{message}{a short message, describing the stopping condition} \item{outer.iterations}{number of iterations in the SUMT step} } } } \section{Note}{ In case of equality constraints, it may be more efficient to enclose the function in a wrapper function. The wrapper calculates full set of parameters based on a smaller set of parameters, and the constraints. } \author{ Ott Toomet, Arne Henningsen } \seealso{ \code{\link[clue]{sumt}} in package \pkg{clue}. } \examples{ ## We maximize exp(-x^2 - y^2) where x+y = 1 hatf <- function(theta) { x <- theta[1] y <- theta[2] exp(-(x^2 + y^2)) ## Note: you may prefer exp(- theta \%*\% theta) instead } ## use constraints: x + y = 1 A <- matrix(c(1, 1), 1, 2) B <- -1 res <- sumt(hatf, start=c(0,0), maxRoutine=maxNR, constraints=list(eqA=A, eqB=B)) print(summary(res)) } \keyword{optimize} maxLik/man/gradient.Rd0000644000176200001440000000373414077525067014362 0ustar liggesusers\name{gradient} \alias{gradient} \alias{gradient.maxim} \alias{estfun} \alias{estfun.maxLik} \title{Extract Gradients Evaluated at each Observation} \description{ Extract the gradients of the log-likelihood function evaluated at each observation (\sQuote{Empirical Estimating Function}, see \code{\link[sandwich]{estfun}}). } \usage{ \method{estfun}{maxLik}(x, ...) \method{gradient}{maxim}(x, ...) } \arguments{ \item{x}{an object inheriting from class \code{maxim} (for \code{gradient}) or \code{maxLik}. (for \code{estfun}.)} \item{\dots}{further arguments (currently ignored).} } \value{ \item{\code{gradient}}{vector, objective function gradient at estimated maximum (or the last calculated value if the estimation did not converge.)} \item{\code{estfun}}{ matrix, observation-wise log-likelihood gradients at the estimated parameter value evaluated at each observation. Observations in rows, parameters in columns.} } \section{Warnings}{ The \pkg{sandwich} package must be loaded in order to use \code{estfun}. \code{estfun} only works if the observaton-specific gradient information was available for the estimation. This is the case of the observation-specific gradient was supplied (see the \code{grad} argument for \code{\link{maxLik}}), or the log-likelihood function returns a vector of observation-specific values. } \author{ Arne Henningsen, Ott Toomet } \seealso{\code{\link{hessian}}, \code{\link[sandwich]{estfun}}, \code{\link{maxLik}}.} \examples{ ## ML estimation of exponential duration model: t <- rexp(10, 2) loglik <- function(theta) log(theta) - theta*t ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1 ) gradient(a) # Extract the gradients evaluated at each observation library( sandwich ) estfun( a ) ## Estimate with analytic gradient. ## Note: it returns a vector gradlik <- function(theta) 1/theta - t b <- maxLik(loglik, gradlik, start=1) gradient(a) estfun( b ) } \keyword{methods} maxLik/man/maxLik.Rd0000644000176200001440000001320014077525067013777 0ustar liggesusers\name{maxLik} \alias{maxLik} \alias{print.maxLik} \title{Maximum likelihood estimation} \description{ This is the main interface for the \pkg{maxLik} package, and the function that performs Maximum Likelihood estimation. It is a wrapper for different optimizers returning an object of class "maxLik". Corresponding methods handle the likelihood-specific properties of the estimates, including standard errors. } \usage{ maxLik(logLik, grad = NULL, hess = NULL, start, method, constraints=NULL, ...) } \arguments{ \item{logLik}{log-likelihood function. Must have the parameter vector as the first argument. Must return either a single log-likelihood value, or a numeric vector where each component is log-likelihood of the corresponding individual observation.} \item{grad}{gradient of log-likelihood. Must have the parameter vector as the first argument. Must return either a single gradient vector with length equal to the number of parameters, or a matrix where each row is the gradient vector of the corresponding individual observation. If \code{NULL}, numeric gradient will be used.} \item{hess}{hessian of log-likelihood. Must have the parameter vector as the first argument. Must return a square matrix. If \code{NULL}, numeric Hessian will be used.} \item{start}{numeric vector, initial value of parameters. If it has names, these will also be used for naming the results.} \item{method}{maximisation method, currently either "NR" (for Newton-Raphson), "BFGS" (for Broyden-Fletcher-Goldfarb-Shanno), "BFGSR" (for the BFGS algorithm implemented in \R), "BHHH" (for Berndt-Hall-Hall-Hausman), "SANN" (for Simulated ANNealing), "CG" (for Conjugate Gradients), or "NM" (for Nelder-Mead). Lower-case letters (such as "nr" for Newton-Raphson) are allowed. The default method is "NR" for unconstrained problems, and "NM" or "BFGS" for constrained problems, depending on if the \code{grad} argument was provided. "BHHH" is a good alternative given the likelihood is returned observation-wise (see \code{\link{maxBHHH}}). Note that stochastic gradient ascent (SGA) is currently not supported as this method seems to be rarely used for maximum likelihood estimation. } \item{constraints}{either \code{NULL} for unconstrained maximization or a list, specifying the constraints. See \code{\link{maxBFGS}}. } \item{\dots}{further arguments, such as \code{control}, \code{iterlim}, or \code{tol}, are passed to the selected maximisation routine, i.e. \code{\link{maxNR}}, \code{\link{maxBFGS}}, \code{\link{maxBFGSR}}, \code{\link{maxBHHH}}, \code{\link{maxSANN}}, \code{\link{maxCG}}, or \code{\link{maxNM}} (depending on argument \code{method}). Arguments not used by the optimizers are forwarded to \code{logLik}, \code{grad} and \code{hess}. } } \details{ \code{maxLik} supports constrained optimization in the sense that constraints are passed further to the underlying optimization routines, and suitable default method is selected. However, no attempt is made to correct the resulting variance-covariance matrix. Hence the inference may be wrong. A corresponding warning is issued by the summary method. } \value{ object of class 'maxLik' which inherits from class 'maxim'. Useful methods include \itemize{ \item \code{\link[=AIC.maxLik]{AIC}}: estimated parameter value \item \code{\link[=coef.maxLik]{coef}}: estimated parameter value \item \code{\link[=logLik.maxLik]{logLik}}: log-likelihood value \item \code{\link{nIter}}: number of iterations \item \code{\link[=stdEr.maxLik]{stdEr}}: standard errors \item \code{\link[=summary.maxLik]{summary}}: print summary table with estimates, standard errors, p, and z-values. \item \code{\link[=vcov.maxLik]{vcov}}: variance-covariance matrix } } \section{Warning}{The constrained maximum likelihood estimation should be considered experimental. In particular, the variance-covariance matrix is not corrected for constrained parameter space. } \author{Ott Toomet, Arne Henningsen} \seealso{\code{\link{maxNR}}, \code{\link{nlm}} and \code{\link{optim}} for different non-linear optimisation routines, see \code{\link{maxBFGS}} for the constrained maximization examples.} \examples{ ## Estimate the parameter of exponential distribution t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1, control=list(printLevel=2)) summary( a ) ## ## Estimate with analytic gradient and hessian. ## require much smaller tolerance ## setting 'tol=0' or negative essentially disables this stopping criterion a <- maxLik(loglik, gradlik, hesslik, start=1, control=list(tol=-1, reltol=1e-12, gradtol=1e-12)) summary( a ) ## ## Next, we give an example with vector argument: ## fit normal distribution by estimating mean and standard deviation ## by maximum likelihood ## loglik <- function(param) { # param: vector of 2, c(mean, standard deviation) mu <- param[1] sigma <- param[2] ll <- -0.5*N*log(2*pi) - N*log(sigma) - sum(0.5*(x - mu)^2/sigma^2) # can use dnorm(x, mu, sigma, log=TRUE) instead ll } x <- rnorm(100, 1, 2) # use mean=1, stdd=2 N <- length(x) res <- maxLik(loglik, start=c(0,1)) # use 'wrong' start values summary(res) ## ## Same example, but now with named parameters and a fixed value ## resFix <- maxLik(loglik, start=c(mu=0, sigma=1), fixed="sigma") summary(resFix) # 'sigma' is exactly 1.000 now. } \keyword{optimize} maxLik/man/objectiveFn.Rd0000644000176200001440000000145414077525067015020 0ustar liggesusers\name{objectiveFn} \alias{objectiveFn} \alias{objectiveFn.maxim} \title{Optimization Objective Function} \description{ This function returns the optimization objective function from a \sQuote{maxim} object. } \usage{ objectiveFn(x, \dots) \method{objectiveFn}{maxim}(x, \dots) } \arguments{ \item{x}{an optimization result, inheriting from class \sQuote{maxim}} \item{\dots}{other arguments for methods} } \value{ function, the function that was optimized. It can be directly called, given that all necessary variables are accessible from the current environment. } \author{Ott Toomet} \examples{ hatf <- function(theta) exp(- theta \%*\% theta) res <- maxNR(hatf, start=c(0,0)) print(summary(res)) print(objectiveFn(res)) print(objectiveFn(res)(2)) # 0.01832 } \keyword{methods} \keyword{optimize} maxLik/man/maximType.Rd0000644000176200001440000000144214077525067014534 0ustar liggesusers\name{maximType} \alias{maximType} \alias{maximType.default} \alias{maximType.maxim} \alias{maximType.MLEstimate} \title{Type of Minimization/Maximization} \description{ Returns the type of optimization as supplied by the optimisation routine. } \usage{ maximType(x) } \arguments{ \item{x}{object of class 'maxim' or another object which involves numerical optimisation. } } \value{ A text message, describing the involved optimisation algorithm } \author{Ott Toomet} \seealso{\code{\link{maxNR}}} \examples{ ## maximize two-dimensional exponential hat. True maximum c(2,1): f <- function(a) exp(-(a[1] - 2)^2 - (a[2] - 1)^2) m <- maxNR(f, start=c(0,0)) coef(m) maximType(m) ## Now use BFGS maximisation. m <- maxBFGS(f, start=c(0,0)) maximType(m) } \keyword{optimize} \keyword{methods} maxLik/man/numericGradient.Rd0000644000176200001440000000541514077525067015703 0ustar liggesusers\name{numericGradient} \alias{numericGradient} \alias{numericHessian} \alias{numericNHessian} \title{Functions to Calculate Numeric Derivatives} \description{ Calculate (central) numeric gradient and Hessian, including of vector-valued functions. } \usage{ numericGradient(f, t0, eps=1e-06, fixed, \dots) numericHessian(f, grad=NULL, t0, eps=1e-06, fixed, \dots) numericNHessian(f, t0, eps=1e-6, fixed, \dots) } \arguments{ \item{f}{function to be differentiated. The first argument must be the parameter vector with respect to which it is differentiated. For numeric gradient, \code{f} may return a (numeric) vector, for Hessian it should return a numeric scalar} \item{grad}{function, gradient of \code{f}} \item{t0}{vector, the parameter values} \item{eps}{numeric, the step for numeric differentiation} \item{fixed}{logical index vector, fixed parameters. Derivative is calculated only with respect to the parameters for which \code{fixed == FALSE}, \code{NA} is returned for the fixed parameters. If missing, all parameters are treated as active.} \item{\dots}{furter arguments for \code{f}} } \details{ \code{numericGradient} numerically differentiates a (vector valued) function with respect to it's (vector valued) argument. If the functions value is a \eqn{N_{val} \times 1}{\code{N_val * 1}} vector and the argument is \eqn{N_{par} \times 1}{\code{N_par * 1}} vector, the resulting gradient is a \eqn{N_{val} \times N_{par}}{\code{NVal * NPar}} matrix. \code{numericHessian} checks whether a gradient function is present. If yes, it calculates the gradient of the gradient, if not, it calculates the full numeric Hessian (\code{numericNHessian}). } \value{ Matrix. For \code{numericGradient}, the number of rows is equal to the length of the function value vector, and the number of columns is equal to the length of the parameter vector. For the \code{numericHessian}, both numer of rows and columns is equal to the length of the parameter vector. } \section{Warning}{ Be careful when using numerical differentiation in optimization routines. Although quite precise in simple cases, they may work very poorly in more complicated conditions. } \author{Ott Toomet} \seealso{\code{\link{compareDerivatives}}, \code{\link{deriv}}} \examples{ # A simple example with Gaussian bell surface f0 <- function(t0) exp(-t0[1]^2 - t0[2]^2) numericGradient(f0, c(1,2)) numericHessian(f0, t0=c(1,2)) # An example with the analytic gradient gradf0 <- function(t0) -2*t0*f0(t0) numericHessian(f0, gradf0, t0=c(1,2)) # The results should be similar as in the previous case # The central numeric derivatives are often quite precise compareDerivatives(f0, gradf0, t0=1:2) # The difference is around 1e-10 } \keyword{math} \keyword{utilities} maxLik/man/storedValues.Rd0000644000176200001440000000323514077525067015241 0ustar liggesusers\name{storedValues} \alias{storedValues} \alias{storedValues.maxim} \alias{storedParameters} \alias{storedParameters.maxim} \title{Return the stored values of optimization} \description{ Retrieve the objective function value for each iteration if stored during the optimization. } \usage{ storedValues(x, \dots) \method{storedValues}{maxim}(x, \dots) storedParameters(x, \dots) \method{storedParameters}{maxim}(x, \dots) } \arguments{ \item{x}{a result of maximization, created by \code{\link{maxLik}}, \code{\link{maxSGA}} or another optimizer.} \item{\dots}{further arguments for other methods} } \details{ These is a generic method. If asked by control parameter \code{storeValues=TRUE} or \code{storeParameters=TRUE}, certain optimization methods store the objective function value and the parameter value at each epoch. These methods retrieves the stored values. } \value{ \itemize{ \item \code{storedValues}: a numeric vector, one value for each iteration \item \code{storedParameters}: a numeric matrix with rows corresponding to the iterations and columns to the parameter components. } In both cases, the first value stored corresponds to the initial parameter. } \author{Ott Toomet} \seealso{\code{\link{maxSGA}}, \code{\link{maxControl}} } \examples{ ## Estimate the exponential distribution parameter t <- rexp(100, 2) loglik <- function(theta, index) sum(log(theta) - theta*t[index]) ## Estimate with numeric gradient and numeric Hessian a <- maxSGA(loglik, start=1, control=list(storeValues=TRUE, storeParameters=TRUE, iterlim=10), nObs=100) storedValues(a) storedParameters(a) } \keyword{methods} maxLik/man/logLik.maxLik.Rd0000644000176200001440000000221214077525067015220 0ustar liggesusers\name{logLik.maxLik} \alias{logLik.maxLik} \alias{logLik.summary.maxLik} \title{Return the log likelihood value} \description{ Return the log likelihood value of objects of class \code{maxLik} and \code{summary.maxLik}. } \usage{ \method{logLik}{maxLik}( object, \dots ) \method{logLik}{summary.maxLik}( object, \dots ) } \arguments{ \item{object}{object of class \code{maxLik} or \code{summary.maxLik}, usually a model estimated with Maximum Likelihood} \item{...}{additional arguments to methods} } \value{ A scalar numeric, log likelihood of the estimated model. It has attribute \dQuote{df}, number of free parameters. } \author{ Arne Henningsen, Ott Toomet } \seealso{\code{\link{maxLik}}} \examples{ ## ML estimation of exponential duration model: t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 ## Estimate with analytic gradient and hessian a <- maxLik(loglik, gradlik, hesslik, start=1) ## print log likelihood value logLik( a ) ## print log likelihood value of summary object b <- summary( a ) logLik( b ) } \keyword{methods} maxLik/man/maxControl.Rd0000644000176200001440000002452714077525067014716 0ustar liggesusers\name{MaxControl-class} \Rdversion{1.1} \docType{class} \alias{MaxControl-class} \alias{maxControl} \alias{maxControl,MaxControl-method} \alias{maxControl,missing-method} \alias{maxControl,maxim-method} \alias{show,MaxControl-method} \title{Class \code{"MaxControl"}} \description{ This is the structure that holds the optimization control options. The corresponding constructors take the parameters, perform consistency checks, and return the control structure. Alternatively, it overwrites the supplied parameters in an existing \code{MaxControl} structure. There is also a method to extract the control structure from the estimated \sQuote{maxim}-objects. } \section{Slots}{ The default values and definition of the slots: \describe{ \item{tol}{1e-8, stopping condition for \code{\link{maxNR}} and related optimizers. Stop if the absolute difference between successive iterations is less than \code{tol}, returns code 2.} \item{reltol}{sqrt(.Machine$double.eps), relative convergence tolerance (used by \code{\link{maxNR}} related optimizers, and \code{\link{optim}}-based optimizers. The algorithm stops if it iteration increases the value by less than a factor of \code{reltol*(abs(val) + reltol)}. Returns code 2.} \item{gradtol}{1e-6, stopping condition for \code{\link{maxNR}} and related optimizers. Stops if norm of the gradient is less than \code{gradtol}, returns code 1.} \item{steptol}{1e-10, stopping/error condition for \code{\link{maxNR}} and related optimizers. If \code{qac == "stephalving"} and the quadratic approximation leads to a worse, instead of a better value, or to \code{NA}, the step length is halved and a new attempt is made. If necessary, this procedure is repeated until \code{step < steptol}, thereafter code 3 is returned.} % \item{lambdatol}{1e-6, (for \code{\link{maxNR}} related optimizers) controls whether Hessian is treated as negative definite. If the largest of the eigenvalues of the Hessian is larger than \code{-lambdatol} (Hessian is not negative definite), a suitable diagonal matrix is subtracted from the Hessian (quadratic hill-climbing) in order to enforce negative definiteness.} % \item{qac}{"stephalving", character, Qadratic Approximation Correction for \code{\link{maxNR}} related optimizers. When the new guess is worse than the initial one, program attempts to correct it: \code{"stephalving"} decreases the step but keeps the direction. \code{"marquardt"} uses \cite{Marquardt (1963)} method by decreasing the step length while also moving closer to the pure gradient direction. It may be faster and more robust choice in areas where quadratic approximation behaves poorly.} \item{qrtol}{1e-10, QR-decomposition tolerance for Hessian inversion in \code{\link{maxNR}} related optimizers. } \item{marquardt_lambda0}{0.01, a positive numeric, initial correction term for \cite{Marquardt (1963)} correction in \code{\link{maxNR}}-related optimizers} \item{marquardt_lambdaStep}{2, how much the \cite{Marquardt (1963)} correction is decreased/increased at successful/unsuccesful step for \code{\link{maxNR}} related optimizers} \item{marquardt_maxLambda}{1e12, maximum allowed correction term for \code{\link{maxNR}} related optimizers. If exceeded, the algorithm exits with return code 3.} % \item{nm_alpha}{1, Nelder-Mead simplex method reflection factor (see Nelder \& Mead, 1965)} \item{nm_beta}{0.5, Nelder-Mead contraction factor} \item{nm_gamma}{2, Nelder-Mead expansion factor} % SANN \item{sann_cand}{\code{NULL} or a function for \code{"SANN"} algorithm to generate a new candidate point; if \code{NULL}, Gaussian Markov kernel is used (see argument \code{gr} of \code{\link{optim}}).} \item{sann_temp}{10, starting temperature for the \dQuote{SANN} cooling schedule. See \code{\link{optim}}.} \item{sann_tmax}{10, number of function evaluations at each temperature for the \dQuote{SANN} optimizer. See \code{\link{optim}}.} \item{sann_randomSeed}{123, integer to seed random numbers to ensure replicability of \dQuote{SANN} optimization and preserve \code{R} random numbers. Use options like \code{SANN_randomSeed=Sys.time()} or \code{SANN_randomeSeed=sample(1000,1)} if you want stochastic results. } % SG general General options for stochastic gradient methods: \item{SG_learningRate}{0.1, learning rate, numeric} \item{SG_batchSize}{\code{NULL}, batch size for Stochastic Gradient Ascent. A positive integer, or \code{NULL} for full-batch gradent ascent.} \item{SG_clip}{\code{NULL}, gradient clipping threshold. This is the max allowed squared Euclidean norm of the gradient. If the actual norm of the gradient exceeds (square root of) this threshold, the gradient will be scaled back accordingly while preserving its direction. \code{NULL} means no clipping. } \item{SG_patience}{\code{NULL}, or integer. Stopping condition: if the objective function is worse than its largest value so far this many times, the algorithm stops, and returns not the last parameter value but the one that gave the best results so far. This is mostly useful if gradient is computed on training data and the objective function on validation data. } \item{SG_patienceStep}{1L, integer. After how many epochs to check the patience value. 1 means to check (and hence to compute the objective function) at each epoch. } % Stochastic Gradient Ascent Options for SGA: \item{SGA_momentum}{0, numeric momentum parameter for SGA. Must lie in interval \eqn{[0,1]}{[0,1]}. } % Adam Options for Adam: \item{Adam_momentum1}{0.9, numeric in \eqn{[0,1]}{[0,1]}, the first moment momentum} \item{Adam_momentum2}{0.999, numeric in \eqn{[0,1]}{[0,1]}, the second moment momentum} % general General options: \item{iterlim}{150, stopping condition (the default differs for different methods). Stop if more than \code{iterlim} iterations performed. Note that \sQuote{iteration} may mean different things for different optimizers.} \item{max.rows}{20, maximum number of matrix rows to be printed when requesting verbosity in the optimizers. } \item{max.cols}{7, maximum number of columns to be printed. This also applies to vectors that are printed horizontally. } \item{printLevel}{0, the level of verbosity. Larger values print more information. Result depends on the optimizer. Form \code{print.level} is also accepted by the methods for compatibility.} \item{storeParameters}{\code{FALSE}, whether to store and return the parameter values at each epoch. If \code{TRUE}, the stored values can be retrieved with \code{\link{storedParameters}}-method. The parameters are stored as a matrix with rows corresponding to the epochs and columns to the parameter components. } \item{storeValues}{\code{FALSE}, whether to store and return the objective function values at each epoch. If \code{TRUE}, the stored values can be retrieved with \code{\link{storedValues}}-method.} } } \section{Methods}{ \describe{ \item{maxControl}{\code{(\dots)} creates a \dQuote{MaxControl} object. The arguments must be in the form \code{option1 = value1, option2 = value2, ...}. The options should be slot names, but the method also supports selected other parameter forms for compatibility reasons e.g. \dQuote{print.level} instead of \dQuote{printLevel}. In case there are more than one option with similar name, the last one overwrites the previous values. This allows the user to override default parameters in the control list. See example in \link{maxLik-package}. } \item{maxControl}{\code{(x = "MaxControl", \dots)} overwrites parameters of an existing \dQuote{MaxControl} object. The \sQuote{\dots} argument must be in the form \code{option1 = value1, option2 = value2, ...}. In case there are more than one option with similar name, only the last one is taken into account. This allows the user to override default parameters in the control list. See example in \link{maxLik-package}. } \item{maxControl}{\code{(x = "maxim")} extracts \dQuote{MaxControl} structure from an estimated model} \item{show}{shows the parameter values} } } \section{Details}{ Typically, the control options are supplied in the form of a list, in which case the corresponding default values are overwritten by the user-specified ones. However, one may also create the control structure by \code{maxControl(opt1=value1, opt2=value2, ...)} and supply such value directly to the optimizer. In this case the optimization routine takes all the values from the control object. } \references{ \itemize{ \item Nelder, J. A. & Mead, R. A (1965) Simplex Method for Function Minimization \emph{The Computer Journal} \bold{7}, 308--313 \item Marquardt, D. W. (1963) An Algorithm for Least-Squares Estimation of Nonlinear Parameters \emph{Journal of the Society for Industrial and Applied Mathematics} \bold{11}, 431--441 } } \author{ Ott Toomet } \note{ Several control parameters can also be supplied directly to the optimization routines. } \examples{ library(maxLik) ## Create a 'maxControl' object: maxControl(tol=1e-4, sann_tmax=7, printLevel=2) ## Optimize quadratic form t(D) %*% W %*% D with p.d. weight matrix, ## s.t. constraints sum(D) = 1 quadForm <- function(D) { return(-t(D) \%*\% W \%*\% D) } eps <- 0.1 W <- diag(3) + matrix(runif(9), 3, 3)*eps D <- rep(1/3, 3) # initial values ## create control object and use it for optimization co <- maxControl(printLevel=2, qac="marquardt", marquardt_lambda0=1) res <- maxNR(quadForm, start=D, control=co) print(summary(res)) ## Now perform the same with no trace information co <- maxControl(co, printLevel=0) res <- maxNR(quadForm, start=D, control=co) # no tracing information print(summary(res)) # should be the same as above maxControl(res) # shows the control structure } \keyword{utilities} maxLik/man/confint.maxLik.Rd0000644000176200001440000000261314077525067015444 0ustar liggesusers\name{confint.maxLik} \alias{confint.maxLik} \alias{confint} \title{confint method for maxLik objects} \description{ Wald confidence intervals for Maximum Likelihood Estimates } \usage{ \method{confint}{maxLik}(object, parm, level=0.95, ...) } \arguments{ \item{object}{ object of class \dQuote{maxLik} returned by \code{\link{maxLik}} function } \item{parm}{the name of parameters to compute the confidence intervals. If omitted, confidence intervals for all parameters are computed.} \item{level}{the level of confidence interval } \item{\dots}{additional arguments to be passed to the other methods } } \value{ A matrix of lower and upper confidence interval limits (in the first and second column respectively). The matrix rows are labeled by the parameter names (if any) and columns by the corresponding distribution quantiles. } \seealso{ \code{\link[stats]{confint}} for the generic \code{confint} function, \code{\link[=stdEr.maxLik]{stdEr}} for computing standard errors and \code{\link[=summary.maxLik]{summary}} for summary output that includes statistical significance information. } \author{Luca Scrucca} \examples{ ## compute MLE parameters of normal random sample x <- rnorm(100) loglik <- function(theta) { dnorm(x, mean=theta[1], sd=theta[2], log=TRUE) } m <- maxLik(loglik, start=c(mu=0, sd=1)) summary(m) confint(m) confint(m, "mu", level=0.1) } maxLik/man/maxLik-package.Rd0000644000176200001440000000727514077525067015407 0ustar liggesusers\name{maxLik-package} \alias{maxLik-package} \docType{package} \title{ Maximum Likelihood Estimation } \description{ This package contains a set of functions and tools for Maximum Likelihood (ML) estimation. The focus of the package is on non-linear optimization from the ML viewpoint, and it provides several convenience wrappers and tools, like BHHH algorithm, variance-covariance matrix and standard errors. } \details{ \pkg{maxLik} package is a set of convenience tools and wrappers focusing on Maximum Likelihood (ML) analysis, but it also contains tools for other optimization tasks. The package includes a) wrappers for several existing optimizers (implemented by \code{\link[stats:optim]{optim}}); b) original optimizers, including Newton-Raphson and Stochastic Gradient Ascent; and c) several convenience tools to use these optimizers from the ML perspective. Examples are BHHH optimization (\code{\link{maxBHHH}}) and utilities that extract standard errors from the estimates. Other highlights include a unified interface for all included optimizers, tools to test user-provided analytic derivatives, and constrained optimization. A good starting point to learn about the usage of \pkg{maxLik} are the included vignettes \dQuote{Introduction: what is maximum likelihood}, \dQuote{Maximum likelihood estimation with maxLik} and \dQuote{Stochastic Gradient Ascent in maxLik}. Another good source is Henningsen & Toomet (2011), an introductory paper to the package. Use \code{vignette(package="maxLik")} to see the available vignettes, and \code{vignette("using-maxlik")} to read the usage vignette. From the user's perspective, the central function in the package is \code{\link{maxLik}}. In its simplest form it takes two arguments: the log-likelihood function, and a vector of initial parameter values (see the example below). It returns an object of class \sQuote{maxLik} with convenient methods such as \code{\link[=summary.maxLik]{summary}}, \code{\link[=coef.maxLik]{coef}}, and \code{\link[=stdEr.maxLik]{stdEr}}. It also supports a plethora of other arguments, for instance one can supply analytic gradient and Hessian, select the desired optimizer, and control the optimization in different ways. A useful utility functions in the package is \code{\link{compareDerivatives}} that allows one to compare the analytic and numeric derivatives for debugging purposes. Another useful function is \code{\link{condiNumber}} for analyzing multicollinearity problems in the estimated models. In the interest of providing a unified user interface, all the optimizers are implemented as maximizers in this package. This includes the \code{\link{optim}}-based methods, such as \code{\link{maxBFGS}} and \code{\link{maxSGA}}, the maximizer version of popular Stochastic Gradient Descent. } \author{ Ott Toomet , Arne Henningsen , with contributions from Spencer Graves, Yves Croissant and David Hugh-Jones. Maintainer: Ott Toomet } \references{ Henningsen A, Toomet O (2011). \dQuote{maxLik: A package for maximum likelihood estimation in R.} Computational Statistics, 26(3), 443-458. doi: \doi{10.1007/s00180-010-0217-1}. } \keyword{Basics|package} \keyword{Mathematics|optimize} \examples{ ### estimate mean and variance of normal random vector ## create random numbers where mu=1, sd=2 set.seed(123) x <- rnorm(50, 1, 2 ) ## log likelihood function. ## Note: 'param' is a 2-vector c(mu, sd) llf <- function(param) { mu <- param[1] sd <- param[2] llValue <- dnorm(x, mean=mu, sd=sd, log=TRUE) sum(llValue) } ## Estimate it with mu=0, sd=1 as start values ml <- maxLik(llf, start = c(mu=0, sigma=1) ) print(summary(ml)) ## Estimates close to c(1,2) :-) } maxLik/man/maxLik-internal.Rd0000644000176200001440000000101314077525067015610 0ustar liggesusers\name{maxLik-internal} \alias{checkFuncArgs} \alias{constrOptim2} \alias{maximMessage} \alias{maxNRCompute} \alias{observationGradient} \alias{print.summary.maxLik} \alias{returnCode.maxim} % Document the following: %%%% \title{ Internal maxLik Functions } \description{ Internal maxLik Functions } \details{ These are either various methods, or functions, not intended to be called directly by the user (or in some cases are just waiting for proper documentation to be written :). } \keyword{ internal } maxLik/man/maxNR.Rd0000644000176200001440000004166614077525067013620 0ustar liggesusers\name{maxNR} \alias{maxNR} \alias{maxBFGSR} \alias{maxBHHH} \title{Newton- and Quasi-Newton Maximization} \description{ Unconstrained and equality-constrained maximization based on the quadratic approximation (Newton) method. The Newton-Raphson, BFGS (Broyden 1970, Fletcher 1970, Goldfarb 1970, Shanno 1970), and BHHH (Berndt, Hall, Hall, Hausman 1974) methods are available. } \usage{ maxNR(fn, grad = NULL, hess = NULL, start, constraints = NULL, finalHessian = TRUE, bhhhHessian=FALSE, fixed = NULL, activePar = NULL, control=NULL, ... ) maxBFGSR(fn, grad = NULL, hess = NULL, start, constraints = NULL, finalHessian = TRUE, fixed = NULL, activePar = NULL, control=NULL, ... ) maxBHHH(fn, grad = NULL, hess = NULL, start, finalHessian = "BHHH", ... ) } \arguments{ \item{fn}{the function to be maximized. It must have the parameter vector as the first argument and it must return either a single number, or a numeric vector (this is is summed internally). If the BHHH method is used and argument \code{gradient} is not given, \code{fn} must return a numeric vector of observation-specific log-likelihood values. If the parameters are out of range, \code{fn} should return \code{NA}. See details for constant parameters. \code{fn} may also return attributes "gradient" and/or "hessian". If these attributes are set, the algorithm uses the corresponding values as gradient and Hessian. } \item{grad}{gradient of the objective function. It must have the parameter vector as the first argument and it must return either a gradient vector of the objective function, or a matrix, where \emph{columns} correspond to individual parameters. The column sums are treated as gradient components. If \code{NULL}, finite-difference gradients are computed. If BHHH method is used, \code{grad} must return a matrix, where rows corresponds to the gradient vectors for individual observations and the columns to the individual parameters. If \code{fn} returns an object with attribute \code{gradient}, this argument is ignored. } \item{hess}{Hessian matrix of the function. It must have the parameter vector as the first argument and it must return the Hessian matrix of the objective function. If missing, finite-difference Hessian, based on \code{gradient}, is computed. Hessian is used by the Newton-Raphson method only, and eventually by the other methods if \code{finalHessian} is requested.} \item{start}{initial parameter values. If start values are named, those names are also carried over to the results.} \item{constraints}{either \code{NULL} for unconstrained optimization or a list with two components. The components may be either \code{eqA} and \code{eqB} for equality-constrained optimization \eqn{A \theta + B = 0}{A \%*\% theta + B = 0}; or \code{ineqA} and \code{ineqB} for inequality constraints \eqn{A \theta + B > 0}{A \%*\% theta + B > 0}. More than one row in \code{ineqA} and \code{ineqB} corresponds to more than one linear constraint, in that case all these must be zero (equality) or positive (inequality constraints). The equality-constrained problem is forwarded to \code{\link{sumt}}, the inequality-constrained case to \code{\link{constrOptim2}}. } \item{finalHessian}{how (and if) to calculate the final Hessian. Either \code{FALSE} (do not calculate), \code{TRUE} (use analytic/finite-difference Hessian) or \code{"bhhh"}/\code{"BHHH"} for the information equality approach. The latter approach is only suitable for maximizing log-likelihood functions. It requires the gradient/log-likelihood to be supplied by individual observations. Note that computing the (actual, not BHHH) final Hessian does not carry any extra penalty for the NR method, but does for the other methods.} \item{bhhhHessian}{logical. Indicating whether to use the information equality approximation (Bernd, Hall, Hall, and Hausman, 1974) for the Hessian. This effectively transforms \code{maxNR} into \code{maxBHHH} and is mainly designed for internal use.} \item{fixed}{parameters to be treated as constants at their \code{start} values. If present, it is treated as an index vector of \code{start} parameters.} \item{activePar}{this argument is retained for backward compatibility only; please use argument \code{fixed} instead.} \item{control}{list of control parameters. The control parameters used by these optimizers are \describe{ \item{tol}{\eqn{10^{-8}}{1e-8}, stopping condition. Stop if the absolute difference between successive iterations is less than \code{tol}. Return \code{code=2}. If set to a negative value, the criterion is never fulfilled, and hence disabled. } \item{reltol}{sqrt(.Machine$double.eps), stopping condition. Relative convergence tolerance: the algorithm stops if the relative improvement between iterations is less than \sQuote{reltol}. Return code 8. Negative value disables condition. } \item{gradtol}{stopping condition. Stop if norm of the gradient is less than \code{gradtol}. Return code 1. Negative value disables condition.} \item{steptol}{1e-10, stopping/error condition. If \code{qac == "stephalving"} and the quadratic approximation leads to a worse, instead of a better value, or to \code{NA}, the step length is halved and a new attempt is made. If necessary, this procedure is repeated until step < \code{steptol}, thereafter code 3 is returned.} \item{lambdatol}{\eqn{10^{-6}}{1e-6}, controls whether Hessian is treated as negative definite. If the largest of the eigenvalues of the Hessian is larger than \code{-lambdatol} (Hessian is not negative definite), a suitable diagonal matrix is subtracted from the Hessian (quadratic hill-climbing) in order to enforce negative definiteness. } \item{qrtol}{\eqn{10^{-10}}{1e-10}, QR-decomposition tolerance for the Hessian inversion. } \item{qac}{"stephalving", Quadratic Approximation Correction. When the new guess is worse than the initial one, the algorithm attemts to correct it: "stephalving" decreases the step but keeps the direction, "marquardt" uses \cite{Marquardt (1963)} method by decreasing the step length while also moving closer to the pure gradient direction. It may be faster and more robust choice in areas where quadratic approximation behaves poorly. \code{maxNR} and \code{maxBHHH} only. } \item{marquardt_lambda0}{\eqn{10^{-2}}{1e-2}, positive numeric, initial correction term for \cite{Marquardt (1963)} correction. } \item{marquardt_lambdaStep}{2, how much the \cite{Marquardt (1963)} correction term is decreased/increased at each successful/unsuccesful step. \code{maxNR} and \code{maxBHHH} only. } \item{marquardt_maxLambda}{\eqn{10^{12}}{1e12}, maximum allowed \cite{Marquardt (1963)} correction term. If exceeded, the algorithm exits with return code 3. \code{maxNR} and \code{maxBHHH} only. } \item{iterlim}{stopping condition. Stop if more than \code{iterlim} iterations, return \code{code=4}.} \item{printLevel}{this argument determines the level of printing which is done during the optimization process. The default value 0 means that no printing occurs, 1 prints the initial and final details, 2 prints all the main tracing information for every iteration. Higher values will result in even more output. } } } \item{\dots}{further arguments to \code{fn}, \code{grad} and \code{hess}. Further arguments to \code{maxBHHH} are also passed to \code{maxNR}. To maintain compatibility with the earlier versions, \dots also passes a number of control options (\code{tol}, \code{reltol}, \code{gradtol}, \code{steptol}, \code{lambdatol}, \code{qrtol}, \code{iterlim}) to the optimizers. } } \details{ The idea of the Newton method is to approximate the function at a given location by a multidimensional quadratic function, and use the estimated maximum as the start value for the next iteration. Such an approximation requires knowledge of both gradient and Hessian, the latter of which can be quite costly to compute. Several methods for approximating Hessian exist, including BFGS and BHHH. The BHHH (information equality) approximation is only valid for log-likelihood functions. It requires the score (gradient) values by individual observations and hence those must be returned by individual observations by \code{grad} or \code{fn}. The Hessian is approximated as the negative of the sum of the outer products of the gradients of individual observations, or, in the matrix form, \deqn{ \mathsf{H}^{BHHH} = -\frac{1}{N} \sum_{i=1}^N \left[ \frac{\partial \ell(\boldsymbol{\vartheta})} {\boldsymbol{\vartheta}} \frac{\partial \ell(\boldsymbol{\vartheta})} {\boldsymbol{\vartheta}'} \right] }{ \code{H = -t(gradient) \%*\% gradient = - crossprod( gradient )}. } The functions \code{maxNR}, \code{maxBFGSR}, and \code{maxBHHH} can work with constant parameters, useful if a parameter value converges to the boundary of support, or for testing. One way is to put \code{fixed} to non-NULL, specifying which parameters should be treated as constants. The parameters can also be fixed in runtime (only for \code{maxNR} and \code{maxBHHH}) by signaling it with the \code{fn} return value. See Henningsen & Toomet (2011) for details. } \value{ object of class "maxim". Data can be extracted through the following methods: \item{\code{\link{maxValue}}}{\code{fn} value at maximum (the last calculated value if not converged.)} \item{\code{\link[=coef.maxim]{coef}}}{estimated parameter value.} \item{gradient}{vector, last calculated gradient value. Should be close to 0 in case of normal convergence.} \item{estfun}{matrix of gradients at parameter value \code{estimate} evaluated at each observation (only if \code{grad} returns a matrix or \code{grad} is not specified and \code{fn} returns a vector).} \item{hessian}{Hessian at the maximum (the last calculated value if not converged).} \item{returnCode}{return code: \itemize{ \item{1}{ gradient close to zero (normal convergence).} \item{2}{ successive function values within tolerance limit (normal convergence).} \item{3}{ last step could not find higher value (probably not converged). This is related to line search step getting too small, usually because hitting the boundary of the parameter space. It may also be related to attempts to move to a wrong direction because of numerical errors. In some cases it can be helped by changing \code{steptol}.} \item{4}{ iteration limit exceeded.} \item{5}{infinite value.} \item{6}{infinite gradient.} \item{7}{infinite Hessian.} \item{8}{successive function values within relative tolerance limit (normal convergence).} \item{9}{(BFGS) Hessian approximation cannot be improved because of gradient did not change. May be related to numerical approximation problems or wrong analytic gradient.} \item{100}{ Initial value out of range.} } } \item{returnMessage}{ a short message, describing the return code.} \item{activePar}{logical vector, which parameters are optimized over. Contains only \code{TRUE}-s if no parameters are fixed.} \item{nIter}{number of iterations.} \item{maximType}{character string, type of maximization.} \item{maxControl}{the optimization control parameters in the form of a \code{\linkS4class{MaxControl}} object.} The following components can only be extracted directly (with \code{\$}): \item{last.step}{a list describing the last unsuccessful step if \code{code=3} with following components: \itemize{ \item{theta0}{ previous parameter value} \item{f0}{ \code{fn} value at \code{theta0}} \item{climb}{ the movement vector to the maximum of the quadratic approximation} } } \item{constraints}{A list, describing the constrained optimization (\code{NULL} if unconstrained). Includes the following components: \itemize{ \item{type}{ type of constrained optimization} \item{outer.iterations}{ number of iterations in the constraints step} \item{barrier.value}{ value of the barrier function} } } } \section{Warning}{ No attempt is made to ensure that user-provided analytic gradient/Hessian is correct. The users are encouraged to use \code{\link{compareDerivatives}} function, designed for this purpose. If analytic gradient/Hessian are wrong, the algorithm may not converge, or may converge to a wrong point. As the BHHH method uses the likelihood-specific information equality, it is only suitable for maximizing log-likelihood functions! Quasi-Newton methods, including those mentioned above, do not work well in non-concave regions. This is especially the case with the implementation in \code{maxBFGSR}. The user is advised to experiment with various tolerance options to achieve convergence. } \references{ Berndt, E., Hall, B., Hall, R. and Hausman, J. (1974): Estimation and Inference in Nonlinear Structural Models, \emph{Annals of Social Measurement} \bold{3}, 653--665. Broyden, C.G. (1970): The Convergence of a Class of Double-rank Minimization Algorithms, \emph{Journal of the Institute of Mathematics and Its Applications} \bold{6}, 76--90. Fletcher, R. (1970): A New Approach to Variable Metric Algorithms, \emph{Computer Journal} \bold{13}, 317--322. Goldfarb, D. (1970): A Family of Variable Metric Updates Derived by Variational Means, \emph{Mathematics of Computation} \bold{24}, 23--26. Henningsen, A. and Toomet, O. (2011): maxLik: A package for maximum likelihood estimation in R \emph{Computational Statistics} \bold{26}, 443--458 Marquardt, D.W., (1963) An Algorithm for Least-Squares Estimation of Nonlinear Parameters, \emph{Journal of the Society for Industrial & Applied Mathematics} \bold{11}, 2, 431--441 Shanno, D.F. (1970): Conditioning of Quasi-Newton Methods for Function Minimization, \emph{Mathematics of Computation} \bold{24}, 647--656. } \author{Ott Toomet, Arne Henningsen, function \code{maxBFGSR} was originally developed by Yves Croissant (and placed in 'mlogit' package)} \seealso{\code{\link{maxLik}} for a general framework for maximum likelihood estimation (MLE); \code{\link{maxBHHH}} for maximizations using the Berndt, Hall, Hall, Hausman (1974) algorithm (which is a wrapper function to \code{maxNR}); \code{\link{maxBFGS}} for maximization using the BFGS, Nelder-Mead (NM), and Simulated Annealing (SANN) method (based on \code{\link{optim}}), also supporting inequality constraints; \code{\link{nlm}} for Newton-Raphson optimization; and \code{\link{optim}} for different gradient-based optimization methods.} \examples{ ## Fit exponential distribution by ML t <- rexp(100, 2) # create data with parameter 2 loglik <- function(theta) sum(log(theta) - theta*t) ## Note the log-likelihood and gradient are summed over observations gradlik <- function(theta) sum(1/theta - t) hesslik <- function(theta) -100/theta^2 ## Estimate with finite-difference gradient and Hessian a <- maxNR(loglik, start=1, control=list(printLevel=2)) summary(a) ## You would probably prefer 1/mean(t) instead ;-) ## The same example with analytic gradient and Hessian a <- maxNR(loglik, gradlik, hesslik, start=1) summary(a) ## BFGS estimation with finite-difference gradient a <- maxBFGSR( loglik, start=1 ) summary(a) ## For the BHHH method we need likelihood values and gradients ## of individual observations, not the sum of those loglikInd <- function(theta) log(theta) - theta*t gradlikInd <- function(theta) 1/theta - t ## Estimate with analytic gradient a <- maxBHHH(loglikInd, gradlikInd, start=1) summary(a) ## Example with a vector argument: Estimate the mean and ## variance of a random normal sample by maximum likelihood ## Note: you might want to use maxLik instead loglik <- function(param) { # param is a 2-vector of c(mean, sd) mu <- param[1] sigma <- param[2] ll <- -0.5*N*log(2*pi) - N*log(sigma) - sum(0.5*(x - mu)^2/sigma^2) ll } x <- rnorm(100, 1, 2) # use mean=1, sd=2 N <- length(x) res <- maxNR(loglik, start=c(0,1)) # use 'wrong' start values summary(res) ## The previous example with named parameters and a fixed value resFix <- maxNR(loglik, start=c(mu=0, sigma=1), fixed="sigma") summary(resFix) # 'sigma' is exactly 1.000 now. ### Constrained optimization ### ## We maximize exp(-x^2 - y^2) where x+y = 1 hatf <- function(theta) { x <- theta[1] y <- theta[2] exp(-(x^2 + y^2)) ## Note: you may prefer exp(- theta \%*\% theta) instead } ## use constraints: x + y = 1 A <- matrix(c(1, 1), 1, 2) B <- -1 res <- maxNR(hatf, start=c(0,0), constraints=list(eqA=A, eqB=B), control=list(printLevel=1)) print(summary(res)) } \keyword{optimize} maxLik/man/maxBFGS.Rd0000644000176200001440000002172514077525067014014 0ustar liggesusers\name{maxBFGS} \alias{maxBFGS} \alias{maxCG} \alias{maxSANN} \alias{maxNM} \title{BFGS, conjugate gradient, SANN and Nelder-Mead Maximization} \description{ These functions are wrappers for \code{\link{optim}}, adding constrained optimization and fixed parameters. } \usage{ maxBFGS(fn, grad=NULL, hess=NULL, start, fixed=NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ... ) maxCG(fn, grad=NULL, hess=NULL, start, fixed=NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ...) maxSANN(fn, grad=NULL, hess=NULL, start, fixed=NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ... ) maxNM(fn, grad=NULL, hess=NULL, start, fixed=NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ...) } \arguments{ \item{fn}{function to be maximised. Must have the parameter vector as the first argument. In order to use numeric gradient and BHHH method, \code{fn} must return a vector of observation-specific likelihood values. Those are summed internally where necessary. If the parameters are out of range, \code{fn} should return \code{NA}. See details for constant parameters.} \item{grad}{gradient of \code{fn}. Must have the parameter vector as the first argument. If \code{NULL}, numeric gradient is used (\code{maxNM} and \code{maxSANN} do not use gradient). Gradient may return a matrix, where columns correspond to the parameters and rows to the observations (useful for maxBHHH). The columns are summed internally.} \item{hess}{Hessian of \code{fn}. Not used by any of these methods, included for compatibility with \code{\link{maxNR}}.} \item{start}{initial values for the parameters. If start values are named, those names are also carried over to the results.} \item{fixed}{parameters to be treated as constants at their \code{start} values. If present, it is treated as an index vector of \code{start} parameters.} \item{control}{list of control parameters or a \sQuote{MaxControl} object. If it is a list, the default values are used for the parameters that are left unspecified by the user. These functions accept the following parameters: \describe{ \item{reltol}{sqrt(.Machine$double.eps), stopping condition. Relative convergence tolerance: the algorithm stops if the relative improvement between iterations is less than \sQuote{reltol}. Note: for compatibility reason \sQuote{tol} is equivalent to \sQuote{reltol} for optim-based optimizers. } \item{iterlim}{integer, maximum number of iterations. Default values are 200 for \sQuote{BFGS}, 500 (\sQuote{CG} and \sQuote{NM}), and 10000 (\sQuote{SANN}). Note that \sQuote{iteration} may mean different things for different optimizers. } \item{printLevel}{integer, larger number prints more working information. Default 0, no information. } \item{nm_alpha}{1, Nelder-Mead simplex method reflection coefficient (see Nelder & Mead, 1965) } \item{nm_beta}{0.5, Nelder-Mead contraction coefficient} \item{nm_gamma}{2, Nelder-Mead expansion coefficient} % SANN \item{sann_cand}{\code{NULL} or a function for \code{"SANN"} algorithm to generate a new candidate point; if \code{NULL}, Gaussian Markov kernel is used (see argument \code{gr} of \code{\link{optim}}).} \item{sann_temp}{10, starting temperature for the \dQuote{SANN} cooling schedule. See \code{\link{optim}}.} \item{sann_tmax}{10, number of function evaluations at each temperature for the \dQuote{SANN} optimizer. See \code{\link{optim}}.} \item{sann_randomSeed}{123, integer to seed random numbers to ensure replicability of \dQuote{SANN} optimization and preserve \code{R} random numbers. Use options like \code{sann_randomSeed=Sys.time()} or \code{sann_randomSeed=sample(100,1)} if you want stochastic results. } } } \item{constraints}{either \code{NULL} for unconstrained optimization or a list with two components. The components may be either \code{eqA} and \code{eqB} for equality-constrained optimization \eqn{A \theta + B = 0}{A \%*\% theta + B = 0}; or \code{ineqA} and \code{ineqB} for inequality constraints \eqn{A \theta + B > 0}{A \%*\% theta + B > 0}. More than one row in \code{ineqA} and \code{ineqB} corresponds to more than one linear constraint, in that case all these must be zero (equality) or positive (inequality constraints). The equality-constrained problem is forwarded to \code{\link{sumt}}, the inequality-constrained case to \code{\link{constrOptim2}}. } \item{finalHessian}{how (and if) to calculate the final Hessian. Either \code{FALSE} (not calculate), \code{TRUE} (use analytic/numeric Hessian) or \code{"bhhh"}/\code{"BHHH"} for information equality approach. The latter approach is only suitable for maximizing log-likelihood function. It requires the gradient/log-likelihood to be supplied by individual observations, see \code{\link{maxBHHH}} for details. } \item{parscale}{A vector of scaling values for the parameters. Optimization is performed on 'par/parscale' and these should be comparable in the sense that a unit change in any element produces about a unit change in the scaled value. (see \code{\link{optim}})} \item{\dots}{further arguments for \code{fn} and \code{grad}.} } \details{ In order to provide a consistent interface, all these functions also accept arguments that other optimizers use. For instance, \code{maxNM} accepts the \sQuote{grad} argument despite being a gradient-less method. The \sQuote{state} (or \sQuote{seed}) of R's random number generator is saved at the beginning of the \code{maxSANN} function and restored at the end of this function so this function does \emph{not} affect the generation of random numbers although the random seed is set to argument \code{random.seed} and the \sQuote{SANN} algorithm uses random numbers. } \value{ object of class "maxim". Data can be extracted through the following functions: \item{maxValue}{\code{fn} value at maximum (the last calculated value if not converged.)} \item{coef}{estimated parameter value.} \item{gradient}{vector, last calculated gradient value. Should be close to 0 in case of normal convergence.} \item{estfun}{matrix of gradients at parameter value \code{estimate} evaluated at each observation (only if \code{grad} returns a matrix or \code{grad} is not specified and \code{fn} returns a vector).} \item{hessian}{Hessian at the maximum (the last calculated value if not converged).} \item{returnCode}{integer. Success code, 0 is success (see \code{\link{optim}}).} \item{returnMessage}{ a short message, describing the return code.} \item{activePar}{logical vector, which parameters are optimized over. Contains only \code{TRUE}-s if no parameters are fixed.} \item{nIter}{number of iterations. Two-element integer vector giving the number of calls to \code{fn} and \code{gr}, respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to \code{fn} to compute a finite-difference approximation to the gradient.} \item{maximType}{character string, type of maximization.} \item{maxControl}{the optimization control parameters in the form of a \code{\linkS4class{MaxControl}} object.} The following components can only be extracted directly (with \code{\$}): \item{constraints}{A list, describing the constrained optimization (\code{NULL} if unconstrained). Includes the following components: \describe{ \item{type}{type of constrained optimization} \item{outer.iterations}{number of iterations in the constraints step} \item{barrier.value}{value of the barrier function} } } } \author{Ott Toomet, Arne Henningsen} \seealso{\code{\link{optim}}, \code{\link{nlm}}, \code{\link{maxNR}}, \code{\link{maxBHHH}}, \code{\link{maxBFGSR}} for a \code{\link{maxNR}}-based BFGS implementation.} \references{ Nelder, J. A. & Mead, R. A, Simplex Method for Function Minimization, The Computer Journal, 1965, 7, 308-313 } \examples{ # Maximum Likelihood estimation of Poissonian distribution n <- rpois(100, 3) loglik <- function(l) n*log(l) - l - lfactorial(n) # we use numeric gradient summary(maxBFGS(loglik, start=1)) # you would probably prefer mean(n) instead of that ;-) # Note also that maxLik is better suited for Maximum Likelihood ### ### Now an example of constrained optimization ### f <- function(theta) { x <- theta[1] y <- theta[2] exp(-(x^2 + y^2)) ## you may want to use exp(- theta \%*\% theta) instead } ## use constraints: x + y >= 1 A <- matrix(c(1, 1), 1, 2) B <- -1 res <- maxNM(f, start=c(1,1), constraints=list(ineqA=A, ineqB=B), control=list(printLevel=1)) print(summary(res)) } \keyword{optimize} maxLik/man/fnSubset.Rd0000644000176200001440000000506114077525067014351 0ustar liggesusers\name{fnSubset} \alias{fnSubset} \title{ Call fnFull with variable and fixed parameters } \description{ Combine variable parameters with with fixed parameters and pass to \code{fnFull}. Useful for optimizing over a subset of parameters without writing a separate function. Values are combined by name if available. Otherwise, \code{xFull} is constructed by position (the default). } \usage{ fnSubset(x, fnFull, xFixed, xFull=c(x, xFixed), ...) } \arguments{ \item{x}{ Variable parameters to be passed to \code{fnFull}. } \item{fnFull}{ Function whose first argument has length = length(xFull). } \item{xFixed}{ Parameter values to be combined with \code{x} to construct the first argument for a call to \code{fnFull}. } \item{xFull}{ Prototype initial argument for \code{fnFull}. } \item{\dots}{ Optional arguments passed to \code{fnFull}. } } \details{ This function first confirms that \code{length(x) + length(xFixed) == length(xFull)}. Next, \itemize{ \item If \code{xFull} has names, match at least \code{xFixed} by name. \item Else \code{xFull = c(x, xFixes)}, the default. } Finally, call \code{fnFull(xFull, ...)}. } \value{ value returned by \code{fnFull} } %\references{ } \author{ Spencer Graves } \seealso{ \code{\link{optim}} \code{\link[dlm]{dlmMLE}} \code{\link{maxLik}} \code{\link{maxNR}} } \examples{ ## ## Example with 'optim' ## fn <- function(x) (x[2]-2*x[1])^2 # note: true minimum is 0 on line 2*x[1] == x[2] fullEst <- optim(par=c(1,1), method="BFGS", fn=fn) fullEst$par # par = c(0.6, 1.2) at minimum (not convex) # Fix the last component to 4 est4 <- optim(par=1, fn=fnSubset, method="BFGS", fnFull=fn, xFixed=4) est4$par # now there is a unique minimun x[1] = 2 # Fix the first component fnSubset(x=1, fnFull=fn, xFixed=c(a=4), xFull=c(a=1, b=2)) # After substitution: xFull = c(a=4, b=1), # so fn = (1 - 2*4)^2 = (-7)^2 = 49 est4. <- optim(par=1, fn=fnSubset, method="BFGS", fnFull=fn, xFixed=c(a=4), xFull=c(a=1, b=2)) est4.$par # At optimum: xFull=c(a=4, b=8), # so fn = (8 - 2*4)^2 = 0 ## ## Example with 'maxLik' ## fn2max <- function(x) -(x[2]-2*x[1])^2 # -> need to have a maximum max4 <- maxLik(fnSubset, start=1, fnFull=fn2max, xFixed=4) summary(max4) # Similar result using fixed parameters in maxNR, called by maxLik max4. <- maxLik(fn2max, start=c(1, 4), fixed=2) summary(max4.) } \keyword{optimize} \keyword{utilities} maxLik/man/reexports.Rd0000644000176200001440000000047314077525067014615 0ustar liggesusers\docType{import} \name{reexports} \alias{reexports} \alias{tidy} \alias{glance} \title{Objects exported from other packages} \keyword{internal} \description{ These objects are imported from the "generics" package. See \code{\link[generics:tidy]{tidy}} and \code{\link[generics:glance]{glance}} for details. } maxLik/man/compareDerivatives.Rd0000644000176200001440000001076514077525067016423 0ustar liggesusers\name{compareDerivatives} \alias{compareDerivatives} \title{function to compare analytic and numeric derivatives} \description{ This function compares analytic and numerical derivative and prints related diagnostics information. It is intended for testing and debugging code for analytic derivatives for maximization algorithms. } \usage{ compareDerivatives(f, grad, hess=NULL, t0, eps=1e-6, printLevel=1, print=printLevel > 0, max.rows=getOption("max.rows", 20), max.cols=getOption("max.cols", 7), ...) } \arguments{ \item{f}{ function to be differentiated. The parameter (vector) of interest must be the first argument. The function may return a vector, in that case the derivative will be a matrix. } \item{grad}{ analytic gradient. This may be either a function, returning the analytic gradient, or a numeric vector, the pre-computed gradient. The function must use the same set of parameters as \code{f}. If \code{f} is a vector-valued function, grad must return/be a matrix where the number of rows equals the number of components of \code{f}, and the number of columns must equal to the number of components in \code{t0}. } \item{hess}{ function returning the analytic hessian. If present, hessian matrices are compared too. Only appropriate for scalar-valued functions. } \item{t0}{ numeric vector, parameter at which the derivatives are compared. The derivative is taken with respect to this vector. both \code{f}m \code{grad} (if function) and \code{hess} (if present) must accept this value as the first parameter. } \item{eps}{ numeric. Step size for numeric differentiation. Central derivative is used. } \item{printLevel}{ numeric: a positive number prints summary of the comparison. 0 does not do any printing, only returns the comparison results (invisibly). } \item{print}{ deprecated (for backward compatibility only). } \item{max.rows}{maximum number of matrix rows to be printed. } \item{max.cols}{maximum number of columns to be printed. } \item{\dots}{ further arguments to \code{f}, \code{grad} and \code{hess}. } } \details{ Analytic derivatives (and Hessian) substantially improve the estimation speed and reliability. However, these are typically hard to program. This utility compares the programmed result and the (internally calculated) numeric derivative. For every component of \code{f}, it prints the parameter value, analytic and numeric derivative, and their relative difference \deqn{\textrm{rel.diff} = \frac{\textrm{analytic} - \textrm{numeric}}{\frac{1}{2}(|\textrm{analytic}| + |\textrm{numeric}|)}.}{rel.diff = (analytic - numeric)/(0.5*(abs(analytic) + abs(numeric))).} If \eqn{\textrm{analytic} = 0}{analytic == 0} and \eqn{\textrm{numeric} = 0}{numeric == 0}, then rel.diff is also set to 0. If analytic derivatives are correct and the function is sufficiently smooth, expect the relative differences to be less than \eqn{10^{-7}}{1e-7}. } \value{ A list with following components: \item{t0}{the input argument \code{t0}} \item{f.t0}{f(t0)} \item{compareGrad}{ a list with components \code{analytic} = grad(t0), \code{nmeric} = numericGradient(f, t0), and their \code{rel.diff}. } \item{maxRelDiffGrad}{max(abs(rel.diff))} If \code{hess} is also provided, the following optional components are also present: \item{compareHessian}{ a list with components \code{analytic} = hess(t0), \code{numeric} = numericGradient(grad, t0), and their \code{rel.diff}. } \item{maxRelDiffHess}{max(abs(rel.diff)) for the Hessian} } \author{Ott Toomet \email{otoomet@ut.ee} and Spencer Graves} \seealso{ \code{\link{numericGradient}} \code{\link{deriv}} } \examples{ ## A simple example with sin(x)' = cos(x) f <- function(x) c(sin=sin(x)) Dsin <- compareDerivatives(f, cos, t0=c(angle=1)) ## ## Example of normal log-likelihood. Two-parameter ## function. ## x <- rnorm(100, 1, 2) # generate rnorm x l <- function(b) sum(dnorm(x, mean=b[1], sd=b[2], log=TRUE)) gradl <- function(b) { c(mu=sum(x - b[1])/b[2]^2, sigma=sum((x - b[1])^2/b[2]^3 - 1/b[2])) } gradl. <- compareDerivatives(l, gradl, t0=c(mu=1,sigma=2)) ## ## An example with f returning a vector, t0 = a scalar ## trig <- function(x)c(sin=sin(x), cos=cos(x)) Dtrig <- function(x)c(sin=cos(x), cos=-sin(x)) Dtrig. <- compareDerivatives(trig, Dtrig, t0=1) } \keyword{math} \keyword{utilities} maxLik/man/nParam.Rd0000644000176200001440000000236114077525067013776 0ustar liggesusers\name{nParam.maxim} \alias{nParam.maxim} \title{Number of model parameters} \description{ This function returns the number of model parameters. } \usage{ \method{nParam}{maxim}(x, free=FALSE, \dots) } \arguments{ \item{x}{a model returned by a maximisation method from the \pkg{maxLik} package.} \item{free}{logical, whether to report only the free parameters or the total number of parameters (default)} \item{\dots}{other arguments for methods} } \details{ Free parameters are the parameters with no equality restrictions. Some parameters may be jointly restricted (e.g. sum of two probabilities equals unity). In this case the total number of parameters may depend on the normalization. } \value{ Number of parameters in the model } \author{Ott Toomet} \seealso{\code{\link{nObs}} for number of observations} \examples{ ## fit a normal distribution by ML # generate a variable from normally distributed random numbers x <- rnorm( 100, 1, 2 ) # log likelihood function (for individual observations) llf <- function( param ) { return( dnorm( x, mean = param[ 1 ], sd = param[ 2 ], log = TRUE ) ) } ## ML method ml <- maxLik( llf, start = c( mu = 0, sigma = 1 ) ) # return number of parameters nParam( ml ) } \keyword{methods} maxLik/man/condiNumber.Rd0000644000176200001440000000645614077525067015036 0ustar liggesusers\name{condiNumber} \alias{condiNumber} \alias{condiNumber.default} \alias{condiNumber.maxLik} \title{Print matrix condition numbers column-by-column} \description{ This function prints the condition number of a matrix while adding columns one-by-one. This is useful for testing multicollinearity and other numerical problems. It is a generic function with a default method, and a method for \code{maxLik} objects. } \usage{ condiNumber(x, ...) \method{condiNumber}{default}(x, exact = FALSE, norm = FALSE, printLevel=print.level, print.level=1, digits = getOption( "digits" ), ... ) \method{condiNumber}{maxLik}(x, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{x}{numeric matrix, condition numbers of which are to be printed} \item{exact}{logical, should condition numbers be exact or approximations (see \code{\link{kappa}})} \item{norm}{logical, whether the columns should be normalised to have unit norm} \item{printLevel}{numeric, positive value will output the numbers during the calculations. Useful for interactive work.} \item{print.level}{same as \sQuote{printLevel}, for backward compatibility} \item{digits}{minimal number of significant digits to print (only relevant if argument \code{print.level} is larger than zero).} \item{\dots}{Further arguments to \code{condiNumber.default} are currently ignored; further arguments to \code{condiNumber.maxLik} are passed to \code{condiNumber.default}.} } \details{ Statistical model often fail because of a high correlation between the explanatory variables in the linear index (multicollinearity) or because the evaluated maximum of a non-linear model is virtually flat. In both cases, the (near) singularity of the related matrices may help to understand the problem. \code{condiNumber} inspects the matrices column-by-column and indicates which variables lead to a jump in the condition number (cause singularity). If the matrix column name does not immediately indicate the problem, one may run an OLS model by estimating this column using all the previous columns as explanatory variables. Those columns that explain almost all the variation in the current one will have very high \eqn{t}{t}-values. } \value{ Invisible vector of condition numbers by column. If the start values for \code{\link{maxLik}} are named, the condition numbers are named accordingly. } \references{ Greene, W. (2012): \emph{Econometrics Analysis}, 7th edition, p. 130. } \author{Ott Toomet} \seealso{\code{\link{kappa}}} \examples{ set.seed(0) ## generate a simple nearly multicollinear dataset x1 <- runif(100) x2 <- runif(100) x3 <- x1 + x2 + 0.000001*runif(100) # this is virtually equal to x1 + x2 x4 <- runif(100) y <- x1 + x2 + x3 + x4 + rnorm(100) m <- lm(y ~ -1 + x1 + x2 + x3 + x4) print(summary(m)) # note the outlandish estimates and standard errors # while R^2 is 0.88. This suggests multicollinearity condiNumber(model.matrix(m)) # note the value 'explodes' at x3 ## we may test the results further: print(summary(lm(x3 ~ -1 + x1 + x2))) # Note the extremely high t-values and R^2: x3 is (almost) completely # explained by x1 and x2 } \keyword{math} \keyword{utilities} \keyword{debugging} % is it debugging? maxLik/man/nObs.Rd0000644000176200001440000000227314077525067013463 0ustar liggesusers\name{nObs.maxLik} \alias{nObs.maxLik} \title{Number of Observations} \description{ Returns the number of observations for statistical models, estimated by Maximum Likelihood using \code{\link{maxLik}}. } \usage{ \method{nObs}{maxLik}(x, \dots) } \arguments{ \item{x}{a statistical model estimated by Maximum Likelihood using \code{\link{maxLik}}.} \item{\dots}{further arguments (currently ignored).} } \details{ The \code{nObs} method for \dQuote{maxLik} objects can return the number of observations only if log-likelihood function (or the gradient) returns values by individual observation. } \value{ numeric, number of observations } \author{Arne Henningsen, Ott Toomet} \seealso{\code{\link[miscTools]{nObs}}, \code{\link{maxLik}}, \code{\link{nParam}}.} \examples{ ## fit a normal distribution by ML # generate a variable from normally distributed random numbers x <- rnorm( 100, 1, 2 ) # log likelihood function (for individual observations) llf <- function( param ) { return( dnorm( x, mean = param[ 1 ], sd = param[ 2 ], log = TRUE ) ) } ## ML method ml <- maxLik( llf, start = c( mu = 0, sigma = 1 ) ) # return number of onservations nObs( ml ) } \keyword{methods} maxLik/man/summary.maxLik.Rd0000644000176200001440000000436514077525067015507 0ustar liggesusers\name{summary.maxLik} \alias{summary.maxLik} \alias{coef.summary.maxLik} \title{summary the Maximum-Likelihood estimation} \description{ Summary the Maximum-Likelihood estimation including standard errors and t-values. } \usage{ \method{summary}{maxLik}(object, eigentol=1e-12, ... ) \method{coef}{summary.maxLik}(object, \ldots) } \arguments{ \item{object}{ object of class 'maxLik', or 'summary.maxLik', usually a result from Maximum-Likelihood estimation. } \item{eigentol}{ The standard errors are only calculated if the ratio of the smallest and largest eigenvalue of the Hessian matrix is less than \dQuote{eigentol}. Otherwise the Hessian is treated as singular. } \item{\ldots}{currently not used.} } \value{ An object of class 'summary.maxLik' with following components: \describe{ \item{type}{type of maximization.} \item{iterations}{number of iterations.} \item{code}{code of success.} \item{message}{a short message describing the code.} \item{loglik}{the loglik value in the maximum.} \item{estimate}{numeric matrix, the first column contains the parameter estimates, the second the standard errors, third t-values and fourth corresponding probabilities.} \item{fixed}{logical vector, which parameters are treated as constants.} \item{NActivePar}{number of free parameters.} \item{constraints}{information about the constrained optimization. Passed directly further from \code{maxim}-object. \code{NULL} if unconstrained maximization. } } } \author{Ott Toomet, Arne Henningsen} \seealso{ \code{\link{maxLik}} for maximum likelihood estimation, \code{\link{confint}} for confidence intervals, and \code{\link{tidy}} and \code{\link{glance}} for alternative quick summaries of the ML results. } \examples{ ## ML estimation of exponential distribution: t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1, control=list(printLevel=2)) summary(a) ## Estimate with analytic gradient and hessian a <- maxLik(loglik, gradlik, hesslik, start=1, control=list(printLevel=2)) summary(a) } \keyword{models} maxLik/man/activePar.Rd0000644000176200001440000000265714077525067014506 0ustar liggesusers\name{activePar} \alias{activePar} \alias{activePar.default} \title{free parameters under maximization} \description{ Return a logical vector, indicating which parameters were free under maximization, as opposed to the fixed parameters that are treated as constants. See argument \dQuote{fixed} for \code{\link{maxNR}}. } \usage{ activePar(x, \dots) \method{activePar}{default}(x, \dots) } \arguments{ \item{x}{object, created by a maximization routine, such as \code{\link{maxNR}} or \code{\link{maxLik}}, or derived from a maximization object. } \item{\dots}{further arguments for methods} } \details{ Several optimization routines allow the user to fix some parameter values (or do it automatically in some cases). For gradient or Hessian based inference one has to know which parameters carry optimization-related information. } \value{ A logical vector, indicating whether the parameters were free to change during optimization algorithm. } \author{Ott Toomet} \seealso{\code{\link{maxNR}}, \code{\link{nObs}}} \examples{ ## a two-dimensional exponential hat f <- function(a) exp(-a[1]^2 - a[2]^2) ## maximize wrt. both parameters free <- maxNR(f, start=1:2) summary(free) # results should be close to (0,0) activePar(free) ## keep the first parameter constant cons <- maxNR(f, start=1:2, fixed=c(TRUE,FALSE)) summary(cons) # result should be around (1,0) activePar(cons) } \keyword{methods} \keyword{optimize} maxLik/man/vcov.maxLik.Rd0000644000176200001440000000265214077525067014764 0ustar liggesusers\name{vcov.maxLik} \alias{vcov.maxLik} \title{Variance Covariance Matrix of maxLik objects} \description{ Extract variance-covariance matrices from \code{\link{maxLik}} objects. } \usage{ \method{vcov}{maxLik}( object, eigentol=1e-12, ... ) } \arguments{ \item{object}{a \sQuote{maxLik} object.} \item{eigentol}{ eigenvalue tolerance, controlling when the Hessian matrix is treated as numerically singular. } \item{\dots}{further arguments (currently ignored).} } \value{ the estimated variance covariance matrix of the coefficients. In case of the estimated Hessian is singular, it's values are \code{Inf}. The values corresponding to fixed parameters are zero. } \details{ The standard errors are only calculated if the ratio of the smallest and largest eigenvalue of the Hessian matrix is less than \dQuote{eigentol}. Otherwise the Hessian is treated as singular. } \author{ Arne Henningsen, Ott Toomet } \seealso{\code{\link[stats]{vcov}}, \code{\link{maxLik}}.} \examples{ ## ML estimation of exponential random variables t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1, control=list(printLevel=2)) vcov(a) ## Estimate with analytic gradient and hessian a <- maxLik(loglik, gradlik, hesslik, start=1) vcov(a) } \keyword{methods} maxLik/man/maxSGA.Rd0000644000176200001440000003567614077525067013717 0ustar liggesusers\name{maxSGA} \alias{maxSGA} \alias{maxAdam} \title{Stochastic Gradient Ascent} \description{ Stochastic Gradient Ascent--based optimizers } \usage{ maxSGA(fn = NULL, grad = NULL, hess = NULL, start, nObs, constraints = NULL, finalHessian = FALSE, fixed = NULL, control=NULL, ... ) maxAdam(fn = NULL, grad = NULL, hess = NULL, start, nObs, constraints = NULL, finalHessian = FALSE, fixed = NULL, control=NULL, ... ) } \arguments{ \item{fn}{the function to be maximized. As the objective function values are not directly used for optimization, this argument is optional, given \code{grad} is provided. It must have the parameter vector as the first argument, and it must have an argument \code{index} to specify the integer index of the selected observations. It must return either a single number, or a numeric vector (this is is summed internally). If the parameters are out of range, \code{fn} should return \code{NA}. See details for constant parameters. \code{fn} may also return attributes "gradient" and/or "hessian". If these attributes are set, the algorithm uses the corresponding values as gradient and Hessian. } \item{grad}{gradient of the objective function. It must have the parameter vector as the first argument, and it must have an argument \code{index} to specify the integer index of selected observations. It must return either a gradient vector of the objective function, or a matrix, where columns correspond to individual parameters. The column sums are treated as gradient components. If \code{NULL}, finite-difference gradients are computed. If \code{fn} returns an object with attribute \code{gradient}, this argument is ignored. If \code{grad} is not supplied, it is computed by finite-difference method using \code{fn}. However, this is only adviseable for small-scale tests, not for any production run. Obviously, \code{fn} must be correctly defined in that case. } \item{hess}{Hessian matrix of the function. Mainly for compatibility reasons, only used for computing the final Hessian if asked to do so by setting \code{finalHessian} to \code{TRUE}. It must have the parameter vector as the first argument and it must return the Hessian matrix of the objective function. If missing, either finite-difference Hessian, based on \code{gradient} or BHHH approach is computed if asked to do so. } \item{start}{initial parameter values. If these have names, the names are also used for results.} \item{nObs}{number of observations. This is used to partition the data into individual batches. The resulting batch indices are forwarded to the \code{grad} function through the argument \code{index}.} \item{constraints}{either \code{NULL} for unconstrained optimization or a list with two components. The components may be either \code{eqA} and \code{eqB} for equality-constrained optimization \eqn{A \theta + B = 0}{A \%*\% theta + B = 0}; or \code{ineqA} and \code{ineqB} for inequality constraints \eqn{A \theta + B > 0}{A \%*\% theta + B > 0}. More than one row in \code{ineqA} and \code{ineqB} corresponds to more than one linear constraint, in that case all these must be zero (equality) or positive (inequality constraints). The equality-constrained problem is forwarded to \code{\link{sumt}}, the inequality-constrained case to \code{\link{constrOptim2}}. } \item{finalHessian}{how (and if) to calculate the final Hessian. Either \code{FALSE} (do not calculate), \code{TRUE} (use analytic/finite-difference Hessian) or \code{"bhhh"}/\code{"BHHH"} for the information equality approach. The latter approach is only suitable when working with a log-likelihood function, and it requires the gradient/log-likelihood to be supplied by individual observations. Hessian matrix is not often used for optimization problems where one applies SGA, but even if one is not interested in standard errors, it may provide useful information about the model performance. If computed by finite-difference method, the Hessian computation may be very slow. } \item{fixed}{parameters to be treated as constants at their \code{start} values. If present, it is treated as an index vector of \code{start} parameters.} \item{control}{list of control parameters. The ones used by these optimizers are \describe{ \item{SGA_momentum}{0, numeric momentum parameter for SGA. Must lie in interval \eqn{[0,1]}{[0,1]}. See details. } Adam-specific parameters \item{Adam_momentum1}{0.9, numeric in interval \eqn{(0,1)}{(0,1)}, the first moment momentum} \item{Adam_momentum2}{0.999, numeric in interval \eqn{(0,1)}{(0,1)}, the second moment momentum} General stochastic gradient parameters: \item{SG_learningRate}{step size the SGA algorithm takes in the gradient direction. If 1, the step equals to the gradient value. A good value is often 0.01--0.3} \item{SG_batchSize}{SGA batch size, an integer between 1 and \code{nObs}. If \code{NULL} (default), the full batch gradient is computed. } \item{SG_clip}{\code{NULL}, gradient clipping threshold. The algorithm ensures that \eqn{||g(\theta)||_2^2 \le \kappa}{norm(gradient)^2 <= kappa} where \eqn{\kappa}{kappa} is the \code{SG_clip} value. If the actual norm of the gradient exceeds (square root of) \eqn{\kappa}{kappa}, the gradient will be scaled back accordingly while preserving its direction. \code{NULL} means no clipping. } Stopping conditions: \item{gradtol}{stopping condition. Stop if norm of the gradient is less than \code{gradtol}. Default 0, i.e. do not use this condition. This condition is useful if the objective is to drive full batch gradient to zero on training data. It is not a good objective in case of the stochastic gradient, and if the objective is to optimize the objective on validation data. } \item{SG_patience}{\code{NULL}, or integer. Stopping condition: the algorithm counts how many times the objective function has been worse than its best value so far, and if this exceeds \code{SG_patience}, the algorithm stops. } \item{SG_patienceStep}{1L, integer. After how many epochs to check the patience value. \code{1} means to check at each epoch, and hence to compute the objective function. This may be undesirable if the objective function is costly to compute. } \item{iterlim}{stopping condition. Stop if more than \code{iterlim} epochs, return \code{code=4}. Epoch is a set of iterations that cycles through all observations. In case of full batch, iterations and epochs are equivalent. If \code{iterlim = 0}, does not do any learning and returns the initial values unchanged. } \item{printLevel}{this argument determines the level of printing which is done during the optimization process. The default value 0 means that no printing occurs, 1 prints the initial and final details, 2 prints all the main tracing information for every epoch. Higher values will result in even more output. } \item{storeParameters}{logical, whether to store and return the parameter values at each epoch. If \code{TRUE}, the stored values can be retrieved with \code{\link{storedParameters}}-method. The parameters are stored as a matrix with rows corresponding to the epochs and columns to the parameter components. There are \code{iterlim} + 1 rows, where the first one corresponds to the initial parameters. Default \code{FALSE}. } \item{storeValues}{logical, whether to store and return the objective function values at each epoch. If \code{TRUE}, the stored values can be retrieved with \code{\link{storedValues}}-method. There are \code{iterlim} + 1 values, where the first one corresponds to the value at the initial parameters. Default \code{FALSE}. } } See \code{\link{maxControl}} for more information. } \item{\dots}{further arguments to \code{fn}, \code{grad} and \code{hess}. To maintain compatibility with the earlier versions, \dots also passes certain control options to the optimizers. } } \details{ Gradient Ascent (GA) is a optimization method where the algorithm repeatedly takes small steps in the gradient's direction, the parameter vector \eqn{\theta}{theta} is updated as \eqn{\theta \leftarrow theta + \mathrm{learning rate}\cdot \nabla f(\theta)}{theta <- learning rate * gradient f(theta)}. In case of Stochastic GA (SGA), the gradient is not computed on the full set of observations but on a small subset, \emph{batch}, potentially a single observation only. In certain circumstances this converges much faster than when using all observation (see \cite{Bottou et al, 2018}). If \code{SGA_momentum} is positive, the SGA algorithm updates the parameters \eqn{\theta}{theta} in two steps. First, the momentum is used to update the \dQuote{velocity} \eqn{v}{v} as \eqn{v \leftarrow \mathrm{momentum}\cdot v + \mathrm{learning rate}\cdot \nabla f(\theta)}{v <- momentum*v + learning rate* gradient f(theta)}, and thereafter the parameter \eqn{\theta}{theta} is updates as \eqn{\theta \leftarrow \theta + v}{theta <- theta + v}. Initial velocity is set to 0. The Adam algorithm is more complex and uses first and second moments of stochastic gradients to automatically adjust the learning rate. See \cite{Goodfellow et al, 2016, page 301}. The function \code{fn} is not directly used for optimization, only for printing or as a stopping condition. In this sense it is up to the user to decide what the function returns, if anything. For instance, it may be useful for \code{fn} to compute the objective function on either full training data, or on validation data, and just ignore the \code{index} argument. The latter is useful if using \emph{patience}-based stopping. However, one may also choose to select the observations determined by the index to compute the objective function on the current data batch. % Does it support contraints? } \value{ object of class "maxim". Data can be extracted through the following methods: \item{\code{\link{maxValue}}}{\code{fn} value at maximum (the last calculated value if not converged.)} \item{\code{\link{coef}}}{estimated parameter value.} \item{\code{\link{gradient}}}{vector, last calculated gradient value. Should be close to 0 in case of normal convergence.} \item{estfun}{matrix of gradients at parameter value \code{estimate} evaluated at each observation (only if \code{grad} returns a matrix or \code{grad} is not specified and \code{fn} returns a vector).} \item{\code{\link{hessian}}}{Hessian at the maximum (the last calculated value if not converged).} \item{\code{\link{storedValues}}}{return values stored at each epoch} \item{\code{\link{storedParameters}}}{return parameters stored at each epoch} \item{\code{\link{returnCode}}}{ a numeric code that describes the convergence or error. } \item{\code{\link{returnMessage}}}{a short message, describing the return code.} \item{\code{\link{activePar}}}{logical vector, which parameters are optimized over. Contains only \code{TRUE}-s if no parameters are fixed.} \item{\code{\link{nIter}}}{number of iterations.} \item{\code{\link{maximType}}}{character string, type of maximization.} \item{\code{\link{maxControl}}}{the optimization control parameters in the form of a \code{\linkS4class{MaxControl}} object.} } \references{ Bottou, L.; Curtis, F. & Nocedal, J.: Optimization Methods for Large-Scale Machine Learning \emph{SIAM Review}, 2018, \bold{60}, 223--311. Goodfellow, I.; Bengio, Y.; Courville, A. (2016): Deep Learning, \emph{MIT Press} Henningsen, A. and Toomet, O. (2011): maxLik: A package for maximum likelihood estimation in R \emph{Computational Statistics} \bold{26}, 443--458 } \author{Ott Toomet, Arne Henningsen} \seealso{ A good starting point to learn about the usage of stochastic gradient ascent in \pkg{maxLik} package is the vignette \dQuote{Stochastic Gradient Ascent in maxLik}. The other related functions are \code{\link{maxNR}} for Newton-Raphson, a popular Hessian-based maximization; \code{\link{maxBFGS}} for maximization using the BFGS, Nelder-Mead (NM), and Simulated Annealing (SANN) method (based on \code{\link[stats]{optim}}), also supporting inequality constraints; \code{\link{maxLik}} for a general framework for maximum likelihood estimation (MLE); \code{\link{optim}} for different gradient-based optimization methods. } \examples{ ## estimate the exponential distribution parameter by ML set.seed(1) t <- rexp(100, 2) loglik <- function(theta, index) sum(log(theta) - theta*t[index]) ## Note the log-likelihood and gradient are summed over observations gradlik <- function(theta, index) sum(1/theta - t[index]) ## Estimate with full-batch a <- maxSGA(loglik, gradlik, start=1, control=list(iterlim=1000, SG_batchSize=10), nObs=100) # note that loglik is not really needed, and is not used # here, unless more print verbosity is asked summary(a) ## ## demonstrate the usage of index, and using ## fn for computing the objective function on validation data. ## Create a linear model where variables are very unequally scaled ## ## OLS loglik function: compute the function value on validation data only loglik <- function(beta, index) { e <- yValid - XValid \%*\% beta -crossprod(e)/length(y) } ## OLS gradient: compute it on training data only ## Use 'index' to select the subset corresponding to the minibatch gradlik <- function(beta, index) { e <- yTrain[index] - XTrain[index,,drop=FALSE] \%*\% beta g <- t(-2*t(XTrain[index,,drop=FALSE]) \%*\% e) -g/length(index) } N <- 1000 ## two random variables: one with scale 1, the other with 100 X <- cbind(rnorm(N), rnorm(N, sd=100)) beta <- c(1, 1) # true parameter values y <- X \%*\% beta + rnorm(N, sd=0.2) ## training-validation split iTrain <- sample(N, 0.8*N) XTrain <- X[iTrain,,drop=FALSE] XValid <- X[-iTrain,,drop=FALSE] yTrain <- y[iTrain] yValid <- y[-iTrain] ## ## do this without momentum: learning rate must stay small for the gradient not to explode cat(" No momentum:\n") a <- maxSGA(loglik, gradlik, start=c(10,10), control=list(printLevel=1, iterlim=50, SG_batchSize=30, SG_learningRate=0.0001, SGA_momentum=0 ), nObs=length(yTrain)) print(summary(a)) # the first component is off, the second one is close to the true value ## do with momentum 0.99 cat(" Momentum 0.99:\n") a <- maxSGA(loglik, gradlik, start=c(10,10), control=list(printLevel=1, iterlim=50, SG_batchSize=30, SG_learningRate=0.0001, SGA_momentum=0.99 # no momentum ), nObs=length(yTrain)) print(summary(a)) # close to true value } \keyword{optimize} maxLik/man/tidy.maxLik.Rd0000644000176200001440000000352514077525067014760 0ustar liggesusers\name{tidy.maxLik} \alias{tidy.maxLik} \alias{glance.maxLik} \title{tidy and glance methods for maxLik objects} \description{ These methods return summary information about the estimated model. Both require the \pkg{tibble} package to be installed. } \usage{ \method{tidy}{maxLik}(x, ...) \method{glance}{maxLik}(x, ...) } \arguments{ \item{x}{ object of class 'maxLik'. } \item{\ldots}{Not used.} } \value{ For \code{tidy()}, a tibble with columns: \describe{ \item{term}{The name of the estimated parameter (parameters are sequentially numbered if names missing).} \item{estimate}{The estimated parameter.} \item{std.error}{The standard error of the estimate.} \item{statistic}{The \eqn{z}{z}-statistic of the estimate.} \item{p.value}{The \eqn{p}{p}-value.} } This is essentially the same table as \code{summary}-method prints, just in form of a tibble (data frame). For \code{glance()}, a one-row tibble with columns: \describe{ \item{df}{The degrees of freedom of the model.} \item{logLik}{The log-likelihood of the model.} \item{AIC}{Akaike's Information Criterion for the model.} \item{nobs}{The number of observations, if this is available, otherwise \code{NA}.} } } \seealso{ The functions \code{\link[generics:tidy]{tidy}} and \code{\link[generics:glance]{glance}} in package \pkg{generics}, and \code{\link[=summary.maxLik]{summary}} to display the \dQuote{standard} summary information. } \author{David Hugh-Jones} \examples{ ## Example with a single parameter t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t a <- maxLik(loglik, start=2) tidy(a) glance(a) ## Example with a parameter vector x <- rnorm(100) loglik <- function(theta) { dnorm(x, mean=theta[1], sd=theta[2], log=TRUE) } a <- maxLik(loglik, start=c(mu=0, sd=1)) tidy(a) glance(a) } maxLik/man/bread.maxLik.Rd0000644000176200001440000000277014077525067015065 0ustar liggesusers\name{bread.maxLik} \alias{bread} \alias{bread.maxLik} \title{Bread for Sandwich Estimator} \description{ Extracting an estimator for the \sQuote{bread} of the sandwich estimator, see \code{\link[sandwich]{bread}}. } \usage{ \method{bread}{maxLik}( x, ... ) } \arguments{ \item{x}{an object of class \code{maxLik}.} \item{\dots}{further arguments (currently ignored).} } \value{ Matrix, the inverse of the expectation of the second derivative (Hessian matrix) of the log-likelihood function with respect to the parameters. In case of the simple Maximum Likelihood, it is equal to the variance covariance matrix of the parameters, multiplied by the number of observations. } \section{Warnings}{ The \pkg{sandwich} package is required for this function. This method works only if the observaton-specific gradient information was available for the estimation. This is the case if the observation-specific gradient was supplied (see the \code{grad} argument for \code{\link{maxLik}}), or the log-likelihood function returns a vector of observation-specific values. } \author{ Arne Henningsen } \seealso{\code{\link[sandwich]{bread}}, \code{\link{maxLik}}.} \examples{ ## ML estimation of exponential duration model: t <- rexp(100, 2) loglik <- function(theta) log(theta) - theta*t ## Estimate with numeric gradient and hessian a <- maxLik(loglik, start=1 ) # Extract the "bread" library( sandwich ) bread( a ) all.equal( bread( a ), vcov( a ) * nObs( a ) ) } \keyword{methods} maxLik/man/returnCode.Rd0000644000176200001440000000552114077525067014673 0ustar liggesusers\name{returnCode} \alias{returnCode} \alias{returnCode.default} \alias{returnCode.maxLik} \alias{returnMessage} \alias{returnMessage.default} \alias{returnMessage.maxim} \alias{returnMessage.maxLik} \title{Success or failure of the optimization} \description{ These function extract success or failure information from optimization objects. The \code{returnCode} gives a numeric code, and \code{returnMessage} a brief description about the success or failure of the optimization, and point to the problems occured (see documentation for the corresponding functions). } \usage{ returnCode(x, ...) \method{returnCode}{default}(x, ...) \method{returnCode}{maxLik}(x, ...) returnMessage(x, ...) \method{returnMessage}{maxim}(x, ...) \method{returnMessage}{maxLik}(x, ...) } \arguments{ \item{x}{object, usually an optimization result} \item{...}{further arguments for other methods} } \details{ \code{returnMessage} and \code{returnCode} are a generic functions, with methods for various optimisation algorithms. The message should either describe the convergence (stopping condition), or the problem. The known codes and the related messages are: \itemize{ \item{1}{ gradient close to zero (normal convergence).} \item{2}{ successive function values within tolerance limit (normal convergence).} \item{3}{ last step could not find higher value (probably not converged). This is related to line search step getting too small, usually because hitting the boundary of the parameter space. It may also be related to attempts to move to a wrong direction because of numerical errors. In some cases it can be helped by changing \code{steptol}.} \item{4}{ iteration limit exceeded.} \item{5}{ Infinite value.} \item{6}{ Infinite gradient.} \item{7}{ Infinite Hessian.} \item{8}{Successive function values withing relative tolerance limit (normal convergence).} \item{9}{ (BFGS) Hessian approximation cannot be improved because of gradient did not change. May be related to numerical approximation problems or wrong analytic gradient. } \item{10}{ Lost patience: the optimizer has hit an inferior value too many times (see \code{\link{maxSGA}} for more information) } \item{100}{ Initial value out of range.} } } \value{ Integer for \code{returnCode}, character for \code{returnMessage}. Different optimization routines may define it in a different way. } \author{Ott Toomet} \seealso{\code{\link{maxNR}}, \code{\link{maxBFGS}}} \examples{ ## maximise the exponential bell f1 <- function(x) exp(-x^2) a <- maxNR(f1, start=2) returnCode(a) # should be success (1 or 2) returnMessage(a) ## Now try to maximise log() function a <- maxNR(log, start=2) returnCode(a) # should give a failure (4) returnMessage(a) } \keyword{methods} \keyword{utilities} maxLik/man/maxValue.Rd0000644000176200001440000000161114077525067014337 0ustar liggesusers\name{maxValue} \alias{maxValue} \alias{maxValue.maxim} \title{Function value at maximum} \description{ Returns the function value at (estimated) maximum. } \usage{ maxValue(x, ...) \method{maxValue}{maxim}(x, \dots) } \arguments{ \item{x}{a statistical model, or a result of maximisation, created by \code{\link{maxLik}}, \code{\link{maxNR}} or another optimizer.} \item{\dots}{further arguments for other methods} } \value{ numeric, the value of the objective function at maximum. In general, it is the last calculated value in case the process did not converge. } \author{Ott Toomet} \seealso{\code{\link{maxLik}}, \code{\link{maxNR}} } \examples{ ## Estimate the exponential distribution parameter: t <- rexp(100, 2) loglik <- function(theta) sum(log(theta) - theta*t) ## Estimate with numeric gradient and numeric Hessian a <- maxNR(loglik, start=1) maxValue(a) } \keyword{methods} maxLik/man/nIter.Rd0000644000176200001440000000206714077525067013644 0ustar liggesusers\name{nIter} \alias{nIter} \alias{nIter.default} \title{Return number of iterations for iterative models} \description{ Returns the number of iterations for iterative models. The default method assumes presence of a component \code{iterations} in \code{x}. } \usage{ nIter(x, \dots) \method{nIter}{default}(x, \dots) } \arguments{ \item{x}{a statistical model, or a result of maximisation, created by \code{\link{maxLik}}, \code{\link{maxNR}} or another optimizer.} \item{\dots}{further arguments for methods} } \details{ This is a generic function. The default method returns the component \code{x$iterations}. } \value{ numeric, number of iterations. Note that \sQuote{iteration} may mean different things for different optimizers. } \author{Ott Toomet} \seealso{\code{\link{maxLik}}, \code{\link{maxNR}} } \examples{ ## Estimate the exponential distribution parameter: t <- rexp(100, 2) loglik <- function(theta) sum(log(theta) - theta*t) ## Estimate with numeric gradient and numeric Hessian a <- maxNR(loglik, start=1) nIter(a) } \keyword{methods} maxLik/man/summary.maxim.Rd0000644000176200001440000000365114077525067015372 0ustar liggesusers\name{summary.maxim} \alias{summary.maxim} \alias{print.summary.maxim} \title{Summary method for maximization} \description{ Summarizes the general maximization results in a way that does not assume the function is log-likelihood. } \usage{ \method{summary}{maxim}( object, hessian=FALSE, unsucc.step=FALSE, ... ) \method{print}{summary.maxim}(x, max.rows=getOption("max.rows", 20), max.cols=getOption("max.cols", 7), ... ) } \arguments{ \item{object}{optimization result, object of class \code{maxim}. See \code{\link{maxNR}}.} \item{hessian}{logical, whether to display Hessian matrix.} \item{unsucc.step}{logical, whether to describe last unsuccesful step if \code{code} == 3} \item{x}{object of class \code{summary.maxim}, summary of maximization result. } \item{max.rows}{maximum number of rows to be printed. This applies to the resulting coefficients (as those are printed as a matrix where the other column is the gradient), and to the Hessian if requested. } \item{max.cols}{maximum number of columns to be printed. Only Hessian output, if requested, uses this argument. } \item{\ldots}{currently not used.} } \value{ Object of class \code{summary.maxim}, intended to be printed with corresponding print method. } \author{Ott Toomet} \seealso{\code{\link{maxNR}}, \code{\link{returnCode}}, \code{\link{returnMessage}}} \examples{ ## minimize a 2D quadratic function: f <- function(b) { x <- b[1]; y <- b[2]; val <- -(x - 2)^2 - (y - 3)^2 # concave parabola attr(val, "gradient") <- c(-2*x + 4, -2*y + 6) attr(val, "hessian") <- matrix(c(-2, 0, 0, -2), 2, 2) val } ## Note that NR finds the minimum of a quadratic function with a single ## iteration. Use c(0,0) as initial value. res <- maxNR( f, start = c(0,0) ) summary(res) summary(res, hessian=TRUE) } \keyword{methods} \keyword{print} maxLik/man/hessian.Rd0000644000176200001440000000336714077525067014221 0ustar liggesusers\name{hessian} \alias{hessian} \alias{hessian.default} \title{Hessian matrix} \description{ This function extracts the Hessian of the objective function at optimum. The Hessian information should be supplied by the underlying optimization algorithm, possibly by an approximation. } \usage{ hessian(x, \dots) \method{hessian}{default}(x, \dots) } \arguments{ \item{x}{an optimization result of class \sQuote{maxim} or \sQuote{maxLik}} \item{\dots}{other arguments for methods} } \value{ A numeric matrix, the Hessian of the model at the estimated parameter values. If the maximum is flat, the Hessian is singular. In that case you may want to invert only the non-singular part of the matrix. You may also want to fix certain parameters (see \code{\link{activePar}}). } \author{Ott Toomet} \seealso{\code{\link{maxLik}}, \code{\link{activePar}}, \code{\link{condiNumber}}} \examples{ # log-likelihood for normal density # a[1] - mean # a[2] - standard deviation ll <- function(a) sum(-log(a[2]) - (x - a[1])^2/(2*a[2]^2)) x <- rnorm(100) # sample from standard normal ml <- maxLik(ll, start=c(1,1)) # ignore eventual warnings "NaNs produced in: log(x)" summary(ml) # result should be close to c(0,1) hessian(ml) # How the Hessian looks like sqrt(-solve(hessian(ml))) # Note: standard deviations are on the diagonal # # Now run the same example while fixing a[2] = 1 mlf <- maxLik(ll, start=c(1,1), activePar=c(TRUE, FALSE)) summary(mlf) # first parameter close to 0, the second exactly 1.0 hessian(mlf) # Note that now NA-s are in place of passive # parameters. # now invert only the free parameter part of the Hessian sqrt(-solve(hessian(mlf)[activePar(mlf), activePar(mlf)])) # gives the standard deviation for the mean } \keyword{methods} \keyword{optimize} maxLik/man/maxLik-methods.Rd0000644000176200001440000000326014077525067015445 0ustar liggesusers\name{AIC.maxLik} \alias{AIC.maxLik} \alias{coef.maxim} \alias{coef.maxLik} \alias{stdEr.maxLik} \title{Methods for the various standard functions} \description{ These are methods for the maxLik related objects. See also the documentation for the corresponding generic functions } \usage{ \method{AIC}{maxLik}(object, \dots, k=2) \method{coef}{maxim}(object, \dots) \method{coef}{maxLik}(object, \dots) \method{stdEr}{maxLik}(x, eigentol=1e-12, \dots) } \arguments{ \item{object}{a \sQuote{maxLik} object (\code{coef} can also handle \sQuote{maxim} objects)} \item{k}{numeric, the penalty per parameter to be used; the default \sQuote{k = 2} is the classical AIC.} \item{x}{a \sQuote{maxLik} object} \item{eigentol}{ The standard errors are only calculated if the ratio of the smallest and largest eigenvalue of the Hessian matrix is less than \dQuote{eigentol}. Otherwise the Hessian is treated as singular. } \item{\dots}{other arguments for methods} } \details{ \describe{ \item{AIC}{calculates Akaike's Information Criterion (and other information criteria).} \item{coef}{extracts the estimated parameters (model's coefficients).} \item{stdEr}{extracts standard errors (using the Hessian matrix). } } } \examples{ ## estimate mean and variance of normal random vector set.seed(123) x <- rnorm(50, 1, 2) ## log likelihood function. ## Note: 'param' is a vector llf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] return(sum(dnorm(x, mean=mu, sd=sigma, log=TRUE))) } ## Estimate it. Take standard normal as start values ml <- maxLik(llf, start = c(mu=0, sigma=1) ) coef(ml) stdEr(ml) AIC(ml) } \keyword{methods} maxLik/DESCRIPTION0000644000176200001440000000251414077570432013220 0ustar liggesusersPackage: maxLik Version: 1.5-2 Date: 2021-07-26 Title: Maximum Likelihood Estimation and Related Tools Authors@R: c(person("Ott", "Toomet", role=c("aut", "cre"), email="otoomet@gmail.com"), person("Arne", "Henningsen", role=c("aut"), email="arne.henningsen@gmail.com"), person("Spencer", "Graves", role=c("ctb")), person("Yves", "Croissant", role=c("ctb")), person("David", "Hugh-Jones", role=c("ctb")), person("Luca", "Scrucca", role=c("ctb")) ) Depends: R (>= 2.4.0), miscTools (>= 0.6-8), methods Imports: sandwich, generics Suggests: MASS, clue, dlm, plot3D, tibble, tinytest Description: Functions for Maximum Likelihood (ML) estimation, non-linear optimization, and related tools. It includes a unified way to call different optimizers, and classes and methods to handle the results from the Maximum Likelihood viewpoint. It also includes a number of convenience tools for testing and developing your own models. License: GPL (>= 2) ByteCompile: yes NeedsCompilation: no Packaged: 2021-07-26 15:39:56 UTC; otoomet Author: Ott Toomet [aut, cre], Arne Henningsen [aut], Spencer Graves [ctb], Yves Croissant [ctb], David Hugh-Jones [ctb], Luca Scrucca [ctb] Maintainer: Ott Toomet Repository: CRAN Date/Publication: 2021-07-26 17:30:02 UTC maxLik/build/0000755000176200001440000000000014077553514012611 5ustar liggesusersmaxLik/build/vignette.rds0000644000176200001440000000047314077553514015154 0ustar liggesusers‹•RMKÃ@Ý4±Ú‚Pè]÷¨Hò¼ ¥"*”êÁëšÝ$C³»%»!zókœµk¬ 2oÞÛ7 yB$ 0†X†S Cü&®!a>e+[Kö²–q +QB¡5O–ªñ´#cuZ0c!óŠqÊ:Å ¬¾°&µ•;¼ôxßìäÊ™ñ:µ Õ9m f)êéÖÙóÏî./(ô4¶ô‘HimX.¨Îœy^rz»ó¸¶dNO°ÝìýŸC¬yö‡ClYýC8¼oFv2!Óù†G~ȿ̣†À¿·§˜Æ‡ŒæPŠŽpö³ ³¹/ƒ¥/ögb-7¾=¸Ï®°ï*Ý$Ù¡ûß^0´mûú}£´d¦Û¨ÇœY–dê±{{a›ê½±maxLik/build/partial.rdb0000644000176200001440000001361714077553327014750 0ustar liggesusers‹í]}[ÛH’7„·@2yÏd&i ™à‰_0’0! oIØ !1d.{£™¬°£‹,ù$Âr¹opÿßsá>Ñ~Œ{öŸûoçºZ%Ó–%BK‚ÌÞ.ÏcªÀm÷¯ºª«««_ôn 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liggesusers### Run tinytest tests if(requireNamespace("tinytest", quietly=TRUE)) { tinytest::test_package("maxLik") } maxLik/tests/numericGradient.Rout.save0000644000176200001440000000330314077525067017605 0ustar liggesusers R version 3.0.1 (2013-05-16) -- "Good Sport" Copyright (C) 2013 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > > ### test numeric methods, in particular handling of unequal > ### function lengths > library(maxLik) Loading required package: miscTools Please cite the 'maxLik' package as: Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1. If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site: https://r-forge.r-project.org/projects/maxlik/ > > f <- function(x) { + if(x[1] <= 0) + return(NA) + # support of x[1] is (0, Inf) + return(c(log(x[1]),x[2])) + } > > ng <- numericGradient(f, c(0.01,1), eps=0.1) Warning message: In numericGradient(f, c(0.01, 1), eps = 0.1) : Function value at -0.04 1.00 = NA (length 1) does not conform with the length at original value 2 Component 1 set to NA > > nh <- try(numericHessian(f, t0=c(0.01,1), eps=0.1)) There were 13 warnings (use warnings() to see them) > > proc.time() user system elapsed 0.188 0.016 0.192 maxLik/tests/BFGSR.R0000644000176200001440000000357714077525067013660 0ustar liggesusers### BFGSR-related tests ## 1. Test maximization algorithm for convex regions ## ## Optimize quadratic form t(D) %*% W %*% D with p.d. weight matrix ## (ie unbounded problems). ## All solutions should go to large values with a message about successful convergence set.seed(0) options(digits=4) quadForm <- function(D) { C <- seq(1, N) return( - t(D - C) %*% W %*% ( D - C) ) } N <- 3 # 3-dimensional case ## a) test quadratic function t(D) %*% D library(maxLik) W <- diag(N) D <- rep(1/N, N) res <- maxBFGSR(quadForm, start=D) all.equal(coef(res), 1:3, tolerance=1e-4) all.equal(gradient(res), rep(0,3), tolerance=1e-3) all.equal(nIter(res) < 100, TRUE) all.equal(returnCode(res) < 4, TRUE) ## Next, optimize hat function in non-concave region. Does not work well. hat <- function(param) { ## Hat function. Hessian negative definite if sqrt(x^2 + y^2) < 0.5 x <- param[1] y <- param[2] exp(-(x-2)^2 - (y-2)^2) } hatNC <- maxBFGSR(hat, start=c(1,1), tol=0, reltol=0) all.equal(coef(hatNC), rep(2,2), tolerance=1e-4) all.equal(gradient(hatNC), rep(0,2), tolerance=1e-3) all.equal(nIter(hatNC) < 100, TRUE) all.equal(returnCode(hatNC) < 4, TRUE) ## Test BFGSR with fixed parameters and equality constraints ## Optimize 3D hat with one parameter fixed (== 2D hat). ## Add an equality constraint on that hat3 <- function(param) { ## Hat function. Hessian negative definite if sqrt((x-2)^2 + (y-2)^2) < 0.5 x <- param[1] y <- param[2] z <- param[3] exp(-(x-2)^2-(y-2)^2-(z-2)^2) } sv <- c(x=1,y=1,z=1) ## constraints: x + y + z = 8 A <- matrix(c(1,1,1), 1, 3) B <- -8 constraints <- list(eqA=A, eqB=B) hat3CF <- maxBFGSR(hat3, start=sv, constraints=constraints, fixed=3) all.equal(coef(hat3CF), c(x=3.5, y=3.5, z=1), tolerance=1e-4) all.equal(nIter(hat3CF) < 100, TRUE) all.equal(returnCode(hat3CF) < 4, TRUE) all.equal(sum(coef(hat3CF)), 8, tolerance=1e-4) maxLik/tests/numericGradient.R0000644000176200001440000000052414077525067016122 0ustar liggesusers ### test numeric methods, in particular handling of unequal ### function lengths library(maxLik) f <- function(x) { if(x[1] <= 0) return(NA) # support of x[1] is (0, Inf) return(c(log(x[1]),x[2])) } ng <- numericGradient(f, c(0.01,1), eps=0.1) nh <- try(numericHessian(f, t0=c(0.01,1), eps=0.1)) maxLik/tests/constraints.R0000644000176200001440000003177714077525067015367 0ustar liggesusers### Various tests for constrained optimization ### options(digits=4) ### -------------------- Normal mixture likelihood, no additional parameters -------------------- ### param = c(rho, mean1, mean2) ### ### X = N(mean1) w/Pr rho ### X = N(mean2) w/Pr 1-rho ### logLikMix <- function(param) { ## a single likelihood value rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) ll <- sum(ll) ll } gradLikMix <- function(param) { rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] f1 <- dnorm(x - mu1) f2 <- dnorm(x - mu2) L <- rho*f1 + (1 - rho)*f2 g <- matrix(0, length(x), 3) g[,1] <- (f1 - f2)/L g[,2] <- rho*(x - mu1)*f1/L g[,3] <- (1 - rho)*(x - mu2)*f2/L colSums(g) g } hessLikMix <- function(param) { rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] f1 <- dnorm(x - mu1) f2 <- dnorm(x - mu2) L <- rho*f1 + (1 - rho)*f2 dldrho <- (f1 - f2)/L dldmu1 <- rho*(x - mu1)*f1/L dldmu2 <- (1 - rho)*(x - mu2)*f2/L h <- matrix(0, 3, 3) h[1,1] <- -sum(dldrho*(f1 - f2)/L) h[2,1] <- h[1,2] <- sum((x - mu1)*f1/L - dldmu1*dldrho) h[3,1] <- h[1,3] <- sum(-(x - mu2)*f2/L - dldmu2*dldrho) h[2,2] <- sum(rho*(-f1 + (x - mu1)^2*f1)/L - dldmu1^2) h[2,3] <- h[3,2] <- -sum(dldmu1*dldmu2) h[3,3] <- sum((1 - rho)*(-f2 + (x - mu2)^2*f2)/L - dldmu2^2) h } logLikMixInd <- function(param) { ## individual obs-wise likelihood values rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) ll <- sum(ll) ll } gradLikMixInd <- function(param) { rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] f1 <- dnorm(x - mu1) f2 <- dnorm(x - mu2) L <- rho*f1 + (1 - rho)*f2 g <- matrix(0, length(x), 3) g[,1] <- (f1 - f2)/L g[,2] <- rho*(x - mu1)*f1/L g[,3] <- (1 - rho)*(x - mu2)*f2/L colSums(g) g } ### -------------------------- library(maxLik) ## mixed normal set.seed(1) N <- 100 x <- c(rnorm(N, mean=-1), rnorm(N, mean=1)) ## ---------- INEQUALITY CONSTRAINTS ----------- ## First test inequality constraints, numeric/analytical gradients ## Inequality constraints: rho < 0.5, mu1 < -0.1, mu2 > 0.1 A <- matrix(c(-1, 0, 0, 0, -1, 0, 0, 0, 1), 3, 3, byrow=TRUE) B <- c(0.5, 0.1, 0.1) start <- c(0.4, 0, 0.9) ineqCon <- list(ineqA=A, ineqB=B) ## analytic gradient cat("Inequality constraints, analytic gradient & Hessian\n") a <- maxLik(logLikMix, grad=gradLikMix, hess=hessLikMix, start=start, constraints=ineqCon) all.equal(coef(a), c(0.5, -1, 1), tolerance=0.1) # TRUE: relative tolerance 0.045 ## No analytic gradient cat("Inequality constraints, numeric gradient & Hessian\n") a <- maxLik(logLikMix, start=start, constraints=ineqCon) all.equal(coef(a), c(0.5, -1, 1), tolerance=0.1) # should be close to the true values, but N is too small ## NR method with inequality constraints try( maxLik(logLikMix, start = start, constraints = ineqCon, method = "NR" ) ) # Error in maxRoutine(fn = logLik, grad = grad, hess = hess, start = start, : # Inequality constraints not implemented for maxNR ## BHHH method with inequality constraints try( maxLik(logLikMix, start = start, constraints = ineqCon, method = "BHHH" ) ) # Error in maxNR(fn = fn, grad = grad, hess = hess, start = start, finalHessian = finalHessian, : # Inequality constraints not implemented for maxNR ## ---------- EQUALITY CONSTRAINTS ----------------- cat("Test for equality constraints mu1 + 2*mu2 = 0\n") A <- matrix(c(0, 1, 2), 1, 3) B <- 0 eqCon <- list( eqA = A, eqB = B ) ## default, numeric gradient mlEq <- maxLik(logLikMix, start = start, constraints = eqCon, tol=0) # only rely on gradient stopping condition all.equal(coef(mlEq), c(0.33, -1.45, 0.72), tolerance=0.01, scale=1) ## default, individual likelihood mlEqInd <- maxLik(logLikMixInd, start = start, constraints = eqCon, tol=0) # only rely on gradient stopping condition all.equal(coef(mlEq), coef(mlEqInd), tol=1e-4) ## default, analytic gradient mlEqG <- maxLik(logLikMix, grad=gradLikMix, start = start, constraints = eqCon ) all.equal(coef(mlEq), coef(mlEqG), tolerance=1e-4) ## default, analytic gradient, individual likelihood mlEqGInd <- maxLik(logLikMixInd, grad=gradLikMixInd, start = start, constraints = eqCon ) all.equal(coef(mlEqG), coef(mlEqGInd), tolerance=1e-4) ## default, analytic Hessian mlEqH <- maxLik(logLikMix, grad=gradLikMix, hess=hessLikMix, start=start, constraints=eqCon) all.equal(coef(mlEqG), coef(mlEqH), tolerance=1e-4) ## BFGS, numeric gradient eqBFGS <- maxLik(logLikMix, start=start, method="bfgs", constraints=eqCon, SUMTRho0=1) all.equal(coef(eqBFGS), c(0.33, -1.45, 0.72), tolerance=0.01, scale=1) ## BHHH, analytic gradient (numeric does not converge?) eqBHHH <- maxLik(logLikMix, gradLikMix, start=start, method="bhhh", constraints=eqCon, SUMTRho0=1) all.equal(coef(eqBFGS), coef(eqBHHH), tol=1e-4) ### ------------------ Now test additional parameters for the function ---- ### similar mixture as above but rho is give as an extra parameter ### logLikMix2 <- function(param, rho) { mu1 <- param[1] mu2 <- param[2] ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) # ll <- sum(ll) ll } gradLikMix2 <- function(param, rho) { mu1 <- param[1] mu2 <- param[2] f1 <- dnorm(x - mu1) f2 <- dnorm(x - mu2) L <- rho*f1 + (1 - rho)*f2 g <- matrix(0, length(x), 2) g[,1] <- rho*(x - mu1)*f1/L g[,2] <- (1 - rho)*(x - mu2)*f2/L # colSums(g) g } hessLikMix2 <- function(param, rho) { mu1 <- param[1] mu2 <- param[2] f1 <- dnorm(x - mu1) f2 <- dnorm(x - mu2) L <- rho*f1 + (1 - rho)*f2 dldrho <- (f1 - f2)/L dldmu1 <- rho*(x - mu1)*f1/L dldmu2 <- (1 - rho)*(x - mu2)*f2/L h <- matrix(0, 2, 2) h[1,1] <- sum(rho*(-f1 + (x - mu1)^2*f1)/L - dldmu1^2) h[1,2] <- h[2,1] <- -sum(dldmu1*dldmu2) h[2,2] <- sum((1 - rho)*(-f2 + (x - mu2)^2*f2)/L - dldmu2^2) h } ## ---------- Equality constraints & extra parameters ------------ A <- matrix(c(1, 2), 1, 2) B <- 0 start <- c(0, 1) ## We run only a few iterations as we want to test correct handling ## of parameters, not the final value. We also avoid any ## debug information iterlim <- 3 cat("Test for extra parameters for the function\n") ## NR, numeric gradient cat("Newton-Raphson, numeric gradient\n") a <- maxLik(logLikMix2, start=start, method="nr", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## NR, numeric hessian a <- maxLik(logLikMix2, gradLikMix2, start=start, method="nr", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## nr, analytic hessian a <- maxLik(logLikMix2, gradLikMix2, hessLikMix2, start=start, method="nr", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## BHHH cat("BHHH, analytic gradient, numeric Hessian\n") a <- maxLik(logLikMix2, gradLikMix2, start=start, method="bhhh", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## BHHH, analytic a <- maxLik(logLikMix2, gradLikMix2, start=start, method="bhhh", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## bfgs, no analytic gradient a <- maxLik(logLikMix2, start=start, method="bfgs", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## bfgs, analytic gradient a <- maxLik(logLikMix2, start=start, method="bfgs", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## SANN, analytic gradient a <- maxLik(logLikMix2, gradLikMix2, start=start, method="SANN", constraints=list(eqA=A, eqB=B), iterlim=iterlim, SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## NM, numeric a <- maxLik(logLikMix2, start=start, method="nm", constraints=list(eqA=A, eqB=B), iterlim=100, # use more iters for NM SUMTRho0=1, rho=0.5) all.equal(coef(a), c(-1.36, 0.68), tol=0.01) ## -------------------- NR, multiple constraints -------------------- f <- function(theta) exp(-theta %*% theta) # test quadratic function ## constraints: ## theta1 + theta3 = 1 ## theta1 + theta2 = 1 A <- matrix(c(1, 0, 1, 1, 1, 0), 2, 3, byrow=TRUE) B <- c(-1, -1) cat("NR, multiple constraints\n") a <- maxNR(f, start=c(1,1.1,2), constraints=list(eqA=A, eqB=B)) theta <- coef(a) all.equal(c(theta[1] + theta[3], theta[1] + theta[2]), c(1,1), tolerance=1e-4) ## Error handling for equality constraints A <- matrix(c(1, 1), 1, 2) B <- -1 cat("Error handling: ncol(A) != lengths(start)\n") try(a <- maxNR(f, start=c(1, 2, 3), constraints=list(eqA=A, eqB=B))) # ncol(A) != length(start) A <- matrix(c(1, 1), 1, 2) B <- c(-1, 2) try(a <- maxNR(f, start=c(1, 2), constraints=list(eqA=A, eqB=B))) # nrow(A) != nrow(B) ## ## -------------- inequality constraints & extra paramters ---------------- ## ## mu1 < 1 ## mu2 > -1 A <- matrix(c(-1, 0, 0, 1), 2,2, byrow=TRUE) B <- c(1,1) start <- c(0.8, 0.9) ## inEGrad <- maxLik(logLikMix2, gradLikMix2, start=start, method="bfgs", constraints=list(ineqA=A, ineqB=B), rho=0.5) all.equal(coef(inEGrad), c(-0.98, 1.12), tol=0.01) ## inE <- maxLik(logLikMix2, start=start, method="bfgs", constraints=list(ineqA=A, ineqB=B), rho=0.5) all.equal(coef(inEGrad), coef(inE), tol=1e-4) ## inENM <- maxLik(logLikMix2, gradLikMix2, start=start, method="nm", constraints=list(ineqA=A, ineqB=B), rho=0.5) all.equal(coef(inEGrad), coef(inENM), tol=1e-3) # this is further off than gradient-based methods ## ---------- test vector B for inequality -------------- ## mu1 < 1 ## mu2 > 2 A <- matrix(c(-1, 0, 0, 1), 2,2, byrow=TRUE) B1 <- c(1,-2) a <- maxLik(logLikMix2, gradLikMix2, start=c(0.5, 2.5), method="bfgs", constraints=list(ineqA=A, ineqB=B1), rho=0.5) theta <- coef(a) all.equal(c(theta[1] < 1, theta[2] > 2), c(TRUE, TRUE)) # components should be larger than # (-1, -2) ## ## ---- ERROR HANDLING: insert wrong A and B forms ---- ## A2 <- c(-1, 0, 0, 1) try(maxLik(logLikMix2, gradLikMix2, start=start, method="bfgs", constraints=list(ineqA=A2, ineqB=B), print.level=1, rho=0.5) ) # should explain that matrix needed A2 <- matrix(c(-1, 0, 0, 1), 1, 4) try(maxLik(logLikMix2, gradLikMix2, start=start, method="bfgs", constraints=list(ineqA=A2, ineqB=B), print.level=1, rho=0.5) ) # should explain that wrong matrix # dimension B2 <- 1:3 try(maxLik(logLikMix2, gradLikMix2, start=start, method="bfgs", constraints=list(ineqA=A, ineqB=B2), print.level=1, rho=0.5) ) # A & B do not match cat("A & B do not match\n") B2 <- matrix(1,2,2) try(maxLik(logLikMix2, gradLikMix2, start=start, method="bfgs", constraints=list(ineqA=A, ineqB=B2), print.level=1, rho=0.5) ) # B must be a vector ## ---- fixed parameters with constrained optimization ----- ## Thanks to Bob Loos for finding this error. ## Optimize 3D hat with one parameter fixed (== 2D hat). ## Add an equality constraint on that cat("Constraints + fixed parameters\n") hat3 <- function(param) { ## Hat function. Hessian negative definite if sqrt(x^2 + y^2) < 0.5 x <- param[1] y <- param[2] z <- param[3] exp(-x^2-y^2-z^2) } sv <- c(1,1,1) ## constraints: x + y + z >= 2.5 A <- matrix(c(x=1,y=1,z=1), 1, 3) B <- -2.5 constraints <- list(ineqA=A, ineqB=B) res <- maxBFGS(hat3, start=sv, constraints=constraints, fixed=3, iterlim=3) all.equal(coef(res), c(0.770, 0.770, 1), tol=0.01) maxLik/tests/finalHessian.Rout.save0000644000176200001440000002665314077525067017106 0ustar liggesusers R version 4.0.3 (2020-10-10) -- "Bunny-Wunnies Freak Out" Copyright (C) 2020 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > ### Test the 'finalHessian' argument of optimization routines > > library(maxLik) Loading required package: miscTools Please cite the 'maxLik' package as: Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1. If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site: https://r-forge.r-project.org/projects/maxlik/ > set.seed( 4 ) > > # log-likelihood function, gradient, and Hessian for 1-parameter case (exponential distribution) > ll1i <- function(theta) { + if(!all(theta > 0)) + return(NA) + log(theta) - theta*t + } > ll1 <- function(theta) sum( log(theta) - theta*t ) > gr1i <- function(theta) 1/theta - t > gr1 <- function(theta) sum( 1/theta - t ) > hs1 <- function(theta) -100/theta^2 > t <- rexp( 100, 2 ) > > ## the same functions for 2-variable case (normal distribution) > ll2 <- function( param ) { + ## log likelihood function + mu <- param[ 1 ] + sigma <- param[ 2 ] + if(!(sigma > 0)) + return(NA) + # to avoid warnings in the output + N <- length( x ) + llValue <- -0.5 * N * log( 2 * pi ) - N * log( sigma ) - + 0.5 * sum( ( x - mu )^2 / sigma^2 ) + return( llValue ) + } > > ## log likelihood function (individual observations) > ll2i <- function( param ) { + mu <- param[ 1 ] + sigma <- param[ 2 ] + if(!(sigma > 0)) + return(NA) + # to avoid warnings in the output + llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - + 0.5 * ( x - mu )^2 / sigma^2 + return( llValues ) + } > > gr2 <- function( param ) { + ## function to calculate analytical gradients + mu <- param[ 1 ] + sigma <- param[ 2 ] + N <- length( x ) + llGrad <- c( sum( ( x - mu ) / sigma^2 ), + - N / sigma + sum( ( x - mu )^2 / sigma^3 ) ) + return( llGrad ) + } > > ## function to calculate analytical gradients (individual observations) > gr2i <- function( param ) { + mu <- param[ 1 ] + sigma <- param[ 2 ] + llGrads <- cbind( ( x - mu ) / sigma^2, + - 1 / sigma + ( x - mu )^2 / sigma^3 ) + return( llGrads ) + } > > ## function to calculate analytical Hessians > hs2 <- function( param ) { + mu <- param[ 1 ] + sigma <- param[ 2 ] + N <- length( x ) + llHess <- matrix( c( + N * ( - 1 / sigma^2 ), + sum( - 2 * ( x - mu ) / sigma^3 ), + sum( - 2 * ( x - mu ) / sigma^3 ), + N / sigma^2 + sum( - 3 * ( x - mu )^2 / sigma^4 ) ), + nrow = 2, ncol = 2 ) + return( llHess ) + } > x <- rnorm(100, 1, 2) > > > ## NR > # Estimate with only function values (single parameter) > a <- maxLik( ll1i, gr1i, start = 1, method = "NR" ) > summary(a ) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 5 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 2.1159 0.2116 10 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > b <- maxLik( ll1i, gr1i, start = 1, method = "NR", finalHessian="bhhh") > # should issue a warning as BHHH not possible > summary(b ) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 5 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 2.1159 0.2145 9.863 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > c <- maxLik( ll1i, gr1i, start = 1, method = "NR", finalHessian=FALSE) > summary(c) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 5 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate t value Pr(> t) [1,] 2.116 NA NA -------------------------------------------- > ## (vector parameter) > a <- maxLik( ll2, gr2, start = c(0,1), method = "NR" ) > summary(a ) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 7 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 0.8532 0.2031 4.201 2.66e-05 *** [2,] 2.0311 0.1436 14.142 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > b <- maxLik( ll2, gr2, start = c(0,1), method = "NR", finalHessian="bhhh") Warning message: In maxNRCompute(fn = function (theta, fnOrig, gradOrig = NULL, hessOrig = NULL, : For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations > # should issue a warning as BHHH not possible > summary(b ) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 7 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate t value Pr(> t) [1,] 0.8532 NA NA [2,] 2.0311 NA NA -------------------------------------------- > c <- maxLik( ll2, gr2, start = c(0,1), method = "NR", finalHessian=FALSE) > summary(c) -------------------------------------------- Maximum Likelihood estimation Newton-Raphson maximisation, 7 iterations Return code 1: gradient close to zero (gradtol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate t value Pr(> t) [1,] 0.8532 NA NA [2,] 2.0311 NA NA -------------------------------------------- > > ## BFGSR > # Estimate with only function values (single parameter) > a <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR" ) > summary(a ) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 26 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 2.1159 0.2116 10 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > b <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR", finalHessian="bhhh") > # should issue a warning as BHHH not possible > summary(b ) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 26 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 2.1159 0.2145 9.863 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > c <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR", finalHessian=FALSE) > summary(c) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 26 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -25.05386 1 free parameters Estimates: Estimate t value Pr(> t) [1,] 2.116 NA NA -------------------------------------------- > # Estimate with only function values (vector parameter) > a <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR" ) > summary(a ) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 22 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 0.8528 0.2031 4.199 2.68e-05 *** [2,] 2.0309 0.1436 14.144 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > b <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR", finalHessian="bhhh") Warning message: In maxBFGSRCompute(fn = function (theta, fnOrig, gradOrig = NULL, : For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations > # should issue a warning as BHHH not possible > summary(b ) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 22 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate t value Pr(> t) [1,] 0.8528 NA NA [2,] 2.0309 NA NA -------------------------------------------- > c <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR", finalHessian=FALSE) > summary(c) -------------------------------------------- Maximum Likelihood estimation BFGSR maximization, 22 iterations Return code 2: successive function values within tolerance limit (tol) Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate t value Pr(> t) [1,] 0.8528 NA NA [2,] 2.0309 NA NA -------------------------------------------- > > > ### Nelder-Mead > ## Individual observations only > b <- maxLik( ll2i, start = c(0,1), method = "NM", finalHessian="bhhh") > summary(b) -------------------------------------------- Maximum Likelihood estimation Nelder-Mead maximization, 63 iterations Return code 0: successful convergence Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate Std. error t value Pr(> t) [1,] 0.8530 0.2032 4.199 2.69e-05 *** [2,] 2.0312 0.1670 12.163 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 -------------------------------------------- > ## Individual observations, summed gradient > b <- maxLik( ll2i, gr2, start = c(0,1), method = "NM", finalHessian="bhhh") Warning message: In maxOptim(fn = fn, grad = grad, hess = hess, start = start, method = "Nelder-Mead", : For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations > # should issue a warning as BHHH not selected > # (yes, could do it based on individual likelihood and numeric gradient) > summary(b) -------------------------------------------- Maximum Likelihood estimation Nelder-Mead maximization, 63 iterations Return code 0: successful convergence Log-Likelihood: -212.7524 2 free parameters Estimates: Estimate t value Pr(> t) [1,] 0.853 NA NA [2,] 2.031 NA NA -------------------------------------------- > > proc.time() user system elapsed 0.436 0.020 0.447 maxLik/tests/BFGSR.Rout.save0000644000176200001440000000631414077525067015335 0ustar liggesusers R version 3.6.0 (2019-04-26) -- "Planting of a Tree" Copyright (C) 2019 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > ### BFGSR-related tests > > ## 1. Test maximization algorithm for convex regions > ## > ## Optimize quadratic form t(D) %*% W %*% D with p.d. weight matrix > ## (ie unbounded problems). > ## All solutions should go to large values with a message about successful convergence > set.seed(0) > options(digits=4) > quadForm <- function(D) { + C <- seq(1, N) + return( - t(D - C) %*% W %*% ( D - C) ) + } > N <- 3 > # 3-dimensional case > ## a) test quadratic function t(D) %*% D > library(maxLik) Loading required package: miscTools Please cite the 'maxLik' package as: Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1. If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site: https://r-forge.r-project.org/projects/maxlik/ > W <- diag(N) > D <- rep(1/N, N) > res <- maxBFGSR(quadForm, start=D) > all.equal(coef(res), 1:3, tolerance=1e-4) [1] TRUE > all.equal(gradient(res), rep(0,3), tolerance=1e-3) [1] TRUE > all.equal(nIter(res) < 100, TRUE) [1] TRUE > all.equal(returnCode(res) < 4, TRUE) [1] TRUE > > ## Next, optimize hat function in non-concave region. Does not work well. > hat <- function(param) { + ## Hat function. Hessian negative definite if sqrt(x^2 + y^2) < 0.5 + x <- param[1] + y <- param[2] + exp(-(x-2)^2 - (y-2)^2) + } > > hatNC <- maxBFGSR(hat, start=c(1,1), tol=0, reltol=0) > all.equal(coef(hatNC), rep(2,2), tolerance=1e-4) [1] TRUE > all.equal(gradient(hatNC), rep(0,2), tolerance=1e-3) [1] TRUE > all.equal(nIter(hatNC) < 100, TRUE) [1] TRUE > all.equal(returnCode(hatNC) < 4, TRUE) [1] TRUE > > ## Test BFGSR with fixed parameters and equality constraints > ## Optimize 3D hat with one parameter fixed (== 2D hat). > ## Add an equality constraint on that > hat3 <- function(param) { + ## Hat function. Hessian negative definite if sqrt((x-2)^2 + (y-2)^2) < 0.5 + x <- param[1] + y <- param[2] + z <- param[3] + exp(-(x-2)^2-(y-2)^2-(z-2)^2) + } > sv <- c(x=1,y=1,z=1) > ## constraints: x + y + z = 8 > A <- matrix(c(1,1,1), 1, 3) > B <- -8 > constraints <- list(eqA=A, eqB=B) > hat3CF <- maxBFGSR(hat3, start=sv, constraints=constraints, fixed=3) > all.equal(coef(hat3CF), c(x=3.5, y=3.5, z=1), tolerance=1e-4) [1] TRUE > all.equal(nIter(hat3CF) < 100, TRUE) [1] TRUE > all.equal(returnCode(hat3CF) < 4, TRUE) [1] TRUE > all.equal(sum(coef(hat3CF)), 8, tolerance=1e-4) [1] TRUE > > proc.time() user system elapsed 0.562 0.560 0.338 maxLik/tests/constraints.Rout.save0000644000176200001440000004257514077525067017052 0ustar liggesusers R version 3.5.3 (2019-03-11) -- "Great Truth" Copyright (C) 2019 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > ### Various tests for constrained optimization > ### > options(digits=4) > > ### -------------------- Normal mixture likelihood, no additional parameters -------------------- > ### param = c(rho, mean1, mean2) > ### > ### X = N(mean1) w/Pr rho > ### X = N(mean2) w/Pr 1-rho > ### > logLikMix <- function(param) { + ## a single likelihood value + rho <- param[1] + if(rho < 0 || rho > 1) + return(NA) + mu1 <- param[2] + mu2 <- param[3] + ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) + ll <- sum(ll) + ll + } > > gradLikMix <- function(param) { + rho <- param[1] + if(rho < 0 || rho > 1) + return(NA) + mu1 <- param[2] + mu2 <- param[3] + f1 <- dnorm(x - mu1) + f2 <- dnorm(x - mu2) + L <- rho*f1 + (1 - rho)*f2 + g <- matrix(0, length(x), 3) + g[,1] <- (f1 - f2)/L + g[,2] <- rho*(x - mu1)*f1/L + g[,3] <- (1 - rho)*(x - mu2)*f2/L + colSums(g) + g + } > > hessLikMix <- function(param) { + rho <- param[1] + if(rho < 0 || rho > 1) + return(NA) + mu1 <- param[2] + mu2 <- param[3] + f1 <- dnorm(x - mu1) + f2 <- dnorm(x - mu2) + L <- rho*f1 + (1 - rho)*f2 + dldrho <- (f1 - f2)/L + dldmu1 <- rho*(x - mu1)*f1/L + dldmu2 <- (1 - rho)*(x - mu2)*f2/L + h <- matrix(0, 3, 3) + h[1,1] <- -sum(dldrho*(f1 - f2)/L) + h[2,1] <- h[1,2] <- sum((x - mu1)*f1/L - dldmu1*dldrho) + h[3,1] <- h[1,3] <- sum(-(x - mu2)*f2/L - dldmu2*dldrho) + h[2,2] <- sum(rho*(-f1 + (x - mu1)^2*f1)/L - dldmu1^2) + h[2,3] <- h[3,2] <- -sum(dldmu1*dldmu2) + h[3,3] <- sum((1 - rho)*(-f2 + (x - mu2)^2*f2)/L - dldmu2^2) + h + } > > logLikMixInd <- function(param) { + ## individual obs-wise likelihood values + rho <- param[1] + if(rho < 0 || rho > 1) + return(NA) + mu1 <- param[2] + mu2 <- param[3] + ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) + ll <- sum(ll) + ll + } > > gradLikMixInd <- function(param) { + rho <- param[1] + if(rho < 0 || rho > 1) + return(NA) + mu1 <- param[2] + mu2 <- param[3] + f1 <- dnorm(x - mu1) + f2 <- dnorm(x - mu2) + L <- rho*f1 + (1 - rho)*f2 + g <- matrix(0, length(x), 3) + g[,1] <- (f1 - f2)/L + g[,2] <- rho*(x - mu1)*f1/L + g[,3] <- (1 - rho)*(x - mu2)*f2/L + colSums(g) + g + } > > ### -------------------------- > library(maxLik) Loading required package: miscTools Please cite the 'maxLik' package as: Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1. If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site: https://r-forge.r-project.org/projects/maxlik/ > ## mixed normal > set.seed(1) > N <- 100 > x <- c(rnorm(N, mean=-1), rnorm(N, mean=1)) > > ## ---------- INEQUALITY CONSTRAINTS ----------- > ## First test inequality constraints, numeric/analytical gradients > ## Inequality constraints: rho < 0.5, mu1 < -0.1, mu2 > 0.1 > A <- matrix(c(-1, 0, 0, + 0, -1, 0, + 0, 0, 1), 3, 3, byrow=TRUE) > B <- c(0.5, 0.1, 0.1) > start <- c(0.4, 0, 0.9) > ineqCon <- list(ineqA=A, ineqB=B) > ## analytic gradient > cat("Inequality constraints, analytic gradient & Hessian\n") Inequality constraints, analytic gradient & Hessian > a <- maxLik(logLikMix, grad=gradLikMix, hess=hessLikMix, + start=start, + constraints=ineqCon) > all.equal(coef(a), c(0.5, -1, 1), tolerance=0.1) [1] "Mean relative difference: 0.1624" > # TRUE: relative tolerance 0.045 > ## No analytic gradient > cat("Inequality constraints, numeric gradient & Hessian\n") Inequality constraints, numeric gradient & Hessian > a <- maxLik(logLikMix, + start=start, + constraints=ineqCon) > all.equal(coef(a), c(0.5, -1, 1), tolerance=0.1) [1] "Mean relative difference: 0.2547" > # should be close to the true values, but N is too small > ## NR method with inequality constraints > try( maxLik(logLikMix, start = start, constraints = ineqCon, method = "NR" ) ) Error in maxRoutine(fn = logLik, grad = grad, hess = hess, start = start, : Inequality constraints not implemented for maxNR > # Error in maxRoutine(fn = logLik, grad = grad, hess = hess, start = start, : > # Inequality constraints not implemented for maxNR > > ## BHHH method with inequality constraints > try( maxLik(logLikMix, start = start, constraints = ineqCon, method = "BHHH" ) ) Error in maxNR(fn = fn, grad = grad, hess = hess, start = start, finalHessian = finalHessian, : Inequality constraints not implemented for maxNR > # Error in maxNR(fn = fn, grad = grad, hess = hess, start = start, finalHessian = finalHessian, : > # Inequality constraints not implemented for maxNR > > ## ---------- EQUALITY CONSTRAINTS ----------------- > cat("Test for equality constraints mu1 + 2*mu2 = 0\n") Test for equality constraints mu1 + 2*mu2 = 0 > A <- matrix(c(0, 1, 2), 1, 3) > B <- 0 > eqCon <- list( eqA = A, eqB = B ) > ## default, numeric gradient > mlEq <- maxLik(logLikMix, start = start, constraints = eqCon, tol=0) > # only rely on gradient stopping condition > all.equal(coef(mlEq), c(0.33, -1.45, 0.72), tolerance=0.01, scale=1) [1] "Mean absolute difference: 0.1777" > ## default, individual likelihood > mlEqInd <- maxLik(logLikMixInd, start = start, constraints = eqCon, tol=0) > # only rely on gradient stopping condition > all.equal(coef(mlEq), coef(mlEqInd), tol=1e-4) [1] TRUE > ## default, analytic gradient > mlEqG <- maxLik(logLikMix, grad=gradLikMix, + start = start, constraints = eqCon ) > all.equal(coef(mlEq), coef(mlEqG), tolerance=1e-4) [1] TRUE > ## default, analytic gradient, individual likelihood > mlEqGInd <- maxLik(logLikMixInd, grad=gradLikMixInd, + start = start, constraints = eqCon ) > all.equal(coef(mlEqG), coef(mlEqGInd), tolerance=1e-4) [1] TRUE > ## default, analytic Hessian > mlEqH <- maxLik(logLikMix, grad=gradLikMix, hess=hessLikMix, + start=start, + constraints=eqCon) > all.equal(coef(mlEqG), coef(mlEqH), tolerance=1e-4) [1] TRUE > > > ## BFGS, numeric gradient > eqBFGS <- maxLik(logLikMix, + start=start, method="bfgs", + constraints=eqCon, + SUMTRho0=1) > all.equal(coef(eqBFGS), c(0.33, -1.45, 0.72), tolerance=0.01, scale=1) [1] "Mean absolute difference: 0.1777" > > ## BHHH, analytic gradient (numeric does not converge?) > eqBHHH <- maxLik(logLikMix, gradLikMix, + start=start, method="bhhh", + constraints=eqCon, + SUMTRho0=1) > all.equal(coef(eqBFGS), coef(eqBHHH), tol=1e-4) [1] "Mean relative difference: 0.003536" > > > ### ------------------ Now test additional parameters for the function ---- > ### similar mixture as above but rho is give as an extra parameter > ### > logLikMix2 <- function(param, rho) { + mu1 <- param[1] + mu2 <- param[2] + ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) + # ll <- sum(ll) + ll + } > > gradLikMix2 <- function(param, rho) { + mu1 <- param[1] + mu2 <- param[2] + f1 <- dnorm(x - mu1) + f2 <- dnorm(x - mu2) + L <- rho*f1 + (1 - rho)*f2 + g <- matrix(0, length(x), 2) + g[,1] <- rho*(x - mu1)*f1/L + g[,2] <- (1 - rho)*(x - mu2)*f2/L + # colSums(g) + g + } > > hessLikMix2 <- function(param, rho) { + mu1 <- param[1] + mu2 <- param[2] + f1 <- dnorm(x - mu1) + f2 <- dnorm(x - mu2) + L <- rho*f1 + (1 - rho)*f2 + dldrho <- (f1 - f2)/L + dldmu1 <- rho*(x - mu1)*f1/L + dldmu2 <- (1 - rho)*(x - mu2)*f2/L + h <- matrix(0, 2, 2) + h[1,1] <- sum(rho*(-f1 + (x - mu1)^2*f1)/L - dldmu1^2) + h[1,2] <- h[2,1] <- -sum(dldmu1*dldmu2) + h[2,2] <- sum((1 - rho)*(-f2 + (x - mu2)^2*f2)/L - dldmu2^2) + h + } > > ## ---------- Equality constraints & extra parameters ------------ > A <- matrix(c(1, 2), 1, 2) > B <- 0 > start <- c(0, 1) > ## We run only a few iterations as we want to test correct handling > ## of parameters, not the final value. We also avoid any > ## debug information > iterlim <- 3 > cat("Test for extra parameters for the function\n") Test for extra parameters for the function > ## NR, numeric gradient > cat("Newton-Raphson, numeric gradient\n") Newton-Raphson, numeric gradient > a <- maxLik(logLikMix2, + start=start, method="nr", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3619" > ## NR, numeric hessian > a <- maxLik(logLikMix2, gradLikMix2, + start=start, method="nr", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3619" > ## nr, analytic hessian > a <- maxLik(logLikMix2, gradLikMix2, hessLikMix2, + start=start, method="nr", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3619" > ## BHHH > cat("BHHH, analytic gradient, numeric Hessian\n") BHHH, analytic gradient, numeric Hessian > a <- maxLik(logLikMix2, gradLikMix2, + start=start, method="bhhh", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3512" > ## BHHH, analytic > a <- maxLik(logLikMix2, gradLikMix2, + start=start, method="bhhh", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3512" > ## bfgs, no analytic gradient > a <- maxLik(logLikMix2, + start=start, method="bfgs", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3381" > ## bfgs, analytic gradient > a <- maxLik(logLikMix2, + start=start, method="bfgs", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3381" > ## SANN, analytic gradient > a <- maxLik(logLikMix2, gradLikMix2, + start=start, method="SANN", + constraints=list(eqA=A, eqB=B), + iterlim=iterlim, SUMTRho0=1, rho=0.5) Warning message: In (function (fn, grad = NULL, hess = NULL, start, maxRoutine, constraints, : problem in imposing equality constraints: the constraints are not satisfied (barrier value = 0.00173566161904632). Try setting 'SUMTTol' to 0 > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.2285" > ## NM, numeric > a <- maxLik(logLikMix2, + start=start, method="nm", + constraints=list(eqA=A, eqB=B), + iterlim=100, + # use more iters for NM + SUMTRho0=1, rho=0.5) > all.equal(coef(a), c(-1.36, 0.68), tol=0.01) [1] "Mean relative difference: 0.3621" > > ## -------------------- NR, multiple constraints -------------------- > f <- function(theta) exp(-theta %*% theta) > # test quadratic function > ## constraints: > ## theta1 + theta3 = 1 > ## theta1 + theta2 = 1 > A <- matrix(c(1, 0, 1, + 1, 1, 0), 2, 3, byrow=TRUE) > B <- c(-1, -1) > cat("NR, multiple constraints\n") NR, multiple constraints > a <- maxNR(f, start=c(1,1.1,2), constraints=list(eqA=A, eqB=B)) > theta <- coef(a) > all.equal(c(theta[1] + theta[3], theta[1] + theta[2]), c(1,1), tolerance=1e-4) [1] TRUE > ## Error handling for equality constraints > A <- matrix(c(1, 1), 1, 2) > B <- -1 > cat("Error handling: ncol(A) != lengths(start)\n") Error handling: ncol(A) != lengths(start) > try(a <- maxNR(f, start=c(1, 2, 3), constraints=list(eqA=A, eqB=B))) Error in sumt(fn = function (theta) : Equality constraint matrix A must have the same number of columns as the parameter length (currently 2 and 3) > # ncol(A) != length(start) > A <- matrix(c(1, 1), 1, 2) > B <- c(-1, 2) > try(a <- maxNR(f, start=c(1, 2), constraints=list(eqA=A, eqB=B))) Error in sumt(fn = function (theta) : Equality constraint matrix A must have the same number of rows as the matrix B (currently 1 and 2) > # nrow(A) != nrow(B) > ## > ## -------------- inequality constraints & extra paramters ---------------- > ## > ## mu1 < 1 > ## mu2 > -1 > A <- matrix(c(-1, 0, + 0, 1), 2,2, byrow=TRUE) > B <- c(1,1) > start <- c(0.8, 0.9) > ## > inEGrad <- maxLik(logLikMix2, gradLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A, ineqB=B), + rho=0.5) > all.equal(coef(inEGrad), c(-0.98, 1.12), tol=0.01) [1] "Mean relative difference: 0.2716" > ## > inE <- maxLik(logLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A, ineqB=B), + rho=0.5) > all.equal(coef(inEGrad), coef(inE), tol=1e-4) [1] TRUE > ## > inENM <- maxLik(logLikMix2, gradLikMix2, + start=start, method="nm", + constraints=list(ineqA=A, ineqB=B), + rho=0.5) > all.equal(coef(inEGrad), coef(inENM), tol=1e-3) [1] TRUE > # this is further off than gradient-based methods > ## ---------- test vector B for inequality -------------- > ## mu1 < 1 > ## mu2 > 2 > A <- matrix(c(-1, 0, + 0, 1), 2,2, byrow=TRUE) > B1 <- c(1,-2) > a <- maxLik(logLikMix2, gradLikMix2, + start=c(0.5, 2.5), method="bfgs", + constraints=list(ineqA=A, ineqB=B1), + rho=0.5) > theta <- coef(a) > all.equal(c(theta[1] < 1, theta[2] > 2), c(TRUE, TRUE)) [1] TRUE > # components should be larger than > # (-1, -2) > > ## > ## ---- ERROR HANDLING: insert wrong A and B forms ---- > ## > A2 <- c(-1, 0, 0, 1) > try(maxLik(logLikMix2, gradLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A2, ineqB=B), + print.level=1, rho=0.5) + ) Error in maxOptim(fn = fn, grad = grad, hess = hess, start = start, method = "BFGS", : Inequality constraint A must be a matrix Current dimension > # should explain that matrix needed > A2 <- matrix(c(-1, 0, 0, 1), 1, 4) > try(maxLik(logLikMix2, gradLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A2, ineqB=B), + print.level=1, rho=0.5) + ) Error in maxOptim(fn = fn, grad = grad, hess = hess, start = start, method = "BFGS", : Inequality constraint A must have the same number of columns as length of the parameter. Currently 4 and 2. > # should explain that wrong matrix > # dimension > B2 <- 1:3 > try(maxLik(logLikMix2, gradLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A, ineqB=B2), + print.level=1, rho=0.5) + ) Error in maxOptim(fn = fn, grad = grad, hess = hess, start = start, method = "BFGS", : Inequality constraints A and B suggest different number of constraints: 2 and 3 > # A & B do not match > cat("A & B do not match\n") A & B do not match > B2 <- matrix(1,2,2) > try(maxLik(logLikMix2, gradLikMix2, + start=start, method="bfgs", + constraints=list(ineqA=A, ineqB=B2), + print.level=1, rho=0.5) + ) Error in maxOptim(fn = fn, grad = grad, hess = hess, start = start, method = "BFGS", : Inequality constraint B must be a vector (or Nx1 matrix). Currently 2 columns > # B must be a vector > > ## ---- fixed parameters with constrained optimization ----- > ## Thanks to Bob Loos for finding this error. > ## Optimize 3D hat with one parameter fixed (== 2D hat). > ## Add an equality constraint on that > cat("Constraints + fixed parameters\n") Constraints + fixed parameters > hat3 <- function(param) { + ## Hat function. Hessian negative definite if sqrt(x^2 + y^2) < 0.5 + x <- param[1] + y <- param[2] + z <- param[3] + exp(-x^2-y^2-z^2) + } > sv <- c(1,1,1) > ## constraints: x + y + z >= 2.5 > A <- matrix(c(x=1,y=1,z=1), 1, 3) > B <- -2.5 > constraints <- list(ineqA=A, ineqB=B) > res <- maxBFGS(hat3, start=sv, constraints=constraints, fixed=3, + iterlim=3) > all.equal(coef(res), c(0.770, 0.770, 1), tol=0.01) [1] TRUE > > proc.time() user system elapsed 1.676 0.329 1.571 maxLik/tests/finalHessian.R0000644000176200001440000000772014077525067015413 0ustar liggesusers### Test the 'finalHessian' argument of optimization routines library(maxLik) set.seed( 4 ) # log-likelihood function, gradient, and Hessian for 1-parameter case (exponential distribution) ll1i <- function(theta) { if(!all(theta > 0)) return(NA) log(theta) - theta*t } ll1 <- function(theta) sum( log(theta) - theta*t ) gr1i <- function(theta) 1/theta - t gr1 <- function(theta) sum( 1/theta - t ) hs1 <- function(theta) -100/theta^2 t <- rexp( 100, 2 ) ## the same functions for 2-variable case (normal distribution) ll2 <- function( param ) { ## log likelihood function mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output N <- length( x ) llValue <- -0.5 * N * log( 2 * pi ) - N * log( sigma ) - 0.5 * sum( ( x - mu )^2 / sigma^2 ) return( llValue ) } ## log likelihood function (individual observations) ll2i <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - 0.5 * ( x - mu )^2 / sigma^2 return( llValues ) } gr2 <- function( param ) { ## function to calculate analytical gradients mu <- param[ 1 ] sigma <- param[ 2 ] N <- length( x ) llGrad <- c( sum( ( x - mu ) / sigma^2 ), - N / sigma + sum( ( x - mu )^2 / sigma^3 ) ) return( llGrad ) } ## function to calculate analytical gradients (individual observations) gr2i <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] llGrads <- cbind( ( x - mu ) / sigma^2, - 1 / sigma + ( x - mu )^2 / sigma^3 ) return( llGrads ) } ## function to calculate analytical Hessians hs2 <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] N <- length( x ) llHess <- matrix( c( N * ( - 1 / sigma^2 ), sum( - 2 * ( x - mu ) / sigma^3 ), sum( - 2 * ( x - mu ) / sigma^3 ), N / sigma^2 + sum( - 3 * ( x - mu )^2 / sigma^4 ) ), nrow = 2, ncol = 2 ) return( llHess ) } x <- rnorm(100, 1, 2) ## NR # Estimate with only function values (single parameter) a <- maxLik( ll1i, gr1i, start = 1, method = "NR" ) summary(a ) b <- maxLik( ll1i, gr1i, start = 1, method = "NR", finalHessian="bhhh") # should issue a warning as BHHH not possible summary(b ) c <- maxLik( ll1i, gr1i, start = 1, method = "NR", finalHessian=FALSE) summary(c) ## (vector parameter) a <- maxLik( ll2, gr2, start = c(0,1), method = "NR" ) summary(a ) b <- maxLik( ll2, gr2, start = c(0,1), method = "NR", finalHessian="bhhh") # should issue a warning as BHHH not possible summary(b ) c <- maxLik( ll2, gr2, start = c(0,1), method = "NR", finalHessian=FALSE) summary(c) ## BFGSR # Estimate with only function values (single parameter) a <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR" ) summary(a ) b <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR", finalHessian="bhhh") # should issue a warning as BHHH not possible summary(b ) c <- maxLik( ll1i, gr1i, start = 1, method = "BFGSR", finalHessian=FALSE) summary(c) # Estimate with only function values (vector parameter) a <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR" ) summary(a ) b <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR", finalHessian="bhhh") # should issue a warning as BHHH not possible summary(b ) c <- maxLik( ll2, gr2, start = c(0,1), method = "BFGSR", finalHessian=FALSE) summary(c) ### Nelder-Mead ## Individual observations only b <- maxLik( ll2i, start = c(0,1), method = "NM", finalHessian="bhhh") summary(b) ## Individual observations, summed gradient b <- maxLik( 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Copyright 2019 Radical Eye Software probability-density_.dvi endstream endobj 2 0 obj <>endobj xref 0 25 0000000000 65535 f 0000002913 00000 n 0000010646 00000 n 0000002854 00000 n 0000002697 00000 n 0000000332 00000 n 0000002677 00000 n 0000002978 00000 n 0000003842 00000 n 0000008108 00000 n 0000003685 00000 n 0000007368 00000 n 0000003341 00000 n 0000004870 00000 n 0000003112 00000 n 0000004301 00000 n 0000003019 00000 n 0000003049 00000 n 0000004510 00000 n 0000005179 00000 n 0000007581 00000 n 0000008332 00000 n 0000003255 00000 n 0000004204 00000 n 0000009177 00000 n trailer << /Size 25 /Root 1 0 R /Info 2 0 R /ID [<0EBD16C126AFE1F871F9B1F51F37959F><0EBD16C126AFE1F871F9B1F51F37959F>] >> startxref 10865 %%EOF maxLik/vignettes/probability-density.asy0000644000176200001440000000403614077525067020242 0ustar liggesusersunitsize(25mm,65mm); defaultpen(fontsize(9)); real xLeft = -2.2; real xRight = 2.2; real yTop = 0.5; // normal density real dnorm(real x) { return 1/sqrt(2*pi)*exp(-1/2*x^2); } // compute normal curve, plot later path normalCurve; for(real x = xLeft + 0.1; x < xRight - 0.1; x += 0.15) { normalCurve = normalCurve..(x, dnorm(x)); } // Example points real xs[] = {-1.695, 0.3}; real delta = 0.15; int i = 1; for(real x : xs) { real fx = dnorm(x); real xl = x - delta/2; real xr = x + delta/2; real tl = times(normalCurve, xl)[0]; real tr = times(normalCurve, xr)[0]; path striptop = subpath(normalCurve, tl, tr); path area = (xl, 0)--striptop--(xr, 0)--cycle; filldraw(area, lightgray, linewidth(0.2)); draw((x, 0)--(x, dnorm(x)), dashed); label("$x_" + string(i) + " = " + format("%f", x) +"$", (x, 0), S + 0.2E); // width marks and width real barheight = dnorm(x) + 0.06; Label widthLabel = Label("width $\delta$", MidPoint, 2N); draw(widthLabel, (xl, barheight)--(xr, barheight), linewidth(0.4), Bars); arrow((xl, barheight), W, length=50delta, margin=DotMargin, linewidth(0.4)); arrow((xr, barheight), E, length=50delta, margin=DotMargin); // mark the function value real xmarker = x + 1.5delta; draw((x, fx)--(xmarker,fx), dotted); Label valueLabel = Label("$f(x_" + string(i) + ") = " + format("%5.3f", fx) + "$", position=EndPoint, E); path valuePath = (xmarker, fx)--(xmarker+delta, fx); draw(valueLabel, valuePath, linewidth(0.4)); pair barx = relpoint(valuePath, 0.5); draw((barx.x, 0)--barx, Arrow(4)); // ++i; } // add normal curve later as filling area cuts into the curve otherwise draw(normalCurve, linewidth(0.7)); // Add Axes after are to avoid cutting into it path xaxis = (xLeft,0)--(xRight,0); path yaxis = (0,0)--(0,yTop); draw(xaxis, Arrow(TeXHead, 1)); draw(yaxis, Arrow(TeXHead, 1)); label("$x$", point(xaxis, 1), 2S); // Axis labels real tickLength = 0.05*yTop; for(int x = (int)xLeft; x <= (int)xRight; ++x) { draw((x,0)--(x,-tickLength)); label(string(x), (x,-tickLength), 3S); } maxLik/vignettes/maxlik.bib0000644000176200001440000001237714077525067015501 0ustar liggesusers% Encoding: UTF-8 @Article{bottou2018SIAM, author = {Bottou, L. and Curtis, F. and Nocedal, J.}, title = {Optimization Methods for Large-Scale Machine Learning}, journal = {SIAM Review}, year = {2018}, volume = {60}, number = {2}, pages = {223-311}, doi = {10.1137/16M1080173}, eprint = {https://doi.org/10.1137/16M1080173}, owner = {otoomet}, review = {A long review of different optimization methods from ML perspective. Revolves around SGD and a lot of space is devoted to show how other popular methods are related to SGD. A lot about convergence speed. Very little about non-smooth objective functions, just l1 norm optimization.}, timestamp = {2019.08.06}, url = { https://doi.org/10.1137/16M1080173 }, } @Article{keskar+2016ArXiv, author = {Nitish Shirish Keskar and Dheevatsa Mudigere and Jorge Nocedal and Mikhail Smelyanskiy and Ping Tak Peter Tang}, title = {On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima}, journal = {ArXiv}, year = {2016}, volume = {abs/1609.04836}, abstract = {The stochastic gradient descent (SGD) method and its variants are algorithms of choice for many Deep Learning tasks. These methods operate in a small-batch regime wherein a fraction of the training data, say $32$-$512$ data points, is sampled to compute an approximation to the gradient. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. We investigate the cause for this generalization drop in the large-batch regime and present numerical evidence that supports the view that large-batch methods tend to converge to sharp minimizers of the training and testing functions - and as is well known, sharp minima lead to poorer generalization. In contrast, small-batch methods consistently converge to flat minimizers, and our experiments support a commonly held view that this is due to the inherent noise in the gradient estimation. We discuss several strategies to attempt to help large-batch methods eliminate this generalization gap}, owner = {siim}, review = {Analyze the sharpness of obtained minima in loss function when using small/large batches for SGD. Incorporate several mid-size neural networks for image processing, including fully connected and convolutional. Show both using graphs and computing sharpness that small batches lead to flat minima while large ones to sharp minima. Speculate it is because the small batches are more noisy, will jump out of the sharp basing but get stuck in flat ones. Give some evidence that small batch followed by large batch may improve the results.}, timestamp = {2020.04.08}, } @Book{goodfellow+2016DL, title = {Deep Learning}, publisher = {MIT Press}, year = {2016}, author = {Ian J. Goodfellow and Yoshua Bengio and Aaron Courville}, editor = {Thomas Dietterich}, isbn = {9780262035613}, owner = {siim}, timestamp = {2020.06.02}, } @Article{henningsen+toomet2011, author = {Henningsen, Arne and Toomet, Ott}, title = {maxLik: A package for maximum likelihood estimation in R}, journal = {Computational Statistics}, year = {2011}, volume = {26}, pages = {443-458}, issn = {0943-4062}, note = {10.1007/s00180-010-0217-1}, affiliation = {Institute of Food and Resource Economics, University of Copenhagen, Rolighedsvej 25, 1958 Frederiksberg C, Denmark}, issue = {3}, keyword = {Computer Science}, owner = {siim}, publisher = {Physica Verlag, An Imprint of Springer-Verlag GmbH}, timestamp = {2020.06.02}, url = {http://dx.doi.org/10.1007/s00180-010-0217-1}, } @Article{smith+2018arXiv, author = {Samuel L. Smith and Pieter-Jan Kindermans and Quoc V. Le}, title = {Don't Decay the Learning Rate, Increase the Batch Size}, journal = {ArXiv}, year = {2018}, volume = {abs/1711.00489}, abstract = {It is common practice to decay the learning rate. Here we show one can usually obtain the same learning curve on both training and test sets by instead increasing the batch size during training. This procedure is successful for stochastic gradient descent (SGD), SGD with momentum, Nesterov momentum, and Adam. It reaches equivalent test accuracies after the same number of training epochs, but with fewer parameter updates, leading to greater parallelism and shorter training times. We can further reduce the number of parameter updates by increasing the learning rate ϵ and scaling the batch size BâˆÏµ. Finally, one can increase the momentum coefficient m and scale Bâˆ1/(1−m), although this tends to slightly reduce the test accuracy. Crucially, our techniques allow us to repurpose existing training schedules for large batch training with no hyper-parameter tuning. We train ResNet-50 on ImageNet to 76.1% validation accuracy in under 30 minutes.}, owner = {siim}, review = {Employ Smith and Le (2017) scaling result that noise scale ~ B/n(1-m). Instead of decreasing the learning rate, they propose to increase the batch size. Show that this works well, mostly in SGD with momentum framework.}, timestamp = {2020.05.04}, } @Comment{jabref-meta: databaseType:bibtex;} maxLik/vignettes/using-maxlik.Rnw0000644000176200001440000010626014077525067016631 0ustar liggesusers\documentclass[a4paper]{article} \usepackage{amsmath} \usepackage{bbm} \usepackage[inline]{enumitem} \usepackage[T1]{fontenc} \usepackage[bookmarks=TRUE, colorlinks, pdfpagemode=none, pdfstartview=FitH, citecolor=black, filecolor=black, linkcolor=blue, urlcolor=black, ]{hyperref} \usepackage{graphicx} \usepackage{icomma} \usepackage[utf8]{inputenc} \usepackage{mathtools} % for extended pderiv arguments \usepackage{natbib} \usepackage{xargs} % for extended pderiv arguments \usepackage{xspace} % \SweaveUTF8 \newcommand{\COii}{\ensuremath{\mathit{CO}_{2}}\xspace} \DeclareMathOperator*{\E}{\mathbbm{E}}% expectation \newcommand*{\mat}[1]{\mathsf{#1}} \newcommand{\likelihood}{\mathcal{L}}% likelihood \newcommand{\loglik}{\ell}% log likelihood \newcommand{\maxlik}{\texttt{maxLik}\xspace} \newcommand{\me}{\mathrm{e}} % Konstant e=2,71828 \newcommandx{\pderiv}[3][1={}, 2={}]{\frac{\partial^{#2}{#1}}{\mathmbox{\partial{#3}}^{#2}}} % #1: function to differentiate (optional, empty = write after the formula) % #2: the order of differentiation (optional, empty=1) % #3: the variable to differentiate wrt (mandatory) \newcommand{\R}{\texttt{R}\xspace} \newcommand*{\transpose}{^{\mkern-1.5mu\mathsf{T}}} \renewcommand*{\vec}[1]{\boldsymbol{#1}} % \VignetteIndexEntry{Maximum likelihood estimation with maxLik} \title{Maximum Likelihood Estimation with \emph{maxLik}} \author{Ott Toomet} \begin{document} \maketitle <>= library(maxLik) set.seed(6) @ \section{Introduction} \label{sec:introduction} This vignette is intended for users who are familiar with concepts of likelihood and with the related methods, such as information equality and BHHH approximation, and with \R language. The vignette focuses on \maxlik usage and does not explain the underlying mathematical concepts. Potential target group includes researchers, graduate students, and industry practitioners who want to apply their own custom maximum likelihood estimators. If you need a refresher, consult the accompanied vignette ``Getting started with maximum likelihood and \maxlik''. The next section introduces the basic usage, including the \maxlik function, the main entry point for the package; gradients; different optimizers; and how to control the optimization behavior. These are topics that are hard to avoid when working with applied ML estimation. Section~\ref{sec:advanced-usage} contains a selection of more niche topics, including arguments to the log-likelihood function, other types of optimization, testing condition numbers, and constrained optimization. \section{Basic usage} \label{sec:basic-usage} \subsection{The maxLik function} \label{sec:maxlik-function} The main entry point to \maxlik functionality is the function of the same name, \verb|maxLik|. It is a wrapper around the underlying optimization algorithms that ensures that the returned object is of the right class so one can use the convenience methods, such as \verb|summary| or \verb|logLik|. It is important to keep in mind that \maxlik \emph{maximizes}, not minimizes functions. The basic usage of the function is very simple: just pass the log-likelihood function (argument \verb|logLik|) and the start value (argument \verb|start|). Let us demonstrate the basic usage by estimating the normal distribution parameters. We create 100 standard normals, and estimate the best fit mean and standard deviation. Instead of explicitly coding the formula for log-likelihood, we rely on the \R function \verb|dnorm| instead (see Section~\ref{sec:different-optimizers} for a version that does not use \verb|dnorm|): <<>>= x <- rnorm(100) # data. true mu = 0, sigma = 1 loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=2)) # give start value somewhat off summary(m) @ The algorithm converged in 7 iterations and one can check that the results are equal to the sample mean and variance.\footnote{Note that \R function \texttt{var} returns the unbiased estimator by using denominator $n-1$, the ML estimator is biased with denominator $n$. } This example demonstrates a number of key features of \verb|maxLik|: \begin{itemize} \item The first argument of the likelihood must be the parameter vector. In this example we define it as $\vec{\theta} = (\mu, \sigma)$, and the first lines of \verb|loglik| are used to extract these values from the vector. \item The \verb|loglik| function returns a single number, sum of individual log-likelihood contributions of individual $x$ components. (It may also return the components individually, see BHHH method in Section~\ref{sec:different-optimizers} below.) \item Vector of start values must be of correct length. If its components are named, those names are also displayed in \verb|summary| (and for \verb|coef| and \verb|stdEr|, see below). \item \verb|summary| method displays a handy summary of the results, including the convergence message, the estimated values, and statistical significance. \item \verb|maxLik| (and other auxiliary optimizers in the package) is a \emph{maximizer}, not minimizer. \end{itemize} As we did not specify the optimizer, \verb|maxLik| picked Newton-Raphson by default, and computed the necessary gradient and Hessian matrix numerically. \bigskip Besides summary, \verb|maxLik| also contains a number of utility functions to simplify handling of estimated models: \begin{itemize} \item \verb|coef| extracts the model coefficients: <<>>= coef(m) @ \item \verb|stdEr| returns the standard errors (by inverting Hessian): <<>>= stdEr(m) @ \item Other functions include \verb|logLik| to return the log-likelihood value, \verb|returnCode| and \verb|returnMessage| to return the convergence code and message respectively, and \verb|AIC| to return Akaike's information criterion. See the respective documentation for more information. \item One can also query the number of observations with \verb|nObs|, but this requires likelihood values to be supplied by observation (see the BHHH method in Section~\ref{sec:different-optimizers} below). \end{itemize} \subsection{Supplying analytic gradient} \label{sec:supplying-gradients} The simple example above worked fast and well. In particular, the numeric gradient \verb|maxLik| computed internally did not pose any problems. But users are strongly advised to supply analytic gradient, or even better, both the gradient and the Hessian matrix. More complex problems may be intractably slow, converge to a sub-optimal solution, or not converge at all if numeric gradients are noisy. Needless to say, unreliable Hessian also leads to unreliable inference. Here we show how to supply gradient to the \verb|maxLik| function. We demonstrate this with a linear regression example. Non-linear optimizers perform best in regions where level sets (contours) are roughly circular. In the following example we use data in a very different scale and create the log-likelihood function with extremely elongated elliptical contours. Now Newton-Raphson algorithm fails to converge when relying on numeric derivatives, but works well with analytic gradient. % using matrix notation We combine three vectors, $\vec{x}_{1}$, $\vec{x}_{2}$ and $\vec{x}_{3}$, created at a very different scale, into the design matrix $\mat{X} = \begin{pmatrix} \vec{x}_{1} & \vec{x}_{2} & \vec{x}_{3} \end{pmatrix}$ and compute $\vec{y}$ as \begin{equation} \label{eq:linear-regression-matrix} \vec{y} = \mat{X} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + \vec{\epsilon}. \end{equation} We create $\vec{x}_{1}$, $\vec{x}_{2}$ and $\vec{x}_{3}$ as random normals with standard deviation of 1, 1000 and $10^{7}$ respectively, and let $\vec{\epsilon}$ be standard normal disturbance term: <<>>= ## create 3 variables with very different scale X <- cbind(rnorm(100), rnorm(100, sd=1e3), rnorm(100, sd=1e7)) ## note: correct coefficients are 1, 1, 1 y <- X %*% c(1,1,1) + rnorm(100) @ Next, we maximize negative of sum of squared errors \emph{SSE} (remember, \verb|maxLik| is a maximizer not minimizer) \begin{equation} \label{eq:ols-sse-matrix} \mathit{SSE}(\vec{\beta}) = (\vec{y} - \mat{X} \cdot \vec{\beta})^{\transpose} (\vec{y} - \mat{X} \cdot \vec{\beta}) \end{equation} as this is equivalent to likelihood maximization: <<>>= negSSE <- function(beta) { e <- y - X %*% beta -crossprod(e) # note '-': we are maximizing } m <- maxLik(negSSE, start=c(0,0,0)) # give start values a bit off summary(m, eigentol=1e-15) @ As one can see, the algorithm gets stuck and fails to converge, the last parameter value is also way off from the correct value $(1, 1, 1)$. We have amended summary with an extra argument, \verb|eigentol=1e-15|. Otherwise \maxlik refuses to compute standard errors for near-singular Hessian, see the documentation of \verb|summary.maxLik|. It makes no difference right here but we want to keep it consistent with the two following examples. Now let's improve the model performance with analytic gradient. The gradient of \emph{SSE} can be written as \begin{equation} \label{eq:ols-sse-gradient-matrix} \pderiv{\vec{\beta}}\mathit{SSE}(\vec{\beta}) = -2(\vec{y} - \mat{X}\vec{\beta})^{\transpose} \mat{X}. \end{equation} \maxlik uses numerator layout, i.e. the derivative of the scalar log-likelihood with respect to the column vector of parameters is a row vector. We can code the negative of it as <<>>= grad <- function(beta) { 2*t(y - X %*% beta) %*% X } @ We can add gradient to \verb|maxLik| as an additional argument \verb|grad|: <<>>= m <- maxLik(negSSE, grad=grad, start=c(0,0,0)) summary(m, eigentol=1e-15) @ Now the algorithm converges rapidly, and the estimate is close to the true value. Let us also add analytic Hessian, in this case it is \begin{equation} \label{eq:ols-sse-hessian-matrix} \frac{\partial^{2}}{\partial\vec{\beta}\,\partial\vec{\beta}^{\transpose}} \mathit{SSE}(\vec{\beta}) = 2\mat{X}^{\transpose}\mat{X} \end{equation} and we implement the negative of it as <<>>= hess <- function(beta) { -2*crossprod(X) } @ Analytic Hessian matrix can be included with the argument \verb|hess|, and now the results are <>= m <- maxLik(negSSE, grad=grad, hess=hess, start=c(0,0,0)) summary(m, eigentol=1e-15) @ Analytic Hessian did not change the convergence behavior here. Note that as the loss function is quadratic, Newton-Raphson should provide the correct solution in a single iteration only. However, this example has numerical issues when inverting near-singular Hessian. One can easily check that when creating covariates in a less extreme scale, then the convergence is indeed immediate. While using separate arguments \texttt{grad} and \texttt{hess} is perhaps the most straightforward way to supply gradients, \maxlik also supports gradient and Hessian supplied as log-likelihood attributes. This is motivated by the fact that computing gradient often involves a number of similar computations as computing log-likelihood, and one may want to re-use some of the results. We demonstrate this on the same example, by writing a version of log-likelihood function that also computes the gradient and Hessian: <>= negSSEA <- function(beta) { ## negative SSE with attributes e <- y - X %*% beta # we will re-use 'e' sse <- -crossprod(e) # note '-': we are maximizing attr(sse, "gradient") <- 2*t(e) %*% X attr(sse, "Hessian") <- -2*crossprod(X) sse } m <- maxLik(negSSEA, start=c(0,0,0)) summary(m, eigentol=1e-15) @ The log-likelihood with ``gradient'' and ``Hessian'' attributes, \verb|negSSEA|, computes log-likelihood as above, but also computes its gradient, and adds it as attribute ``gradient'' to the log-likelihood. This gives a potential efficiency gain as the residuals $\vec{e}$ are re-used. \maxlik checks the presence of the attribute, and if it is there, it uses the provided gradient. In real applications the efficiency gain will depend on the amount of computations re-used, and the number of likelihood calls versus gradient calls. While analytic gradients are always helpful and often necessary, they may be hard to derive and code. In order to help to derive and debug the analytic gradient, another provided function, \verb|compareDerivatives|, takes the log-likelihood function, analytic gradent, and compares the numeric and analytic gradient. As an example, we compare the log-likelihood and gradient functions we just coded: <<>>= compareDerivatives(negSSE, grad, t0=c(0,0,0)) # 't0' is the parameter value @ The function prints the analytic gradient, numeric gradient, their relative difference, and the largest relative difference value (in absolute value). The latter is handy in case of large gradient vectors where it may be hard to spot a lonely component that is off. In case of reasonably smooth functions, expect the relative difference to be smaller than $10^{-7}$. But in this example the numerical gradients are clearly problematic. \verb|compareDerivatives| supports vector functions, so one can test analytic Hessian in the same way by calling \verb|compareDerivatives| with \verb|gradlik| as the first argument and the analytic hessian as the second argument. \subsection{Different optimizers} \label{sec:different-optimizers} By default, \maxlik uses Newton-Raphson optimizer but one can easily swap the optimizer by \verb|method| argument. The supported optimizers include ``NR'' for the default Newton-Raphson, ``BFGS'' for gradient-only Broyden-Fletcher-Goldfarb-Shannon, ``BHHH'' for the information-equality based Berndt-Hall-Hall-Hausman, and ``NM'' for gradient-less Nelder-Mead. Different optimizers may be based on a very different approach, and certain concepts, such as \emph{iteration}, may mean quite different things. For instance, although Newton-Raphson is a simple, fast and intuitive method that approximates the function with a parabola, it needs to know the Hessian matrix (the second derivatives). This is usually even harder to program than gradient, and even slower and more error-prone when computed numerically. Let us replace NR with gradient-only BFGS method. It is a quasi-Newton method that computes its own internal approximation of the Hessian while relying only on gradients. We re-use the data and log-likelihood function from the first example where we estimated normal distribution parameters: <>= m <- maxLik(loglik, start=c(mu=1, sigma=2), method="BFGS") summary(m) @ One can see that the results were identical, but while NR converged in 7 iterations, it took 20 iterations for BFGS. In this example the BFGS approximation errors were larger than numeric errors when computing Hessian, but this may not be true for more complex objective functions. In a similar fashion, one can simply drop in most other provided optimizers. One method that is very popular for ML estimation is BHHH. We discuss it here at length because that method requires both log-likelihood and gradient function to return a somewhat different value. The essence of BHHH is information equality, the fact that in case of log-likelihood function $\loglik(\theta)$, the expected value of Hessian at the true parameter value $\vec{\theta}_{0}$ can be expressed through the expected value of the outer product of the gradient: \begin{equation} \label{eq:information-equality} \E \left[ \frac{\partial^2 l(\vec{\theta})} {\partial\vec{\theta}\, \partial\vec{\theta}^{\transpose}} \right]_{\vec{\theta} = \vec{\theta}_0} = - \E \left[ \left. \frac{\partial l(\vec{\theta})} {\partial\vec{\theta}^{\transpose}} \right|_{\vec{\theta} = \vec{\theta}_0} \cdot \left. \frac{\partial l(\vec{\theta})} {\partial\vec{\theta}} \right|_{\vec{\theta} = \vec{\theta}_0} \right]. \end{equation} Hence we can approximate Hessian by the average outer product of the gradient. Obviously, this is only an approximation, and it is less correct when we are far from the true value $\vec{\theta}_{0}$. Note also that when approximating expected value with average we rely on the assumption that the observations are independent. This may not be true for certain type of data, such as time series. However, in order to compute the average outer product, we need to compute gradient \emph{by observation}. Hence it is not enough to just return a single gradient vector, we have to compute a matrix where rows correspond to individual data points and columns to the gradient components. We demonstrate BHHH method by replicating the normal distribution example from above. Remember, the normal probability density is \begin{equation} \label{eq:normal-pdf} f(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \, \me^{ -\displaystyle\frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} }. \end{equation} and hence the log-likelihood contribution of $x$ is \begin{equation} \label{eq:normal-loglik} \loglik(\mu, \sigma; x) = - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} \end{equation} and its gradient \begin{equation} \label{eq:normal-loglik-gradient} \begin{split} \pderiv{\mu} \loglik(\mu, \sigma; x) &= \frac{1}{\sigma^{2}}(x - \mu) \\ \pderiv{\sigma} \loglik(\mu, \sigma; x) &= -\frac{1}{\sigma} + \frac{1}{\sigma^{2}}(x - \mu)^{2}. \end{split} \end{equation} We can code these two functions as <<>>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- length(x) -N*log(sqrt(2*pi)) - N*log(sigma) - sum(0.5*(x - mu)^2/sigma^2) # sum over observations } gradlikB <- function(theta) { ## BHHH-compatible gradient mu <- theta[1] sigma <- theta[2] N <- length(x) # number of observations gradient <- matrix(0, N, 2) # gradient is matrix: # N datapoints (rows), 2 components gradient[, 1] <- (x - mu)/sigma^2 # first column: derivative wrt mu gradient[, 2] <- -1/sigma + (x - mu)^2/sigma^3 # second column: derivative wrt sigma gradient } @ Note that in this case we do not sum over the individual values in the gradient function (but we still do in log-likelihood). Instead, we fill the rows of the $N\times2$ gradient matrix with the values observation-wise. The results are similar to what we got above and the convergence speed is in-between that of Newton-Raphson and BFGS: \label{code:bhhh-example} <<>>= m <- maxLik(loglik, gradlikB, start=c(mu=1, sigma=2), method="BHHH") summary(m) @ In case we do not have time and energy to code the analytic gradient, we can let \maxlik compute the numeric one for BHHH too. In this case we have to supply the log-likelihood by observation. This essentially means we remove summing from the original likelihood function: <<>>= loglikB <- function(theta) { mu <- theta[1] sigma <- theta[2] -log(sqrt(2*pi)) - log(sigma) - 0.5*(x - mu)^2/sigma^2 # no summing here # also no 'N*' terms as we work by # individual observations } m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH") summary(m) @ Besides of relying on information equality, BHHH is essentially the same algorithm as NR. As the Hessian is just approximated, its is converging at a slower pace than NR with analytic Hessian. But when relying on numeric derivatives only, BHHH may be more reliable. For convenience, the other methods also support observation-wise gradients and log-likelihood values, those numbers are just summed internally. So one can just code the problem in an BHHH-compatible manner and use it for all supported optimizers. \maxlik package also includes stochastic gradient ascent optimizer. As that method is rarely used for ML estimation, it cannot be supplied through the ``method'' argument. Consult the separate vignette ``Stochastic gradient ascent in \maxlik''. \subsection{Control options} \label{sec:control-options} \maxlik supports a number of control options, most of which can be supplied through \verb|control=list(...)| method. Some of the most important options include \verb|printLevel| to control debugging information, \verb|iterLim| to control the maximum number of iterations, and various \verb|tol|-parameters to control the convergence tolerances. For instance, we can limit the iterations to two, while also printing out the parameter estimates at each step. We use the previous example with BHHH optimizer: <<>>= m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH", control=list(printLevel=3, iterlim=2)) summary(m) @ The first option, \verb|printLevel=3|, make \verb|maxLik| to print out parameters, gradient a few other bits of information at every step. Larger levels output more information, printlevel 1 only prints the first and last parameter values. The output from \maxlik-implemented optimizers is fairly consistent, but methods that call optimizers in other packages, such as BFGS, may output debugging information in a quite different way. The second option, \verb|iterLim=2| stops the algorithm after two iterations. It returns with code 4: iteration limit exceeded. Other sets of handy options are the convergence tolerances. There are three convergence tolerances: \begin{description} \item[tol] This measures the absolute convergence tolerance. Stop if successive function evaluations differ by less than \emph{tol} (default $10^{-8}$). \item[reltol] This is somewhat similar to \emph{tol}, but relative to the function value. Stop if successive function evaluations differ by less than $\mathit{reltol}\cdot (\loglik(\vec{\theta}) + \mathit{reltol})$ (default \verb|sqrt(.Machine[["double.eps"]])|, may be approximately \Sexpr{formatC(sqrt(.Machine[["double.eps"]]), digits=1)} on a modern computer). \item[gradtol] stop if the (Euclidean) norm of the gradient is smaller than this value (default $10^{-6}$). \end{description} Default tolerance values are typically good enough, but in certain cases one may want to adjust these. For instance, in case of function values are very large, one may rely only on tolerance, and ignore relative tolerance and gradient tolerance criteria. A simple way to achieve this is to set both \emph{reltol} and \emph{gradtol} to zero. In that case these two conditions are never satisfied and the algorithm stops only when the absolute convergence criterion is fulfilled. For instance, in the previous case we get: <<>>= m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH", control=list(reltol=0, gradtol=0)) summary(m) @ When comparing the result with that on Page~\pageref{code:bhhh-example} we can see that the optimizer now needs more iterations and it stops with a return code that is related to tolerance, not relative tolerance. Note that BFGS and other optimizers that are based on the \verb|stats::optim| does not report the convergence results in a similar way as BHHH and NR, the algorithms provided by the \maxlik package. Instead of tolerance limits or gradient close to zero message, we hear about ``successful convergence''. Stochastic gradient ascent relies on completely different convergence criteria. See the dedicated vignette ``Stochastic Gradient Ascent in \maxlik''. \section{Advanced usage} \label{sec:advanced-usage} This section describes more advanced and less frequently used aspects of \maxlik. \subsection{Additional arguments to the log-likelihood function} \label{sec:additional-arguments-loglik} \maxlik expects the first argument of log-likelihood function to be the parameter vector. But the function may have more arguments. Those can be passed as additional named arguments to \verb|maxLik| function. For instance, let's change the log-likelihood function in a way that it expects data $\vec{x}$ to be passed as an argument \verb|x|. Now we have to call \maxlik with an additional argument \verb|x=...|: <<>>= loglik <- function(theta, x) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=2), x=x) # named argument 'x' will be passed # to loglik summary(m) @ This approach only works if the argument names do not overlap with \verb|maxLik|'s arguments' names. If that happens, it prints an informative error message. \subsection{Maximizing other functions} \label{sec:maximizing-other-functions} \verb|maxLik| function is basically a wrapper around a number of maximization algorithms, and a set of likelihood-related methods, such as standard errors. However, from time-to-time we need to optimize other functions where inverting the Hessian to compute standard errors is not applicable. In such cases one can call the included optimizers directly, using the form \verb|maxXXX| where \verb|XXX| stands for the name of the method, e.g. \verb|maxNR| for Newton-Rapshon (\verb|method="NR"|) and \verb|maxBFGS| for BFGS. There is also \verb|maxBHHH| although the information equality--based BHHH is not correct if we do not work with log-likelihood functions. The arguments for \verb|maxXXX|-functions are largely similar to those for \maxlik, the first argument is the function, and one also has to supply start values. Let us demonstrate this functionality by optimizing 2-dimensional bell curve, \begin{equation} \label{eq:2d-bell-curve} f(x, y) = \me^{-x^{2} - y^{2}}. \end{equation} We code this function and just call \verb|maxBFGS| on it: <<>>= f <- function(theta) { x <- theta[1] y <- theta[2] exp(-x^2 - y^2) # optimum at (0, 0) } m <- maxBFGS(f, start=c(1,1)) # give start value a bit off summary(m) @ Note that the summary output is slightly different: it reports the parameter and gradient value, appropriate for a task that is not likelihood optimization. Behind the scenes, this is because the \verb|maxXXX|-functions return an object of \emph{maxim}-class, not \emph{maxLik}-class. \subsection{Testing condition numbers} \label{sec:testing-condition-numbers} Analytic gradient we demonstrated in Section~\ref{sec:supplying-gradients} helps to avoid numerical problems. But not all problems can or should be solved by analytic gradients. For instance, multicollinearity should be addressed on data or model level. \maxlik provides a helper function, \verb|condiNumbers|, to detect such problems. We demonstrate this by creating a highly multicollinear dataset and estimating a linear regression model. We re-use the regression code from Section~\ref{sec:supplying-gradients} but this time we create multicollinear data in similar scale. <<>>= ## create 3 variables, two independent, third collinear x1 <- rnorm(100) x2 <- rnorm(100) x3 <- x1 + x2 + rnorm(100, sd=1e-6) # highly correlated w/x1, x2 X <- cbind(x1, x2, x3) y <- X %*% c(1, 1, 1) + rnorm(100) m <- maxLik(negSSEA, start=c(x1=0, x2=0, x3=0)) # negSSEA: negative sum of squared errors # with gradient, hessian attribute summary(m) @ As one can see, the model converges but the standard errors are missing (because Hessian is not negative definite). In such case we may learn more about the problem by testing the condition numbers $\kappa$ of either the design matrix $\mat{X}$ or of the Hessian matrix. It is instructive to test not just the whole matrix, but to do it column-by-column, and see where the number suddenly jumps. This hints which variable does not play nicely with the rest of data. \verb|condiNumber| provides such functionality. First, we test the condition number of the design matrix: <<>>= condiNumber(X) @ We can see that when only including $\vec{x}_{1}$ and $\vec{x}_{2}$ into the design, the condition number is 1.35, far from any singularity-related problems. However, adding $\vec{x}_{3}$ to the matrix causes $\kappa$ to jump to over 5 millions. This suggests that $\vec{x}_{3}$ is highly collinear with $\vec{x}_{1}$ and $\vec{x}_{2}$. In this example the problem is obvious as this is how we created $\vec{x}_{3}$, in real applications one often needs further analysis. For instance, the problem may be in categorical values that contain too few observations or complex fixed effects that turn out to be perfectly multicollinear. A good suggestion is to estimate a linear regression model where one explains the offending variable using all the previous variables. In this example we might estimate \verb|lm(x3 ~ x1 + x2)| and see which variables help to explain $\vec{x}_{3}$ perfectly. Sometimes the design matrix is fine but the problem arises because data and model do not match. In that case it may be more informative to test condition number of Hessian matrix instead. The example below creates a linearly separated set of observations and estimates this with logistic regression. As a refresher, the log-likelihood of logistic regression is \begin{equation} \label{eq:logistic-loglik} \loglik(\beta) = \sum_{i: y_{i} = 1} \log\Lambda(\vec{x}_{i}^{\transpose} \vec{\beta}) + \sum_{i: y_{i} = 0} \log\Lambda(-\vec{x}_{i}^{\transpose} \vec{\beta}) \end{equation} where $\Lambda(x) = 1/(1 + \exp(-x))$ is the logistic cumulative distribution function. We implement it using \R function \verb|plogis| <<>>= x1 <- rnorm(100) x2 <- rnorm(100) x3 <- rnorm(100) X <- cbind(x1, x2, x3) y <- X %*% c(1, 1, 1) > 0 # y values 1/0 linearly separated loglik <- function(beta) { link <- X %*% beta sum(ifelse(y > 0, plogis(link, log=TRUE), plogis(-link, log=TRUE))) } m <- maxLik(loglik, start=c(x1=0, x2=0, x3=0)) summary(m) @ Not surprisingly, all coefficients tend to infinity and inference is problematic. In this case the design matrix does not show any issues: <<>>= condiNumber(X) @ But the Hessian reveals that including $\vec{x}_{3}$ in the model is still problematic: <<>>= condiNumber(hessian(m)) @ Now the problem is not multicollinearity but the fact that $\vec{x}_{3}$ makes the data linearly separable. In such cases we may want to adjust our model or estimation strategy. \subsection{Fixed parameters and constrained optimization} \label{sec:fixed-parameters} \maxlik supports three types of constrains. The simplest case just keeps certain parameters' values fixed. The other two, general linear equality and inequality constraints are somewhat more complex. Occasionally we want to treat one of the model parameters as constant. This can be achieved in a very simple manner, just through the argument \verb|fixed|. It must be an index vector, either numeric, such as \verb|c(2,4)|, logical as \verb|c(FALSE, TRUE, FALSE, TRUE)|, or character as \verb|c("beta2", "beta4")| given \verb|start| is a named vector. We revisit the first example of this vignette and estimate the normal distribution parameters again. However, this time we fix $\sigma = 1$: <<>>= x <- rnorm(100) loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=1), fixed="sigma") # fix the component named 'sigma' summary(m) @ The result has $\sigma$ exactly equal to $1$, it's standard error $0$, and $t$ value undefined. The fixed components are ignored when computing gradients and Hessian in the optimizer, essentially reducing the problem from 2-dimensional to 1-dimensional. Hence the inference for $\mu$ is still correct. Next, we demonstrate equality constraints. We take the two-dimensional function we used in Section~\ref{sec:maximizing-other-functions} and add constraints $x + y = 1$. The constraint must be described in matrix form $\mat{A}\,\vec{\theta} + \vec{B} = 0$ where $\vec{\theta}$ is the parameter vector and matrix $\mat{A}$ and vector $\vec{B}$ describe the constraints. In this case we can write \begin{equation} \label{eq:equality-constraints} \begin{pmatrix} 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \end{pmatrix} = 0, \end{equation} i.e. $\mat{A} = (1 \; 1)$ and $\vec{B} = -1$. These values must be supplied to the optimizer argument \verb|constraints|. This is a list with components names \verb|eqA| and \verb|eqB| for $\mat{A}$ and $\vec{B}$ accordingly. We do not demonstrate this with a likelihood example as no corrections to the Hessian matrix is done and hence the standard errors are incorrect. But if you are not interested in likelihood-based inference, it works well: <<>>= f <- function(theta) { x <- theta[1] y <- theta[2] exp(-x^2 - y^2) # optimum at (0, 0) } A <- matrix(c(1, 1), ncol=2) B <- -1 m <- maxNR(f, start=c(1,1), constraints=list(eqA=A, eqB=B)) summary(m) @ The problem is solved using sequential unconstrained maximization technique (SUMT). The idea is to add a small penalty for the constraint violation, and to slowly increase the penalty until violations are prohibitively expensive. As the example indicates, the solution is extremely close to the constraint line. The usage of inequality constraints is fairly similar. We have to code the inequalities as $\mat{A}\,\vec{\theta} + \vec{B} > 0$ where the matrices $\mat{A}$ and $\vec{B}$ are defined as above. Let us optimize the function over the region $x + y > 1$. In matrix form this will be \begin{equation} \label{eq:inequality-constraints-1} \begin{pmatrix} 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \end{pmatrix} > 0. \end{equation} Supplying the constraints is otherwise similar to the equality constraints, just the constraints-list components must be called \verb|ineqA| and \verb|ineqB|. As \verb|maxNR| does not support inequality constraints, we use \verb|maxBFGS| instead. The corresponding code is <<>>= A <- matrix(c(1, 1), ncol=2) B <- -1 m <- maxBFGS(f, start=c(1,1), constraints=list(ineqA=A, ineqB=B)) summary(m) @ Not surprisingly, the result is exactly the same as in case of equality constraints, in this case the optimum is found at the boundary line, the same line what we specified when demonstrating the equality constraints. One can supply more than one set of constraints, in that case these all must be satisfied at the same time. For instance, let's add another condition, $x - y > 1$. This should be coded as another line of $\mat{A}$ and another component of $\vec{B}$, in matrix form the constraint is now \begin{equation} \label{eq:inequality-constraints-2} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \\ -1 \end{pmatrix} > \begin{pmatrix} 0 \\ 0 \end{pmatrix} \end{equation} where ``>'' must be understood as element-wise operation. We also have to ensure the initial value satisfies the constraint, so we choose $\vec{\theta}_{0} = (2, 0)$. The code will be accordingly: <<>>= A <- matrix(c(1, 1, 1, -1), ncol=2) B <- c(-1, -1) m <- maxBFGS(f, start=c(2, 0), constraints=list(ineqA=A, ineqB=B)) summary(m) @ The solution is $(1, 0)$ the closest point to the origin where both constraints are satisfied. \bigskip This example concludes the \maxlik usage introduction. For more information, consult the fairly extensive documentation, and the other vignettes. % \bibliographystyle{apecon} % \bibliography{maxlik} \end{document} maxLik/vignettes/intro-to-maximum-likelihood.Rnw0000644000176200001440000010672614077525067021577 0ustar liggesusers\documentclass[a4paper]{article} \usepackage{graphics} \usepackage{amsmath} \usepackage{amssymb} \usepackage[font={small,sl}]{caption} \usepackage[inline]{enumitem} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage{natbib} \usepackage{siunitx} \usepackage{xspace} % \SweaveUTF8 \newcommand{\COii}{\ensuremath{\mathit{CO}_{2}}\xspace} \newcommand*{\mat}[1]{\mathsf{#1}} \newcommand{\likelihood}{\mathcal{L}}% likelihood \newcommand{\loglik}{\ell}% log likelihood \newcommand{\maxlik}{\texttt{maxLik}\xspace} \newcommand{\me}{\mathrm{e}} % Konstant e=2,71828 \newcommand{\R}{\texttt{R}\xspace} \newcommand*{\transpose}{^{\mkern-1.5mu\mathsf{T}}} \renewcommand*{\vec}[1]{\boldsymbol{#1}} % \VignetteIndexEntry{Introduction: what is maximum likelihood} \begin{document} <>= options(keep.source = TRUE, width = 60, try.outFile=stdout() # make try to produce error messages ) set.seed(34) @ \title{Getting started with maximum likelihood and \texttt{maxLik}} \author{Ott Toomet} \maketitle \section{Introduction} This vignette is intended for readers who are unfamiliar with the concept of likelihood, and for those who want a quick intuitive brush-up. The potential target group includes advanced undergraduate students in technical fields, such as statistics or economics, graduate students in social sciences and engineering who are devising their own estimators, and researchers and practitioners who have little previous experience with ML. However, one should have basic knowledge of \R language. If you are familiar enough with the concept of likelihood and maximum likelihood, consult instead the other vignette ``Maximum Likelihood Estimation with \maxlik''. Maximum Likelihood (ML) in its core is maximizing the \emph{likelihood} over the parameters of interest. We start with an example of a random experiment that produces discrete values to explain what is likelihood and how it is related to probability. The following sections cover continuous values, multiple parameters in vector form, and we conclude with a linear regression example. The final section discusses the basics of non-linear optimization. The examples are supplemented with very simple code and assume little background besides basic statistics and basic \R knowledge. \section{Discrete Random Values} \label{sec:discrete-random-variables} We start with a discrete case. ``Discrete'' refers to random experiments or phenomena with only limited number of possible outcomes, and hence we can compute and tabulate every single outcome separately. Imagine you are flipping a fair coin. What are the possible outcomes and what are the related probabilities? Obviously, in case of a coin there are only two outcomes, heads $H$ and tails $T$. If the coin is fair, both of these will have probability of exactly 0.5. Such random experiment is called \emph{Bernoulli process}. More specifically, this is \emph{Bernoulli(0.5)} process as for the fair coin the probability of ``success'' is 0.5 (below we consider success to be heads, but you can choose tails as well). If the coin is not fair, we denote the corresponding process Bernoulli($p$), where $p$ is the probability of heads. Now let us toss the coin two times. What is the probability that we end up with one heads and one tails? As the coin flips are independent,\footnote{Events are independent when outcome of one event does not carry information about the outcome of the other event. Here the result of the second toss is not related to the outcome of the first toss.} we can just multiply the probabilities: $0.5$ for a single heads and $0.5$ for a single tails equals $0.25$ when multiplied. However, this is not the whole story--there are two ways to get one heads and one tails, either $H$ first and $T$ thereafter or $T$ first and $H$ thereafter. Both of these events are equally likely, so the final answer will be 0.5. But now imagine we do not know if the coin is fair. Maybe we are not tossing a coin but an object of a complex shape. We can still label one side as ``heads'' and the other as ``tails''. But how can we tell what is the probability of heads? Let's start by denoting this probability with $p$. Hence the probability of tails will be $1-p$, and the probability to receive one heads, one tails when we toss the object two times will be $2 p (1-p)$: $p$ for one heads, $1-p$ for one tails, and ``2'' takes into account the fact that we can get this outcome in two different orders. This probability is essentially likelihood. We denote likelihood with $\likelihood(p)$, stressing that it depends on the unknown probability $p$. So in this example we have \begin{equation} \label{eq:2-coin-likelihood} \likelihood(p) = 2 \, p \, (1-p). \end{equation} $p$ is the \emph{model parameter}, the unknown number we want to compute with the help of likelihood. Let's repeat here what did we do above: \begin{enumerate} \item We observe data. In this example data contains the counts: one heads, one tails. \item We model the coin toss experiment, the data generating process, as Bernoulli($p$) random variable. $p$, the probability of heads, is the model parameter we want to calculate. Bernoulli process has only a single parameter, but more complex processes may contain many more. \item Thereafter we compute the probability to observe the data based on the model. Here it is equation~\eqref{eq:2-coin-likelihood}. This is why we need a probability model. As the model contains unknown parameters, the probability will also contain parameters. \item And finally we just call this probability \emph{likelihood} $\likelihood(p)$. We write it as a function of the parameter to stress that the parameter is what we are interested in. Likelihood also depends on data (the probability will look different for e.g. two heads instead of a head and a tail) but we typically do not reflect this in notation. \end{enumerate} The next task is to use this likelihood function to \emph{estimate} the parameter, to use data to find the best possible parameter value. \emph{Maximum likelihood} (ML) method finds such parameter value that maximizes the likelihood function. It can be shown that such parameter value has a number of desirable properties, in particular it will become increasingly similar to the ``true value'' on an increasingly large dataset (given that our probability model is correct).\footnote{This property is formally referred to as \emph{consistency}. ML is a consistent estimator.} These desirable properties, and relative simplicity of the method, have made ML one of the most widely used statistical estimators. Let us generalize the example we did above for an arbitrary number of coin flips. Assume the coin is of unknown ``fairness'' where we just denote the probability to receive heads with $p$. Further, assume that out of $N$ trials, $N_{H}$ trials were heads and $N_{T}$ trials were tails. The probability of this occuring is \begin{equation} \label{eq:general-cointoss-probability} \binom{N}{N_{H}} \, p^{N_{H}} \, (1 - p)^{N_{T}} \end{equation} $p^{N_{H}}$ is the probability to get $N_{H}$ heads, $(1 - p)^{N_{T}}$ is the probability to get $N_{T}$ tails, and the binomial coefficient $\displaystyle\binom{N}{N_{H}} = \displaystyle\frac{N!}{N_{H}! (N - N_{H})!}$ takes into account that there are many ways how heads and tail can turn up while still resulting $N_{H}$ heads and $N_{T}$ tails. In the previous example $N=2$, $N_{H} = 1$ and there were just two possible combinations as $\displaystyle\binom{2}{1} = 2$. The probability depends on both the parameter $p$ and data--the corresponding counts $N_{H}$ and $N_{T}$. Equation~\eqref{eq:general-cointoss-probability} is essentially likelihood--probability to observe data. We are interested how does it depend on $p$ and stress this by writing $p$ in the first position followed by semicolon and data as we care less about the dependency on data: \begin{equation} \label{eq:general-cointoss-likelihood} \likelihood(p; N_{H}, N_{T}) = \binom{N}{N_{H}} \, p^{N_{H}} \, (1 - p)^{N_{T}} \end{equation} Technically, it is easier to work with log-likelihood instead of likelihood (as log is monotonic function, maximum of likelihood and maximum of log-likelihood occur at the same parameter value). We denote log-likelihood by $\loglik$ and write \begin{equation} \label{eq:general-cointoss-loglik} \loglik(p; N_{H}, N_{T}) = \log\likelihood(p; N_{H}, N_{T}) = \log \binom{N}{N_{H}} + N_{H} \log p + N_{T} \log (1 - p). \end{equation} ML estimator of $p$ is the value that maximizes this expression. Fortunately, in this case the binomial coefficient $\displaystyle\binom{N}{N_{H}}$ depends only on data but not on the $p$. Intuitively, $p$ determines the probability of various combinations of heads and tails, but \emph{what kind of combinations are possible} does not depend on $p$. Hence we can ignore the first term on the right hand side of~\eqref{eq:general-cointoss-loglik} when maximizing the log-likelihood. Such approach is very common in practice, many terms that are invariant with respect to parameters are often ignored. Hence, with we can re-define the log-likelihood as \begin{equation} \label{eq:general-cointoss-partial-loglik} \loglik(p; N_{H}, N_{T}) = N_{H} \log p + N_{T} \log (1 - p). \end{equation} It is easy to check that the solution, the value of $p$ that maximizes log-likelihood~\eqref{eq:general-cointoss-partial-loglik} is\footnote{Just differentiate $\loglik(p)$ with respect to $p$, set the result to zero, and isolate $p$.} \begin{equation} \label{eq:general-cointoss-solution} p^{*} = \frac{N_{H}}{N_{H} + N_{T}} = \frac{N_{H}}{N}. \end{equation} This should be surprise to no-one: the intuitive ``fairness'' of the coin is just the average percentage of heads we get. Now it is time to try this out on computer with \texttt{maxLik}. Let's assume we toss a coin and receive $H_{H} = 3$ heads and $H_{T} = 7$ tails: <<>>= NH <- 3 NT <- 7 @ Next, we have to define the log-likelihood function. It has to be a function of the parameter, and the parameter must be its first argument. We can access data in different ways, for instance through the \R workspace environment. So we can write the log-likelihood as <<>>= loglik <- function(p) { NH*log(p) + NT*log(1-p) } @ And finally, we can use \texttt{maxLik} function to compute the likelihood. In its simplest form, \texttt{maxLik} requires two arguments: the log-likelihood function, and the start value for the iterative algorithm (see Section~\ref{sec:non-linear-optimization}, and the documentation and vignette \textsl{Maximum Likelihood Estimation with \maxlik} for more detailed explanations). The start value must be a valid parameter value (the loglik function must not give errors when called with the start value). We can choose $p_{0} = 0.5$ as the initial value, and let the algorithm find the best possible $p$ from there: <<>>= library(maxLik) m <- maxLik(loglik, start=0.5) summary(m) @ As expected, the best bet for $p$ is 0.3. Our intuitive approach--the percentage of heads in the experiment--turns also out to be the ML estimate. Next, we look at an example with continuous outcomes. \section{Continuous case: probability density and likelihood} \label{sec:continuous-outcomes} In the example above we looked at a discrete random process, a case where there were only a small number of distinct possibilities (heads and tails). Discrete cases are easy to understand because we can actually compute the respective probabilities, such as the probability to receive one heads and one tails in our experiment. Now we consider continuous random variables where the outcome can be any number in a certain interval. Unfortunately, in continuous case we cannot compute probability of any particular outcome. Or more precisely--we can do it, but the answer is always 0. This may sound a little counter-intuitive but perhaps the following example helps. If you ask the computer to generate a single random number between 0 and 1, you may receive \Sexpr{x <- runif(1); x}. What is the probability to get the same number again? You can try, you will get close but you won't get exactly the same number.\footnote{As computers operate with finite precision, the actual chances to repeat any particular random number are positive, although small. The exact answer depends on the numeric precision and the quality of random number generator. } But despite the probability to receive this number is zero, we somehow still produced it in the first place. Clearly, zero probability does not mean the number was impossible. However, if we want to receive a negative number from the same random number generator, it will be impossible (because we chose a generator that only produces numbers between 0 and 1). So probability 0-events may be possible and they may also be impossible. And to make matter worse, they may also be more likely and less likely. For instance, in case of standard normal random numbers (these numbers are distributed according to ``bell curve'') the values near $0$ are much more likely than values around $-2$, despite of the probability to receive any particular number still being 0 (see Figure~\ref{fig:standard-normal-intervals}). The solution is to look not at the individual numbers but narrow interval near these numbers. Consider the number of interest $x_{1}$, and compute the probability that the random outcome $X$ falls into the narrow interval of width $\delta$, $[x_{1} - \delta/2,\, x_{1} + \delta/2]$, around this number (Figure~\ref{fig:standard-normal-intervals}). Obviously, the smaller the width $\delta$, the less likely it is that $X$ falls into this narrow interval. But it turns out that when we divide the probability by the width, we get a stable value at the limit which we denote by $f(x_{1})$: \begin{equation} \label{eq:probability-density} f(x_{1}) = \lim_{\delta\to0} \frac{\Pr(X \in [x_{1} - \delta/2,\, x_{1} + \delta/2])}{\delta}. \end{equation} In the example on the Figure the values around $x_{1}$ are less likely than around $x_{2}$ and hence $f(x_{1}) < f(x_{2})$. The result, $f(x)$, is called \emph{probability density function}, often abbreviated as \emph{pdf}. In case of continuous random variables, we have to work with pdf-s instead of probabilities. \begin{figure}[ht] \centering \includegraphics{probability-density.pdf} \caption{Standard normal probability density (thick black curve). While $\Pr(X = x_{1}) = 0$, i.e. the probability to receive a random number exactly equal to $x_{1}$ is 0, the probability to receive a random number in the narrow interval of width $\delta$ around $x_{1}$ is positive. In this example, the probability to get a random number in the interval around $x_{2}$ is four times larger than for the interval around $x_{1}$. } \label{fig:standard-normal-intervals} \end{figure} Consider the following somewhat trivial example: we have sampled two independent datapoints $x_{1}$ and $x_{2}$ from normal distribution with variance 1 and mean (expected value) equal to $\mu$. Say, $x_{1} = \Sexpr{x1 <- rnorm(1); round(x1, 3)}$ and $x_{2} = \Sexpr{x1 <- rnorm(1); round(x1, 3)}$. Assume we do not know $\mu$ and use ML to estimate it. We can proceed in a similar steps as what we did for the discrete case: \begin{enumerate*}[label=\roman*)] \item observe data, in this case $x_{1}$ and $x_{2}$; \item set up the probability model; \item use the model to compute probability to observe the data; \item write the probability as $\loglik(\mu)$, log-likelihood function of the parameter $\mu$; \item and finally, find $\mu^{*}$, the $\mu$ value that maximizes the corresponding log-likelihood. \end{enumerate*} This will be our best estimate for the true mean. As we already have our data points $x_{1}$ and $x_{2}$, our next step is the probability model. The probability density function (pdf) for normal distribution with mean $\mu$ and variance 1 is \begin{equation} \label{eq:standard-normal-pdf} f(x; \mu) = \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x - \mu)^{2} } \end{equation} (This is the thick curve in Figure~\ref{fig:standard-normal-intervals}). We write it as $f(x; \mu)$ as pdf is usually written as a function of data. But as our primary interest is $\mu$, we also add this as an argument. Now we use this pdf and~\eqref{eq:probability-density} to find the probability that we observe a datapoint in the narrow interval around $x$. Here it is just $f(x; \mu)\cdot \delta$. As $x_{1}$ and $x_{2}$ are independent, we can simply multiply the corresponding probabilities to find the combined probability that both random numbers are near their corresponding values: \begin{multline} \label{eq:two-normal-probability-likelihood} \Pr{\Big(X_{1} \in [x_{1} - \delta/2, x_{1} + \delta/2] \quad\text{and}\quad X_{2} \in [x_{2} - \delta/2, x_{2} + \delta/2]\Big)} =\\[2ex]= \underbrace{ \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{1} - \mu)^{2} } \cdot\delta\ }_{ \text{First random value near $x_{1}$} } \times \underbrace{ \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{2} - \mu)^{2} } \cdot\delta }_{ \text{Second random value near $x_{2}$} } \equiv\\[2ex]\equiv \tilde\likelihood(\mu; x_{1}, x_{2}). \end{multline} The interval width $\delta$ must be small for the equation to hold precisely. We denote this probability with $\tilde\likelihood$ to stress that it is essentially the likelihood, just not written in the way it is usually done. As in the coin-toss example above, we write it as a function of the parameter $\mu$, and put data $x_{1}$ and $x_{2}$ after semicolon. Now we can estimate $\mu$ by finding such a value $\mu^{*}$ that maximizes the expression~\eqref{eq:two-normal-probability-likelihood}. But note that $\delta$ plays no role in maximizing the likelihood. It is just a multiplicative factor, and it cannot be negative because it is a width. So for our maximization problem we can just ignore it. This is what is normally done when working with continuous random variables. Hence we write the likelihood as \begin{equation} \label{eq:two-normal-likelihood} \likelihood(\mu; x_{1}, x_{2}) = \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{1} - \mu)^{2} } \times \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{2} - \mu)^{2} }. \end{equation} We denote this by $\likelihood$ instead of $\tilde\likelihood$ to stress that this is how likelihood function for continuous random variables is usually written. Exactly as in the discrete case, it is better to use log-likelihood instead of likelihood to actually compute the maximum. From~\eqref{eq:two-normal-likelihood} we get log-likelihood as \begin{multline} \label{eq:two-standard-normal-loglik} \loglik(\mu; x_{1}, x_{2}) = -\log{\sqrt{2\pi}} -\frac{1}{2} (x_{1} - \mu)^{2} + (- \log{\sqrt{2\pi}}) -\frac{1}{2} (x_{2} - \mu)^{2} =\\[2ex]= - 2\log{\sqrt{2\pi}} - \frac{1}{2} \sum_{i=1}^{2} (x_{i} - \mu)^{2}. \end{multline} The first term, $- 2\log{\sqrt{2\pi}}$, is just an additive constant and plays no role in the actual maximization but it is typically still included when defining the likelihood function.\footnote{Additive or multiplicative constants do not play any role for optimization, but they are important when comparing different log-likelihood values. This is often needed for likelihood-based statistical tests. } One can easily check by differentiating the log-likelihood function that the maximum is achieved at $\mu^{*} = \frac{1}{2}(x_{1} + x_{2})$. It is not surprising, our intuitive understanding of mean value carries immediately over to the normal distribution context. Now it is time to demonstrate these results with \texttt{maxLik} package. First, create our ``data'', just two normally distributed random numbers: <<>>= x1 <- rnorm(1) # centered around 0 x2 <- rnorm(1) x1 x2 @ and define the log-likelihood function. We include all the terms as in the final version of~\eqref{eq:two-standard-normal-loglik}: <<>>= loglik <- function(mu) { -2*log(sqrt(2*pi)) - 0.5*((x1 - mu)^2 + (x2 - mu)^2) } @ We also need the parameter start value--we can pick $0$. And we use \texttt{maxLik} to find the best $\mu$: <<>>= m <- maxLik(loglik, start=0) summary(m) @ The answer is the same as sample mean: <<>>= (x1 + x2)/2 @ \section{Vector arguments} \label{sec:vector-arguments} The previous example is instructive but it does have very few practical applications. The problem is that we wrote the probability model as normal density with unknown mean $\mu$ but standard deviation $\sigma$ equal to one. However, in practice we hardly ever know that we are dealing with unit standard deviation. More likely both mean and standard deviation are unknown. So we have to incorporate the unknown $\sigma$ into the model. The more general normal pdf with standard deviation $\sigma$ is \begin{equation} \label{eq:normal-pdf} f(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \, \me^{ -\displaystyle\frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} }. \end{equation} Similar reasoning as what we did above will give the log-likelihood \begin{equation} \label{eq:two-normal-loglik} \loglik(\mu, \sigma; x_{1}, x_{2}) = - 2\log{\sqrt{2\pi}} - 2\log \sigma - \frac{1}{2} \sum_{i=1}^{2} \frac{(x_{i} - \mu)^{2}}{\sigma^{2}}. \end{equation} We write the log-likelihood as function of both parameters, $\mu$ and $\sigma$; the semicolon that separates data $x_{1}$ and $x_{2}$ shows that though the log-likelihood depends on data too, we are not much interested in that dependency for now. This formula immediately extends to the case of $N$ datapoints as \begin{equation} \label{eq:normal-loglik} \loglik(\mu, \sigma) = - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{(x_{i} - \mu)^{2}}{\sigma^{2}} \end{equation} where we have dropped the dependency on data in the notation. In this case we can actually do the optimization analytically, and derive the well-known intuitive results: the best estimator for mean $\mu$ is the sample average, and the best estimator for $\sigma^{2}$ is the sample variance. However, in general the expression cannot be solved analytically. We have to use numeric optimization to search for the best $\mu$ and $\sigma$ combination. The common multi-dimensional optimizers rely on linear algebra and expect all the parameters submitted as a single vector. So we can write the log-likelihood as \begin{equation} \label{eq:normal-loglik-vector} \loglik(\vec{\theta}) \quad\text{where}\quad \vec{\theta} = (\mu, \sigma). \end{equation} Here we denote both parameters $\mu$ and $\sigma$ as components of a single parameter vector $\vec{\theta}$. (Traditionally vectors are denoted by bold symbols.) We have also dropped dependency on data in notation, but remember that in practical applications log-likelihood always depends on data. This notation can be converted to computer code almost verbatim, just remember to extract the parameters $\mu$ and $\sigma$ from $\vec{\theta}$ in the log-likelihood function. Let us illustrate this using the \emph{CO2} dataset (in package \emph{datasets}). It describes \COii uptake (\si{\micro\mol\per\meter\squared\sec}, variable \emph{uptake}) by different grasses in various conditions. Let us start by plotting the histogram of uptake: <>= data(CO2) hist(CO2$uptake) @ Let us model the uptake as a normal random variable with expected value $\mu$ and standard deviation $\sigma$. We code~\eqref{eq:normal-loglik} while keeping both parameters in a single vector as in~\eqref{eq:normal-loglik-vector}: <<>>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- nrow(CO2) -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } @ The function is similar to the function \texttt{loglik} we used in Section~\ref{sec:continuous-outcomes}. There are just two main differences: \begin{itemize} \item both arguments, $\mu$ and $\sigma$ are passed as components of $\vec{\theta}$, and hence the function starts by unpacking the values. \item instead of using variables \texttt{x1} and \texttt{x2}, we now extract data directly from the data frame. \end{itemize} Besides these two differences, the formula now also includes $\sigma$ and sums over all observations, not just over two observations. As our parameter vector now contains two components, the start vector must also be of length two. Based on the figure we guess that a good starting value might be $\mu=30$ and $\sigma=10$: <<>>= m <- maxLik(loglik, start=c(mu=30, sigma=10)) summary(m) @ Indeed, our guess was close. \section{Final Example: Linear Regression} \label{sec:linear-regression} Now we have the main tools in place to extend the example above to a real statistical model. Let us build the previous example into linear regression. We describe \COii uptake (variable \emph{uptake}) by \COii concentration in air (variable \emph{conc}). We can write the corresponding regression model as \begin{equation} \label{eq:co2-regression} \mathit{uptake}_{i} = \beta_{0} + \beta_{1} \cdot \mathit{conc}_{i} + \epsilon_{i}. \end{equation} In order to turn this regression model into a ML problem, we need a probability model. Assume that the disturbance term $\epsilon$ is normally distributed with mean 0 and (unknown) variance $\sigma^{2}$ (this is a standard assumption in linear regression). Now we can follow~\eqref{eq:two-normal-loglik} and write log of pdf for a single observation as \begin{equation} \label{eq:co2-epsilon-loglik} \loglik(\sigma; \epsilon_{i}) = - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{\epsilon_{i}^{2}}{\sigma^{2}}. \end{equation} Here we have replaced $x_{i}$ by the random outcome $\epsilon_{i}$. As the expected value $\mu=0$ by assumption, we do not include $\mu$ in~\eqref{eq:co2-epsilon-loglik} and hence we drop it also from the argument list of $\loglik$. We do not know $\epsilon_{i}$ but we can express it using linear regression model~\eqref{eq:co2-regression}: \begin{equation} \label{eq:co2-epsilon} \epsilon_{i} = \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i}. \end{equation} This expression depends on two additional unknown parameters, $\beta_{0}$ and $\beta_{1}$. These are the linear regression coefficients we want to find. Now we plug this into~\eqref{eq:co2-epsilon-loglik}: \begin{multline} \label{eq:co2-single-loglik} \loglik(\beta_{0}, \beta_{1}, \sigma; \mathit{uptake}_{i}, \mathit{conc}_{i}) =\\= - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i} )^{2}}{\sigma^{2}}. \end{multline} We have designed log-likelihood formula for a single linear regression observation. It depends on three parameters, $\beta_{0}$, $\beta_{1}$ and $\sigma$. For $N$ observations we have \begin{multline} \label{eq:co2-loglik} \loglik(\beta_{0}, \beta_{1}, \sigma; \vec{\mathit{uptake}}, \vec{\mathit{conc}}) =\\= - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i})^{2}}{\sigma^{2}} \end{multline} where vectors $\vec{\mathit{uptake}}$ and $\vec{\mathit{conc}}$ contain the data values for all the observations. This is a fully specified log-likelihood function that we can use for optimization. Let us repeat what we have done: \begin{itemize} \item We wrote log-likelihood as a function of parameters $\beta_{0}$, $\beta_{1}$ and $\sigma$. Note that in case of linear regression we typically do not call $\sigma$ a parameter. But it is still a parameter, although one we usually do not care much about (sometimes called ``nuisance parameter''). \item The likelihood function also depends on data, here the vectors $\vec{\mathit{uptake}}$ and $\vec{\mathit{conc}}$. \item The function definition itself is just sum of log-likelihood contributions of individual normal disturbance terms, but as we do not observe the disturbance terms, we express those through the regression equation in~\eqref{eq:co2-single-loglik}. \end{itemize} Finally, we combine the three parameters into a single vector $\vec{\theta}$, suppress dependency on data in the notation, and write \begin{equation} \label{eq:co2-loglik-simplified} \loglik(\vec{\theta}) = - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i})^{2}}{\sigma^{2}}. \end{equation} This is the definition we can easily code and estimate. We guess start values $\beta_{0} = 30$ (close to the mean), $\beta_{1} = 0$ (uptake does not depend on concentration) and $\sigma=10$ (close to sample standard deviation). We can convert~\eqref{eq:co2-loglik-simplified} into code almost verbatim, below we choose to compute the expected uptake $\mu$ as an auxiliary variable: <<>>= loglik <- function(theta) { beta0 <- theta[1] beta1 <- theta[2] sigma <- theta[3] N <- nrow(CO2) ## compute new mu based on beta1, beta2 mu <- beta0 + beta1*CO2$conc ## use this mu in a similar fashion as previously -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } m <- maxLik(loglik, start=c(beta0=30, beta1=0, sigma=10)) summary(m) @ These are the linear regression estimates: $\beta_{0} = \Sexpr{round(coef(m)["beta0"], 3)}$ and $\beta_{1} = \Sexpr{round(coef(m)["beta1"], 3)}$. Note that \maxlik output also provides standard errors, $z$-values and $p$-values, hence we see that the results are highly statistically significant. One can check that a linear regression model will give similar results: <<>>= summary(lm(uptake ~ conc, data=CO2)) @ Indeed, the results are close although not identical. \section{Non-linear optimization} \label{sec:non-linear-optimization} Finally, we discuss the magic inside \texttt{maxLik} that finds the optimal parameter values. Although not necessary in everyday work, this knowledge helps to understand the issues and potential solutions when doing non-linear optimization. So how does the optimization work? Consider the example in Section~\ref{sec:vector-arguments} where we computed the normal distribution parameters for \COii intake. There are two parameters, $\mu$ and $\sigma$, and \maxlik returns the combination that gives the largest possible log-likelihood value. We can visualize the task by plotting the log-likelihood value for different combinations of $\mu$, $\sigma$ (Figure~\ref{fig:mu-sigma-plot}). \begin{figure}[ht] \centering <>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- nrow(CO2) -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } m <- maxLik(loglik, start=c(mu=30, sigma=10)) params <- coef(m) np <- 33 # number of points mu <- seq(6, 36, length.out=np) sigma <- seq(5, 50, length.out=np) X <- as.matrix(expand.grid(mu=mu, sigma=sigma)) ll <- matrix(apply(X, 1, loglik), nrow=np) levels <- quantile(ll, c(0.05, 0.4, 0.6, 0.8, 0.9, 0.97)) # where to draw the contours colors <- colorRampPalette(c("Blue", "White"))(30) par(mar=c(0,0,0,0), mgp=2:0) ## Perspective plot if(require(plot3D)) { persp3D(mu, sigma, ll, xlab=expression(mu), ylab=expression(sigma), zlab=expression(log-likelihood), theta=40, phi=30, colkey=FALSE, col=colors, alpha=0.5, facets=TRUE, shade=1, lighting="ambient", lphi=60, ltheta=0, image=TRUE, bty="b2", contour=list(col="gray", side=c("z"), levels=levels) ) ## add the dot for maximum scatter3D(rep(coef(m)[1], 2), rep(coef(m)[2], 2), c(maxValue(m), min(ll)), col="red", pch=16, facets=FALSE, bty="n", add=TRUE) ## line from max on persp to max at bottom surface segments3D(coef(m)[1], coef(m)[2], maxValue(m), coef(m)[1], coef(m)[2], min(ll), col="red", lty=2, bty="n", add=TRUE) ## contours for the bottom image contour3D(mu, sigma, z=min(ll) + 0.1, colvar=ll, col="black", levels=levels, add=TRUE) } else { plot(1:2, type="n") text(1.5, 1.5, "This figure requires 'plot3D' package", cex=1.5) } @ \caption{Log-likelihood surface as a function of $\mu$ and $\sigma$. The optimum, denoted as the red dot, is at $\mu=\Sexpr{round(coef(m)[1], 3)}$ and $\sigma=\Sexpr{round(coef(m)[2], 3)}$. The corresponding countour plot is shown at the bottom of the figure box. } \label{fig:mu-sigma-plot} \end{figure} So how does the algorithm find the optimal parameter value $\vec{\theta}^*$, the red dot on the figure? All the common methods are iterative, i.e. they start with a given start value (that's why we need the start value), and repeatedly find a new and better parameter that gives a larger log-likelihood value. While humans can look at the figure and immediately see where is its maximum, computers cannot perceive the image in this way. And more importantly--even humans cannot visualize the function in more than three dimensions. This visualization is so helpful for us because we can intuitively understand the 3-dimensional structure of the surface. It is 3-D because we have two parameters, $\mu$ and $\sigma$, and a single log-likelihood value. Add one more parameter as we did in Section~\ref{sec:linear-regression}, and visualization options are very limited. In case of 5 parameters, it is essentially impossible to solve the problem by just visualizations. Non-linear optimization is like climbing uphill in whiteout conditions where you cannot distinguish any details around you--sky is just a white fog and the ground is covered with similar white snow. But you can still feel which way the ground goes up and so you can still go uphill. This is what the popular algorithms do. They rely on the slope of the function, the gradient, and follow the direction suggested by gradient. Most optimizers included in the \texttt{maxLik} package need gradients, including the default Newton-Raphson method. But how do we know the gradient if the log-likelihood function only returns a single value? There are two ways: \begin{enumerate*}[label=\roman*)] \item provide a separate function that computes gradient; \item compute the log-likelihood value in multiple points nearby and deduce the gradient from that information. \end{enumerate*} The first option is superior, in high dimensions it is much faster and much less error prone. But computing and coding gradient can easily be days of work. The second approach, numeric gradient, forces the computer to do more work and hence it is slower. Unfortunately importantly, it may also unreliable for more complex cases. In practice you may notice how the algorithm refuses to converge for thousands of iterations. But numeric gradient works very well in simple cases we demonstrated here. This also hints why it is useful to choose good start values. The closer we start to our final destination, the less work the computer has to do. And while we may not care too much about a few seconds of computer's work, we also help the algorithm to find the correct maximum. The less the algorithm has to work, the less likely it is that it gets stuck in a wrong place or just keeps wandering around in a clueless manner. If this is the case, you may see how the algorithm gets slow, does not converge (returns the ``maximum number of iterations exceeded'' message), how the results look weird, or standard errors are extremely large. % \bibliographystyle{apecon} % \bibliography{maxlik} \end{document} maxLik/vignettes/stochastic-gradient-maxLik.Rnw0000644000176200001440000007613714077525067021414 0ustar liggesusers\documentclass{article} \usepackage{graphics} \usepackage{amsmath} \usepackage{amssymb} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage{natbib} \usepackage{xspace} \newcommand{\elemProd}{\ensuremath{\odot}} % elementwise product of matrices \newcommand*{\mat}[1]{\mathsf{#1}} \newcommand{\maxlik}{\texttt{maxLik}\xspace} \newcommand*{\transpose}{^{\mkern-1.5mu\mathsf{T}}} %\newcommand{\transpose}{\intercal} \renewcommand*{\vec}[1]{\boldsymbol{#1}} % \VignetteIndexEntry{SGA introduction: the basic usage of maxSGA} \begin{document} <>= options(keep.source = TRUE, width = 60, try.outFile=stdout() # make try to produce error messages ) foo <- packageDescription("maxLik") @ \title{Stochastic Gradient Ascent in maxLik} \author{Ott Toomet} \maketitle \section{\texttt{maxLik} and Stochastic Gradient Ascent} \texttt{maxLik} is a package, primarily intended for Maximum Likelihood and related estimations. It includes several optimizers and associated tools for a typical Maximum Likelihood workflow. However, as predictive modeling and complex (deep) models have gained popularity in the recend decade, \texttt{maxLik} also includes a few popular algorithms for stochastic gradient ascent, the mirror image for the more widely known stochastic gradient descent. This vignette gives a brief overview of these methods, and their usage in \texttt{maxLik}. \section{Stochastic Gradient Ascent} \label{sec:stochastic-gradient-ascent} In machine learning literature, it is more common to describe the optimization problems as minimization and hence to talk about gradient descent. As \texttt{maxLik} is primarily focused on maximizing likelihood, it implements the maximization version of the method, stochastic gradient ascent (SGA). The basic method is simple and intuitive, it is essentially just a careful climb in the gradient's direction. Given and objective function $f(\vec{\theta})$, and the initial parameter vector $\vec{\theta}_{0}$, the algorithm will compute the gradient $\vec{g}(\vec{\theta}_{0}) = \nabla_{\vec{\theta}} f(\vec{\theta})\big|_{\vec{\theta} = \vec{\theta}_{0}}$, and update the parameter vector as $\vec{\theta}_{1} = \vec{\theta}_{0} + \rho \vec{g}(\vec{\theta}_{0})$. Here $\rho$, the \emph{learning rate}, is a small positive constant to ensure we do not overshoot the optimum. Depending on the task it is typically of order $0.1 \dots 0.001$. In common tasks, the objective function $f(\vec{\theta})$ depends on data, ``predictors'' $\mat{X}$ and ``outcome'' $\vec{y}$ in an additive form $f(\vec{\theta}; \mat{X}, \vec{y}) = \sum_{i} f(\vec{\theta}; \vec{x}_{i}, y_{i})$ where $i$ denotes ``observations'', typically arranged as the rows of the design matrix $\mat{X}$. Observations are often considered to be independent of each other. The overview above does not specify how to compute the gradient $\vec{g}(\vec{\theta}_{0})$ in a sense of which observations $i$ to include. A natural approach is to include the complete data and compute \begin{equation} \label{eq:full-batch-gradient} \vec{g}_{N}(\vec{\theta}_{0}) = \frac{1}{N}\sum_{i=1}^{N} \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}}. \end{equation} In SGA context, this approach is called ``full batch'' and it has a number of advantages. In particular, it is deterministic (given data $\mat{X}$ and $\vec{y}$), and computing of the sum can be done in parallel. However, there are also a number of reasons why full-batch approach may not be desirable \citep[see][]{bottou2018SIAM}: \begin{itemize} \item Data over different observations is often more or less redundant. If we use all the observations to compute the update then we spend a substantial effort on redundant calculations. \item Full-batch gradient is deterministic and hence there is no stochastic noise. While advantageous in the latter steps of optimization, the noise helps the optimizer to avoid local optima and overcome flat areas in the objective function early in the process. \item SGA achieves much more rapid initial convergence compared to the full batch method (although full-batch methods may achieve better final result). \item Cost of computing the full-batch gradient grows with the sample size but that of minibatch gradient does not grow. \item It is empirically known that large-batch optimization tend to find sharp optima \citep[see][]{keskar+2016ArXiv} that do not generalize well to validation data. Small batch approach leads to a better validation performance. \end{itemize} In contrast, SGA is an approach where the gradient is computed on just a single observation as \begin{equation} \label{eq:stochastic-gradient} \vec{g}_{1}(\vec{\theta}_{0}) = \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i}, y_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}} \end{equation} where $i$ is chosen randomly. In applications, all the observations are usually walked through in a random order, to ensure that each observation is included once, and only once, in an \emph{epoch}. Epoch is a full walk-through of the data, and in many ways similar to iteration in a full-batch approach. As SGA only accesses a single observation at time, it suffers from other kind of performance issues. In particular, one cannot parallelize the gradient function \eqref{eq:stochastic-gradient}, operating on individual data vectors may be inefficient compared to larger matrices, and while we gain in terms of gradient computation speed, we lose by running the optimizer for many more loops. \emph{Minibatch} approach offers a balance between the full-batch and SGA. In case of minibatch, we compute gradient not on individual observations but on \emph{batches} \begin{equation} \label{eq:minibatch-gradient} \vec{g}_{m}(\vec{\theta}_{0}) = \frac{1}{|\mathcal{B}|}\sum_{i\in\mathcal{B}} \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i}, y_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}} \end{equation} where $\mathcal{B}$ is the batch, a set of observations that are included in the gradient computation. Normally the full data is partitioned into a series of minibatches and walked through sequentially in one epoch. \section{SGA in \texttt{maxLik} package} \label{sec:sga-in-maxlik} \maxlik implements two different optimizers: \texttt{maxSGA} for simple SGA (including momentum), and \texttt{maxAdam} for the Adaptive Moments method \citep[see][p. 301]{goodfellow+2016DL}. The usage of both methods mostly follows that of the package's main workhorse, \texttt{maxNR} \citep[see][]{henningsen+toomet2011}, but their API has some important differences due to the different nature of SGA. The basic usage of the maxSGA is as follows: <>= maxSGA(fn, grad, start, nObs, control) @ where \texttt{fn} is the objective function, \texttt{grad} is the gradient function, \texttt{nObs} is number of observations, and \texttt{control} is a list of control parameters. From the user's perspective, \texttt{grad} is typically the most important (and the most complex) argument. Next, we describe the API and explain the differences between the \texttt{maxSGA} API and \texttt{maxNR} API, and thereafter give a few toy examples that demonstrate how to use \texttt{maxSGA} in practice. \subsection{The objective function} Unlike in \texttt{maxNR} and the related optimizers, SGA does not directly need the objective function \texttt{fn}. The function can still be provided (and perhaps will in most cases), but one can run the optimizer without it. If provided, the function can be used for printing the value at each epoch (by setting a suitable \texttt{printLevel} control option), and for stopping through \emph{patience} stopping condition. If \texttt{fn} is not provided, do not forget to add the argument name for the gradient, \texttt{grad=}, as otherwise the gradient will be treated as the objective function with unexpected results! If provided, the function should accept two (or more) arguments: the first must be the numeric parameter vector, and another one, named \texttt{index}, is the list of indices in the current minibatch. As the function is not needed by the optimizer itself, it is up to the user to decide what it does. An obvious option is to compute the objective function value on the same minibatch as used for the gradient computation. But one can also opt for something else, for instance to compute the value on the validation data instead (and ignore the provided \emph{index}). The latter may be a useful option if one wants to employ the patience-based stopping criteria. \subsection{Gradient function} \label{sec:gradient-function} Gradient is the work-horse of the SGA methods. Although \maxlik can also compute numeric gradient using the finite difference method (this will be automatically done if the objective function is provided but the gradient isn't), this is not advisable, and may be very slow in high-dimensional problems. \texttt{maxLik} uses the numerator layout, i.e. the gradient should be a $1\times K$ matrix where columns correspond to the components of the parameter vector $\vec{\theta}$. For compatibility with other optimizers in \texttt{maxLik} it also accepts a observation-wise matrix where rows correspond to the individual observations and columns to the parameter vector components. The requirements for the gradient function arguments are the same as for \texttt{fn}: the first formal argument must be the parameter vector, and it must also have an argument \texttt{index}, a numeric index for the observations to be included in the minibatch. \subsection{Stopping Conditions} \label{sec:stopping-conditions} \texttt{maxSGA} uses three stopping criteria: \begin{itemize} \item Number of epochs (control option \texttt{iterlim}): number of times all data is iterated through using the minibatches. \item Gradient norm. However, in case of stochastic approach one cannot expect the gradient at optimum to be close to zero, and hence the corresponding criterion (control option \texttt{gradtol}) is set to zero by default. If interested, one may make it positive. \item Patience. Normally, each new iteration has better (higher) value of the objective function. However, in certain situations this may not be the case. In such cases the algorithm does not stop immediately, but continues up to \emph{patience} more epochs. It also returns the best parameters, not necessarily the last parameters. Patience can be controlled with the options \texttt{SG\_patience} and \texttt{SG\_patienceStep}. The former controls the patience itself--how many times the algorithm is allowed to produce an inferior result (default value \texttt{NULL} means patience criterion is not used). The latter controls how often the patience criterion is checked. If computing the objective function is costly, it may be useful to increase the patience step and decrease the patience. \end{itemize} \subsection{Optimizers} \label{sec:optimizers} \texttt{maxLik} currently implements two optimizers: \emph{SGA}, the stock gradient ascent (including momentum), and \emph{Adam}. Here we give some insight into the momentum, and into the Adam method, the basic gradient-only based optimization technique was explained in Section~\ref{sec:stochastic-gradient-ascent}. It is easy and intuitive to extend the SGA method with momentum. As implemented in \texttt{maxSGA}, the momentum $\mu$ ($0 < \mu < 1$) is incorporated into the the gradient update as \begin{equation} \label{eq:gradient-update-momentum} \vec{\theta}_{t+1} = \vec{\theta}_{t} + \vec{v}_{t} \quad\text{where}\quad \vec{v}_{t} = \mu \vec{v}_{t-1} + \rho \vec{g}(\vec{\theta}_{t}). \end{equation} See \citet[p. 288]{goodfellow+2016DL}. The algorithm takes the initial ``velocity'' $\vec{v}_{0} = \vec{0}$. It is easy to see that $\mu=0$ is equivalent to no-momentum case, and if $\vec{g}(\vec{\theta})$ is constant, $\vec{v}_{t} \to \rho \vec{g}(\vec{\theta})/(1 - \mu)$. So the movement speeds up in a region with stable gradient. As a downside, it is also easier overshoot a maximum. But this behavior makes momentum-equipped SGA less prone of getting stuck in a local optimum. Momentum can be set by the control option \texttt{SG\_momentum}, the default value is 0. Adaptive Moments method, usually referred to as \emph{Adam}, \citep[p. 301]{goodfellow+2016DL} adapts the learning rate by variance of the gradient--if gradient components are unstable, it slows down, and if they are stable, it speeds up. The adaptation is proportional to the weighted average of the gradient divided by the square root of the weighted average of the gradient squared, all operations done component-wise. In this way a stable gradient component (where moving average is similar to the gradient value) will have higher speed than a fluctuating gradient (where the components frequently shift the sign and the average is much smaller). More specifically, the algorithm is as follows: \begin{enumerate} \item Initialize the first and second moment averages $\vec{s} = \vec{0}$ and $\vec{r} = \vec{0}$. \item Compute the gradient $\vec{g}_{t} = \vec{g}(\vec{\theta}_{t})$. \item Update the average first moment: $\vec{s}_{t+1} = \mu_{1} \vec{s}_{t} + (1 - \mu_{1}) \vec{g}_{t}$. $\mu_{1}$ is the decay parameter, the larger it is, the longer memory does the method have. It can be adjusted with the control parameter \texttt{Adam\_momentum1}, the default value is 0.9. \item Update the average second moment: $\vec{r}_{t+1} = \mu_{2} \vec{r}_{t} + (1 - \mu_{2}) \vec{g}_{t} \elemProd \vec{g}_{t}$ where $\elemProd$ denotes element-wise multiplication. The control parameter for the $\mu_{2}$ is \texttt{Adam\_momentum2}, the default value is 0.999. \item As the algorithm starts with the averages $\vec{s}_{0} = \vec{r}_{0}= 0$, we also correct the resulting bias: $\hat{\vec{s}} = \vec{s}/(1 - \mu_{1}^{t})$ and $\hat{\vec{r}} = \vec{r}/(1 - \mu_{2}^{t})$. \item Finally, update the estimate: $\vec{\theta}_{t+1} = \vec{\theta}_{t} + \rho \hat{\vec{s}}/(\delta + \sqrt{\hat{\vec{r}}})$ where division and square root are done element-wise and $\delta=10^{-8}$ takes care of numerical stabilization. \end{enumerate} Adam optimizer can be used with \texttt{maxAdam}. \subsection{Controlling Optimizers} \label{sec:control-options} Both \texttt{maxSGA} and \texttt{maxAdam} are designed to be similar to \texttt{maxNR}, and mostly expect similar arguments. In particular, both functions expect the objective function \texttt{fn}, gradient \texttt{grad} and Hessian function \texttt{hess}, and the initial parameter start values \texttt{start}. As these optimizers only need gradient, one can leave out both \texttt{fn} and \texttt{hess}. The Hessian is mainly included for compatibility reasons and only used to compute the final Hessian, if requested by the user. As SGA methods are typically used in contexts where Hessian is not needed, by default the algorithms do not return Hessian matrix and hence do not use the \texttt{hess} function even if provided. Check out the argument \texttt{finalHessian} if interested. An important SGA-specific control options is \texttt{SG\_batchSize}. This determines the batch size, or \texttt{NULL} for the full-batch approach. Finally, unlike the traditional optimizers, stochastic optimizers need to know the size of data (argument \texttt{nObs}) in order to calculate the batches. \section{Example usage: Linear regression} \label{sec:example-usage-cases} \subsection{Setting Up} \label{sec:setting-up} We demonstrate the usage of \texttt{maxSGA} and \texttt{maxAdam} to solve a linear regression (OLS) problem. Although OLS is not a task where one commonly relies on stochastic optimization, it is a simple and easy-to understand model. We use the Boston housing data, a popular dataset where one traditionally attempts to predict the median house price across 500 neighborhoods using a number of neighborhood descriptors, such as mean house size, age, and proximity to Charles river. All variables in the dataset are numeric, and there are no missing values. The data is provided in \emph{MASS} package. First, we create the design matrix $\mat{X}$ and extract the house price $y$: <<>>= i <- which(names(MASS::Boston) == "medv") X <- as.matrix(MASS::Boston[,-i]) X <- cbind("const"=1, X) # add constant y <- MASS::Boston[,i] @ Although the model and data are simple, it is not an easy task for stock gradient ascent. The problem lies in different scaling of variables, the means are <<>>= colMeans(X) @ One can see that \emph{chas} has an average value \Sexpr{round(mean(X[,"chas"]), 3)} while that of \emph{tax} is \Sexpr{round(mean(X[,"tax"]), 3)}. This leads to extremely elongated contours of the loss function: <>= eigenvals <- eigen(crossprod(X))$values @ One can see that the ratio of the largest and the smallest eigenvalue is $\mat{X}^{\transpose} \mat{X} = \Sexpr{round(eigenvals[1]/eigenvals[14], -5)}$. Solely gradient-based methods, such as SGA, have trouble working in the resulting narrow valleys. For reference, let's also compute the analytic solution to this linear regression model (reminder: $\hat{\vec{\beta}} = (\mat{X}^{\transpose}\,\mat{X})^{-1}\,\mat{X}^{\transpose}\,\vec{y}$): <<>>= betaX <- solve(crossprod(X)) %*% crossprod(X, y) betaX <- drop(betaX) # matrix to vector betaX @ Next, we provide the gradient function. As a reminder, OLS gradient in numerator layout can be expressed as \begin{equation} \label{eq:ols-gradient} \vec{g}_{m}(\vec{\theta}) = -\frac{2}{|\mathcal{B}|} \sum_{i\in\mathcal{B}} \left(y_{i} - \vec{x}_{i}^{\transpose} \cdot \vec{\theta} \right) \vec{x}_{i}^{\transpose} = -\frac{2}{|\mathcal{B}|} \left(y_{\mathcal{B}} - \mat{X}_{\mathcal{B}} \cdot \vec{\theta} \right)^{\transpose} \mat{X}_{\mathcal{B}} \end{equation} where $y_{\mathcal{B}}$ and $\mat{X}_{\mathcal{B}}$ denote the elements of the outcome vector and the slice of the design matrix that correspond to the minibatch $\mathcal{B}$. We choose to divide the value by batch size $|\mathcal{B}|$ in order to have gradient values of roughly similar size, independent of the batch size. We implement it as: <<>>= gradloss <- function(theta, index) { e <- y[index] - X[index,,drop=FALSE] %*% theta g <- t(e) %*% X[index,,drop=FALSE] 2*g/length(index) } @ The \texttt{gradloss} function has two arguments: \texttt{theta} is the parameter vector, and \texttt{index} tells which observations belong to the current minibatch. The actual argument will be an integer vector, and hence we can use \texttt{length(index)} to find the size of the minibatch. Finally, we return the negative of~\eqref{eq:ols-gradient} as \texttt{maxSGA} performs maximization, not minimization. First, we demonstrate how the models works without the objective function. We have to supply the gradient function, initial parameter values (we use random normals below), and also \texttt{nObs}, number of observations to select the batches from. The latter is needed as the optimizer itself does not have access to data but still has to partition it into batches. Finally, we may also provide various control parameters, such as number of iterations, stopping conditions, and batch size. We start with only specifying the iteration limit, the only stopping condition we use here: <>= library(maxLik) set.seed(3) start <- setNames(rnorm(ncol(X), sd=0.1), colnames(X)) # add names for better reference res <- try(maxSGA(grad=gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000) ) ) @ This run was a failure. We encountered a run-away growth of the gradient because the default learning rate $\rho=0.1$ is too big for such strongly curved objective function. But before we repeat the exercise with a smaller learning rate, let's incorporate gradient clipping. Gradient clipping, performed with \texttt{SG\_clip} control option, caps the $L_{2}$-norm of the gradient while keeping it's direction. We clip the squared norm at 10,000, i.e. the gradient norm cannot exceed 100: <<>>= res <- maxSGA(grad=gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000, SG_clip=1e4) # limit ||g|| <= 100 ) summary(res) @ This time the gradient did not explode and we were able to get a result. But the estimates are rather far from the analytic solution shown above, e.g. the constant estimate \Sexpr{round(coef(res)[1], 3)} is very different from the corresponding analytic value \Sexpr{round(betaX[1], 3)}. Let's analyze what is happening inside the optimizer. We can ask for both the parameter values and the objective function value to be stored for each epoch. But before we can store its value, in this case mean squared error (MSE), we have to supply an objective function to maxSGA. We compute MSE on the same minibatch as <<>>= loss <- function(theta, index) { e <- y[index] - X[index,] %*% theta -crossprod(e)/length(index) } @ Now we can store the values with the control options \texttt{storeParameters} and \texttt{storeValues}. The corresponding numbers can be retrieved with \texttt{storedParameters} and \texttt{storedValues} methods. For \texttt{iterlim=R}, the former returns a $(R+1) \times K$ matrix, one row for each epoch and one column for each parameter component, and the latter returns a numeric vector of length $R+1$ where $R$ is the number of epochs. The first value in both cases is the initial value, so we have $R+1$ values in total. Let's retrieve the values and plot both. We decrease the learning rate to $0.001$ using the \texttt{SG\_learningRate} control. Note that although we maximize negative loss, we plot positive loss. \setkeys{Gin}{width=\textwidth, height=80mm} <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000, # will misbehave with larger numbers SG_clip=1e4, SG_learningRate=0.001, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") ## add some arrows to see which way the parameters move iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 10 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) ## plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We can see how the parameters (the first and the second components, ``const'' and ``crim'' in this figure) evolve through the iterations while the loss is rapidly falling. One can see an initial jump where the loss is falling very fast, followed but subsequent slow movement. It is possible the initial jump be limited by gradient clipping. \subsection{Training and Validation Sets} \label{sec:training-validation} However, as we did not specify the batch size, \texttt{maxSGA} will automatically pick the full batch (equivalent to control option \texttt{SG\_batchSize = NULL}). So there was nothing stochastic in what we did above. Let us pick a small batch size--a single observation at time. However, as smaller batch sizes introduce more noise to the gradient, we also make the learning rate smaller and choose \texttt{SG\_learningRate = 1e-5}. But now the existing loss function, calculated just at the single observation, carries little meaning. Instead, we split the data into training and validation sets and feed batches of training data to gradient descent while calculating the loss on the complete validation set. This can be achieved with small modifications in the \texttt{gradloss} and \texttt{loss} function. But as the first step, we split the data: <<>>= i <- sample(nrow(X), 0.8*nrow(X)) # training indices, 80% of data Xt <- X[i,] # training data yt <- y[i] Xv <- X[-i,] # validation data yv <- y[-i] @ Thereafter we modify \texttt{gradloss} to only use the batches of training data while \texttt{loss} will use the complete validation data and just ignore \texttt{index}: <<>>= gradloss <- function(theta, index) { e <- yt[index] - Xt[index,,drop=FALSE] %*% theta g <- -2*t(e) %*% Xt[index,,drop=FALSE] -g/length(index) } loss <- function(theta, index) { e <- yv - Xv %*% theta -crossprod(e)/length(yv) } @ Note that because the optimizer only uses training data, the \texttt{nObs} argument now must equal to the size of training data in this case. Another thing to discuss is the computation speed. \texttt{maxLik} implements SGA in a fairly complex loop that does printing, storing, and complex function calls, computes stopping conditions and does many other checks. Hence a smaller batch size leads to many more such auxiliary computations per epoch and the algorithm gets considerably slower. This is less of a problem for complex objective functions or larger batch sizes, but for linear regression, the slow-down is very large. For demonstration purposes we lower the number of epochs from 1000 to 100. How do the convergence properties look now with the updated approach? <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(Xt), # note: only training data now control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-5, SG_clip=1e4, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We can see the parameters evolving and loss decreasing over epochs. The convergence seems to be smooth and not ruptured by gradient clipping. Next, we try to improve the convergence by introducing momentum. We add momentum $\mu = 0.95$ to the gradient and decrease the learning rate down to $1\cdot10^{-6}$: <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, SGA_momentum = 0.99, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We achieved a lower loss but we are still far from the correct solution. As the next step, we use Adam optimizer. Adam has two momentum parameters but we leave those untouched at the initial values. \texttt{SGA\_momentum} is not used, so we remove that argument. <>= res <- maxAdam(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ As visible from the figure, Adam was marching toward the solution without any stability issues. \subsection{Sequence of Batch Sizes } \label{sec:sequence-batch-sizes} The OLS' loss function is globally convex and hence there is no danger to get stuck in a local maximum. However, when the objective function is more complex, the noise that is generated by the stochastic sampling helps the algorithm to leave local maxima. A suggested strategy is to increase the batch size over time to achieve good exploratory properties early in the process and stable convergence later \citep[see][for more information]{smith+2018arXiv}. This approach is in some ways similar to Simulated Annealing. Here we introduce such an approach by using batch sizes $B=1$, $B=10$ and $B=100$ in succession. We also introduce patience stopping condition. If the objective function value is worse than the best value so far for more than \emph{patience} times then the algorithm stops. Here we use patience value 5. We also store the loss values from all the batch sizes into a single vector \texttt{val}. If the algorithm stops early, some of the stored values are left uninitialized (\texttt{NA}-s), hence we use \texttt{na.omit} to include only the actual values in the final \texttt{val}-vector. We allow the algorithm to run for 200 epochs, but as we now have introduced early stopping through patience, the actual number of epochs may be less than that. \setkeys{Gin}{width=\textwidth, height=110mm} <>= val <- NULL # loop over batch sizes for(B in c(1,10,100)) { res <- maxAdam(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=200, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, SG_patience=5, # worse value allowed only 5 times storeValues=TRUE ) ) cat("Batch size", B, ",", nIter(res), "epochs, function value", maxValue(res), "\n") val <- c(val, na.omit(storedValues(res))) start <- coef(res) } plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") summary(res) @ Two first batch sizes run through all 200 epochs, but the last run stopped early after 7 epochs only. The figure shows that Adam works well for approximately 170 epochs, thereafter the steady pace becomes uneven. It may be advantageous to slow down the movement further. As explained above, this dataset is not an easy task for methods that are solely gradient-based, and so we did not achieve a result that is close to the analytic solution. But our task here is to demonstrate the usage of the package, not to solve a linear regression exercise. We believe every \emph{R}-savy user can adapt the method to their needs. \bibliographystyle{apecon} \bibliography{maxlik} \end{document} maxLik/NEWS0000644000176200001440000002370614077525067012223 0ustar liggesusersTHIS IS THE CHANGELOG OF THE "maxLik" PACKAGE Please note that only the most significant user visible changes are reported here. A full ChangeLog is available in the log messages of the SVN repository on R-Forge. CHANGES IN VERSION 1.5-1 (2020-07-26) * CHANGES IN VERSION 1.5-0 (2020-07-26) * maxLik methods for 'tidy' and 'glance' generics (by David Hugh-Jones) * maxLik method for 'confint' (by Luca Scrucca) * most tests moved to 'tinytest' package * fixed an issue with negative reltol values CHANGES IN VERSION 1.4-8 (2020-03-22) * added two vignettes: "Getting started with maximum likelihood and maxLik" and "maximum likelihood estimation with maxLik" CHANGES IN VERSION 1.4-6 (2020-11-24) * changed the name of internal function head... to headDots to avoid issues with perforce VCS * maxNR and friends now correctly return code 8 if reltol stopping condition invoked * documentation fixes and clean-ups CHANGES IN VERSION 1.4-4 (2020-07-08) * fixed another issue with CRAN tests on ATLAS CHANGES IN VERSION 1.4-2 (2020-07-08) * fixed CRAN test issues CHANGES IN VERSION 1.4-0 (2020-07-07) * includes stochastic gradient ascent and Adam optimizer CHANGES IN VERSION 1.3-10 (2020-05-13) * fixed an issue where maxControl() silently ignored a number of parameters * print.summary.maxim accepts parameters max.rows and max.cols, and only prints this many columns/rows of output matrices CHANGES IN VERSION 1.3-8 (2019-05-18) * better handling of matrix class CHANGES IN VERSION 1.3-8 (2020-01-01) * better handling of matrix class CHANGES IN VERSION 1.3-6 (2019-05-18) * 'maxim' objects now support 'maxValue' and 'gradient' methods. * tests cleaned and give fewer notes on check CHANGES IN VERSION 1.3-4 (2015-11-08) * If Hessian is not negative definite in maxNRCompute, the program now attempts to correct this repeatedly, but not infinite number of times. If Marquardt selected, it uses Marquardt lambda and it's update method. * Fixed an issue where summary.maxLik did not use 'eigentol' option for displaying standard errors CHANGES IN VERSION 1.3-2 (2015-10-28) * Corrected a bug that did not permit maxLik to pass additional arguments to the likelihood function CHANGES IN VERSION 1.3-0 (2015-10-24) * maxNR & friends now support argument 'qac' (quadratic approximation correction) option that allows to choose the behavior if the next guess performs worse than the previous one. This includes the original step halving while keeping direction, and now also Marquardt's (1963) shift toward the steepest gradient. * all max** functions now take control options in the form as 'control=list(...)', analogously as 'optim'. The former method of directly supplying options is preserved for compatibility reasons. * sumt, and stdEr method for 'maxLik' are now in namespace * the preferred way to specify the amount of debugging information is now 'printLevel', not 'print.level'. CHANGES IN VERSION 1.2-4 (2014-12-31) * Equality constraints (SUMT) checks conformity of the matrices * coef.maxim() is now exported * added argument "digits" to print.summary.maxLik() * added argument "digits" to condiNumber.default() * further arguments to condiNumber.maxLik() are now passed to condiNumber.default() rather than to hessian() CHANGES IN VERSION 1.2-0 (2013-10-22) * Inequality constraints now support multiple constraints (B may be a vector). * Fixed a bug in documentation, inequality constraint requires A %*% theta + B > 0, not >= 0 as stated earlier. * function sumKeepAttr() is imported from the miscTools package now (before maxLik() could not be used by another package when this package imported (and not depended on) the maxLik package) (bug reported and solution provided by Martin Becker) CHANGES IN VERSION 1.1-8 (2013-09-17) * fixed bug that could occur in the Newton-Raphson algorithm if the log-likelihood function returns a vector with observation-specific values or if there are NAs in the function values, gradients, or Hessian CHANGES IN VERSION 1.1-4 (2013-09-16) * the package code is byte-compiled * if the log-likelihood function contains NA, the gradient is not calculated; if components of the gradient contain NA, the Hessian is not calculated * slightly improved documentation * improved warning messages and error messages when doing constrained optimisation * added citation information * added start-up message CHANGES IN VERSION 1.1-2 (2012-03-04) * BHHH only considers free parameters when analysing the size of gradient * numericGradient and numericHessian check for the length of vector function CHANGES IN VERSION 1.1-0 (2012-01-...) * Conjugate-gradient (CG) optimization method included. * it is guaranteed now that the variance covariance matrix returned by the vcov() method is always symmetric. * summary.maxLik is guaranteed to use maxLik specific methods, even if corresponding methods for derived classes have higher priority. CHANGES IN VERSION 1.0-2 (2011-10-16) This is mainly bugfix release. * maxBFGSR works with fixed parameters. * maxBFGS and other optim-based routines work with both fixed parameters and inequality constraints. * constrOptim2 removed from API. Names of it's formal arguments are changed. CHANGES IN VERSION 1.0-0 (2010-10-15) * moved the generic function stdEr() including a default method and a method for objects of class "lm" to the "miscTools" package (hence, this package now depends on the version 0.6-8 of the "miscTools" package that includes stdEr() * if argument print.level is 0 (the default) and some parameters are automatically fixed during the estimation, because the returned log-likelihood value has attributes "constPar" and "newVal", the adjusted "starting values" are no longer printed. CHANGES IN VERSION 0.8-0 * fixed bug that occured in maxBFGS(), mxNM(), and maxSANN if the model had only one parameter and the function specified by argument "grad" returned a vector with the analytical gradients at each observation * maxNR() now performs correctly with argument "iterlim" set to 0 * maxNR, maxBHHH(), maxBFGS(), maxNM(), and maxSANN() now use attributes "gradient" and "hessian" of the object returned by the log-likelihood function; if supplied, these are used instead of arguments "grad" and "hess" * added function maxBFGSR() that implements the BFGS algorithm (in R); this function was originally developed by Yves Croissant and placed in the "mlogit" package * maxNR() now has an argument "bhhhHessian" (defaults to FALSE): if this argument is TRUE, the Hessian is approximated by the BHHH method (using information equality), i.e. the BHHH optimization algorithm is used * maxLik() now has an argument 'finalHessian'; if it is TRUE, the final Hessian is returned; if it is the character string "BHHH", the BHHH approximation of the Hessian matrix (using information equality) with attribute "type" set to "BHHH" is returned * maxNR(), maxBHHH(), maxBFGS(), maxNM(), and maxSANN() now additionally return a component "gradientObs" that is the matrix of gradients evaluated at each observation if argument "grad" returns a matrix or argument "grad" is not specified and argument "fn" returns a vector * the definitions of the generic functions nObs() and nParam() have been moved to the "miscTools" package * added methods bread() and estfun() for objects of class "maxLik" (see documentation of the generic functions bread() and estfun() defined in package "sandwich") * replaced argument "activePar" of numericGradient(), numericHessian(), and numericNHessian() by argument "fixed" to be consistent with maxLik(), maxNR(), and the other maxXXX() functions * maxNR(), maxBHHH(), maxBFGSYC(), maxBFGS(), maxNM(), maxSANN(), and summary.maxLik() now return component "fixed" instead of component "activePar" CHANGES IN VERSION 0.7-2 * corrected negative definiteness correction of Hessian in maxNR() which led to infinite loops * changed stopping condition in sumt(): instead of checking whether estimates are stimilar, we check for penalty being low now CHANGES IN VERSION 0.7-0 * Holding parameters fixed in maxNR() (and hence, also in maxBHHH()) should now be done by the new (optional) argument "fixed", because it is convenient to use than the "old" argument "activePar" in many situations. However, the "old" argument "activePar" is kept for backward-compatibility. * added (optional) argument "fixed" to functions maxBFGS(), maxNM(), and maxSANN(), which can be used for holding parameters fixed at their starting values * added function constrOptim2(), which is a modified copy of constrOptim() from the "stats" package, but which includes a bug fix * added optional argument "cand" to function maxSANN(), which can be used to specify a function for generating a new candidate point (passed to argument "gr" of optim()) * added argument "random.seed" to maxSANN() to ensure replicability * several mainly smaller improvements in ML estimations with linear equality and inequality constraints (via sumt() and constrOptim2(), respectively) * several internal changes that make the code easier to maintain CHANGES IN VERSION 0.6-0 * maxLik() can perform maximum likelihood estimations under linear equality and inequality constraints on the parameters now (see documentation of the new argument "constraints"). Please note that estimations under constraints are experimental and have not been thoroughly tested yet. * a new method "stdEr" to extract standard errors of the estimates has been introduced * added a "coef" method for objects of class "summary.maxLik" that extracts the matrix of the estimates, standard errors, t-values, and P-values * some minor bugs have been fixed * we did some general polishing of the returned object and under the hood CHANGES IN VERSION 0.5-12 AND BEFORE * please take a look at the log messages of the SVN repository on R-Forge maxLik/R/0000755000176200001440000000000014077525067011715 5ustar liggesusersmaxLik/R/logLikAttr.R0000644000176200001440000001447514077525067014127 0ustar liggesusers### this function returns the log-likelihood value with gradient and Hessian as ### attributes. If the log-likelihood function provided by the user does not add ### these attributes, this functions uses the functions provided by the user ### as arguments "grad" and "hess" or (if they are not provided) uses the ### finite-difference method to obtain the gradient and Hessian logLikAttr <- function(theta, fnOrig, gradOrig=NULL, hessOrig=NULL, fixed, sumObs = FALSE, returnHessian = TRUE, ...) { ## fixed: logical, which parameters to keep fixed ## # large initial indentation to be able to diff to previous version # that was defined in maxNR() / maxNR.R. ## number of parameters nParam <- length( theta ) ## value of log-likelihood function f <- fnOrig(theta, ...) ## if there are NA-s in the function value, do not ## compute gradient and Hessian if(any(is.na(f))) { attr(f, "gradient") <- NA attr(f, "hessian") <- NA return(f) } ## gradient of log-likelihood function gr <- attr( f, "gradient" ) if( is.null( gr ) ) { if( !is.null( gradOrig ) ) { gr <- gradOrig(theta, ...) } else { gr <- numericGradient(f = fnOrig, t0 = theta, fixed=fixed, ...) } } ## if there are NA-s in active gradient, do not compute Hessian if(is.matrix(gr)) { if(ncol(gr) != length(theta)) { stop(paste0("if gradient is a matrix, it must have length(parameter) colums (currently ", length(theta), "), not ", ncol(gr))) } activeGr <- gr[,!fixed] } else { activeGr <- gr[!fixed] } if(any(is.na(activeGr))) { attr(f, "gradient") <- gr attr(f, "hessian") <- NA return(f) } # if gradients are observation-specific, they must be stored in a matrix if(observationGradient(gr, length(theta))) { gr <- as.matrix(gr) } ## Set gradients of fixed parameters to NA so that they are always NA ## (no matter if they are analytical or finite-difference gradients) if( is.null( dim( gr ) ) ) { gr[ fixed ] <- NA } else { gr[ , fixed ] <- NA } ## Hessian of log-likelihood function if( isTRUE( returnHessian ) ) { h <- attr( f, "hessian" ) if( is.null( h ) ) { if(!is.null(hessOrig)) { h <- as.matrix(hessOrig(theta, ...)) } else { llFunc <- function( theta, ... ) { return( sum( fnOrig( theta, ... ) ) ) } if( !is.null( attr( f, "gradient" ) ) ) { gradFunc <- function( theta, ... ) { return( sumGradients( attr( fnOrig( theta, ... ), "gradient" ), nParam ) ) } } else if( !is.null( gradOrig ) ) { gradFunc <- function( theta, ... ) { return( sumGradients( gradOrig( theta, ... ), nParam ) ) } } else { gradFunc <- NULL } h <- numericHessian(f = llFunc, grad = gradFunc, t0 = theta, fixed=fixed, ...) } } ## Check the correct size of Hessian. if((dim(h)[1] != nParam) | (dim(h)[2] != nParam)) { stop("Wrong hessian dimension. Needed ", nParam, "x", nParam, " but supplied ", dim(h)[1], "x", dim(h)[2]) } else { ## Set elements of the Hessian corresponding to the ## fixed parameters ## to NA so that they are always zero ## (no matter if they are ## calculated analytical or by the finite-difference ## method) h[ fixed, ] <- NA h[ , fixed ] <- NA } } else if( tolower( returnHessian ) == "bhhh" ) { ## We have to return BHHH Hessian. Check if it contains NA in free paramateres, otherwise ## return outer product as Hessian. h <- NULL # to keep track of what we have done if(is.null(dim(gr)) & any(is.na(gr[!fixed]))) { # NA gradient: do not check but send the wrong values to the optimizer. # The optimizer should take corresponding action, such as looking for another value h <- NA } else if(is.matrix(gr)) { if(any(is.na(gr[,!fixed]))) { # NA gradient: do not check but send the wrong values to the optimizer. # The optimizer should take corresponding action, such as looking for another value h <- NA } } if(is.null(h)) { # gr seems not to contain NA-s at free parameters checkBhhhGrad( g = gr, theta = theta, analytic = ( !is.null( attr( f, "gradient" ) ) || !is.null( gradOrig ) ), fixed=fixed) h <- - crossprod( gr ) } attr( h, "type" ) = "BHHH" } else { h <- NULL } ## sum log-likelihood values over observations (if requested) if( sumObs ) { f <- sumKeepAttr( f ) } ## sum gradients over observations (if requested) if( sumObs ) { ## We need just summed gradient gr <- sumGradients( gr, nParam ) } if( !is.null( gradOrig ) && !is.null( attr( f, "gradient" ) ) ) { attr( f, "gradBoth" ) <- TRUE } if( !is.null( hessOrig ) && !is.null( attr( f, "hessian" ) ) ) { attr( f, "hessBoth" ) <- TRUE } attr( f, "gradient" ) <- gr attr( f, "hessian" ) <- h return( f ) } maxLik/R/maxCG.R0000644000176200001440000000266614077525067013051 0ustar liggesusersmaxCG <- function(fn, grad=NULL, hess=NULL, start, fixed = NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ...) { ## Wrapper of optim-based 'Conjugate Gradient' optimization ## ## contraints constraints to be passed to 'constrOptim' ## hessian: how (and if) to calculate the final Hessian: ## FALSE not calculate ## TRUE use analytic/numeric Hessian ## bhhh/BHHH use information equality approach ## ... : further arguments to fn() ## ## Note: grad and hess are for compatibility only, SANN uses only fn values ## if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(iterlim=500), control) # default values } else { mControl <- control } # default, user values mControl <- addControlList(mControl, list(...), check=FALSE) # open values result <- maxOptim( fn = fn, grad = grad, hess = hess, start = start, method = "CG", fixed = fixed, constraints = constraints, finalHessian=finalHessian, parscale = parscale, control=mControl, ... ) return(result) } maxLik/R/addFixedPar.R0000644000176200001440000000025414077525067014214 0ustar liggesusersaddFixedPar <- function( theta, start, fixed, ...) { if( is.null( fixed ) ) { start <- theta } else { start[ !fixed ] <- theta } return( start ) } maxLik/R/maxNR.R0000644000176200001440000001233214077525067013066 0ustar liggesusersmaxNR <- function(fn, grad=NULL, hess=NULL, start, constraints=NULL, finalHessian=TRUE, bhhhHessian=FALSE, fixed=NULL, activePar=NULL, control=NULL, ...) { ## Newton-Raphson maximisation ## Parameters: ## fn - the function to be minimized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## grad - gradient function (numeric used if missing). Must return either ## * vector, length=nParam ## * matrix, dim=c(nObs, 1). Treated as vector ## * matrix, dim=c(M, nParam), where M is arbitrary. In this case the ## rows are simply summed (useful for maxBHHH). ## hess - hessian function (numeric used if missing) ## start - initial parameter vector (eventually w/names) ## ... - extra arguments for fn() ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## activePar - an index vector -- which parameters are taken as ## variable (free). Other paramters are treated as ## fixed constants ## fixed index vector, which parameters to keep fixed ## ## RESULTS: ## a list of class "maxim": ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success: ## 1 - gradient close to zero ## 2 - successive values within tolerance limit ## 3 - could not find a higher point (step error) ## 4 - iteration limit exceeded ## 100 - initial value out of range ## message character message describing the code ## last.step only present if code == 3 (step error). A list with following components: ## theta0 - parameter value which led to the error ## f0 - function value at these parameter values ## climb - the difference between theta0 and the new approximated parameter value (theta1) ## activePar - logical vector, which parameters are active (not constant) ## activePar logical vector, which parameters were treated as free (resp fixed) ## iterations number of iterations ## type "Newton-Raphson maximisation" ## ## ------------------------------ ## Add parameters from ... to control if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(), control) } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## argNames <- c(c("fn", "grad", "hess", "start", "activePar", "fixed", "control"), openParam(mControl)) # Here we allow to submit all parameters outside of the # 'control' list. May eventually include only a # subset here ## checkFuncArgs( fn, argNames, "fn", "maxNR" ) if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxNR" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxNR" ) } ## establish the active parameters. Internally, we just use 'activePar' fixed <- prepareFixed( start = start, activePar = activePar, fixed = fixed ) ## chop off the control args from ... and forward the new ... dddot <- list(...) dddot <- dddot[!(names(dddot) %in% openParam(mControl))] cl <- list(start=start, finalHessian=finalHessian, bhhhHessian=bhhhHessian, fixed=fixed, control=mControl) if(length(dddot) > 0) { cl <- c(cl, dddot) } ## if(is.null(constraints)) { ## call maxNRCompute with the modified ... list cl <- c(quote(maxNRCompute), fn=logLikAttr, fnOrig = fn, gradOrig = grad, hessOrig = hess, cl) result <- eval(as.call(cl)) } else { if(identical(names(constraints), c("ineqA", "ineqB"))) { stop("Inequality constraints not implemented for maxNR") } else if(identical(names(constraints), c("eqA", "eqB"))) { # equality constraints: A %*% beta + B = 0 cl <- c(quote(sumt), fn=fn, grad=grad, hess=hess, maxRoutine=maxNR, constraints=list(constraints), cl) result <- eval(as.call(cl)) } else { stop("maxNR only supports the following constraints:\n", "constraints=list(ineqA, ineqB)\n", "\tfor A %*% beta + B >= 0 linear inequality constraints\n", "current constraints:", paste(names(constraints), collapse=" ")) } } ## Save the objective function result$objectiveFn <- fn ## return( result ) } maxLik/R/logLik.maxLik.R0000644000176200001440000000047314077525067014511 0ustar liggesusers### Methods for accessing loglik value maximum likelihood estimates logLik.summary.maxLik <- function( object, ...) { ll <- object$loglik attr(ll, "df") <- sum(activePar(object)) ll } logLik.maxLik <- function( object, ...) { ll <- maxValue(object) attr(ll, "df") <- sum(activePar(object)) ll } maxLik/R/nParam.R0000644000176200001440000000040214077525067013252 0ustar liggesusers## Return the #of parameters of model nParam.maxim <- function(x, free=FALSE, ...) { if(!inherits(x, "maxim")) { stop("'nParam.maxim' called on non-'maxim' object") } if(free) sum( activePar( x ) ) else length( x$estimate ) } maxLik/R/condiNumber.R0000644000176200001440000000273114077525067014310 0ustar liggesusers### condiNumber: print matrix' condition number adding columns one by one. ### In this way user may investigate the which columns cause problems with singularity condiNumber <- function(x, ...) UseMethod("condiNumber") condiNumber.default <- function(x, exact=FALSE, norm=FALSE, printLevel=print.level, print.level=1, digits = getOption( "digits" ), ... ) { ## x: a matrix, condition number of which are to be printed ## exact: whether the condition number have to be exact or approximated (see 'kappa') ## norm: whether to normalise the matrix' columns. ## printLevel: whether to print the condition numbers while calculating. Useful for interactive testing. savedDigits <- getOption("digits") options( digits = digits ) if(dim(x)[2] > dim(x)[1]) { warning(paste(dim(x)[1], "rows and", dim(x)[2], "columns, use transposed matrix")) x <- t(x) } cn <- numeric(ncol(x)) if(norm) { # Now normalise column vectors x <- apply(x, 2, FUN=function(v) v/sqrt(sum(v*v))) } for(i in seq(length=ncol(x))) { m <- x[,1:i] cn[i] <- kappa(m, exact=exact) if(printLevel > 0) cat(colnames(x)[i], "\t", cn[i], "\n") } names(cn) <- colnames(x) options( digits = savedDigits ) invisible(cn) } condiNumber.maxLik <- function(x, ...) condiNumber.default( x = hessian(x)[activePar(x), activePar(x),drop=FALSE], ... ) maxLik/R/tidyMethods.R0000644000176200001440000000142014077525067014332 0ustar liggesusers require_tibble_package <- function () { if (! requireNamespace("tibble", quietly = TRUE)) { stop("The `tibble` package must be installed to use tidy() or glance() methods") } } tidy.maxLik <- function (x, ...) { require_tibble_package() s <- summary(x) ret <- tibble::as_tibble(s$estimate, rownames = "term") colnames(ret) <- c("term", "estimate", "std.error", "statistic", "p.value") ret } glance.maxLik <- function (x, ...) { require_tibble_package() ll <- logLik(x) nobs <- tryCatch(nObs(x), error = function(e) NA) # nobs = NA in case of error ret <- tibble::tibble( df = attr(ll, "df"), logLik = as.numeric(ll), AIC = AIC(x), nobs = nobs ) ret } maxLik/R/callWithoutArgs.R0000644000176200001440000000051214077525067015152 0ustar liggesusers## strip arguments "args" and call the function with name "fName" thereafter callWithoutArgs <- function(theta, fName, args, ...) { f <- match.call() f[ args ] <- NULL f[[1]] <- as.name(fName) names(f)[2] <- "" f[["fName"]] <- NULL f[["args"]] <- NULL f1 <- eval(f, sys.frame(sys.parent())) return( f1 ) } maxLik/R/observationGradient.R0000644000176200001440000000053714077525067016056 0ustar liggesusers ### The function tests whether a given gradient is given ### observation-wise. It tests essentially the # of rows ### in the gradient observationGradient <- function(g, nParam) { if(is.null(dim(g))) { if(nParam == 1 & length(g) > 1) return(TRUE) return(FALSE) } if(nrow(g) == 1) return(FALSE) return(TRUE) } maxLik/R/maxSANN.R0000644000176200001440000000356614077525067013317 0ustar liggesusersmaxSANN <- function(fn, grad=NULL, hess=NULL, start, fixed = NULL, control=NULL, constraints = NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ... ) { ## Wrapper of optim-based 'SANN' optimization ## ## contraints constraints to be passed to 'constrOptim' ## finalHessian: how (and if) to calculate the final Hessian: ## FALSE not calculate ## TRUE use analytic/numeric Hessian ## bhhh/BHHH use information equality approach ## ## ... : further arguments to fn() ## ## Note: grad and hess are for compatibility only, SANN uses only fn values if(!inherits(control, "MaxControl")) { mControl <- maxControl(iterlim=10000L) mControl <- addControlList(mControl, control) # default values } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## save seed of the random number generator if( exists( ".Random.seed" ) ) { savedSeed <- .Random.seed } # set seed for the random number generator (used by 'optim( method="SANN" )') set.seed(slot(mControl, "sann_randomSeed")) # restore seed of the random number generator on exit # (end of function or error) if( exists( "savedSeed" ) ) { on.exit( assign( ".Random.seed", savedSeed, envir = sys.frame() ) ) } else { on.exit( rm( .Random.seed, envir = sys.frame() ) ) } result <- maxOptim( fn = fn, grad = grad, hess = hess, start = start, method = "SANN", fixed = fixed, constraints = constraints, finalHessian=finalHessian, parscale = parscale, control=mControl, ... ) return(result) } maxLik/R/zzz.R0000644000176200001440000000157414077525067012704 0ustar liggesusers.onAttach <- function( libname, pkgname ) { packageStartupMessage( paste0( "\nPlease cite the 'maxLik' package as:\n", "Henningsen, Arne and Toomet, Ott (2011). ", "maxLik: A package for maximum likelihood estimation in R. ", "Computational Statistics 26(3), 443-458. ", "DOI 10.1007/s00180-010-0217-1.\n\n", "If you have questions, suggestions, or comments ", "regarding the 'maxLik' package, ", "please use a forum or 'tracker' at maxLik's R-Forge site:\n", "https://r-forge.r-project.org/projects/maxlik/"), domain = NULL, appendLF = TRUE ) } .onLoad <- function(libname, pkgname) { ## max rows and columns to output when printing matrices/vectors options(max.rows = 20L, max.cols = 7L) } .onUnload <- function(libpath) { .Options$max.rows <- NULL .Options$max.cols <- NULL } maxLik/R/headDots.R0000644000176200001440000000034414077525067013574 0ustar liggesusers### paste head of vector, and if some of it is left out, add '...' to it. headDots <- function(x, max.cols) { s <- paste(head(x, max.cols), collapse=", ") if(length(x) > max.cols) { s <- paste(s, "...") } s } maxLik/R/summary.maxim.R0000644000176200001440000000612714077525067014655 0ustar liggesusersprint.summary.maxim <- function( x, max.rows=getOption("max.rows", 20), max.cols=getOption("max.cols", 7), ... ) { summary <- x cat("--------------------------------------------\n") cat(summary$type, "\n") cat("Number of iterations:", summary$iterations, "\n") cat("Return code:", summary$code, "\n") cat(summary$message, "\n") if(!is.null(summary$unsucc.step)) { cat("Last (unsuccessful) step: function value", summary$unsucc.step$value, "\n") print(summary$unsucc.step$parameters) } if(!is.null(summary$estimate)) { cat("Function value:", summary$maximum, "\n") cat("Estimates:\n") printRowColLimits(summary$estimate, max.rows, max.cols, ...) if(!is.null(summary$hessian)) { cat("Hessian:\n") printRowColLimits(summary$hessian, max.rows, max.cols, ...) } } if(!is.null(summary$constraints)) { cat("\nConstrained optimization based on", summary$constraints$type, "\n") if(!is.null(summary$constraints$code)) cat("Return code:", summary$constraints$code, "\n") # note: this is missing for 'constrOptim' if(!is.null(summary$constraints$message)) cat(summary$constraints$message, "\n") # note: this is missing for 'constrOptim' cat(summary$constraints$outer.iterations, " outer iterations, barrier value", summary$constraints$barrier.value, "\n") } cat("--------------------------------------------\n") } summary.maxim <- function(object, hessian=FALSE, unsucc.step=FALSE, ... ) { ## The object of class "maxim" should include following components: ## maximum : function value at optimum ## estimate : matrix, estimated parameter values and gradient at optimum ## hessian : hessian ## code : code of convergence ## message : message, description of the code ## last.step : information about last step, if unsuccessful ## iterations : number of iterations ## type : type of optimisation ## nParam <- length(object$estimate) if(object$code == 3 & unsucc.step) { a <- cbind(object$last.step$theta0, object$last.step$theta1) dimnames(a) <- list(parameter=object$names, c("current par", "new par")) unsucc.step <- list(value=object$last.step$f0, parameters=a) } else { unsucc.step <- NULL } estimate <- cbind("estimate"=object$estimate, "gradient"=object$gradient) if(hessian) { H <- object$hessian } else { H <- NULL } summary <- list(maximum=object$maximum, type=object$type, iterations=object$iterations, code=object$code, message=object$message, unsucc.step=unsucc.step, estimate=estimate, hessian=H, constraints=object$constraints) class(summary) <- c("summary.maxim", class(summary)) summary } maxLik/R/maxSGACompute.R0000644000176200001440000003047114077525067014522 0ustar liggesusersmaxSGACompute <- function(fn, grad, hess, start, nObs, finalHessian=FALSE, bhhhHessian = FALSE, fixed=NULL, control=maxControl(), optimizer="SGA", # type of optimizer: SGA, Adam ...) { ## Stochastic Gradient Ascent: implements ## * SGA with momentum ## * Adam ## Parameters: ## fn - the function to be maximized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## start - initial parameter vector (eventually w/names) ## control MaxControl object: ## The stopping criteria ## tol - maximum allowed absolute difference between sequential values ## reltol - maximum allowed reltive difference (stops if < reltol*(abs(fn) + reltol) ## gradtol - maximum allowed norm of gradient vector ## ## iterlim - maximum # of iterations ## ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## fixed - a logical vector -- which parameters are taken as fixed. ## Other paramters are treated as variable (free). ## ... additional argument to 'fn'. This may include ## 'fnOrig', 'gradOrig', 'hessOrig' if called fromm ## 'maxNR'. ## ## RESULTS: ## an object of class 'maxim' ## ## ------------------------------------------------- maximType <- "Stochastic Gradient Ascent" iterlim <- slot(control, "iterlim") nParam <- length(start) start1 <- start storeParameters <- slot(control, "storeParameters") storeValues <- slot(control, "storeValues") learningRate <- slot(control, "SG_learningRate") clip <- slot(control, "SG_clip") max.rows <- slot(control, "max.rows") max.cols <- slot(control, "max.cols") patience <- slot(control, "SG_patience") patienceStep <- slot(control, "SG_patienceStep") printLevel <- slot(control, "printLevel") batchSize <- slot(control, "SG_batchSize") if(optimizer == "Adam") { maximType <- "Stochastic Gradient Ascent/Adam" Adam.momentum1 <- slot(control, "Adam_momentum1") Adam.momentum2 <- slot(control, "Adam_momentum2") Adam.delta <- 1e-8 # maybe make it a parameter in the future Adam.s <- 0 Adam.r <- 0 Adam.time <- 0 } else if(optimizer == "SGA") { momentum <- slot(control, "SGA_momentum") v <- 0 # velocity that retains the momentum } else { stop(paste("unknown optimizer", optimizer)) } ## ---------- How many batches if(is.null(batchSize)) { nBatches <- 1 index <- seq(from=1, to=nObs, by=nBatches) } else { nBatches <- max(1L, nObs %/% batchSize) # ensure that we get at least one batch if batchSize set too large shuffledIndex <- sample(nObs, nObs) index <- shuffledIndex[seq(from=1, to=nObs, by=nBatches)] } ## f1 <- NULL # mark that we haven't computed the fcn value if(printLevel > 0) { f1 <- fn(start, fixed = fixed, sumObs = TRUE, index=index, ...) cat("Initial function value:", f1, "\n") if( isTRUE( attr( f1, "gradBoth" ) ) ) { warning( "the gradient is provided both as attribute 'gradient' and", " as argument 'grad': ignoring argument 'grad'" ) } if( isTRUE( attr( f1, "hessBoth" ) ) ) { warning( "the Hessian is provided both as attribute 'hessian' and", " as argument 'hess': ignoring argument 'hess'" ) } } if(!is.null(patience)) { if(is.null(f1)) { f1 <- fn(start, fixed = fixed, sumObs = TRUE, index=index, ...) } fBest <- f1 # remember the previous best value paramBest <- start patienceCount <- 0 # how many times have we hit a worse outcome } G1 <- grad(start, fixed = fixed, sumObs = TRUE, index=index, ...) # have to compute fn as we cannot get gradient otherwise if(any(is.na(G1[!fixed]))) { stop("NA in the initial gradient") } if(any(is.infinite(G1[!fixed]))) { stop("Infinite initial gradient") } if(length(G1) != nParam) { stop( "length of gradient (", length(G1), ") not equal to the no. of parameters (", nParam, ")" ) } if(length(clip) > 0) { if((norm2 <- sum(G1*G1)) > clip) G1 <- G1/sqrt(norm2)*sqrt(clip) } if(storeValues) { valueStore <- rep(NA_real_, iterlim + 1) if(is.null(f1)) { f1 <- fn(start, fixed = fixed, sumObs = TRUE, index=index, ...) } valueStore[1] <- f1 } if(storeParameters) { parameterStore <- matrix(NA_real_, iterlim + 1, nParam) dimnames(parameterStore) <- list(epoch=c("start", 1:iterlim), parameter=names(start)) parameterStore[1,] <- start } if(printLevel > 1) { cat( "----- Initial parameters: -----\n") cat( "fcn value:", as.vector(f1), "\n") a <- cbind(start1, G1, as.integer(!fixed)) dimnames(a) <- list(names(start1), c("parameter", "initial gradient", "free")) printRowColLimits(a, max.rows, max.cols) } ## ---------------- Main interation loop ------------------------ iter <- 0L ## we do not need to compute the function itself here, except for ## printing repeat { # repeat over epochs ## break here if iterlim == 0 if( iter >= iterlim) { code <- 4; break } ## break here to avoid potentially costly gradient computation if( iter >= slot(control, "iterlim")) { code <- 4; break } iter <- iter + 1L if(printLevel > 1) { cat( "----- epoch", iter, "-----\n") } for(iBatch in 1:nBatches) { # repeat over minibatches if(!is.null(batchSize)) { index <- shuffledIndex[seq(from=iBatch, to=nObs, by=nBatches)] } start0 <- start1 G0 <- G1 if(any(is.na(G0[!fixed]))) { stop("NA in gradient") } if(optimizer == "SGA") { v <- momentum*v + learningRate*G0 start1 <- start0 + v } else if(optimizer == "Adam") { Adam.time <- Adam.time + 1 Adam.s <- Adam.momentum1*Adam.s + (1 - Adam.momentum1)*G0 Adam.r <- Adam.momentum2*Adam.r + (1 - Adam.momentum2)*G0*G0 Adam.shat <- Adam.s/(1 - Adam.momentum1^Adam.time) Adam.rhat <- Adam.r/(1 - Adam.momentum2^Adam.time) v <- learningRate*Adam.shat/(sqrt(Adam.rhat) + Adam.delta) start1 <- start0 + v } f1 <- NULL # we are at a new location, mark that we haven't computed the f1 values ## still iterations to go, hence compute gradient G1 <- grad(start1, fixed = fixed, sumObs = TRUE, index=index, ...) if(any(is.na(G1[!fixed])) || any(is.infinite(G1[!fixed]))) { cat("Iteration", iter, "\n") cat("Parameter:\n") print(headDots(start1, max.cols), quote=FALSE) cat("Gradient:\n") printRowColLimits(G1, max.rows, max.cols) stop("NA/Inf in gradient") } if(length(clip) > 0) { if((norm2 <- sum(G1*G1)) > clip) # compute norm w/o cross-product as grad may not be a vector G1 <- G1/sqrt(norm2*clip) } ## print every batch if someone wants... if(printLevel > 4) { f1 <- fn(start1, fixed = fixed, sumObs = TRUE, index=index, ...) cat(" - batch", iBatch, "index", index, "learning rate", learningRate, " fcn value:", formatC(as.vector(f1), digits=8, format="f"), "\n") a <- cbind(learningRate*G0, start1, G1, as.integer(!fixed)) dimnames(a) <- list(names(start0), c("delta-v", "param", "gradient", "active")) printRowColLimits(a, max.rows, max.cols) } if(any(is.infinite(G1))) { code <- 6; break; } } # end of repeat over batches if(storeValues) { ## store last value of the epoch if(is.null(f1)) { f1 <- fn(start1, fixed = fixed, sumObs = TRUE, index=index, ...) } valueStore[iter + 1L] <- c(f1) # c removes dimensions and attributes } if(storeParameters) { ## store last value of the epoch parameterStore[iter + 1L,] <- c(start1) # c removes dimensions and attributes } if(slot(control, "printLevel") > 2) { if(is.null(f1)) { f1 <- fn(start1, fixed = fixed, sumObs = TRUE, index=index, ...) } cat(" learning rate", learningRate, " fcn value:", formatC(as.vector(f1), digits=8, format="f"), "\n") a <- cbind(learningRate*G0, start1, G1, as.integer(!fixed)) dimnames(a) <- list(names(start0), c("amount", "param", "gradient", "active")) printRowColLimits(a, max.rows, max.cols) } ## stopping criteria if( sqrt( crossprod( G1[!fixed] ) ) < slot(control, "gradtol") ) { code <-1; break } if(!is.null(patience) && (iter %% patienceStep == 0)) { if(is.null(f1)) { f1 <- fn(start1, fixed = fixed, sumObs = TRUE, index=index, ...) } if(f1 < fBest) { patienceCount <- patienceCount + 1 } else { patienceCount <- 0 fBest <- f1 paramBest <- start1 } if(patienceCount > patience) { code <- 10 f1 <- fBest start1 <- paramBest break } } } # main iteration loop over epochs if(printLevel > 0) { cat( "--------------\n") cat( maximMessage( code), "\n") cat( iter, " iterations\n") cat( "estimate:", headDots(start1, max.cols), "\n") if(is.null(f1)) { f1 <- fn(start1, fixed = fixed, sumObs = TRUE, index=index, ...) } cat( "Function value:", f1, "\n") } if(finalHessian & !bhhhHessian) { G1 <- grad( start1, fixed = fixed, sumObs = FALSE, index=index, ... ) } if(observationGradient(G1, length(start1))) { gradientObs <- G1 colnames( gradientObs ) <- names(start1) G1 <- colSums(as.matrix(G1 )) } else { gradientObs <- NULL } names( G1 ) <- names(start1) ## calculate (final) Hessian if(tolower(finalHessian) == "bhhh") { if(!is.null(gradientObs)) { hessian <- - crossprod( gradientObs ) attr(hessian, "type") <- "BHHH" } else { hessian <- NULL warning("For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations") } } else if( finalHessian != FALSE ) { hessian <- hess( start1, fixed = fixed, index=index, ... ) } else { hessian <- NULL } if( !is.null( hessian ) ) { rownames( hessian ) <- colnames( hessian ) <- names(start1) } ## remove attributes from final value of objective (likelihood) function attributes( f1 )$gradient <- NULL attributes( f1 )$hessian <- NULL attributes( f1 )$gradBoth <- NULL attributes( f1 )$hessBoth <- NULL ## result <- list( maximum = unname( drop( f1 ) ), estimate=start1, gradient=drop(G1), hessian=hessian, code=code, message=maximMessage( code), fixed=fixed, iterations=iter, type=maximType, valueStore = if(storeValues) valueStore else NULL, parameterStore = if(storeParameters) parameterStore else NULL ) if( exists( "gradientObs" ) ) { result$gradientObs <- gradientObs } result <- c(result, control=control) # attach the control parameters ## class(result) <- c("maxim", class(result)) invisible(result) } maxLik/R/logLikFunc.R0000644000176200001440000000251314077525067014076 0ustar liggesusersif( getRversion() >= "2.15.1" ) { globalVariables( c( "lastFuncGrad", "lastFuncParam" ) ) } ## objective function: ## sum over possible individual likelihoods logLikFunc <- function(theta, fnOrig, # the original user-supplied function we wrap here gradOrig, hessOrig, # Arguments "gradOrig" and "hessOrig" are just for compatibility with # logLikGrad() and logLikHess() start = NULL, fixed = NULL, sumObs = TRUE, ...) { if(missing(fnOrig)) { stop("Cannot compute the objective function value: no objective function supplied") } theta <- addFixedPar( theta = theta, start = start, fixed = fixed, ...) result <- fnOrig( theta, ... ) ## save gradients and the corresponding parameter values assign( "lastFuncGrad", attr( result, "gradient" ), inherits = TRUE ) assign( "lastFuncParam", theta, inherits = TRUE ) if( sumObs ) { result <- sumKeepAttr( result ) g <- attributes( result )$gradient if( !is.null( g ) ) { g <- sumGradients( g, length( theta ) ) names( g ) <- names( theta ) if( !is.null( fixed ) ) { g <- g[ !fixed ] } attributes( result )$gradient <- g } } return( result ) } maxLik/R/showMaxControl.R0000644000176200001440000000111114077525067015021 0ustar liggesusers showMaxControl <- function(object) { cat("A 'MaxControl' object with slots:\n") for(s in slotNames(object)) { if(s == "sann_cand") { ## This is a function or NULL, handle with care: if(is.null(slot(object, s))) { cat("sann_cand = \n") } else { cat("sann_cand =\n") print(str(slot(object, s))) } } else { ## Just print cat(s, "=", slot(object, s), "\n") } } } setMethod("show", "MaxControl", showMaxControl) maxLik/R/maxNM.R0000644000176200001440000000253514077525067013065 0ustar liggesusersmaxNM <- function(fn, grad=NULL, hess=NULL, start, fixed = NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ...) { ## Wrapper of optim-based 'Nelder-Mead' optimization ## ## contraints constraints to be passed to 'constrOptim' ## hessian: how (and if) to calculate the final Hessian: ## FALSE not calculate ## TRUE use analytic/numeric Hessian ## bhhh/BHHH use information equality approach ## ... : further arguments to fn() ## ## Note: grad and hess are for compatibility only, SANN uses only fn values if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(iterlim=500L), control) # default values } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## result <- maxOptim( fn = fn, grad = grad, hess = hess, start = start, method = "Nelder-Mead", fixed = fixed, constraints = constraints, finalHessian=finalHessian, parscale = parscale, control=mControl, ... ) return(result) } maxLik/R/maxValue.R0000644000176200001440000000014714077525067013624 0ustar liggesusersmaxValue <- function(x, ...) UseMethod("maxValue") maxValue.maxim <- function(x, ...) x$maximum maxLik/R/20-maxControl.R0000644000176200001440000000114014077525067014401 0ustar liggesusers ### Default constructor of MaxControl object: ### take a list of parameters and overwrite the default values maxControl.default <- function(...) { result <- new("MaxControl") result <- addControlDddot(result, ...) return(result) } ### Standard method for any arguments setGeneric("maxControl", function(x, ...) standardGeneric("maxControl") ) ### Method for 'maxim' objects: fetch the stored MaxControl setMethod("maxControl", "maxim", function(x, ...) x$control) ### Method for missing arguments: just default values setMethod("maxControl", "missing", maxControl.default) maxLik/R/coef.maxLik.R0000644000176200001440000000036514077525067014204 0ustar liggesuserscoef.maxim <- function( object, ... ) { return( object$estimate ) } coef.maxLik <- function( object, ... ) { return( object$estimate ) } coef.summary.maxLik <- function( object, ... ) { result <- object$estimate return( result ) } maxLik/R/storedValues.R0000644000176200001440000000033314077525067014517 0ustar liggesusers## Return the stored values in 'maxim' object storedValues <- function(x, ...) ## stored optimization values at each iteration UseMethod("storedValues") storedValues.maxim <- function(x, ...) x$valueStore maxLik/R/maxBFGS.R0000644000176200001440000000244214077525067013271 0ustar liggesusersmaxBFGS <- function(fn, grad=NULL, hess=NULL, start, fixed = NULL, control=NULL, constraints=NULL, finalHessian=TRUE, parscale=rep(1, length=length(start)), ## sumt parameters ...) { ## Wrapper of optim-based 'BFGS' optimization ## ## contraints constraints to be passed to 'constrOptim' ## finalHessian: how (and if) to calculate the final Hessian: ## FALSE not calculate ## TRUE use analytic/numeric Hessian ## bhhh/BHHH use information equality approach ## ## ... further arguments to fn() and grad() if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(iterlim=200), control) # default values } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) result <- maxOptim( fn = fn, grad = grad, hess = hess, start = start, method = "BFGS", fixed = fixed, constraints = constraints, finalHessian=finalHessian, parscale = parscale, control=mControl, ... ) return(result) } maxLik/R/maxLik.R0000644000176200001440000000616414077525067013274 0ustar liggesusersmaxLik <- function(logLik, grad=NULL, hess=NULL, start, method, constraints=NULL, ...) { ## Maximum Likelihood estimation. ## ## Newton-Raphson maximisation ## Parameters: ## logLik log-likelihood function. First argument must be the vector of parameters. ## grad gradient of log-likelihood. If NULL, numeric gradient is used. Must return either ## * vector, length=nParam ## * matrix, dim=c(nObs, 1). Treated as vector ## * matrix, dim=c(nObs, nParam). In this case the rows are simply ## summed (useful for maxBHHH). ## hess Hessian function (numeric used if NULL) ## start initial vector of parameters (eventually w/names) ## method maximisation method (Newton-Raphson) ## constraints constrained optimization: a list (see below) ## ... additional arguments for the maximisation routine ## ## RESULTS: ## list of class c("maxLik", "maxim"). This is in fact equal to class "maxim", just the ## methods are different. ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success, depends on the optimization ## method ## message character message describing the code ## type character, type of optimization ## ## there may be more components, depending on the choice of ## the algorith. ## argNames <- c( "logLik", "grad", "hess", "start", "method", "constraints" ) checkFuncArgs( logLik, argNames, "logLik", "maxLik" ) if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxLik" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxLik" ) } ## Constrained optimization. We can two possibilities: ## * linear equality constraints ## * linear inequality constraints ## if(missing(method)) { if(is.null(constraints)) { method <- "nr" } else if(identical(names(constraints), c("ineqA", "ineqB"))) { if(is.null(grad)) method <- "Nelder-Mead" else method <- "BFGS" } else method <- "nr" } maxRoutine <- switch(tolower(method), "newton-raphson" =, "nr" = maxNR, "bfgs" = maxBFGS, "bfgsr" =, "bfgs-r" = maxBFGSR, "bhhh" = maxBHHH, "conjugate-gradient" =, "cg" = maxCG, "nelder-mead" =, "nm" = maxNM, "sann" = maxSANN, stop( "Maxlik: unknown maximisation method ", method ) ) result <- maxRoutine(fn=logLik, grad=grad, hess=hess, start=start, constraints=constraints, ...) class(result) <- c("maxLik", class(result)) result } maxLik/R/printRowColLimits.R0000644000176200001440000000164714077525067015514 0ustar liggesusers### print vector/matrix while limiting the number of rows/columns printed printRowColLimits <- function(x, max.rows=getOption("max.rows", 20), max.cols=getOption("max.cols", 7), ... # other arguments to 'print.matrix' ) { x1 <- x msg <- NULL if(is.null(dim(x))) { x1 <- matrix(x, nrow=1) colnames(x1) <- names(x) x <- x1 } ## we have a matrix (higher-D arrays not supported) if(ncol(x) > max.cols) { x1 <- x[, seq(length=max.cols), drop=FALSE] msg <- paste(msg, "reached getOption(\"max.cols\") -- omitted", ncol(x) - max.cols, "columns\n") } print(head(x1, max.rows), ...) if(nrow(x) > max.rows) { msg <- paste(msg, "reached getOption(\"max.rows\") -- omitted", nrow(x) - max.rows, "rows\n") } cat(msg) } maxLik/R/prepareFixed.R0000644000176200001440000000572614077525067014470 0ustar liggesusersprepareFixed <- function( start, activePar, fixed ) { nParam <- length( start ) ## establish the active parameters. if(!is.null(fixed)) { if(!is.null(activePar)) { if(!all(activePar)) { warning("Both 'activePar' and 'fixed' specified. 'activePar' ignored") } } if( is.logical( fixed ) ) { if( length ( fixed ) != length( start ) || !is.null( dim( fixed ) ) ) { stop( "if fixed parameters are specified using logical values,", " argument 'fixed' must be a logical vector", " with one element for each parameter", " (number of elements in argument 'start')" ) } activePar <- !fixed } else if( is.numeric( fixed ) ) { if( length ( fixed ) >= length( start ) || !is.null( dim( fixed ) ) ) { stop( "if fixed parameters are specified using their positions,", " argument 'fixed' must be a numerical vector", " with less elements than the number of parameters", " (number of elements in argument 'start'" ) } else if( min( fixed ) < 1 || max(fixed ) > length( start ) ) { stop( "if fixed parameters are specified using their positions,", " argument 'fixed' must have values between 1 and", " the total number of parameter", " (number of elements in argument 'start'" ) } activePar <- ! c( 1:length( start ) ) %in% fixed } else if( is.character( fixed ) ) { if( length ( fixed ) >= length( start ) || !is.null( dim( fixed ) ) ) { stop( "if fixed parameters are specified using their names,", " argument 'fixed' must be a vector of character strings", " with less elements than the number of parameters", " (number of elements in argument 'start'" ) } else if( is.null( names( start ) ) ) { stop( "if fixed parameters are specified using their names,", " parameter names have to be specified in argument 'start'" ) } else if( any( ! names( fixed ) %in% names( start ) ) ) { stop( "if fixed parameters are specified using their names,", " all parameter names specified in argument 'fixed'", " must be specified in argument 'start'" ) } activePar <- ! names( start ) %in% fixed } else { stop( "argument 'fixed' must be either a logical vector,", " a numeric vector, or a vector of character strings" ) } } else { if( is.null( activePar ) ) { activePar <- rep( TRUE, length( start ) ) } else if(is.numeric(activePar)) { a <- rep(FALSE, nParam) a[activePar] <- TRUE activePar <- a } } names( activePar ) <- names( start ) if( all( !activePar ) ){ stop( "At least one parameter must not be fixed", " using argument 'fixed'" ) } return( !activePar ) }maxLik/R/nObs.R0000644000176200001440000000131614077525067012742 0ustar liggesusers## Return #of observations for models nObs.maxLik <- function(x, ...) { if( is.null( x$gradientObs ) ) { stop( "cannot return the number of observations:", " please re-run 'maxLik' and", " provide a gradient function using argument 'grad' or", " (if no gradient function is specified) a log-likelihood function", " using argument 'logLik'", " that return the gradients or log-likelihood values, respectively,", " at each observation" ) } else if( is.matrix( x$gradientObs ) ) { return( nrow( x$gradientObs ) ) } else { stop( "internal error: component 'gradientObs' is not a matrix.", " Please contact the developers." ) } } maxLik/R/maximMessage.R0000644000176200001440000000230214077525067014455 0ustar liggesusersmaximMessage <- function(code) { message <- switch(code, "1" = "gradient close to zero (gradtol)", "2" = "successive function values within tolerance limit (tol)", "3" = paste("Last step could not find a value above the", "current.\nBoundary of parameter space?", " \nConsider switching to a more robust optimisation method temporarily."), "4" = "Iteration limit exceeded (iterlim)", "5" = "Infinite value", "6" = "Infinite gradient", "7" = "Infinite Hessian", "8" = "successive function values within relative tolerance limit (reltol)", "9" = paste("Gradient did not change,", "cannot improve BFGS approximation for the Hessian.\n", "Use different optimizer and/or analytic gradient."), "10" = "Lost patience (SG_patience)", "100" = "Initial value out of range.", paste("Code", code)) return(message) } maxLik/R/estfun.maxLik.R0000644000176200001440000000104514077525067014570 0ustar liggesusersestfun.maxLik <- function( x, ... ) { if( is.null( x$gradientObs ) ) { stop( "cannot return the gradients of the log-likelihood function", " evaluated at each observation: please re-run 'maxLik' and", " provide a gradient function using argument 'grad' or", " (if no gradient function is specified) a log-likelihood function", " using argument 'logLik'", " that return the gradients or log-likelihood values, respectively,", " at each observation" ) } return( x$gradientObs ) } maxLik/R/05-classes.R0000644000176200001440000000007114077525067013715 0ustar liggesusers## first to be loaded: setOldClass(c("maxLik", "maxim")) maxLik/R/sumt.R0000644000176200001440000001751214077525067013036 0ustar liggesusers### SUMT (Sequential Unconstrained Maximization Technique) ### borrowed from package 'clue' ### ### Adapted for linear constraints sumt <- function(fn, grad=NULL, hess=NULL, start, maxRoutine, constraints, SUMTTol = sqrt(.Machine$double.eps), # difference between estimates for successive outer iterations SUMTPenaltyTol = sqrt(.Machine$double.eps), # maximum allowed penalty SUMTQ = 10, SUMTRho0 = NULL, printLevel=print.level, print.level=0, SUMTMaxIter=100, ...) { ## constraints list w/components eqA and eqB. Maximization will ## be performed wrt to the constraint ## A %*% theta + B = 0 ## The user must ensure the matrices are in correct ## form ## maxSUMTiter how many SUMT iterations to perform max ## penalty <- function(theta) { p <- A %*% theta + B sum(p*p) } ## Penalty gradient and Hessian are used only if corresponding function ## for the likelihood function is provided gPenalty <- function(theta) { 2*(t(theta) %*% t(A) %*% A - t(B) %*% A) } hessPenalty <- function(theta) { 2*t(A) %*% A } ## strip possible arguments of maxRoutine and call the function thereafter callWithoutMaxArgs <- function(theta, fName, ...) { return( callWithoutArgs( theta, fName = fName, args = names(formals(maxRoutine)), ... ) ) } SUMTMessage <- function(code) { message <- switch(code, "1" = "penalty close to zero", "2" = "successive function values within tolerance limit", "4" = "Outer iteration limit exceeded (increase SUMTMaxIter ?).", paste("Code", code)) return(message) } ## the penalized objective function Phi <- function(theta, ...) { llVal <- callWithoutMaxArgs( theta, "logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess, sumObs = FALSE, ... ) llVal <- llVal - rho * penalty( theta ) / length( llVal ) g <- attributes( llVal )$gradient if( !is.null( g ) ) { if( is.matrix( g ) ) { g <- g - matrix( rep( rho * gPenalty( theta ) / nrow( g ), each = nrow( g ) ), nrow = nrow( g ), ncol = ncol( g ) ) } else { g <- g - rho * gPenalty( theta ) } attributes( llVal )$gradient <- g } h <- attributes( llVal )$hessian if( !is.null( h ) ) { attributes( llVal )$hessian <- h - rho * hessPenalty( theta ) } return( llVal ) } ## gradient of the penalized objective function if(!is.null(grad)) { gradPhi<- function(theta, ...) { g <- grad(theta, ...) if(is.matrix(g)) { g <- g - matrix( rep( rho * gPenalty( theta ) / nrow( g ), each = nrow( g ) ), nrow = nrow( g ), ncol = ncol( g ) ) } else { g <- g - rho * gPenalty( theta ) } return( g ) } } else { gradPhi <- NULL } ## Hessian of the penalized objective function if(!is.null(hess)) { hessPhi <- function(theta, ...) { return( hess(theta, ...) - rho*hessPenalty(theta) ) } } else { hessPhi <- NULL } ## -------- SUMT Main code --------- ## Note also that currently we do not check whether optimization was ## "successful" ... A <- constraints$eqA B <- as.matrix(constraints$eqB) ## Check if the matrices conform if(ncol(A) != length(start)) { stop("Equality constraint matrix A must have the same number\n", "of columns as the parameter length ", "(currently ", ncol(A), " and ", length(start), ")") } if(nrow(A) != nrow(B)) { stop("Equality constraint matrix A must have the same number\n", "of rows as the matrix B ", "(currently ", nrow(A), " and ", nrow(B), ")") } ## Find a suitable inital value for rho if not specified if(is.null(SUMTRho0)) { rho <- 0 result <- maxRoutine(fn=Phi, grad=gradPhi, hess=hessPhi, start=start, printLevel=max(printLevel - 1, 0), ...) theta <- coef(result) # Note: this may be a bad idea, # if unconstrained function is unbounded # from above. In that case rather specify SUMTRho0. if(printLevel > 0) { cat("SUMT initial: rho = ", rho, ", function = ", callWithoutMaxArgs( theta, "logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess, ... ), ", penalty = ", penalty(theta), "\n") cat("Estimate:") print(theta) } ## Better upper/lower bounds for rho? rho <- max( callWithoutMaxArgs( theta, "logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess, ... ), 1e-3) / max(penalty(start), 1e-3) } ## if rho specified, simply pick that and use previous initial values else { rho <- SUMTRho0 theta <- start } ## iter <- 1L repeat { thetaOld <- theta result <- maxRoutine(fn=Phi, grad=gradPhi, hess=hessPhi, start=thetaOld, printLevel=max(printLevel - 1, 0), ...) theta <- coef(result) if(printLevel > 0) { cat("SUMT iteration ", iter, ": rho = ", rho, ", function = ", callWithoutMaxArgs( theta, "logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess, ... ), ", penalty = ", penalty(theta), "\n", sep="") cat("Estimate:") print(theta) } if(max(abs(thetaOld - theta)) < SUMTTol) { SUMTCode <- 2 break } if(penalty(theta) < SUMTPenaltyTol) { SUMTCode <- 1 break } if(iter >= SUMTMaxIter) { SUMTCode <- 4 break } iter <- iter + 1L rho <- SUMTQ * rho } ## Now we replace the resulting gradient and Hessian with those, ## calculated on the original function llVal <- callWithoutMaxArgs( theta, "logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess, sumObs = FALSE, ... ) gradient <- attr( llVal, "gradient" ) if( is.null( gradient ) ) { gradient <- callWithoutMaxArgs( theta, "logLikGrad", fnOrig = fn, gradOrig = grad, hessOrig = hess, sumObs = FALSE, ... ) } if( !is.null( dim( gradient ) ) ) { if( nrow( gradient ) > 1 ) { gradientObs <- gradient } gradient <- colSums( gradient ) } else if( length( start ) == 1 && length( gradient ) > 1 ) { gradientObs <- matrix( gradient, ncol = 1 ) gradient <- sum( gradient ) } result$gradient <- gradient names( result$gradient ) <- names( result$estimate ) result$hessian <- callWithoutMaxArgs( theta, "logLikHess", fnOrig = fn, gradOrig = grad, hessOrig = hess, ... ) result$constraints <- list(type="SUMT", barrier.value=penalty(theta), code=SUMTCode, message=SUMTMessage(SUMTCode), outer.iterations=iter ) if( exists( "gradientObs" ) ) { result$gradientObs <- gradientObs colnames( result$gradientObs ) <- names( result$estimate ) } if( result$constraints$barrier.value > 0.001 ) { warning( "problem in imposing equality constraints: the constraints", " are not satisfied (barrier value = ", result$constraints$barrier.value, "). Try setting 'SUMTTol' to 0" ) } return(result) } maxLik/R/numericGradient.R0000644000176200001440000000452514077525067015166 0ustar liggesusersnumericGradient <- function(f, t0, eps=1e-6, fixed, ...) { ## numeric gradient of a vector-valued function ## f function, return Nval x 1 vector of values ## t0 NPar x 1 vector of parameters ## fixed calculate the gradient based on these parameters only ## return: ## NvalxNPar matrix, gradient ## gradient along parameters which are not active are NA warnMessage <- function(theta, value, i) { ## issue a warning if the function value at theta is not a scalar max.print <- 10 if(length(value) != nVal) { warnMsg <- "Function value at\n" warnMsg <- c(warnMsg, paste(format(theta[seq(length=min(max.print,length(theta)))]), collapse=" "), "\n") if(max.print < length(theta)) warnMsg <- c(warnMsg, "...\n") warnMsg <- c(warnMsg, " =\n") warnMsg <- c(warnMsg, paste(format(value[seq(length=min(max.print,length(value)))]), collapse=" "), "\n") if(max.print < length(value)) warnMsg <- c(warnMsg, "...\n") warnMsg <- c(warnMsg, "(length ", length(value), ") does not conform with ", "the length at original value ", nVal, "\n") warnMsg <- c(warnMsg, "Component ", i, " set to NA") return(warnMsg) } if(!all(is.na(value)) & !is.numeric(value)) stop("The function value must be numeric for 'numericGradient'") return(NULL) } NPar <- length(t0) nVal <- length(f0 <- f(t0, ...)) grad <- matrix(NA, nVal, NPar) row.names(grad) <- names(f0) colnames(grad) <- names(t0) if(missing(fixed)) fixed <- rep(FALSE, NPar) for(i in 1:NPar) { if(fixed[i]) next t2 <- t1 <- t0 t1[i] <- t0[i] - eps/2 t2[i] <- t0[i] + eps/2 ft1 <- f(t1, ...) ft2 <- f(t2, ...) ## give meaningful error message if the functions give vectors ## of different length at t1, t2 if(!is.null(msg <- warnMessage(t1, ft1, i))) { warning(msg) ft1 <- NA } if(!is.null(msg <- warnMessage(t2, ft2, i))) { warning(msg) ft2 <- NA } grad[,i] <- (ft2 - ft1)/eps } return(grad) } maxLik/R/maxNRCompute.R0000644000176200001440000004272414077525067014433 0ustar liggesusersmaxNRCompute <- function(fn, start, # maximum lambda for Marquardt (1963) finalHessian=TRUE, bhhhHessian = FALSE, fixed=NULL, control=maxControl(), ...) { ## Newton-Raphson maximisation ## Parameters: ## fn - the function to be maximized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## fn must return the value with attributes 'gradient' ## and 'hessian' ## fn must have an argument sumObs ## start - initial parameter vector (eventually w/names) ## control MaxControl object: ## steptol - minimum step size ## lambda0 initial Hessian corrector (see Marquardt, 1963, p 438) ## lambdaStep how much Hessian corrector lambda is changed between ## two lambda trials ## (nu in Marquardt (1963, p 438) ## maxLambda largest possible lambda (if exceeded will give step error) ## lambdatol - max lowest eigenvalue when forcing pos. definite H ## qrtol - tolerance for qr decomposition ## qac How to handle the case where new function value is ## smaller than the original one: ## "stephalving" smaller step in the same direction ## "marquardt" Marquardt (1963) approach ## ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## fixed - a logical vector -- which parameters are taken as fixed. ## Other paramters are treated as variable (free). ## ... additional argument to 'fn'. This may include ## 'fnOrig', 'gradOrig', 'hessOrig' if called fromm ## 'maxNR'. ## ## RESULTS: ## a list of class "maxim": ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success, see maximMessage ## message character message describing the code ## last.step only present if code == 3 (step error). A list with following components: ## theta0 - parameter value which led to the error ## f0 - function value at these parameter values ## climb - the difference between theta0 and the new approximated parameter value (theta1) ## fixed - logical vector, which parameters are constant (fixed, inactive, non-free) ## fixed logical vector, which parameters were treated as constant (fixed, inactive, non-free) ## iterations number of iterations ## type "Newton-Raphson maximisation" ## ## References: ## Marquardt (1963), "An algorithm for least-squares estimation of nonlinear ## parameters", J. Soc. Indust. Appl. Math 11(2), 431-441 ## max.eigen <- function( M) { ## return maximal eigenvalue of (symmetric) matrix val <- eigen(M, symmetric=TRUE, only.values=TRUE)$values val[1] ## L - eigenvalues in decreasing order, [1] - biggest in abs value } ## ------------------------------------------------- if(slot(control, "qac") == "marquardt") marquardt <- TRUE else marquardt <- FALSE ## maximType <- "Newton-Raphson maximisation" if(marquardt) { maximType <- paste(maximType, "with Marquardt (1963) Hessian correction") } nimed <- names(start) nParam <- length(start) samm <- NULL # data for the last step that could not find a better # value I <- diag(rep(1, nParam)) # I is unit matrix start1 <- start iter <- 0L returnHessian <- ifelse( bhhhHessian, "BHHH", TRUE ) f1 <- fn(start1, fixed = fixed, sumObs = TRUE, returnHessian = returnHessian, ...) if(slot(control, "printLevel") > 2) { cat("Initial function value:", f1, "\n") } if(any(is.na( f1))) { result <- list(code=100, message=maximMessage("100"), iterations=0, type=maximType) class(result) <- "maxim" return(result) } if(any(is.infinite( f1)) && sum(f1) > 0) { # we stop at +Inf but not at -Inf result <- list(code=5, message=maximMessage("5"), iterations=0, type=maximType) class(result) <- "maxim" return(result) } if( isTRUE( attr( f1, "gradBoth" ) ) ) { warning( "the gradient is provided both as attribute 'gradient' and", " as argument 'grad': ignoring argument 'grad'" ) } if( isTRUE( attr( f1, "hessBoth" ) ) ) { warning( "the Hessian is provided both as attribute 'hessian' and", " as argument 'hess': ignoring argument 'hess'" ) } G1 <- attr( f1, "gradient" ) if(slot(control, "printLevel") > 2) { cat("Initial gradient value:\n") print(G1) } if(any(is.na(G1[!fixed]))) { stop("NA in the initial gradient") } if(any(is.infinite(G1[!fixed]))) { stop("Infinite initial gradient") } if(length(G1) != nParam) { stop( "length of gradient (", length(G1), ") not equal to the no. of parameters (", nParam, ")" ) } H1 <- attr( f1, "hessian" ) if(slot(control, "printLevel") > 3) { cat("Initial Hessian value:\n") print(H1) } if(length(H1) == 1) { # Allow the user program to return a # single NA in case of out of support or # other problems if(is.na(H1)) stop("NA in the initial Hessian") } if(any(is.na(H1[!fixed, !fixed]))) { stop("NA in the initial Hessian") } if(any(is.infinite(H1))) { stop("Infinite initial Hessian") } if( slot(control, "printLevel") > 1) { cat( "----- Initial parameters: -----\n") cat( "fcn value:", as.vector(f1), "\n") a <- cbind(start, G1, as.integer(!fixed)) dimnames(a) <- list(nimed, c("parameter", "initial gradient", "free")) print(a) cat( "Condition number of the (active) hessian:", kappa( H1[!fixed, !fixed]), "\n") if( slot(control, "printLevel") > 3) { print( H1) } } lambda1 <- slot(control, "marquardt_lambda0") step <- 1 ## ---------------- Main interation loop ------------------------ repeat { if( iter >= slot(control, "iterlim")) { code <- 4; break } iter <- iter + 1L if(!marquardt) { lambda1 <- 0 # assume the function is concave at start0 } start0 <- start1 f0 <- f1 G0 <- G1 if(any(is.na(G0[!fixed]))) { stop("NA in gradient (at the iteration start)") } H0 <- H1 if(any(is.na(H0[!fixed, !fixed]))) { stop("NA in Hessian (at the iteration start)") } if(marquardt) { lambda1 <- lambda1/slot(control, "marquardt_lambdaStep") # initially we try smaller lambda # lambda1: current lambda for calculations H <- H0 - lambda1*I } else { step <- 1 H <- H0 } ## check whether hessian is positive definite aCount <- 0 # avoid inifinite number of attempts because of # numerical problems while((me <- max.eigen( H[!fixed,!fixed,drop=FALSE])) >= -slot(control, "lambdatol") | (qRank <- qr(H[!fixed,!fixed], tol=slot(control, "qrtol"))$rank) < sum(!fixed)) { # maximum eigenvalue -> negative definite # qr()$rank -> singularity if(marquardt) { lambda1 <- lambda1*slot(control, "marquardt_lambdaStep") } else { lambda1 <- abs(me) + slot(control, "lambdatol") + min(abs(diag(H)[!fixed]))/1e7 # The third term corrects numeric singularity. If diag(H) only contains large values, # (H - (a small number)*I) == H because of finite precision } H <- (H - lambda1*I) # could we multiply it with something like (for stephalving) # *abs(me)*lambdatol # -lambda*I makes the Hessian (barely) # negative definite. # *me*lambdatol keeps the scale roughly # the same as it was before -lambda*I aCount <- aCount + 1 if(aCount > 100) { # should be enough even in the worst case break } } amount <- vector("numeric", nParam) inv <- try(qr.solve(H[!fixed,!fixed,drop=FALSE], G0[!fixed], tol=slot(control, "qrtol"))) if(inherits(inv, "try-error")) { # could not get the Hessian to negative definite samm <- list(theta0=start0, f0=f0, climb=amount) code <- 3 break } amount[!fixed] <- inv start1 <- start0 - step*amount # note: step is always 1 for Marquardt method f1 <- fn(start1, fixed = fixed, sumObs = TRUE, returnHessian = returnHessian, ...) # The call calculates new function, # gradient, and Hessian values ## Are we requested to fix some of the parameters? constPar <- attr(f1, "constPar") if(!is.null(constPar)) { if(any(is.na(constPar))) { stop("NA in the list of constants") } fixed <- rep(FALSE, nParam) fixed[constPar] <- TRUE } ## Are we asked to write in a new value for some of the parameters? if(is.null(newVal <- attr(f1, "newVal"))) { ## no ... if(marquardt) { stepOK <- lambda1 <= slot(control, "marquardt_maxLambda") } else { stepOK <- step >= slot(control, "steptol") } while( any(is.na(f1)) || ( ( sum(f1) < sum(f0) ) & stepOK)) { # We end up in a NA or a higher value. # try smaller step if(marquardt) { lambda1 <- lambda1*slot(control, "marquardt_lambdaStep") H <- (H0 - lambda1*I) amount[!fixed] <- qr.solve(H[!fixed,!fixed,drop=FALSE], G0[!fixed], tol=slot(control, "qrtol")) } else { step <- step/2 } start1 <- start0 - step*amount if(slot(control, "printLevel") > 2) { if(slot(control, "printLevel") > 3) { cat("Try new parameters:\n") print(start1) } cat("function value difference", f1 - f0) if(marquardt) { cat(" -> lambda", lambda1, "\n") } else { cat(" -> step", step, "\n") } } f1 <- fn(start1, fixed = fixed, sumObs = TRUE, returnHessian = returnHessian, ...) # WTF does the 'returnHessian' do here ? ## Find out the constant parameters -- these may be other than ## with full step constPar <- attr(f1, "constPar") if(!is.null(constPar)) { if(any(is.na(constPar))) { stop("NA in the list of constants") } fixed[constPar] <- TRUE ## Any new values requested? if(!is.null(newVal <- attr(f1, "newVal"))) { ## Yes. Write them to parameters and go for ## next iteration start1[newVal$index] <- newVal$val break; } } } if(marquardt) { stepOK <- lambda1 <= slot(control, "marquardt_maxLambda") } else { stepOK <- step >= slot(control, "steptol") } if(!stepOK) { # we did not find a better place to go... start1 <- start0 f1 <- f0 samm <- list(theta0=start0, f0=f0, climb=amount) } } else { ## Yes, indeed. New values given to some of the params. ## Note, this may result in a lower function value, ## hence we do not check f1 > f0 start1[newVal$index] <- newVal$val if( slot(control, "printLevel") > 0 ) { cat( "Keeping parameter(s) ", paste( newVal$index, collapse = ", " ), " at the fixed values ", paste( newVal$val, collapse = ", " ), ", as the log-likelihood function", " returned attributes 'constPar' and 'newVal'\n", sep = "" ) } } G1 <- attr( f1, "gradient" ) if(any(is.na(G1[!fixed]))) { cat("Iteration", iter, "\n") cat("Parameter:\n") print(start1) cat("Gradient (first 30 components):\n") print(head(G1, n=30)) stop("NA in gradient") } if(any(is.infinite(G1))) { code <- 6; break; } H1 <- attr( f1, "hessian" ) if( slot(control, "printLevel") > 1) { cat( "-----Iteration", iter, "-----\n") } if(any(is.infinite(H1))) { code <- 7; break } if(slot(control, "printLevel") > 2) { cat( "lambda ", lambda1, " step", step, " fcn value:", formatC(as.vector(f1), digits=8, format="f"), "\n") a <- cbind(amount, start1, G1, as.integer(!fixed)) dimnames(a) <- list(names(start0), c("amount", "new param", "new gradient", "active")) print(a) if( slot(control, "printLevel") > 3) { cat("Hessian\n") print( H1) } if(!any(is.na(H1[!fixed, !fixed]))) { cat( "Condition number of the hessian:", kappa(H1[!fixed,!fixed,drop=FALSE]), "\n") } } if( step < slot(control, "steptol")) { # wrong guess in step halving code <- 3; break } if(lambda1 > slot(control, "marquardt_maxLambda")) { # wrong guess in Marquardt method code <- 3; break } if( sqrt( crossprod( G1[!fixed] ) ) < slot(control, "gradtol") ) { code <- 1; break } if(is.null(newVal) && ((sum(f1) - sum(f0)) < slot(control, "tol"))) { code <- 2; break # } if(is.null(newVal) && (sum(f1) - sum(f0) < slot(control, "reltol")*abs(sum(f1) + slot(control, "reltol"))) # We need abs(f1) to ensure RHS is positive # (as long as reltol is positive) ) { code <- 8; break } if(any(is.infinite(f1)) && sum(f1) > 0) { code <- 5; break } } if( slot(control, "printLevel") > 0) { cat( "--------------\n") cat( maximMessage( code), "\n") cat( iter, " iterations\n") cat( "estimate:", start1, "\n") cat( "Function value:", f1, "\n") } names(start1) <- nimed F1 <- fn( start1, fixed = fixed, sumObs = FALSE, returnHessian = ( finalHessian == TRUE ), ... ) G1 <- attr( F1, "gradient" ) if(observationGradient(G1, length(start1))) { gradientObs <- G1 colnames( gradientObs ) <- nimed G1 <- colSums(as.matrix(G1 )) } else { gradientObs <- NULL } names( G1 ) <- nimed ## calculate (final) Hessian if(tolower(finalHessian) == "bhhh") { if(!is.null(gradientObs)) { hessian <- -crossprod( gradientObs ) attr(hessian, "type") <- "BHHH" } else { hessian <- NULL warning("For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations") } } else if( finalHessian != FALSE ) { hessian <- attr( F1, "hessian" ) } else { hessian <- NULL } if( !is.null( hessian ) ) { rownames( hessian ) <- colnames( hessian ) <- nimed } ## remove attributes from final value of objective (likelihood) function attributes( f1 )$gradient <- NULL attributes( f1 )$hessian <- NULL attributes( f1 )$gradBoth <- NULL attributes( f1 )$hessBoth <- NULL ## result <-list( maximum = unname( drop( f1 ) ), estimate=start1, gradient=drop(G1), hessian=hessian, code=code, message=maximMessage( code), last.step=samm, # only when could not find a # lower point fixed=fixed, iterations=iter, type=maximType) if( exists( "gradientObs" ) ) { result$gradientObs <- gradientObs } result <- c(result, control=control) # attach the control parameters ## class(result) <- c("maxim", class(result)) invisible(result) } maxLik/R/maxOptim.R0000644000176200001440000002777514077525067013660 0ustar liggesusersmaxOptim <- function(fn, grad, hess, start, method, fixed, constraints, finalHessian=TRUE, parscale, control=maxControl(), ...) { ## Wrapper of optim-based optimization methods ## ## finalHessian: how (and if) to calculate the final Hessian: ## FALSE not calculate ## TRUE use analytic/numeric Hessian ## bhhh/BHHH use information equality approach ## if( method == "Nelder-Mead" ) { maxMethod <- "maxNM" } else { maxMethod <- paste( "max", method, sep = "" ) } ## ## Add parameters from ... to control if(!inherits(control, "MaxControl")) { stop("'control' must be a 'MaxControl' object, created by 'maxControl()'") } control <- addControlList(control, list(...), check=FALSE) ## Any forbidden arguments in fn? argNames <- c( "fn", "grad", "hess", "start", "print.level", "iterlim", "constraints", "tol", "reltol", "parscale", "alpha", "beta", "gamma", "cand", "temp", "tmax" ) checkFuncArgs( fn, argNames, "fn", maxMethod ) if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", maxMethod ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", maxMethod ) } ## check argument 'fixed' fixed <- prepareFixed( start = start, activePar = NULL, fixed = fixed ) message <- function(c) { switch(as.character(c), "0" = "successful convergence", "1" = "iteration limit exceeded", "10" = "degeneracy in Nelder-Mead simplex", "51" = "warning from the 'L-BFGS-B' method; see the corresponding component 'message' for details", "52" = "error from the 'L-BFGS-B' method; see the corresponding component 'message' for details" ) } ## initialize variables for saving gradients provided as attributes ## and the corresponding parameter values lastFuncGrad <- NULL lastFuncParam <- NULL ## chop off the control args from '...' and forward the new '...' dddot <- list(...) dddot <- dddot[!(names(dddot) %in% openParam(control))] # unfortunately now you have to do # do.call(function, args, dddot) instead of just calling # func(args, ...) ## strip possible SUMT parameters and call the function thereafter environment( callWithoutSumt ) <- environment() maximType <- paste( method, "maximization" ) parscale <- rep(parscale, length.out=length(start)) oControl <- list(trace=max(slot(control, "printLevel"), 0), REPORT=1, fnscale=-1, reltol=slot(control, "tol"), maxit=slot(control, "iterlim"), parscale=parscale[ !fixed ], alpha=slot(control, "nm_alpha"), beta=slot(control, "nm_beta"), gamma=slot(control, "nm_gamma"), temp=slot(control, "sann_temp"), tmax=slot(control, "sann_tmax") ) oControl$reltol <- slot(control, "reltol") argList <- list(theta=start, fName="logLikFunc", fnOrig = fn, gradOrig = grad, hessOrig = hess) if(length(dddot) > 0) { argList <- c(argList, dddot) } f1 <- do.call(callWithoutSumt, argList) if(is.na( f1)) { result <- list(code=100, message=maximMessage("100"), iterations=0, type=maximType) class(result) <- "maxim" return(result) } if(slot(control, "printLevel") > 2) { cat("Initial function value:", f1, "\n") } hasGradAttr <- !is.null( attr( f1, "gradient" ) ) if( hasGradAttr && !is.null( grad ) ) { grad <- NULL warning( "the gradient is provided both as attribute 'gradient' and", " as argument 'grad': ignoring argument 'grad'" ) } hasHessAttr <- !is.null( attr( f1, "hessian" ) ) if( hasHessAttr && !is.null( hess ) ) { hess <- NULL warning( "the Hessian is provided both as attribute 'hessian' and", " as argument 'hess': ignoring argument 'hess'" ) } if( method == "BFGS" ) { argList <- list(theta=start, fName="logLikGrad", fnOrig = fn, gradOrig = grad, hessOrig = hess) if(length(dddot) > 0) { argList <- c(argList, dddot) } G1 <- do.call(callWithoutSumt, argList) if(slot(control, "printLevel") > 2) { cat("Initial gradient value:\n") print(G1) } if(any(is.na(G1))) { stop("NA in the initial gradient") } if(any(is.infinite(G1))) { stop("Infinite initial gradient") } if(length(G1) != length(start)) { stop( "length of gradient (", length(G1), ") not equal to the no. of parameters (", length(start), ")" ) } } ## function to return the gradients (BFGS, CG) or the new candidate point (SANN) if( method == "BFGS" ) { gradOptim <- logLikGrad } else if( method == "SANN" ) { if( is.null(slot(control, "sann_cand") ) ) { gradOptim <- NULL } else { gradOptim <- function( theta, fnOrig, gradOrig, hessOrig, start, fixed, ... ) { return(control@sann_cand( theta, ... ) ) } } } else if( method == "CG" ) { gradOptim <- logLikGrad } else if( method == "Nelder-Mead" ) { gradOptim <- NULL } else { stop( "internal error: unknown method '", method, "'" ) } ## A note about return value: ## We can the return from 'optim' in a object of class 'maxim'. ## However, as 'sumt' already returns such an object, we return the ## result of 'sumt' directly, without the canning if(is.null(constraints)) { cl <- list(quote(optim), par = start[ !fixed ], fn = logLikFunc, control = oControl, method = method, gr = gradOptim, fnOrig = fn, gradOrig = grad, hessOrig = hess, start = start, fixed = fixed) if(length(dddot) > 0) { cl <- c(cl, dddot) } result <- eval(as.call(cl)) resultConstraints <- NULL } else { ## linear equality and inequality constraints # inequality constraints: A %*% beta + B >= 0 if(identical(names(constraints), c("ineqA", "ineqB"))) { nra <- nrow(constraints$ineqA) nrb <- nrow(as.matrix(constraints$ineqB)) ncb <- ncol(as.matrix(constraints$ineqB)) if(ncb != 1) { stop("Inequality constraint B must be a vector ", "(or Nx1 matrix). Currently ", ncb, " columns") } if(length(dim(constraints$ineqA)) != 2) { stop("Inequality constraint A must be a matrix\n", "Current dimension", dim(constraints$ineqA)) } if(ncol(constraints$ineqA) != length(start)) { stop("Inequality constraint A must have the same ", "number of columns as length of the parameter.\n", "Currently ", ncol(constraints$ineqA), " and ", length(start), ".") } if(ncol(constraints$ineqA) != length(start)) { stop("Inequality constraint A cannot be matrix multiplied", " with the start value.\n", "A is a ", nrow(constraints$ineqA), "x", ncol(constraints$ineqA), " matrix,", " start value has lenght ", length(start)) } if(nra != nrb) { stop("Inequality constraints A and B suggest different number ", "of constraints: ", nra, " and ", nrb) } cl <- list(quote(constrOptim2), theta = start, f = logLikFunc, grad = gradOptim, ineqA=constraints$ineqA, ineqB=constraints$ineqB, control=oControl, method = method, fnOrig = fn, gradOrig = grad, hessOrig = hess, fixed = fixed, start=start) # 'start' argument is needed for adding fixed parameters later in the call chain if(length(dddot) > 0) { cl <- c(cl, dddot) } result <- eval(as.call(cl)) resultConstraints <- list(type="constrOptim", barrier.value=result$barrier.value, outer.iterations=result$outer.iterations ) } else if(identical(names(constraints), c("eqA", "eqB"))) { # equality constraints: A %*% beta + B = 0 argList <- list(fn=fn, grad=grad, hess=hess, start=start, fixed = fixed, maxRoutine = get( maxMethod ), constraints=constraints, parscale = parscale, control=control) # recursive evaluation-> pass original (possibly # supplemented) control if(length(dddot) > 0) { argList <- c(argList, dddot) } result <- do.call( sumt, argList[ !sapply( argList, is.null ) ] ) return(result) # this is already maxim object } else { stop( maxMethod, " only supports the following constraints:\n", "constraints=list(ineqA, ineqB)\n", "\tfor A %*% beta + B >= 0 linear inequality constraints\n", "current constraints:", paste(names(constraints), collapse=" ")) } } # estimates (including fixed parameters) estimate <- start estimate[ !fixed ] <- result$par ## Calculate the final gradient argList <- list(estimate, "logLikGrad", fnOrig = fn, gradOrig = grad, hessOrig = hess, sumObs = FALSE) if(length(dddot) > 0) { argList <- c(argList, dddot) } gradient <- do.call(callWithoutSumt, argList) if(observationGradient(gradient, length(start))) { gradientObs <- gradient gradient <- colSums(as.matrix(gradient )) } else { gradientObs <- NULL } ## calculate (final) Hessian if(tolower(finalHessian) == "bhhh") { if(!is.null(gradientObs)) { hessian <- - crossprod( gradientObs ) attr(hessian, "type") <- "BHHH" } else { hessian <- NULL warning("For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations") } } else if(finalHessian != FALSE) { argList <- list( estimate, fnOrig = fn, gradOrig = grad, hessOrig = hess) if(length(dddot) > 0) { argList <- c(argList, dddot) } hessian <- as.matrix( do.call(logLikHess, argList) ) } else { hessian <- NULL } if( !is.null( hessian ) ) { rownames( hessian ) <- colnames( hessian ) <- names( estimate ) } result <- list( maximum=result$value, estimate=estimate, gradient=drop(gradient), # ensure the final (non-observation) gradient is just a vector hessian=hessian, code=result$convergence, message=paste(message(result$convergence), result$message), last.step=NULL, fixed = fixed, iterations=result$counts[1], type=maximType, constraints=resultConstraints ) if( exists( "gradientObs" ) ) { result$gradientObs <- gradientObs } result <- c(result, control=control, objectiveFn=fn) # attach the control parameters class(result) <- "maxim" return(result) } maxLik/R/logLikGrad.R0000644000176200001440000000415614077525067014065 0ustar liggesusers## gradient function: ## sum over possible individual gradients logLikGrad <- function(theta, fnOrig, gradOrig=NULL, hessOrig=NULL, start = NULL, fixed = NULL, sumObs = TRUE, gradAttr = NULL, ...) { # Argument "hessOrig" is just for compatibility with logLikHess() # argument "gradAttr" should be # - FALSE if the gradient is not provided as attribute of the log-lik value # - TRUE if the gradient is provided as attribute of the log-lik value # - NULL if this is not known theta <- addFixedPar( theta = theta, start = start, fixed = fixed, ...) if(!is.null(gradOrig)) { g <- gradOrig(theta, ...) } else if( isTRUE( gradAttr ) || is.null( gradAttr ) ) { if( exists( "lastFuncGrad" ) && exists( "lastFuncParam" ) ) { if( identical( theta, lastFuncParam ) ) { g <- lastFuncGrad } else { g <- "different parameters" } } else { g <- "'lastFuncGrad' or 'lastFuncParam' does not exist" } if( is.character( g ) ) { # do not call fnOrig() if 'lastFuncGrad' is NULL g <- attr( fnOrig( theta, ... ), "gradient" ) } } else { g <- NULL } if( is.null( g ) ) { g <- numericGradient(logLikFunc, theta, fnOrig = fnOrig, sumObs = sumObs, ...) } if( sumObs ) { ## We were requested a single (summed) gradient. Return a vector g <- sumGradients( g, length( theta ) ) names( g ) <- names( theta ) if( !is.null( fixed ) ) { g <- g[ !fixed ] } } else { ## we were requested individual gradients (if possible). Ensure g is a matrix if(observationGradient(g, length(theta))) { ## it was indeed by observations g <- as.matrix(g) colnames( g ) <- names( theta ) if( !is.null( fixed ) ) { g <- g[ , !fixed ] } } else { ## it wasn't g <- drop(g) names(g) <- names(theta) if( !is.null( fixed ) ) { g <- g[ !fixed ] } } } return( g ) } maxLik/R/print.maxLik.R0000644000176200001440000000064014077525067014420 0ustar liggesusersprint.maxLik <- function( x, ... ) { cat("Maximum Likelihood estimation\n") cat(maximType(x), ", ", nIter(x), " iterations\n", sep="") cat("Return code ", returnCode(x), ": ", returnMessage(x), "\n", sep="") if(!is.null(x$estimate)) { cat("Log-Likelihood:", x$maximum ) cat( " (", sum( activePar( x ) ), " free parameter(s))\n", sep = "" ) cat("Estimate(s):", x$estimate, "\n" ) } } maxLik/R/fnSubset.R0000644000176200001440000000272014077525067013632 0ustar liggesusersfnSubset <- function(x, fnFull, xFixed, xFull=c(x, xFixed), ...){ ## ## 1. Confirm length(x)+length(xFixed) = length(xFull) ## nx <- length(x) nFixed <- length(xFixed) nFull <- length(xFull) if((nx+nFixed) != nFull) stop("length(x)+length(xFixed) != length(xFull): ", nx, " + ", nFixed, " != ", nFull) ## ## 2. names(xFull)? ## # 2.1. is.null(names(xFull)) if(is.null(names(xFull))) return(fnFull(c(x, xFixed), ...)) # 2.2. xFull[names(xFixed)] <- xFixed, ... { if(is.null(names(xFixed))){ if(is.null(names(x))) xFull <- c(x, xFixed) else { x. <- (names(xFull) %in% names(x)) if(sum(x.) != nx){ print(x) print(xFull) stop("x has names not in xFull.") } xFull[names(x)] <- x xFull[!x.] <- xFixed } } else { Fixed <- (names(xFull) %in% names(xFixed)) if(sum(Fixed) != nFixed){ print(xFixed) print(xFull) stop("xFixed has names not in xFull.") } xFull[names(xFixed)] <- xFixed { if(is.null(names(x))) xFull[!Fixed] <- x else { x. <- (names(xFull) %in% names(x)) if(sum(x.) != nx){ print(x) print(xFull) stop("x has names not in xFull.") } xFull[names(x)] <- x } } } } ## ## 3. fnFull(...) ## fnFull(xFull, ...) } maxLik/R/sumGradients.R0000644000176200001440000000044114077525067014504 0ustar liggesusers### Sum the observation-wise gradient sumGradients <- function( gr, nParam ) { if( !is.null(dim(gr))) { gr <- colSums(gr) } else { ## ... or vector if only one parameter if( nParam == 1 && length( gr ) > 1 ) { gr <- sum(gr) } } return( gr ) } maxLik/R/maxBFGSR.R0000644000176200001440000001245614077525067013421 0ustar liggesusers maxBFGSR <- function(fn, grad=NULL, hess=NULL, start, constraints=NULL, finalHessian=TRUE, fixed=NULL, activePar=NULL, control=NULL, ...) { ## Newton-Raphson maximization ## Parameters: ## fn - the function to be minimized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## grad - gradient function (numeric used if missing). Must return either ## * vector, length=nParam ## * matrix, dim=c(nObs, 1). Treated as vector ## * matrix, dim=c(M, nParam), where M is arbitrary. In this case the ## rows are simply summed (useful for maxBHHH). ## hess - hessian function (numeric used if missing) ## start - initial parameter vector (eventually w/names) ## ... - extra arguments for fn() ## The maxControl structure: ## The stopping criteria ## tol - maximum allowed absolute difference between sequential values ## reltol - maximum allowed reltive difference (stops if < reltol*(abs(fn) + reltol) ## gradtol - maximum allowed norm of gradient vector ## steptol - minimum step size ## iterlim - maximum # of iterations ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## activePar - an index vector -- which parameters are taken as ## variable (free). Other paramters are treated as ## fixed constants ## fixed index vector, which parameters to keep fixed ## ## RESULTS: ## a list of class "maxim": ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success: ## 1 - gradient close to zero ## 2 - successive values within tolerance limit ## 3 - could not find a higher point (step error) ## 4 - iteration limit exceeded ## 100 - initial value out of range ## message character message describing the code ## last.step only present if code == 3 (step error). A list with following components: ## theta0 - parameter value which led to the error ## f0 - function value at these parameter values ## climb - the difference between theta0 and the new approximated parameter value (theta1) ## activePar - logical vector, which parameters are active (not constant) ## activePar logical vector, which parameters were treated as free (resp fixed) ## iterations number of iterations ## type "Newton-Raphson maximization" ## ## ------------------------------ ## Add parameters from ... to control if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(), control) } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## argNames <- c(c( "fn", "grad", "hess", "start", "activePar", "fixed", "control"), openParam(mControl)) checkFuncArgs( fn, argNames, "fn", "maxBFGSR" ) if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxBFGSR" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxBFGSR" ) } ## establish the active parameters. Internally, we just use 'activePar' fixed <- prepareFixed( start = start, activePar = activePar, fixed = fixed ) ## chop off the control args from ... and forward the new ... dddot <- list(...) dddot <- dddot[!(names(dddot) %in% openParam(mControl))] cl <- list(start=start, finalHessian=finalHessian, fixed=fixed, control=mControl) if(length(dddot) > 0) { cl <- c(cl, dddot) } if(is.null(constraints)) { cl <- c(quote(maxBFGSRCompute), fn=logLikAttr, fnOrig = fn, gradOrig = grad, hessOrig = hess, cl) result <- eval(as.call(cl)) } else { if(identical(names(constraints), c("ineqA", "ineqB"))) { stop("Inequality constraints not implemented for maxBFGSR") } else if(identical(names(constraints), c("eqA", "eqB"))) { # equality constraints: A %*% beta + B = 0 cl <- c(quote(sumt), fn=fn, grad=grad, hess=hess, maxRoutine=maxBFGSR, constraints=list(constraints), cl) result <- eval(as.call(cl)) } else { stop("maxBFGSR only supports the following constraints:\n", "constraints=list(ineqA, ineqB)\n", "\tfor A %*% beta + B >= 0 linear inequality constraints\n", "current constraints:", paste(names(constraints), collapse=" ")) } } result$objectiveFn <- fn return( result ) } maxLik/R/25-addControlList.R0000644000176200001440000000661214077525067015216 0ustar liggesusers ## Function overwrite parameters of an existing MaxControl object using ## parameters supplied in a single list. ## We do not make it to a method: the signature would be indistinguishable ## from add(maxControl, ...) where ... is a single list addControlList <- function(x, y, check=TRUE) { ## add list y to the control x ## ## x: a maxcontrol object ## y: a named list of additional maxControl parameters ## ## check only accept known control options. ## useful if attaching known control list ## if false, no checks performed and can add arbitrary list ## setSlot <- function(openName, slotName=openName[1], convert=function(x) x ) { ## Store potentially differently named value in slot ## ## openName vector of accepted name forms ## slotName corresponding actual slot name ## convert how to convert the value ## if(!any(openName %in% names(y))) { return(NULL) } i <- tail(which(names(y) %in% openName), 1) # pick the last occurrence: allow user to overwrite defaults slot(x, slotName) <- convert(y[[i]]) assign("x", x, envir=parent.frame()) # save modified x into parent frame } if(!inherits(x, "MaxControl")) { stop("'x' must be of class 'MaxControl'") } if(is.null(y)) { return(x) } if(!inherits(y, "list")) { stop("Control arguments to 'maxControl' must be supplied in the form of a list") } if(check) { knownNames <- union(openParam(x), slotNames(x)) if(any(uNames <- !(names(y) %in% knownNames))) { cat("Unknown control options:\n") print(names(y)[uNames]) stop("Unknown options not accepted") } } ## setSlot("tol") setSlot("reltol") setSlot("gradtol") setSlot("lambdatol") setSlot("qrtol") ## QAC setSlot(c("qac", "QAC"), "qac") setSlot(c("marquardt_lambda0", "Marquardt_lambda0")) setSlot(c("marquardt_lambdaStep", "Marquardt_lambdaStep")) setSlot(c("marquardt_maxLambda", "Marquardt_maxLambda")) ## NM setSlot(c("nm_alpha", "NM_alpha", "alpha")) setSlot(c("nm_beta", "NM_beta", "beta")) setSlot(c("nm_gamma", "NM_gamma", "gamma")) ## SANN setSlot(c("sann_cand", "SANN_cand", "cand")) setSlot(c("sann_temp", "SANN_temp", "temp")) setSlot(c("sann_tmax", "SANN_tmax", "tmax"), convert=as.integer) setSlot(c("sann_randomSeed", "SANN_randomSeed", "random.seed"), convert=as.integer) ## SGA setSlot("SGA_momentum") ## Adam setSlot("Adam_momentum1", convert=as.numeric) setSlot("Adam_momentum2", convert=as.numeric) ## SG general setSlot("SG_learningRate") setSlot("SG_batchSize", convert=as.integer) setSlot("SG_clip", convert=as.numeric) setSlot("SG_patience", convert=as.integer) setSlot("SG_patienceStep", convert=as.integer) ## setSlot("iterlim", convert=as.integer) setSlot("max.rows", convert=as.integer) setSlot("max.cols", convert=as.integer) setSlot(c("printLevel", "print.level"), convert=as.integer) setSlot("storeValues", convert=as.logical) setSlot("storeParameters", convert=as.logical) ## validObject(x) return(x) } ### Method for 'MaxControl' objects: add the second argument, list setMethod("maxControl", signature("MaxControl"), addControlList) maxLik/R/10-MaxControl_class.R0000644000176200001440000002247614077525067015544 0ustar liggesusers ### should move checkMaxControl to a separate file but how to do it? setClassUnion("functionOrNULL", c("function", "NULL")) setClassUnion("integerOrNULL", c("integer", "NULL")) setClassUnion("numericOrNULL", c("numeric", "NULL")) checkMaxControl <- function(object) { ## check validity of MaxControl objects if(!inherits(object, "MaxControl")) { stop("'MaxControl' object required. Currently '", paste(class(object), sep=", "), "'") } ## errors <- character(0) ## Check length of componenents for(s in slotNames(object)) { if(s == "sann_cand") { if(length(slot(object, s)) > 1) { errors <- c(errors, paste("'", s, "' must be either 'NULL' or ", "a function of length 1, not of length ", length(slot(object, s)), sep="")) } } else if(s %in% c("SG_batchSize", "SG_clip", "SG_patience")) { # integerOrNULL if(length(slot(object, s)) > 1) { errors <- c(errors, paste("'", s, "' must be either 'NULL' or ", "of length 1, not of length ", length(slot(object, s)), sep="")) } } else if(length(slot(object, s)) != 1) { # length 1 errors <- c(errors, paste("'", s, "' must be of length 1, not ", length(slot(object, s)), sep="")) } } ## check missings for(s in slotNames(object)) { if(is.vector(slot(object, s)) && any(is.na(slot(object, s)))) { # is.na only works for vectors errors <- c(errors, paste0("NA in '", s, "'") ) return(errors) # return errors here as otherwise NA-s will interfere the # block of if-s below } } ## if(slot(object, "steptol") < 0) { errors <- c(errors, paste("'steptol' must be non-negative, not", slot(object, "steptol"))) } if(slot(object, "lambdatol") < 0) { errors <- c(errors, paste("'lambdatol' must be non-negative, not", slot(object, "lambdatol"))) } ## qac valid values--only check if length 1 if(length(slot(object, "qac")) == 1 && !pmatch(slot(object, "qac"), c("stephalving", "marquardt"))) { errors <- c(errors, paste("'qac' must be 'stephalving' or 'marquadt', not", slot(object, "qac"))) } if(slot(object, "qrtol") < 0) { errors <- c(errors, paste("'qrtol' must be non-negative, not", slot(object, "qrtol"))) } if(slot(object, "marquardt_lambda0") < 0) { errors <- c(errors, paste("'lambda0' must be non-negative, not", slot(object, "lambda0"))) } if(slot(object, "marquardt_lambdaStep") <= 1) { errors <- c(errors, paste("'lambdaStep' must be > 1, not", slot(object, "lambdaStep"))) } if(slot(object, "marquardt_maxLambda") < 0) { errors <- c(errors, paste("'maxLambda' must be non-negative, not", slot(object, "maxLambda"))) } ## NM if(slot(object, "nm_alpha") < 0) { errors <- c(errors, paste("Nelder-Mead reflection factor 'alpha' ", "must be non-negative, not", slot(object, "nm_alpha"))) } if(slot(object, "nm_beta") < 0) { errors <- c(errors, paste("Nelder-Mead contraction factor 'beta' ", "must be non-negative, not", slot(object, "nm_beta"))) } if(slot(object, "nm_gamma") < 0) { errors <- c(errors, paste("Nelder-Mead expansion factor 'gamma' ", "must be non-negative, not", slot(object, "nm_gamma"))) } ## SANN if(!inherits(slot(object, "sann_cand"), c("function", "NULL"))) { # errors <- c(errors, paste("'SANN_cand' must be either NULL or a function, not", slot(object, "SANN_cand"))) } if(slot(object, "sann_tmax") < 1) { errors <- c(errors, paste("SANN number of calculations at each temperature ", "'tmax' ", "must be positive, not", slot(object, "sann_tmax"))) } ## SGA if(slot(object, "SGA_momentum") < 0 || slot(object, "SGA_momentum") > 1) { errors <- c(errors, paste("SGA momentum parameter must be in [0,1], not", slot(object, "SGA_momentum"))) } ## Adam if(slot(object, "Adam_momentum1") < 0 || slot(object, "Adam_momentum1") > 1) { errors <- c(errors, paste("Adam momentum1 parameter must be in [0,1], not", slot(object, "Adam_momentum1"))) } if(slot(object, "Adam_momentum2") < 0 || slot(object, "Adam_momentum2") > 1) { errors <- c(errors, paste("Adam momentum2 parameter must be in [0,1], not", slot(object, "Adam_momentum2"))) } ## SG general if(slot(object, "SG_learningRate") <= 0) { errors <- c(errors, paste("learning rate for SGA must be positive, not", slot(object, "SG_learningRate"))) } if(length(slot(object, "SG_batchSize")) > 0 && slot(object, "SG_batchSize") <= 0L) { errors <- c(errors, paste("SGA batch size must be positive, not", slot(object, "SG_batchSize"))) } if(length(slot(object, "SG_clip")) > 0 && slot(object, "SG_clip") <= 0L) { errors <- c(errors, paste("SGA gradient clip norm threshold must be positive, not", slot(object, "SG_clip"))) } if(length(slot(object, "SG_patience")) > 0 && slot(object, "SG_patience") <= 0L) { errors <- c(errors, paste("SG patience must be positive (or NULL), not", slot(object, "SG_patience"))) } if(slot(object, "SG_patienceStep") <= 0L) { errors <- c(errors, paste("SG patience step must be positive, not", slot(object, "SG_patienceStep"))) } ## general if(slot(object, "iterlim") < 0) { errors <- c(errors, paste("'iterlim' must be non-negative, not", slot(object, "iterlim"))) } if(slot(object, "max.rows") < 0) { errors <- c(errors, paste("'max.rows' must be non-negative, not", slot(object, "max.rows"))) } if(slot(object, "max.cols") < 0) { errors <- c(errors, paste("'max.cols' must be non-negative, not", slot(object, "max.cols"))) } if(length(errors) > 0) return(errors) return(TRUE) } ### MaxControls contains all control parameters for max* family setClass("MaxControl", slots=representation( tol="numeric", reltol="numeric", gradtol="numeric", steptol="numeric", # lambdatol="numeric", qrtol="numeric", ## Qadratic Approximation Control qac="character", marquardt_lambda0="numeric", marquardt_lambdaStep="numeric", marquardt_maxLambda="numeric", ## Optim Nelder-Mead: nm_alpha="numeric", nm_beta="numeric", nm_gamma="numeric", ## SANN sann_cand="functionOrNULL", sann_temp="numeric", sann_tmax="integer", sann_randomSeed="integer", ## SGA SGA_momentum = "numeric", ## Adam Adam_momentum1 = "numeric", Adam_momentum2 = "numeric", ## SG general SG_patience = "integerOrNULL", # NULL: don't care about patience SG_patienceStep = "integer", # check patience at every epoch SG_learningRate="numeric", SG_batchSize = "integerOrNULL", # NULL: full batch SG_clip="numericOrNULL", # NULL: do not clip ## iterlim="integer", max.rows="integer", max.cols="integer", printLevel="integer", storeValues="logical", storeParameters="logical" ), ## prototype=prototype( tol=1e-8, reltol=sqrt(.Machine$double.eps), gradtol=1e-6, steptol=1e-10, # lambdatol=1e-6, # qac="stephalving", qrtol=1e-10, marquardt_lambda0=1e-2, marquardt_lambdaStep=2, marquardt_maxLambda=1e12, ## Optim Nelder-Mead nm_alpha=1, nm_beta=0.5, nm_gamma=2, ## SANN sann_cand=NULL, sann_temp=10, sann_tmax=10L, sann_randomSeed=123L, ## SGA SGA_momentum = 0, ## Adam Adam_momentum1 = 0.9, Adam_momentum2 = 0.999, ## SG_learningRate=0.1, SG_batchSize=NULL, SG_clip=NULL, SG_patience = NULL, SG_patienceStep = 1L, ## iterlim=150L, max.rows=as.integer(getOption("max.rows", 20L)), max.cols=as.integer(getOption("max.cols", 7L)), printLevel=0L, storeValues=FALSE, storeParameters=FALSE), ## validity=checkMaxControl ) maxLik/R/confint.maxLik.R0000644000176200001440000000126414077525067014727 0ustar liggesusers## confint method by Lucca Scrucca confint.maxLik <- function(object, parm, level = 0.95, ...) { cf <- coef(object) if(missing(parm)) parm <- seq_along(cf) pnames <- names(cf) if(is.null(pnames)) pnames <- parm else if(is.numeric(parm)) parm <- pnames[parm] a <- (1 - level)/2 a <- c(a, 1 - a) pct <- format.perc(a, 3) q <- qnorm(a) ci <- array(NA, dim = c(length(parm), 2L), dimnames = list(parm, pct)) se <- sqrt(diag(vcov(object)))[parm] ci[] <- cf[parm] + se %o% q return(ci) } format.perc <- function(probs, digits) paste(format(100 * probs, trim = TRUE, scientific = FALSE, digits = digits), "%") maxLik/R/nIter.R0000644000176200001440000000030214077525067013114 0ustar liggesusers## Return #of iterations for maxim objects nIter <- function(x, ...) ## Number of iterations for iterative models UseMethod("nIter") nIter.default <- function(x, ...) x$iterations maxLik/R/gradient.R0000644000176200001440000000021314077525067013631 0ustar liggesusers## Return gradient of an object gradient <- function(x, ...) UseMethod("gradient") gradient.maxim <- function(x, ...) x$gradient maxLik/R/constrOptim2.R0000644000176200001440000001227614077525067014453 0ustar liggesusers# This file is a modified copy of src/library/stats/R/constrOptim.R # Part of the R package, http://www.R-project.org ### This foutine is not intended for end-user use. ### API is subject to change. constrOptim2<-function(theta, f,grad=NULL, ineqA,ineqB, mu=0.0001,control=list(), method=if(is.null(grad)) "Nelder-Mead" else "BFGS", outer.iterations=100,outer.eps=0.00001, ...){ ## Optimize with inequality constraint using SUMT/logarithmic ## barrier ## ## start initial value of parameters, included the fixed ones ## ## This function has to operate with free parameter components ## only as 'optim' cannot handle ## fixed parameters. However, for computing constraints in ## 'R' and 'dR' we have to use the complete parameter vector. ## R <- function(thetaFree, thetaFree.old, ...) { ## Wrapper for the function. As this will be feed to the ## 'optim', we have to call it with free parameters only ## (thetaFree) and internally expand it to the full (theta) ## ## Were we called with 'fixed' argument in ... ? dotdotdot <- list(...) # can this be made better? fixed <- dotdotdot[["fixed"]] theta <- addFixedPar( theta = thetaFree, start = theta0, fixed = fixed) theta.old <- addFixedPar( theta = thetaFree.old, start = theta0, fixed = fixed) ineqA.theta<-ineqA%*%theta gi<- ineqA.theta + ineqB if(any(gi < 0)) ## at least one of the constraints not fulfilled return(NaN) gi.old <- ineqA%*%theta.old + ineqB bar <- sum(gi.old*log(gi) - ineqA.theta) # logarithmic barrier value: sum over # components if(!is.finite(bar)) bar<- -Inf result <- f(thetaFree, ...)-mu*bar # do not send 'fixed' and 'start' to the # function here -- we have already # expanded theta to the full parameter result } dR<-function(thetaFree, thetaFree.old, ...){ ## Wrapper for the function. As this will be feed to the 'optim', ## we have to call it with free parameters only (thetaFree) and ## internally expand it to the full (theta) ## ## Were we called with 'fixed' argument in ... ? dotdotdot <- list(...) # can this be made better? fixed <- dotdotdot[["fixed"]] theta <- addFixedPar( theta = thetaFree, start = theta0, fixed = fixed) theta.old <- addFixedPar( theta = thetaFree.old, start = theta0, fixed = fixed) ineqA.theta<-ineqA%*%theta gi<-drop(ineqA.theta + ineqB) gi.old<-drop(ineqA%*%theta.old + ineqB) dbar<-colSums( ineqA*gi.old/gi-ineqA) if(!is.null(fixed)) gr <- grad(thetaFree,...)- (mu*dbar)[!fixed] # grad only gives gradient for the free parameters in order to maintain # compatibility with 'optim'. Hence we compute barrier gradient # for the free parameters only as well. else gr <- grad(thetaFree,...)- (mu*dbar) return(gr) } if (!is.null(control$fnscale) && control$fnscale<0) mu <- -mu ##maximizing if(any(ineqA%*%theta + ineqB < 0)) stop("initial value not the feasible region") theta0 <- theta # inital value, for keeping the fixed params ## Were we called with 'fixed' argument in ... ? fixed <- list(...)[["fixed"]] if(!is.null(fixed)) thetaFree <- theta[!fixed] else thetaFree <- theta ## obj<-f(thetaFree, ...) r<-R(thetaFree,thetaFree,...) for(i in 1L:outer.iterations){ obj.old<-obj r.old<-r thetaFree.old<-thetaFree fun<-function(thetaFree,...){ R(thetaFree,thetaFree.old,...)} if( method == "SANN" ) { if( is.null( grad ) ) { gradient <- NULL } else { gradient <- grad } } else { gradient <- function(thetaFree, ...) { dR(thetaFree, thetaFree.old, ...) } } ## As 'optim' does not directly support fixed parameters, a<-optim(par=thetaFree.old,fn=fun,gr=gradient,control=control,method=method,...) r<-a$value if (is.finite(r) && is.finite(r.old) && abs(r-r.old)/(outer.eps+abs(r-r.old))obj.old) break } if (i==outer.iterations){ a$convergence<-7 a$message<-"Barrier algorithm ran out of iterations and did not converge" } if (mu>0 && obj>obj.old){ a$convergence<-11 a$message<-paste("Objective function increased at outer iteration",i) } if (mu<0 && obj (eigentol*max(hessev))) { ## If hessian is not singular, fill in the free parameter values varcovar[activePar,activePar] <- solve(-hessian(object)[activePar,activePar]) # guarantee that the returned variance covariance matrix is symmetric varcovar <- ( varcovar + t( varcovar ) ) / 2 } else { ## If singular, the free parameter values will be Inf varcovar[activePar,activePar] <- Inf } return(varcovar) } else return(NULL) } maxLik/R/maxAdam.R0000644000176200001440000001063614077525067013416 0ustar liggesusers maxAdam <- function(fn=NULL, grad=NULL, hess=NULL, start, nObs, constraints=NULL, finalHessian=FALSE, fixed=NULL, control=NULL, ...) { ## Adam stochastic gradient ascent ## Parameters: ## fn - the function to be minimized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## grad - gradient function (numeric used if missing). Must return either ## * vector, length=nParam ## * matrix, dim=c(nObs, 1). Treated as vector ## * matrix, dim=c(M, nParam), where M is arbitrary. In this case the ## rows are simply summed (useful for maxBHHH). ## hess - hessian function (used only for finalHessian, otherwise ignored) ## start - initial parameter vector (eventually w/names) ## ... - extra arguments for fn() ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## fixed index vector, which parameters to keep fixed ## ## RESULTS: ## an object of class "maxim": ## ------------------------------ ## Add parameters from ... to control if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(gradtol=0, SG_learningRate=0.001), control) } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## argNames <- c(c("fn", "grad", "hess", "start", "fixed", "control"), openParam(mControl)) # Here we allow to submit all parameters outside of the # 'control' list. May eventually include only a # subset here ## ensure that 'fn', 'grad', and 'hess' do not take any arguments that maxSGA eats up if(!is.null(fn)) { checkFuncArgs( fn, argNames, "fn", "maxAdam" ) } if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxAdam" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxAdam" ) } ## ensure that at least 'fn' or 'grad' are supplied if(is.null(fn) & is.null(grad)) { stop("maxAdam requires at least 'fn' or 'grad' to be supplied") } if(length(start) < 1) { stop("'start' must be of positive length!") } ## establish the active parameters. Internally, we just use 'activePar' fixed <- prepareFixed( start = start, activePar = NULL, fixed = fixed ) ## chop off the control args from ... and forward the new ... dddot <- list(...) dddot <- dddot[!(names(dddot) %in% openParam(mControl))] cl <- list(start=start, finalHessian=finalHessian, fixed=fixed, control=mControl, optimizer="Adam") if(length(dddot) > 0) { cl <- c(cl, dddot) } ## if(is.null(constraints)) { ## call maxSGACompute with the modified ... list cl <- c(quote(maxSGACompute), fn=logLikFunc, grad=logLikGrad, hess=logLikHess, fnOrig = fn, gradOrig = grad, hessOrig = hess, # these are forwarded to the logLikAttr nObs=nObs, cl) result <- eval(as.call(cl)) } else { if(identical(names(constraints), c("ineqA", "ineqB"))) { stop("Inequality constraints not implemented for maxSGA") } else if(identical(names(constraints), c("eqA", "eqB"))) { # equality constraints: A %*% beta + B = 0 cl <- c(quote(sumt), fn=fn, grad=grad, hess=hess, maxRoutine=maxSGA, constraints=list(constraints), cl) result <- eval(as.call(cl)) } else { stop("maxNR only supports the following constraints:\n", "constraints=list(ineqA, ineqB)\n", "\tfor A %*% beta + B >= 0 linear inequality constraints\n", "current constraints:", paste(names(constraints), collapse=" ")) } } ## Save the objective function result$objectiveFn <- fn ## return( result ) } maxLik/R/bread.maxLik.R0000644000176200001440000000011314077525067014334 0ustar liggesusersbread.maxLik <- function( x, ... ) { return( vcov( x ) * nObs( x ) ) } maxLik/R/numericHessian.R0000644000176200001440000000543714077525067015026 0ustar liggesusersnumericHessian <- function(f, grad=NULL, t0, eps=1e-6, fixed, ...) { a <- f(t0, ...) if(is.null(grad)) { numericNHessian( f = f, t0 = t0, eps = eps, fixed=fixed, ...) # gradient not provided -> everything numerically } else { numericGradient( f = grad, t0 = t0, eps = eps, fixed=fixed, ...) # gradient is provided -> Hessian is grad grad } } numericNHessian <- function( f, t0, eps=1e-6, fixed, ...) { ## Numeric Hessian without gradient ## Assume f() returns a scalar ## ## fixed calculate the Hessian only for the non-fixed parameters warnMessage <- function(theta, value) { ## issue a warning if the function value at theta is not a scalar max.print <- 10 if(length(value) != 1) { warnMsg <- "Function value at\n" warnMsg <- c(warnMsg, paste(format(theta[seq(length=min(max.print,length(theta)))]), collapse=" "), "\n") if(max.print < length(theta)) warnMsg <- c(warnMsg, "...\n") warnMsg <- c(warnMsg, " =\n") warnMsg <- c(warnMsg, paste(format(value[seq(length=min(max.print,length(value)))]), collapse=" "), "\n") if(max.print < length(value)) warnMsg <- c(warnMsg, "...\n") warnMsg <- c(warnMsg, "but numeric Hessian only works on numeric scalars\n", "Component set to NA") return(warnMsg) } if(!is.numeric(value)) stop("The function value must be numeric") return(NULL) } f00 <- f( t0, ...) if(!is.null(msg <- warnMessage(t0, f00))) { warning(msg) f00 <- NA } eps2 <- eps*eps N <- length( t0) H <- matrix(NA, N, N) if(missing(fixed)) fixed <- rep(FALSE, length(t0)) for( i in 1:N) { if(fixed[i]) next for( j in 1:N) { if(fixed[j]) next t01 <- t0 t10 <- t0 t11 <- t0 # initial point t01[i] <- t01[i] + eps t10[j] <- t10[j] + eps t11[i] <- t11[i] + eps t11[j] <- t11[j] + eps f01 <- f( t01, ...) if(!is.null(msg <- warnMessage(t01, f01))) { warning(msg) f01 <- NA } f10 <- f( t10, ...) if(!is.null(msg <- warnMessage(t10, f10))) { warning(msg) f10 <- NA } f11 <- f( t11, ...) if(!is.null(msg <- warnMessage(t11, f11))) { warning(msg) f11 <- NA } H[i,j] <- ( f11 - f01 - f10 + f00)/eps2 } } return( H ) } maxLik/R/stdEr.maxLik.R0000644000176200001440000000077414077525067014355 0ustar liggesusers stdEr.maxLik <- function(x, eigentol=1e-12, ...) { ## if(!inherits(x, "maxLik")) ## stop("'stdEr.maxLik' called on a non-'maxLik' object") ## Here we should actually coerce the object to a 'maxLik' object, dropping all the subclasses... ## Instead, we force the program to use maxLik-related methods if(!is.null(vc <- vcov(x, eigentol=eigentol))) { s <- sqrt(diag(vc)) names(s) <- names(coef(x)) return(s) } # if vcov is not working, return NULL return(NULL) } maxLik/R/summary.maxLik.R0000644000176200001440000000611514077525067014764 0ustar liggesusersprint.summary.maxLik <- function( x, digits = max( 3L, getOption("digits") - 3L ), ... ) { cat("--------------------------------------------\n") cat("Maximum Likelihood estimation\n") cat(maximType(x), ", ", nIter(x), " iterations\n", sep="") cat("Return code ", returnCode(x), ": ", returnMessage(x), "\n", sep="") if(!is.null(x$estimate)) { cat("Log-Likelihood:", x$loglik, "\n") cat(x$NActivePar, " free parameters\n") cat("Estimates:\n") printCoefmat( x$estimate, digits = digits ) } if(!is.null(x$constraints)) { cat("\nWarning: constrained likelihood estimation.", "Inference is probably wrong\n") cat("Constrained optimization based on", x$constraints$type, "\n") if(!is.null(x$constraints$code)) cat("Return code:", x$constraints$code, "\n") # note: this is missing for 'constrOptim' if(!is.null(x$constraints$message)) cat(x$constraints$message, "\n") # note: this is missing for 'constrOptim' cat(x$constraints$outer.iterations, " outer iterations, barrier value", x$constraints$barrier.value, "\n") } cat("--------------------------------------------\n") } summary.maxLik <- function(object, eigentol=1e-12,... ) { ## object object of class "maxLik" ## ## RESULTS: ## list of class "summary.maxLik" with following components: ## maximum : function value at optimum ## estimate : estimated parameter values at optimum ## gradient : gradient at optimum ## code : code of convergence ## message : message, description of the code ## iterations : number of iterations ## type : type of optimisation ## if(!inherits(object, "maxLik")) stop("'summary.maxLik' called on a non-'maxLik' object") ## Here we should actually coerce the object to a 'maxLik' object, dropping all the subclasses... ## Instead, we force the program to use maxLik-related methods result <- object$maxim nParam <- length(coef.maxLik(object)) activePar <- activePar( object ) if((object$code < 100) & !is.null(coef.maxLik(object))) { # in case of infinity at initial values, the coefs are not provided t <- coef( object ) / stdEr( object, eigentol = eigentol ) p <- 2*pnorm( -abs( t)) t[!activePar(object)] <- NA p[!activePar(object)] <- NA results <- cbind("Estimate" = coef( object ), "Std. error" = stdEr( object, eigentol = eigentol ), "t value" = t, "Pr(> t)" = p ) } else { results <- NULL } summary <- list(maximType=object$type, iterations=object$iterations, returnCode=object$code, returnMessage=object$message, loglik=object$maximum, estimate=results, fixed=!activePar, NActivePar=sum(activePar), constraints=object$constraints) class(summary) <- "summary.maxLik" summary } maxLik/R/compareDerivatives.R0000644000176200001440000000651614077525067015704 0ustar liggesuserscompareDerivatives <- function(f, grad, hess=NULL, t0, eps=1e-6, printLevel=1, print=printLevel > 0, max.rows=getOption("max.rows", 20), max.cols=getOption("max.cols", 7), ...) { ### t0 - initial parameter vector ## ## 1. Initial function and grad eval ## if(print) cat("-------- compare derivatives -------- \n") f0 <- f(t0, ...) attributes(f0) <- NULL # keep only array data when printing if(is.function(grad)) analytic <- grad(t0, ...) else if(is.numeric(grad)) analytic = grad else stop("Argument 'grad' must be either gradient function or ", "pre-computed numeric gradient matrix") out <- list(t0=t0, f.t0=f0, compareGrad = list(analytic=analytic)) # if(is.null(dim(analytic))) { if(print) cat("Note: analytic gradient is vector. ", "Transforming into a matrix form\n") if(length(f0) > 1) analytic <- matrix(analytic, length(analytic), 1) # Note: we assume t0 is a simple vector -> hence gradient # will be a column vector else analytic <- matrix(analytic, 1, length(analytic)) # f returns a scalar -> we have row vector along t0 } if(print) { cat("Function value:\n") print(f0) } if(print) cat("Dim of analytic gradient:", dim(analytic), "\n") numeric <- numericGradient(f, t0, eps, ...) out$compareGrad$numeric = numeric if(print) cat(" numeric :", dim(numeric), "\n") rDiff <- ((analytic - numeric) / (0.5*(abs(analytic) + abs(numeric))) ) rDiff[(analytic==0) & (numeric==0)] <- 0 rDiff. <- max(abs(rDiff), na.rm=TRUE) out$compareGrad$rel.diff <- rDiff out$maxRelDiffGrad <- rDiff. # if(print){ if(ncol(analytic) < 2) { a <- cbind(t0, analytic, numeric, rDiff) dimnames(a) <- list(param=names(f0), c("theta 0", "analytic", "numeric", "rel.diff")) printRowColLimits(a, max.rows, max.cols) } else { cat("t0\n") printRowColLimits(t0, max.rows, max.cols) cat("analytic gradient\n") printRowColLimits(analytic, max.rows, max.cols) cat("numeric gradient\n") printRowColLimits(numeric, max.rows, max.cols) cat(paste("(anal-num)/(0.5*(abs(anal)+abs(num)))\n")) printRowColLimits(rDiff, max.rows, max.cols) a=list(t0=t0, analytic=analytic, numeric=numeric, rel.diff=rDiff) } cat("Max relative difference:", rDiff., "\n") } # out <- list(t0=t0, f.t0=f0, compareGrad=a, maxRelDiffGrad=rDiff.) ## ## Hessian? ## if(!is.null(hess)) { if(print) cat("Comparing hessians: relative dfference\n") anHess <- hess(t0, ...) numHess <- numericGradient(grad, t0, eps, ...) rDifHess <- (anHess-numHess) / (0.5*(abs(anHess)+abs(numHess))) rDifHess[(anHess==0) & (numHess==0)] <- 0 rDifHess. <- max(abs(rDifHess), na.rm=TRUE) if(print) printRowColLimits(rDifHess., max.rows, max.cols) out$compareHessian <- list(analytic = anHess, numeric = numHess, rel.diff = rDifHess) out$maxRelDiffHess = rDifHess. } if(print) cat("-------- END of compare derivatives -------- \n") invisible(out) } maxLik/R/activePar.R0000644000176200001440000000061514077525067013760 0ustar liggesusers## activePar: returns parameters which are free under maximisation (not fixed as constants) activePar <- function(x, ...) UseMethod("activePar") activePar.default <- function(x, ...) { if( !is.null( x$fixed ) ) { result <- !x$fixed } else { result <- x$activePar } if( is.null( result ) ) { result <- rep( TRUE, length( coef( x ) ) ) } return( result ) } maxLik/R/AIC.R0000644000176200001440000000021214077525067012427 0ustar liggesusers## Akaike (and other) information criteria AIC.maxLik <- function(object, ..., k = 2) -2*logLik(object) + k*nParam(object, free=TRUE) maxLik/R/checkBhhhGrad.R0000644000176200001440000000645414077525067014516 0ustar liggesuserscheckBhhhGrad <- function( g, theta, analytic, fixed=NULL) { ## This function controls if the user-supplied analytic or ## numeric gradient of the right dimension. ## If not, signals an error. ## ## analytic: logical, do we have a user-supplied analytic ## gradient? if(is.null(fixed)) { activePar <- rep(T, length=length(theta)) } else { activePar <- !fixed } if( analytic ) { ## Gradient supplied by the user. ## Check whether the gradient has enough rows (about enough ## observations in data) if( !is.matrix( g ) ) { stop("gradient is not a matrix but of class '", class( g ), "';\n", "the BHHH method requires that the gradient function\n", "(argument 'grad') returns a numeric matrix,\n", "where each row must correspond to the gradient(s)\n", "of the log-likelihood function at an individual\n", "(independent) observation and each column must\n", "correspond to a parameter" ) } else if( nrow( g ) < length( theta[activePar] ) ) { stop( "the matrix returned by the gradient function", " (argument 'grad') must have at least as many", " rows as the number of parameters (", length( theta ), "),", " where each row must correspond to the gradients", " of the log-likelihood function of an individual", " (independent) observation:\n", " currently, there are (is) ", length( theta ), " parameter(s)", " but the gradient matrix has only ", nrow( g ), " row(s)" ) } else if( ncol( g ) != length( theta ) ) { stop( "the matrix returned by the gradient function", " (argument 'grad') must have exactly as many columns", " as the number of parameters:\n", " currently, there are (is) ", length( theta ), " parameter(s)", " but the gradient matrix has ", ncol( g ), " columns" ) } } else { ## numeric gradient ## Check whether the gradient has enough rows. This is the case ## if and only if loglik has enough rows, hence the error message ## about loglik. if( !is.matrix( g ) || nrow( g ) == 1 ) { stop( "if the gradients (argument 'grad') are not provided by the user,", " the BHHH method requires that the log-likelihood function", " (argument 'fn') returns a numeric vector,", " where each element must be the log-likelihood value corresponding", " to an individual (independent) observation" ) } if( nrow( g ) < length( theta ) ) { stop( "the vector returned by the log-likelihood function", " (argument 'fn') must have at least as many elements", " as the number of parameters,", " where each element must be the log-likelihood value corresponding", " to an individual (independent) observation:\n", " currently, there are (is) ", length( theta ), " parameter(s)", " but the log likelihood function return only ", nrow( g ), " element(s)" ) } } return( NULL ) } maxLik/R/hessian.R0000644000176200001440000000021014077525067013463 0ustar liggesusers## Return Hessian of an object hessian <- function(x, ...) UseMethod("hessian") hessian.default <- function(x, ...) x$hessian maxLik/R/maxBHHH.R0000644000176200001440000000155214077525067013262 0ustar liggesusersmaxBHHH <- function(fn, grad=NULL, hess=NULL, start, finalHessian="BHHH", ...) { ## hess: Hessian, not used, for compatibility with the other methods ## check if arguments of user-provided functions have reserved names argNames <- c( "fn", "grad", "hess", "start", "print.level", "iterlim" ) checkFuncArgs( fn, argNames, "fn", "maxBHHH" ) if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxBHHH" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxBHHH" ) } ## using the Newton-Raphson algorithm with BHHH method for Hessian a <- maxNR( fn=fn, grad = grad, hess = hess, start=start, finalHessian = finalHessian, bhhhHessian = TRUE, ...) a$type = "BHHH maximisation" invisible(a) } maxLik/R/returnMessage.R0000644000176200001440000000036014077525067014663 0ustar liggesusers returnMessage <- function(x, ...) UseMethod("returnMessage") returnMessage.default <- function(x, ...) x$returnMessage returnMessage.maxim <- function(x, ...) x$message returnMessage.maxLik <- function(x, ...) x$message maxLik/R/openParam.R0000644000176200001440000000130414077525067013760 0ustar liggesusersopenParam <- function(object) { ## Return character list of 'open parameters', parameters that can ## be supplied to max* outside of 'control' list ## if(!inherits(object, "MaxControl")) { stop("'MaxControl' object required. Currently ", class(object)) } c("tol", "reltol", "gradtol", "steptol", # "lambdatol", ## Qadratic Approximation Control "qac", "qrtol", "lambda0", "lambdaStep", "maxLambda", ## optim Nelder-Mead "alpha", "beta", "gamma", ## SANN (open versions) "cand", "temp", "tmax", "random.seed", ## SGA ## - none ## "iterlim", "printLevel", "print.level") } maxLik/R/returnCode.R0000644000176200001440000000054714077525067014160 0ustar liggesusers### Returns return code of maxim objects ### This is tells either error, or other cause the iterations ended, ### such as the result converged returnCode <- function(x, ...) UseMethod("returnCode") returnCode.default <- function(x, ...) x$returnCode returnCode.maxim <- function(x, ...) x$code returnCode.maxLik <- function(x, ...) x$code maxLik/R/callWithoutSumt.R0000644000176200001440000000032414077525067015207 0ustar liggesusers## strip possible SUMT parameters and call the function thereafter callWithoutSumt <- function(theta, fName, ...) { return( callWithoutArgs( theta, fName = fName, args = names(formals(sumt)), ... ) ) } maxLik/R/maxSGA.R0000644000176200001440000001174514077525067013170 0ustar liggesusers maxSGA <- function(fn=NULL, grad=NULL, hess=NULL, start, nObs, constraints=NULL, finalHessian=FALSE, fixed=NULL, control=NULL, ...) { ## Newton-Raphson maximisation ## Parameters: ## fn - the function to be maximized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## grad - gradient function (numeric used if missing). Must return either ## * vector, length=nParam ## * matrix, dim=c(nObs, 1). Treated as vector ## * matrix, dim=c(M, nParam), where M is arbitrary. In this case the ## rows are simply summed (useful for maxBHHH). ## hess - hessian function (used only for finalHessian, otherwise ignored) ## start - initial parameter vector (eventually w/names) ## ... - extra arguments for fn() ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## fixed index vector, which parameters to keep fixed ## ## RESULTS: ## a list of class "maxim": ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success: ## 1 - gradient close to zero ## 2 - successive values within tolerance limit ## 3 - could not find a higher point (step error) ## 4 - iteration limit exceeded ## 100 - initial value out of range ## message character message describing the code ## iterations number of iterations ## type "Newton-Raphson maximisation" ## ## ------------------------------ ## Add parameters from ... to control if(!inherits(control, "MaxControl")) { mControl <- addControlList(maxControl(gradtol=0), control) } else { mControl <- control } mControl <- addControlList(mControl, list(...), check=FALSE) ## argNames <- c(c("fn", "grad", "hess", "start", "fixed", "control"), openParam(mControl)) # Here we allow to submit all parameters outside of the # 'control' list. May eventually include only a # subset here ## ensure that 'fn', 'grad', and 'hess' do not take any arguments that maxSGA eats up if(!is.null(fn)) { checkFuncArgs( fn, argNames, "fn", "maxSGA" ) } if( !is.null( grad ) ) { checkFuncArgs( grad, argNames, "grad", "maxSGA" ) } if( !is.null( hess ) ) { checkFuncArgs( hess, argNames, "hess", "maxSGA" ) } ## ensure that at least 'fn' or 'grad' are supplied if(is.null(fn) & is.null(grad)) { stop("maxSGA/maxAdam requires at least 'fn' or 'grad' to be supplied") } if(length(start) < 1) { stop("'start' must be of positive length!") } ## establish the active parameters. Internally, we just use 'activePar' fixed <- prepareFixed( start = start, activePar = NULL, fixed = fixed ) ## chop off the control args from ... and forward the new ... dddot <- list(...) dddot <- dddot[!(names(dddot) %in% openParam(mControl))] cl <- list(start=start, finalHessian=finalHessian, fixed=fixed, control=mControl, optimizer="SGA") if(length(dddot) > 0) { cl <- c(cl, dddot) } ## if(is.null(constraints)) { ## call maxSGACompute with the modified ... list cl <- c(quote(maxSGACompute), fn=logLikFunc, grad=logLikGrad, hess=logLikHess, fnOrig = fn, gradOrig = grad, hessOrig = hess, # these are forwarded to the logLikAttr nObs=nObs, cl) result <- eval(as.call(cl)) } else { if(identical(names(constraints), c("ineqA", "ineqB"))) { stop("Inequality constraints not implemented for maxSGA") } else if(identical(names(constraints), c("eqA", "eqB"))) { # equality constraints: A %*% beta + B = 0 cl <- c(quote(sumt), fn=fn, grad=grad, hess=hess, maxRoutine=maxSGA, constraints=list(constraints), cl) result <- eval(as.call(cl)) } else { stop("maxNR only supports the following constraints:\n", "constraints=list(ineqA, ineqB)\n", "\tfor A %*% beta + B >= 0 linear inequality constraints\n", "current constraints:", paste(names(constraints), collapse=" ")) } } ## Save the objective function result$objectiveFn <- fn ## return( result ) } maxLik/R/maximType.R0000644000176200001440000000022114077525067014010 0ustar liggesusersmaximType <- function(x) UseMethod("maximType") maximType.default <- function(x) x$maximType maximType.maxim <- function(x) x$type maxLik/R/logLikHess.R0000644000176200001440000000360214077525067014105 0ustar liggesusers## Calculate the Hessian of the function, either by analytic or numeric method logLikHess <- function( theta, fnOrig, gradOrig=NULL, hessOrig=NULL, start = NULL, fixed = NULL, gradAttr = NULL, hessAttr = NULL, ... ) { # argument "gradAttr" should be # - FALSE if the gradient is not provided as attribute of the log-lik value # - TRUE if the gradient is provided as attribute of the log-lik value # - NULL if this is not known # argument "hessAttr" should be # - FALSE if the Hessian is not provided as attribute of the log-lik value # - TRUE if the Hessian is provided as attribute of the log-lik value # - NULL if this is not known theta <- addFixedPar( theta = theta, start = start, fixed = fixed, ...) if(!is.null(hessOrig)) { hessian <- as.matrix(hessOrig( theta, ... )) } else { if( is.null( hessAttr ) || hessAttr || is.null( gradAttr ) ) { llVal <- fnOrig( theta, ... ) gradient <- attr( llVal, "gradient" ) hessian <- attr( llVal, "hessian" ) gradAttr <- !is.null( gradient ) hessAttr <- !is.null( hessian ) } if( !hessAttr ) { if( !is.null( gradOrig ) ) { grad2 <- logLikGrad } else if( gradAttr ) { grad2 <- function( theta, fnOrig = NULL, gradOrig = NULL, ... ) { gradient <- attr( fnOrig( theta, ... ), "gradient" ) gradient <- sumGradients( gradient, length( theta ) ) return( gradient ) } } else { grad2 <- NULL } hessian <- numericHessian( f = logLikFunc, grad = grad2, t0 = theta, fnOrig = fnOrig, gradOrig = gradOrig, ... ) } } rownames( hessian ) <- colnames( hessian ) <- names( theta ) if( !is.null( fixed ) ) { hessian <- hessian[ !fixed, !fixed, drop = FALSE ] } return( hessian ) } maxLik/R/objectiveFn.R0000644000176200001440000000025714077525067014302 0ustar liggesusers## Return the objective function, used for optimization objectiveFn <- function(x, ...) UseMethod("objectiveFn") objectiveFn.maxim <- function(x, ...) x$objectiveFn maxLik/R/storedParameters.R0000644000176200001440000000036414077525067015367 0ustar liggesusers## Return the stored parameters in a 'maxim' object storedParameters <- function(x, ...) ## stored parameter values at each epoch/iteration UseMethod("storedParameters") storedParameters.maxim <- function(x, ...) x$parameterStore maxLik/R/maxBFGSRCompute.R0000644000176200001440000003452014077525067014752 0ustar liggesusersmaxBFGSRCompute <- function(fn, start, finalHessian=TRUE, fixed=NULL, control=maxControl(), ...) { ## This function is originally developed by Yves Croissant (and placed in 'mlogit' package). ## Fitted for 'maxLik' by Ott Toomet, and revised by Arne Henningsen ## ## BFGS maximisation, implemented by Yves Croissant ## Parameters: ## fn - the function to be minimized. Returns either scalar or ## vector value with possible attributes ## constPar and newVal ## fn must return the value with attribute 'gradient' ## (and also attribute 'hessian' if it should be returned) ## fn must have an argument sumObs ## start - initial parameter vector (eventually w/names) ## finalHessian include final Hessian? As computing final hessian does not carry any extra penalty for NR method, this option is ## mostly for compatibility reasons with other maxXXX functions. ## TRUE/something else include ## FALSE do not include ## fixed - a logical vector -- which parameters are taken as fixed. ## control MaxControl object: ## steptol - minimum step size ## lambdatol - max lowest eigenvalue when forcing pos. definite H ## qrtol - tolerance for qr decomposition ## qac How to handle the case where new function value is ## smaller than the original one: ## "stephalving" smaller step in the same direction ## "marquardt" Marquardt (1963) approach ## The stopping criteria ## tol - maximum allowed absolute difference between sequential values ## reltol - maximum allowed reltive difference (stops if < reltol*(abs(fn) + reltol) ## gradtol - maximum allowed norm of gradient vector ## ## iterlim - maximum # of iterations ## ## Other paramters are treated as variable (free). ## ## RESULTS: ## a list of class "maxim": ## maximum function value at maximum ## estimate the parameter value at maximum ## gradient gradient ## hessian Hessian ## code integer code of success: ## 1 - gradient close to zero ## 2 - successive values within tolerance limit ## 3 - could not find a higher point (step error) ## 4 - iteration limit exceeded ## 100 - initial value out of range ## message character message describing the code ## last.step only present if code == 3 (step error). A list with following components: ## theta0 - parameter value which led to the error ## f0 - function value at these parameter values ## climb - the difference between theta0 and the new approximated parameter value (theta1) ## fixed - logical vector, which parameters are constant (fixed, inactive, non-free) ## fixed logical vector, which parameters were treated as constant (fixed, inactive, non-free) ## iterations number of iterations ## type "BFGSR maximisation" ## ## max.eigen <- function( M) { ## return maximal eigenvalue of (symmetric) matrix val <- eigen(M, symmetric=TRUE, only.values=TRUE)$values val[1] ## L - eigenvalues in decreasing order, [1] - biggest in abs value } ## maxim.type <- "BFGSR maximization" param <- start nimed <- names(start) nParam <- length(param) ## chi2 <- 1E+10 iter <- 0L # eval a first time the function, the gradient and the hessian x <- sumKeepAttr( fn( param, fixed = fixed, sumObs = FALSE, returnHessian = FALSE, ... ) ) # sum of log-likelihood value but not sum of gradients if (slot(control, "printLevel") > 0) cat( "Initial value of the function :", x, "\n" ) if(is.na(x)) { result <- list(code=100, message=maximMessage("100"), iterations=0, type=maxim.type) class(result) <- "maxim" return(result) } if(is.infinite(x) & (x > 0)) { # we stop at +Inf but not at -Inf result <- list(code=5, message=maximMessage("5"), iterations=0, type=maxim.type) class(result) <- "maxim" return(result) } if( isTRUE( attr( x, "gradBoth" ) ) ) { warning( "the gradient is provided both as attribute 'gradient' and", " as argument 'grad': ignoring argument 'grad'" ) } if( isTRUE( attr( x, "hessBoth" ) ) ) { warning( "the Hessian is provided both as attribute 'hessian' and", " as argument 'hess': ignoring argument 'hess'" ) } ## ## gradient by individual observations, used for BHHH approximation of initial Hessian. ## If not supplied by observations, we use the summed gradient. gri <- attr( x, "gradient" ) gr <- sumGradients( gri, nParam = length( param ) ) if(slot(control, "printLevel") > 2) { cat("Initial gradient value:\n") print(gr) } if(any(is.na(gr[!fixed]))) { stop("NA in the initial gradient") } if(any(is.infinite(gr[!fixed]))) { stop("Infinite initial gradient") } if(length(gr) != nParam) { stop( "length of gradient (", length(gr), ") not equal to the no. of parameters (", nParam, ")" ) } ## initial approximation for inverse Hessian. We only work with the non-fixed part if(observationGradient(gri, length(param))) { invHess <- -solve(crossprod(gri[,!fixed])) # initial approximation of inverse Hessian (as in BHHH), if possible if(slot(control, "printLevel") > 3) { cat("Initial inverse Hessian by gradient crossproduct\n") if(slot(control, "printLevel") > 4) { print(invHess) } } } else { invHess <- -1e-5*diag(1, nrow=length(gr[!fixed])) # ... if not possible (Is this OK?). Note we make this negative definite. if(slot(control, "printLevel") > 3) { cat("Initial inverse Hessian is diagonal\n") if(slot(control, "printLevel") > 4) { print(invHess) } } } if( slot(control, "printLevel") > 1) { cat("-------- Initial parameters: -------\n") cat( "fcn value:", as.vector(x), "\n") a <- cbind(start, gr, as.integer(!fixed)) dimnames(a) <- list(nimed, c("parameter", "initial gradient", "free")) print(a) cat("------------------------------------\n") } samm <- NULL # this will be returned in case of step getting too small I <- diag(nParam - sum(fixed)) direction <- rep(0, nParam) ## ----------- Main loop --------------- repeat { iter <- iter + 1L if( iter > slot(control, "iterlim")) { code <- 4; break } if(any(is.na(invHess))) { cat("Error in the approximated (free) inverse Hessian:\n") print(invHess) stop("NA in Hessian") } if(slot(control, "printLevel") > 0) { cat("Iteration ", iter, "\n") if(slot(control, "printLevel") > 3) { cat("Eigenvalues of approximated inverse Hessian:\n") print(eigen(invHess, only.values=TRUE)$values) if(slot(control, "printLevel") > 4) { cat("inverse Hessian:\n") print(invHess) } } } ## Next, ensure that the approximated inverse Hessian is negative definite for computing ## the new climbing direction. However, retain the original, potentially not negative definite ## for computing the following approximation. ## This procedure seems to work, but unfortunately I have little idea what I am doing :-( approxHess <- invHess # approxHess is used for computing climbing direction, invHess for next approximation while((me <- max.eigen( approxHess)) >= -slot(control, "lambdatol") | (qRank <- qr(approxHess, tol=slot(control, "qrtol"))$rank) < sum(!fixed)) { # maximum eigenvalue -> negative definite # qr()$rank -> singularity lambda <- abs(me) + slot(control, "lambdatol") + min(abs(diag(approxHess)))/1e7 # The third term corrects numeric singularity. If diag(H) only contains # large values, (H - (a small number)*I) == H because of finite precision approxHess <- approxHess - lambda*I if(slot(control, "printLevel") > 4) { cat("Not negative definite. Subtracting", lambda, "* I\n") cat("Eigenvalues of new approximation:\n") print(eigen(approxHess, only.values=TRUE)$values) if(slot(control, "printLevel") > 5) { cat("new Hessian approximation:\n") print(approxHess) } } # how to make it better? } ## next, take a step of suitable length to the suggested direction step <- 1 direction[!fixed] <- as.vector(approxHess %*% gr[!fixed]) oldx <- x oldgr <- gr oldparam <- param param[!fixed] <- oldparam[!fixed] - step * direction[!fixed] x <- sumKeepAttr( fn( param, fixed = fixed, sumObs = FALSE, returnHessian = FALSE, ... ) ) # sum of log-likelihood value but not sum of gradients ## did we end up with a larger value? while((is.na(x) | x < oldx) & step > slot(control, "steptol")) { step <- step/2 if(slot(control, "printLevel") > 2) { cat("Function decreased. Function values: old ", oldx, ", new ", x, ", difference ", x - oldx, "\n") if(slot(control, "printLevel") > 3) { resdet <- cbind(param = param, gradient = gr, direction=direction, active=!fixed) cat("Attempted parameters:\n") print(resdet) } cat(" -> step ", step, "\n", sep="") } param[!fixed] <- oldparam[!fixed] - step * direction[!fixed] x <- sumKeepAttr( fn( param, fixed = fixed, sumObs = FALSE, returnHessian = FALSE, ... ) ) # sum of log-likelihood value but not sum of gradients } if(step < slot(control, "steptol")) { # we did not find a better place to go... samm <- list(theta0=oldparam, f0=oldx, climb=direction) } gri <- attr( x, "gradient" ) # observation-wise gradient. We only need it in order to compute the BHHH Hessian, if asked so. gr <- sumGradients( gri, nParam = length( param ) ) incr <- step * direction y <- gr - oldgr if(all(y == 0)) { # gradient did not change -> cannot proceed code <- 9; break } ## Compute new approximation for the inverse hessian update <- outer( incr[!fixed], incr[!fixed]) * (sum(y[!fixed] * incr[!fixed]) + as.vector( t(y[!fixed]) %*% invHess %*% y[!fixed])) / sum(incr[!fixed] * y[!fixed])^2 + (invHess %*% outer(y[!fixed], incr[!fixed]) + outer(incr[!fixed], y[!fixed]) %*% invHess)/ sum(incr[!fixed] * y[!fixed]) invHess <- invHess - update ## chi2 <- - crossprod(direction[!fixed], oldgr[!fixed]) if (slot(control, "printLevel") > 0){ cat("step = ",step, ", lnL = ", x,", chi2 = ", chi2, ", function increment = ", x - oldx, "\n",sep="") if (slot(control, "printLevel") > 1){ resdet <- cbind(param = param, gradient = gr, direction=direction, active=!fixed) print(resdet) cat("--------------------------------------------\n") } } if( step < slot(control, "steptol")) { code <- 3; break } if( sqrt( crossprod( gr[!fixed] ) ) < slot(control, "gradtol") ) { code <- 1; break } if(x - oldx < slot(control, "tol")) { code <- 2; break } if(x - oldx < slot(control, "reltol")*(x + slot(control, "reltol"))) { code <- 8; break } if(is.infinite(x) & x > 0) { code <- 5; break } } if( slot(control, "printLevel") > 0) { cat( "--------------\n") cat( maximMessage( code), "\n") cat( iter, " iterations\n") cat( "estimate:", param, "\n") cat( "Function value:", x, "\n") } if( is.matrix( gr ) ) { if( dim( gr )[ 1 ] == 1 ) { gr <- gr[ 1, ] } } names(gr) <- names(param) # calculate (final) Hessian if(tolower(finalHessian) == "bhhh") { if(observationGradient(gri, length(param))) { hessian <- - crossprod( gri ) attr(hessian, "type") <- "BHHH" } else { hessian <- NULL warning("For computing the final Hessian by 'BHHH' method, the log-likelihood or gradient must be supplied by observations") } } else if(finalHessian) { hessian <- attr( fn( param, fixed = fixed, returnHessian = TRUE, ... ) , "hessian" ) } else { hessian <- NULL } if( !is.null( hessian ) ) { rownames( hessian ) <- colnames( hessian ) <- nimed } ## remove attributes from final value of objective (likelihood) function attributes( x )$gradient <- NULL attributes( x )$hessian <- NULL attributes( x )$gradBoth <- NULL attributes( x )$hessBoth <- NULL ## result <-list( maximum = unname( drop( x ) ), estimate=param, gradient=gr, hessian=hessian, code=code, message=maximMessage( code), last.step=samm, # only when could not find a # lower point fixed=fixed, iterations=iter, type=maxim.type) if(observationGradient(gri, length(param))) { colnames( gri ) <- names( param ) result$gradientObs <- gri } result <- c(result, control=control) # attach the control parameters class(result) <- c("maxim", class(result)) invisible(result) } maxLik/R/checkFuncArgs.R0000644000176200001440000000231214077525067014544 0ustar liggesusers### check of any of the args to the function that calls 'func' ### match arguments of 'func' ### checkFuncArgs <- function( func, checkArgs, argName, funcName ) { ## is the 'func' a function? if( !is.function( func ) ) { stop( "argument '", argName, "' of function '", funcName, "' is not a function" ) } funcArgs <- names( formals( func ) ) if( length( funcArgs ) > 1 ) { a <- charmatch( funcArgs[ -1 ], checkArgs ) if( sum( !is.na( a ) ) == 1 ) { stop( "argument '", funcArgs[ -1 ][ !is.na( a ) ], "' of the function specified in argument '", argName, "' of function '", funcName, "' (partially) matches the argument names of function '", funcName, "'. Please change the name of this argument" ) } else if( sum( !is.na( a ) ) > 1 ) { stop( "arguments '", paste( funcArgs[ -1 ][ !is.na( a ) ], collapse = "', '" ), "' of the function specified in argument '", argName, "' of function '", funcName, "' (partially) match the argument names of function '", funcName, "'. 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endstream endobj startxref 285514 %%EOF maxLik/inst/doc/using-maxlik.Rnw0000644000176200001440000010626014077525067016343 0ustar liggesusers\documentclass[a4paper]{article} \usepackage{amsmath} \usepackage{bbm} \usepackage[inline]{enumitem} \usepackage[T1]{fontenc} \usepackage[bookmarks=TRUE, colorlinks, pdfpagemode=none, pdfstartview=FitH, citecolor=black, filecolor=black, linkcolor=blue, urlcolor=black, ]{hyperref} \usepackage{graphicx} \usepackage{icomma} \usepackage[utf8]{inputenc} \usepackage{mathtools} % for extended pderiv arguments \usepackage{natbib} \usepackage{xargs} % for extended pderiv arguments \usepackage{xspace} % \SweaveUTF8 \newcommand{\COii}{\ensuremath{\mathit{CO}_{2}}\xspace} \DeclareMathOperator*{\E}{\mathbbm{E}}% expectation \newcommand*{\mat}[1]{\mathsf{#1}} \newcommand{\likelihood}{\mathcal{L}}% likelihood \newcommand{\loglik}{\ell}% log likelihood \newcommand{\maxlik}{\texttt{maxLik}\xspace} \newcommand{\me}{\mathrm{e}} % Konstant e=2,71828 \newcommandx{\pderiv}[3][1={}, 2={}]{\frac{\partial^{#2}{#1}}{\mathmbox{\partial{#3}}^{#2}}} % #1: function to differentiate (optional, empty = write after the formula) % #2: the order of differentiation (optional, empty=1) % #3: the variable to differentiate wrt (mandatory) \newcommand{\R}{\texttt{R}\xspace} \newcommand*{\transpose}{^{\mkern-1.5mu\mathsf{T}}} \renewcommand*{\vec}[1]{\boldsymbol{#1}} % \VignetteIndexEntry{Maximum likelihood estimation with maxLik} \title{Maximum Likelihood Estimation with \emph{maxLik}} \author{Ott Toomet} \begin{document} \maketitle <>= library(maxLik) set.seed(6) @ \section{Introduction} \label{sec:introduction} This vignette is intended for users who are familiar with concepts of likelihood and with the related methods, such as information equality and BHHH approximation, and with \R language. The vignette focuses on \maxlik usage and does not explain the underlying mathematical concepts. Potential target group includes researchers, graduate students, and industry practitioners who want to apply their own custom maximum likelihood estimators. If you need a refresher, consult the accompanied vignette ``Getting started with maximum likelihood and \maxlik''. The next section introduces the basic usage, including the \maxlik function, the main entry point for the package; gradients; different optimizers; and how to control the optimization behavior. These are topics that are hard to avoid when working with applied ML estimation. Section~\ref{sec:advanced-usage} contains a selection of more niche topics, including arguments to the log-likelihood function, other types of optimization, testing condition numbers, and constrained optimization. \section{Basic usage} \label{sec:basic-usage} \subsection{The maxLik function} \label{sec:maxlik-function} The main entry point to \maxlik functionality is the function of the same name, \verb|maxLik|. It is a wrapper around the underlying optimization algorithms that ensures that the returned object is of the right class so one can use the convenience methods, such as \verb|summary| or \verb|logLik|. It is important to keep in mind that \maxlik \emph{maximizes}, not minimizes functions. The basic usage of the function is very simple: just pass the log-likelihood function (argument \verb|logLik|) and the start value (argument \verb|start|). Let us demonstrate the basic usage by estimating the normal distribution parameters. We create 100 standard normals, and estimate the best fit mean and standard deviation. Instead of explicitly coding the formula for log-likelihood, we rely on the \R function \verb|dnorm| instead (see Section~\ref{sec:different-optimizers} for a version that does not use \verb|dnorm|): <<>>= x <- rnorm(100) # data. true mu = 0, sigma = 1 loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=2)) # give start value somewhat off summary(m) @ The algorithm converged in 7 iterations and one can check that the results are equal to the sample mean and variance.\footnote{Note that \R function \texttt{var} returns the unbiased estimator by using denominator $n-1$, the ML estimator is biased with denominator $n$. } This example demonstrates a number of key features of \verb|maxLik|: \begin{itemize} \item The first argument of the likelihood must be the parameter vector. In this example we define it as $\vec{\theta} = (\mu, \sigma)$, and the first lines of \verb|loglik| are used to extract these values from the vector. \item The \verb|loglik| function returns a single number, sum of individual log-likelihood contributions of individual $x$ components. (It may also return the components individually, see BHHH method in Section~\ref{sec:different-optimizers} below.) \item Vector of start values must be of correct length. If its components are named, those names are also displayed in \verb|summary| (and for \verb|coef| and \verb|stdEr|, see below). \item \verb|summary| method displays a handy summary of the results, including the convergence message, the estimated values, and statistical significance. \item \verb|maxLik| (and other auxiliary optimizers in the package) is a \emph{maximizer}, not minimizer. \end{itemize} As we did not specify the optimizer, \verb|maxLik| picked Newton-Raphson by default, and computed the necessary gradient and Hessian matrix numerically. \bigskip Besides summary, \verb|maxLik| also contains a number of utility functions to simplify handling of estimated models: \begin{itemize} \item \verb|coef| extracts the model coefficients: <<>>= coef(m) @ \item \verb|stdEr| returns the standard errors (by inverting Hessian): <<>>= stdEr(m) @ \item Other functions include \verb|logLik| to return the log-likelihood value, \verb|returnCode| and \verb|returnMessage| to return the convergence code and message respectively, and \verb|AIC| to return Akaike's information criterion. See the respective documentation for more information. \item One can also query the number of observations with \verb|nObs|, but this requires likelihood values to be supplied by observation (see the BHHH method in Section~\ref{sec:different-optimizers} below). \end{itemize} \subsection{Supplying analytic gradient} \label{sec:supplying-gradients} The simple example above worked fast and well. In particular, the numeric gradient \verb|maxLik| computed internally did not pose any problems. But users are strongly advised to supply analytic gradient, or even better, both the gradient and the Hessian matrix. More complex problems may be intractably slow, converge to a sub-optimal solution, or not converge at all if numeric gradients are noisy. Needless to say, unreliable Hessian also leads to unreliable inference. Here we show how to supply gradient to the \verb|maxLik| function. We demonstrate this with a linear regression example. Non-linear optimizers perform best in regions where level sets (contours) are roughly circular. In the following example we use data in a very different scale and create the log-likelihood function with extremely elongated elliptical contours. Now Newton-Raphson algorithm fails to converge when relying on numeric derivatives, but works well with analytic gradient. % using matrix notation We combine three vectors, $\vec{x}_{1}$, $\vec{x}_{2}$ and $\vec{x}_{3}$, created at a very different scale, into the design matrix $\mat{X} = \begin{pmatrix} \vec{x}_{1} & \vec{x}_{2} & \vec{x}_{3} \end{pmatrix}$ and compute $\vec{y}$ as \begin{equation} \label{eq:linear-regression-matrix} \vec{y} = \mat{X} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + \vec{\epsilon}. \end{equation} We create $\vec{x}_{1}$, $\vec{x}_{2}$ and $\vec{x}_{3}$ as random normals with standard deviation of 1, 1000 and $10^{7}$ respectively, and let $\vec{\epsilon}$ be standard normal disturbance term: <<>>= ## create 3 variables with very different scale X <- cbind(rnorm(100), rnorm(100, sd=1e3), rnorm(100, sd=1e7)) ## note: correct coefficients are 1, 1, 1 y <- X %*% c(1,1,1) + rnorm(100) @ Next, we maximize negative of sum of squared errors \emph{SSE} (remember, \verb|maxLik| is a maximizer not minimizer) \begin{equation} \label{eq:ols-sse-matrix} \mathit{SSE}(\vec{\beta}) = (\vec{y} - \mat{X} \cdot \vec{\beta})^{\transpose} (\vec{y} - \mat{X} \cdot \vec{\beta}) \end{equation} as this is equivalent to likelihood maximization: <<>>= negSSE <- function(beta) { e <- y - X %*% beta -crossprod(e) # note '-': we are maximizing } m <- maxLik(negSSE, start=c(0,0,0)) # give start values a bit off summary(m, eigentol=1e-15) @ As one can see, the algorithm gets stuck and fails to converge, the last parameter value is also way off from the correct value $(1, 1, 1)$. We have amended summary with an extra argument, \verb|eigentol=1e-15|. Otherwise \maxlik refuses to compute standard errors for near-singular Hessian, see the documentation of \verb|summary.maxLik|. It makes no difference right here but we want to keep it consistent with the two following examples. Now let's improve the model performance with analytic gradient. The gradient of \emph{SSE} can be written as \begin{equation} \label{eq:ols-sse-gradient-matrix} \pderiv{\vec{\beta}}\mathit{SSE}(\vec{\beta}) = -2(\vec{y} - \mat{X}\vec{\beta})^{\transpose} \mat{X}. \end{equation} \maxlik uses numerator layout, i.e. the derivative of the scalar log-likelihood with respect to the column vector of parameters is a row vector. We can code the negative of it as <<>>= grad <- function(beta) { 2*t(y - X %*% beta) %*% X } @ We can add gradient to \verb|maxLik| as an additional argument \verb|grad|: <<>>= m <- maxLik(negSSE, grad=grad, start=c(0,0,0)) summary(m, eigentol=1e-15) @ Now the algorithm converges rapidly, and the estimate is close to the true value. Let us also add analytic Hessian, in this case it is \begin{equation} \label{eq:ols-sse-hessian-matrix} \frac{\partial^{2}}{\partial\vec{\beta}\,\partial\vec{\beta}^{\transpose}} \mathit{SSE}(\vec{\beta}) = 2\mat{X}^{\transpose}\mat{X} \end{equation} and we implement the negative of it as <<>>= hess <- function(beta) { -2*crossprod(X) } @ Analytic Hessian matrix can be included with the argument \verb|hess|, and now the results are <>= m <- maxLik(negSSE, grad=grad, hess=hess, start=c(0,0,0)) summary(m, eigentol=1e-15) @ Analytic Hessian did not change the convergence behavior here. Note that as the loss function is quadratic, Newton-Raphson should provide the correct solution in a single iteration only. However, this example has numerical issues when inverting near-singular Hessian. One can easily check that when creating covariates in a less extreme scale, then the convergence is indeed immediate. While using separate arguments \texttt{grad} and \texttt{hess} is perhaps the most straightforward way to supply gradients, \maxlik also supports gradient and Hessian supplied as log-likelihood attributes. This is motivated by the fact that computing gradient often involves a number of similar computations as computing log-likelihood, and one may want to re-use some of the results. We demonstrate this on the same example, by writing a version of log-likelihood function that also computes the gradient and Hessian: <>= negSSEA <- function(beta) { ## negative SSE with attributes e <- y - X %*% beta # we will re-use 'e' sse <- -crossprod(e) # note '-': we are maximizing attr(sse, "gradient") <- 2*t(e) %*% X attr(sse, "Hessian") <- -2*crossprod(X) sse } m <- maxLik(negSSEA, start=c(0,0,0)) summary(m, eigentol=1e-15) @ The log-likelihood with ``gradient'' and ``Hessian'' attributes, \verb|negSSEA|, computes log-likelihood as above, but also computes its gradient, and adds it as attribute ``gradient'' to the log-likelihood. This gives a potential efficiency gain as the residuals $\vec{e}$ are re-used. \maxlik checks the presence of the attribute, and if it is there, it uses the provided gradient. In real applications the efficiency gain will depend on the amount of computations re-used, and the number of likelihood calls versus gradient calls. While analytic gradients are always helpful and often necessary, they may be hard to derive and code. In order to help to derive and debug the analytic gradient, another provided function, \verb|compareDerivatives|, takes the log-likelihood function, analytic gradent, and compares the numeric and analytic gradient. As an example, we compare the log-likelihood and gradient functions we just coded: <<>>= compareDerivatives(negSSE, grad, t0=c(0,0,0)) # 't0' is the parameter value @ The function prints the analytic gradient, numeric gradient, their relative difference, and the largest relative difference value (in absolute value). The latter is handy in case of large gradient vectors where it may be hard to spot a lonely component that is off. In case of reasonably smooth functions, expect the relative difference to be smaller than $10^{-7}$. But in this example the numerical gradients are clearly problematic. \verb|compareDerivatives| supports vector functions, so one can test analytic Hessian in the same way by calling \verb|compareDerivatives| with \verb|gradlik| as the first argument and the analytic hessian as the second argument. \subsection{Different optimizers} \label{sec:different-optimizers} By default, \maxlik uses Newton-Raphson optimizer but one can easily swap the optimizer by \verb|method| argument. The supported optimizers include ``NR'' for the default Newton-Raphson, ``BFGS'' for gradient-only Broyden-Fletcher-Goldfarb-Shannon, ``BHHH'' for the information-equality based Berndt-Hall-Hall-Hausman, and ``NM'' for gradient-less Nelder-Mead. Different optimizers may be based on a very different approach, and certain concepts, such as \emph{iteration}, may mean quite different things. For instance, although Newton-Raphson is a simple, fast and intuitive method that approximates the function with a parabola, it needs to know the Hessian matrix (the second derivatives). This is usually even harder to program than gradient, and even slower and more error-prone when computed numerically. Let us replace NR with gradient-only BFGS method. It is a quasi-Newton method that computes its own internal approximation of the Hessian while relying only on gradients. We re-use the data and log-likelihood function from the first example where we estimated normal distribution parameters: <>= m <- maxLik(loglik, start=c(mu=1, sigma=2), method="BFGS") summary(m) @ One can see that the results were identical, but while NR converged in 7 iterations, it took 20 iterations for BFGS. In this example the BFGS approximation errors were larger than numeric errors when computing Hessian, but this may not be true for more complex objective functions. In a similar fashion, one can simply drop in most other provided optimizers. One method that is very popular for ML estimation is BHHH. We discuss it here at length because that method requires both log-likelihood and gradient function to return a somewhat different value. The essence of BHHH is information equality, the fact that in case of log-likelihood function $\loglik(\theta)$, the expected value of Hessian at the true parameter value $\vec{\theta}_{0}$ can be expressed through the expected value of the outer product of the gradient: \begin{equation} \label{eq:information-equality} \E \left[ \frac{\partial^2 l(\vec{\theta})} {\partial\vec{\theta}\, \partial\vec{\theta}^{\transpose}} \right]_{\vec{\theta} = \vec{\theta}_0} = - \E \left[ \left. \frac{\partial l(\vec{\theta})} {\partial\vec{\theta}^{\transpose}} \right|_{\vec{\theta} = \vec{\theta}_0} \cdot \left. \frac{\partial l(\vec{\theta})} {\partial\vec{\theta}} \right|_{\vec{\theta} = \vec{\theta}_0} \right]. \end{equation} Hence we can approximate Hessian by the average outer product of the gradient. Obviously, this is only an approximation, and it is less correct when we are far from the true value $\vec{\theta}_{0}$. Note also that when approximating expected value with average we rely on the assumption that the observations are independent. This may not be true for certain type of data, such as time series. However, in order to compute the average outer product, we need to compute gradient \emph{by observation}. Hence it is not enough to just return a single gradient vector, we have to compute a matrix where rows correspond to individual data points and columns to the gradient components. We demonstrate BHHH method by replicating the normal distribution example from above. Remember, the normal probability density is \begin{equation} \label{eq:normal-pdf} f(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \, \me^{ -\displaystyle\frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} }. \end{equation} and hence the log-likelihood contribution of $x$ is \begin{equation} \label{eq:normal-loglik} \loglik(\mu, \sigma; x) = - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} \end{equation} and its gradient \begin{equation} \label{eq:normal-loglik-gradient} \begin{split} \pderiv{\mu} \loglik(\mu, \sigma; x) &= \frac{1}{\sigma^{2}}(x - \mu) \\ \pderiv{\sigma} \loglik(\mu, \sigma; x) &= -\frac{1}{\sigma} + \frac{1}{\sigma^{2}}(x - \mu)^{2}. \end{split} \end{equation} We can code these two functions as <<>>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- length(x) -N*log(sqrt(2*pi)) - N*log(sigma) - sum(0.5*(x - mu)^2/sigma^2) # sum over observations } gradlikB <- function(theta) { ## BHHH-compatible gradient mu <- theta[1] sigma <- theta[2] N <- length(x) # number of observations gradient <- matrix(0, N, 2) # gradient is matrix: # N datapoints (rows), 2 components gradient[, 1] <- (x - mu)/sigma^2 # first column: derivative wrt mu gradient[, 2] <- -1/sigma + (x - mu)^2/sigma^3 # second column: derivative wrt sigma gradient } @ Note that in this case we do not sum over the individual values in the gradient function (but we still do in log-likelihood). Instead, we fill the rows of the $N\times2$ gradient matrix with the values observation-wise. The results are similar to what we got above and the convergence speed is in-between that of Newton-Raphson and BFGS: \label{code:bhhh-example} <<>>= m <- maxLik(loglik, gradlikB, start=c(mu=1, sigma=2), method="BHHH") summary(m) @ In case we do not have time and energy to code the analytic gradient, we can let \maxlik compute the numeric one for BHHH too. In this case we have to supply the log-likelihood by observation. This essentially means we remove summing from the original likelihood function: <<>>= loglikB <- function(theta) { mu <- theta[1] sigma <- theta[2] -log(sqrt(2*pi)) - log(sigma) - 0.5*(x - mu)^2/sigma^2 # no summing here # also no 'N*' terms as we work by # individual observations } m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH") summary(m) @ Besides of relying on information equality, BHHH is essentially the same algorithm as NR. As the Hessian is just approximated, its is converging at a slower pace than NR with analytic Hessian. But when relying on numeric derivatives only, BHHH may be more reliable. For convenience, the other methods also support observation-wise gradients and log-likelihood values, those numbers are just summed internally. So one can just code the problem in an BHHH-compatible manner and use it for all supported optimizers. \maxlik package also includes stochastic gradient ascent optimizer. As that method is rarely used for ML estimation, it cannot be supplied through the ``method'' argument. Consult the separate vignette ``Stochastic gradient ascent in \maxlik''. \subsection{Control options} \label{sec:control-options} \maxlik supports a number of control options, most of which can be supplied through \verb|control=list(...)| method. Some of the most important options include \verb|printLevel| to control debugging information, \verb|iterLim| to control the maximum number of iterations, and various \verb|tol|-parameters to control the convergence tolerances. For instance, we can limit the iterations to two, while also printing out the parameter estimates at each step. We use the previous example with BHHH optimizer: <<>>= m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH", control=list(printLevel=3, iterlim=2)) summary(m) @ The first option, \verb|printLevel=3|, make \verb|maxLik| to print out parameters, gradient a few other bits of information at every step. Larger levels output more information, printlevel 1 only prints the first and last parameter values. The output from \maxlik-implemented optimizers is fairly consistent, but methods that call optimizers in other packages, such as BFGS, may output debugging information in a quite different way. The second option, \verb|iterLim=2| stops the algorithm after two iterations. It returns with code 4: iteration limit exceeded. Other sets of handy options are the convergence tolerances. There are three convergence tolerances: \begin{description} \item[tol] This measures the absolute convergence tolerance. Stop if successive function evaluations differ by less than \emph{tol} (default $10^{-8}$). \item[reltol] This is somewhat similar to \emph{tol}, but relative to the function value. Stop if successive function evaluations differ by less than $\mathit{reltol}\cdot (\loglik(\vec{\theta}) + \mathit{reltol})$ (default \verb|sqrt(.Machine[["double.eps"]])|, may be approximately \Sexpr{formatC(sqrt(.Machine[["double.eps"]]), digits=1)} on a modern computer). \item[gradtol] stop if the (Euclidean) norm of the gradient is smaller than this value (default $10^{-6}$). \end{description} Default tolerance values are typically good enough, but in certain cases one may want to adjust these. For instance, in case of function values are very large, one may rely only on tolerance, and ignore relative tolerance and gradient tolerance criteria. A simple way to achieve this is to set both \emph{reltol} and \emph{gradtol} to zero. In that case these two conditions are never satisfied and the algorithm stops only when the absolute convergence criterion is fulfilled. For instance, in the previous case we get: <<>>= m <- maxLik(loglikB, start=c(mu=1, sigma=2), method="BHHH", control=list(reltol=0, gradtol=0)) summary(m) @ When comparing the result with that on Page~\pageref{code:bhhh-example} we can see that the optimizer now needs more iterations and it stops with a return code that is related to tolerance, not relative tolerance. Note that BFGS and other optimizers that are based on the \verb|stats::optim| does not report the convergence results in a similar way as BHHH and NR, the algorithms provided by the \maxlik package. Instead of tolerance limits or gradient close to zero message, we hear about ``successful convergence''. Stochastic gradient ascent relies on completely different convergence criteria. See the dedicated vignette ``Stochastic Gradient Ascent in \maxlik''. \section{Advanced usage} \label{sec:advanced-usage} This section describes more advanced and less frequently used aspects of \maxlik. \subsection{Additional arguments to the log-likelihood function} \label{sec:additional-arguments-loglik} \maxlik expects the first argument of log-likelihood function to be the parameter vector. But the function may have more arguments. Those can be passed as additional named arguments to \verb|maxLik| function. For instance, let's change the log-likelihood function in a way that it expects data $\vec{x}$ to be passed as an argument \verb|x|. Now we have to call \maxlik with an additional argument \verb|x=...|: <<>>= loglik <- function(theta, x) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=2), x=x) # named argument 'x' will be passed # to loglik summary(m) @ This approach only works if the argument names do not overlap with \verb|maxLik|'s arguments' names. If that happens, it prints an informative error message. \subsection{Maximizing other functions} \label{sec:maximizing-other-functions} \verb|maxLik| function is basically a wrapper around a number of maximization algorithms, and a set of likelihood-related methods, such as standard errors. However, from time-to-time we need to optimize other functions where inverting the Hessian to compute standard errors is not applicable. In such cases one can call the included optimizers directly, using the form \verb|maxXXX| where \verb|XXX| stands for the name of the method, e.g. \verb|maxNR| for Newton-Rapshon (\verb|method="NR"|) and \verb|maxBFGS| for BFGS. There is also \verb|maxBHHH| although the information equality--based BHHH is not correct if we do not work with log-likelihood functions. The arguments for \verb|maxXXX|-functions are largely similar to those for \maxlik, the first argument is the function, and one also has to supply start values. Let us demonstrate this functionality by optimizing 2-dimensional bell curve, \begin{equation} \label{eq:2d-bell-curve} f(x, y) = \me^{-x^{2} - y^{2}}. \end{equation} We code this function and just call \verb|maxBFGS| on it: <<>>= f <- function(theta) { x <- theta[1] y <- theta[2] exp(-x^2 - y^2) # optimum at (0, 0) } m <- maxBFGS(f, start=c(1,1)) # give start value a bit off summary(m) @ Note that the summary output is slightly different: it reports the parameter and gradient value, appropriate for a task that is not likelihood optimization. Behind the scenes, this is because the \verb|maxXXX|-functions return an object of \emph{maxim}-class, not \emph{maxLik}-class. \subsection{Testing condition numbers} \label{sec:testing-condition-numbers} Analytic gradient we demonstrated in Section~\ref{sec:supplying-gradients} helps to avoid numerical problems. But not all problems can or should be solved by analytic gradients. For instance, multicollinearity should be addressed on data or model level. \maxlik provides a helper function, \verb|condiNumbers|, to detect such problems. We demonstrate this by creating a highly multicollinear dataset and estimating a linear regression model. We re-use the regression code from Section~\ref{sec:supplying-gradients} but this time we create multicollinear data in similar scale. <<>>= ## create 3 variables, two independent, third collinear x1 <- rnorm(100) x2 <- rnorm(100) x3 <- x1 + x2 + rnorm(100, sd=1e-6) # highly correlated w/x1, x2 X <- cbind(x1, x2, x3) y <- X %*% c(1, 1, 1) + rnorm(100) m <- maxLik(negSSEA, start=c(x1=0, x2=0, x3=0)) # negSSEA: negative sum of squared errors # with gradient, hessian attribute summary(m) @ As one can see, the model converges but the standard errors are missing (because Hessian is not negative definite). In such case we may learn more about the problem by testing the condition numbers $\kappa$ of either the design matrix $\mat{X}$ or of the Hessian matrix. It is instructive to test not just the whole matrix, but to do it column-by-column, and see where the number suddenly jumps. This hints which variable does not play nicely with the rest of data. \verb|condiNumber| provides such functionality. First, we test the condition number of the design matrix: <<>>= condiNumber(X) @ We can see that when only including $\vec{x}_{1}$ and $\vec{x}_{2}$ into the design, the condition number is 1.35, far from any singularity-related problems. However, adding $\vec{x}_{3}$ to the matrix causes $\kappa$ to jump to over 5 millions. This suggests that $\vec{x}_{3}$ is highly collinear with $\vec{x}_{1}$ and $\vec{x}_{2}$. In this example the problem is obvious as this is how we created $\vec{x}_{3}$, in real applications one often needs further analysis. For instance, the problem may be in categorical values that contain too few observations or complex fixed effects that turn out to be perfectly multicollinear. A good suggestion is to estimate a linear regression model where one explains the offending variable using all the previous variables. In this example we might estimate \verb|lm(x3 ~ x1 + x2)| and see which variables help to explain $\vec{x}_{3}$ perfectly. Sometimes the design matrix is fine but the problem arises because data and model do not match. In that case it may be more informative to test condition number of Hessian matrix instead. The example below creates a linearly separated set of observations and estimates this with logistic regression. As a refresher, the log-likelihood of logistic regression is \begin{equation} \label{eq:logistic-loglik} \loglik(\beta) = \sum_{i: y_{i} = 1} \log\Lambda(\vec{x}_{i}^{\transpose} \vec{\beta}) + \sum_{i: y_{i} = 0} \log\Lambda(-\vec{x}_{i}^{\transpose} \vec{\beta}) \end{equation} where $\Lambda(x) = 1/(1 + \exp(-x))$ is the logistic cumulative distribution function. We implement it using \R function \verb|plogis| <<>>= x1 <- rnorm(100) x2 <- rnorm(100) x3 <- rnorm(100) X <- cbind(x1, x2, x3) y <- X %*% c(1, 1, 1) > 0 # y values 1/0 linearly separated loglik <- function(beta) { link <- X %*% beta sum(ifelse(y > 0, plogis(link, log=TRUE), plogis(-link, log=TRUE))) } m <- maxLik(loglik, start=c(x1=0, x2=0, x3=0)) summary(m) @ Not surprisingly, all coefficients tend to infinity and inference is problematic. In this case the design matrix does not show any issues: <<>>= condiNumber(X) @ But the Hessian reveals that including $\vec{x}_{3}$ in the model is still problematic: <<>>= condiNumber(hessian(m)) @ Now the problem is not multicollinearity but the fact that $\vec{x}_{3}$ makes the data linearly separable. In such cases we may want to adjust our model or estimation strategy. \subsection{Fixed parameters and constrained optimization} \label{sec:fixed-parameters} \maxlik supports three types of constrains. The simplest case just keeps certain parameters' values fixed. The other two, general linear equality and inequality constraints are somewhat more complex. Occasionally we want to treat one of the model parameters as constant. This can be achieved in a very simple manner, just through the argument \verb|fixed|. It must be an index vector, either numeric, such as \verb|c(2,4)|, logical as \verb|c(FALSE, TRUE, FALSE, TRUE)|, or character as \verb|c("beta2", "beta4")| given \verb|start| is a named vector. We revisit the first example of this vignette and estimate the normal distribution parameters again. However, this time we fix $\sigma = 1$: <<>>= x <- rnorm(100) loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] sum(dnorm(x, mean=mu, sd=sigma, log=TRUE)) } m <- maxLik(loglik, start=c(mu=1, sigma=1), fixed="sigma") # fix the component named 'sigma' summary(m) @ The result has $\sigma$ exactly equal to $1$, it's standard error $0$, and $t$ value undefined. The fixed components are ignored when computing gradients and Hessian in the optimizer, essentially reducing the problem from 2-dimensional to 1-dimensional. Hence the inference for $\mu$ is still correct. Next, we demonstrate equality constraints. We take the two-dimensional function we used in Section~\ref{sec:maximizing-other-functions} and add constraints $x + y = 1$. The constraint must be described in matrix form $\mat{A}\,\vec{\theta} + \vec{B} = 0$ where $\vec{\theta}$ is the parameter vector and matrix $\mat{A}$ and vector $\vec{B}$ describe the constraints. In this case we can write \begin{equation} \label{eq:equality-constraints} \begin{pmatrix} 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \end{pmatrix} = 0, \end{equation} i.e. $\mat{A} = (1 \; 1)$ and $\vec{B} = -1$. These values must be supplied to the optimizer argument \verb|constraints|. This is a list with components names \verb|eqA| and \verb|eqB| for $\mat{A}$ and $\vec{B}$ accordingly. We do not demonstrate this with a likelihood example as no corrections to the Hessian matrix is done and hence the standard errors are incorrect. But if you are not interested in likelihood-based inference, it works well: <<>>= f <- function(theta) { x <- theta[1] y <- theta[2] exp(-x^2 - y^2) # optimum at (0, 0) } A <- matrix(c(1, 1), ncol=2) B <- -1 m <- maxNR(f, start=c(1,1), constraints=list(eqA=A, eqB=B)) summary(m) @ The problem is solved using sequential unconstrained maximization technique (SUMT). The idea is to add a small penalty for the constraint violation, and to slowly increase the penalty until violations are prohibitively expensive. As the example indicates, the solution is extremely close to the constraint line. The usage of inequality constraints is fairly similar. We have to code the inequalities as $\mat{A}\,\vec{\theta} + \vec{B} > 0$ where the matrices $\mat{A}$ and $\vec{B}$ are defined as above. Let us optimize the function over the region $x + y > 1$. In matrix form this will be \begin{equation} \label{eq:inequality-constraints-1} \begin{pmatrix} 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \end{pmatrix} > 0. \end{equation} Supplying the constraints is otherwise similar to the equality constraints, just the constraints-list components must be called \verb|ineqA| and \verb|ineqB|. As \verb|maxNR| does not support inequality constraints, we use \verb|maxBFGS| instead. The corresponding code is <<>>= A <- matrix(c(1, 1), ncol=2) B <- -1 m <- maxBFGS(f, start=c(1,1), constraints=list(ineqA=A, ineqB=B)) summary(m) @ Not surprisingly, the result is exactly the same as in case of equality constraints, in this case the optimum is found at the boundary line, the same line what we specified when demonstrating the equality constraints. One can supply more than one set of constraints, in that case these all must be satisfied at the same time. For instance, let's add another condition, $x - y > 1$. This should be coded as another line of $\mat{A}$ and another component of $\vec{B}$, in matrix form the constraint is now \begin{equation} \label{eq:inequality-constraints-2} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \\ -1 \end{pmatrix} > \begin{pmatrix} 0 \\ 0 \end{pmatrix} \end{equation} where ``>'' must be understood as element-wise operation. We also have to ensure the initial value satisfies the constraint, so we choose $\vec{\theta}_{0} = (2, 0)$. The code will be accordingly: <<>>= A <- matrix(c(1, 1, 1, -1), ncol=2) B <- c(-1, -1) m <- maxBFGS(f, start=c(2, 0), constraints=list(ineqA=A, ineqB=B)) summary(m) @ The solution is $(1, 0)$ the closest point to the origin where both constraints are satisfied. \bigskip This example concludes the \maxlik usage introduction. For more information, consult the fairly extensive documentation, and the other vignettes. % \bibliographystyle{apecon} % \bibliography{maxlik} \end{document} maxLik/inst/doc/intro-to-maximum-likelihood.Rnw0000644000176200001440000010672614077525067021311 0ustar liggesusers\documentclass[a4paper]{article} \usepackage{graphics} \usepackage{amsmath} \usepackage{amssymb} \usepackage[font={small,sl}]{caption} \usepackage[inline]{enumitem} \usepackage{indentfirst} \usepackage[utf8]{inputenc} \usepackage{natbib} \usepackage{siunitx} \usepackage{xspace} % \SweaveUTF8 \newcommand{\COii}{\ensuremath{\mathit{CO}_{2}}\xspace} \newcommand*{\mat}[1]{\mathsf{#1}} \newcommand{\likelihood}{\mathcal{L}}% likelihood \newcommand{\loglik}{\ell}% log likelihood \newcommand{\maxlik}{\texttt{maxLik}\xspace} \newcommand{\me}{\mathrm{e}} % Konstant e=2,71828 \newcommand{\R}{\texttt{R}\xspace} \newcommand*{\transpose}{^{\mkern-1.5mu\mathsf{T}}} \renewcommand*{\vec}[1]{\boldsymbol{#1}} % \VignetteIndexEntry{Introduction: what is maximum likelihood} \begin{document} <>= options(keep.source = TRUE, width = 60, try.outFile=stdout() # make try to produce error messages ) set.seed(34) @ \title{Getting started with maximum likelihood and \texttt{maxLik}} \author{Ott Toomet} \maketitle \section{Introduction} This vignette is intended for readers who are unfamiliar with the concept of likelihood, and for those who want a quick intuitive brush-up. The potential target group includes advanced undergraduate students in technical fields, such as statistics or economics, graduate students in social sciences and engineering who are devising their own estimators, and researchers and practitioners who have little previous experience with ML. However, one should have basic knowledge of \R language. If you are familiar enough with the concept of likelihood and maximum likelihood, consult instead the other vignette ``Maximum Likelihood Estimation with \maxlik''. Maximum Likelihood (ML) in its core is maximizing the \emph{likelihood} over the parameters of interest. We start with an example of a random experiment that produces discrete values to explain what is likelihood and how it is related to probability. The following sections cover continuous values, multiple parameters in vector form, and we conclude with a linear regression example. The final section discusses the basics of non-linear optimization. The examples are supplemented with very simple code and assume little background besides basic statistics and basic \R knowledge. \section{Discrete Random Values} \label{sec:discrete-random-variables} We start with a discrete case. ``Discrete'' refers to random experiments or phenomena with only limited number of possible outcomes, and hence we can compute and tabulate every single outcome separately. Imagine you are flipping a fair coin. What are the possible outcomes and what are the related probabilities? Obviously, in case of a coin there are only two outcomes, heads $H$ and tails $T$. If the coin is fair, both of these will have probability of exactly 0.5. Such random experiment is called \emph{Bernoulli process}. More specifically, this is \emph{Bernoulli(0.5)} process as for the fair coin the probability of ``success'' is 0.5 (below we consider success to be heads, but you can choose tails as well). If the coin is not fair, we denote the corresponding process Bernoulli($p$), where $p$ is the probability of heads. Now let us toss the coin two times. What is the probability that we end up with one heads and one tails? As the coin flips are independent,\footnote{Events are independent when outcome of one event does not carry information about the outcome of the other event. Here the result of the second toss is not related to the outcome of the first toss.} we can just multiply the probabilities: $0.5$ for a single heads and $0.5$ for a single tails equals $0.25$ when multiplied. However, this is not the whole story--there are two ways to get one heads and one tails, either $H$ first and $T$ thereafter or $T$ first and $H$ thereafter. Both of these events are equally likely, so the final answer will be 0.5. But now imagine we do not know if the coin is fair. Maybe we are not tossing a coin but an object of a complex shape. We can still label one side as ``heads'' and the other as ``tails''. But how can we tell what is the probability of heads? Let's start by denoting this probability with $p$. Hence the probability of tails will be $1-p$, and the probability to receive one heads, one tails when we toss the object two times will be $2 p (1-p)$: $p$ for one heads, $1-p$ for one tails, and ``2'' takes into account the fact that we can get this outcome in two different orders. This probability is essentially likelihood. We denote likelihood with $\likelihood(p)$, stressing that it depends on the unknown probability $p$. So in this example we have \begin{equation} \label{eq:2-coin-likelihood} \likelihood(p) = 2 \, p \, (1-p). \end{equation} $p$ is the \emph{model parameter}, the unknown number we want to compute with the help of likelihood. Let's repeat here what did we do above: \begin{enumerate} \item We observe data. In this example data contains the counts: one heads, one tails. \item We model the coin toss experiment, the data generating process, as Bernoulli($p$) random variable. $p$, the probability of heads, is the model parameter we want to calculate. Bernoulli process has only a single parameter, but more complex processes may contain many more. \item Thereafter we compute the probability to observe the data based on the model. Here it is equation~\eqref{eq:2-coin-likelihood}. This is why we need a probability model. As the model contains unknown parameters, the probability will also contain parameters. \item And finally we just call this probability \emph{likelihood} $\likelihood(p)$. We write it as a function of the parameter to stress that the parameter is what we are interested in. Likelihood also depends on data (the probability will look different for e.g. two heads instead of a head and a tail) but we typically do not reflect this in notation. \end{enumerate} The next task is to use this likelihood function to \emph{estimate} the parameter, to use data to find the best possible parameter value. \emph{Maximum likelihood} (ML) method finds such parameter value that maximizes the likelihood function. It can be shown that such parameter value has a number of desirable properties, in particular it will become increasingly similar to the ``true value'' on an increasingly large dataset (given that our probability model is correct).\footnote{This property is formally referred to as \emph{consistency}. ML is a consistent estimator.} These desirable properties, and relative simplicity of the method, have made ML one of the most widely used statistical estimators. Let us generalize the example we did above for an arbitrary number of coin flips. Assume the coin is of unknown ``fairness'' where we just denote the probability to receive heads with $p$. Further, assume that out of $N$ trials, $N_{H}$ trials were heads and $N_{T}$ trials were tails. The probability of this occuring is \begin{equation} \label{eq:general-cointoss-probability} \binom{N}{N_{H}} \, p^{N_{H}} \, (1 - p)^{N_{T}} \end{equation} $p^{N_{H}}$ is the probability to get $N_{H}$ heads, $(1 - p)^{N_{T}}$ is the probability to get $N_{T}$ tails, and the binomial coefficient $\displaystyle\binom{N}{N_{H}} = \displaystyle\frac{N!}{N_{H}! (N - N_{H})!}$ takes into account that there are many ways how heads and tail can turn up while still resulting $N_{H}$ heads and $N_{T}$ tails. In the previous example $N=2$, $N_{H} = 1$ and there were just two possible combinations as $\displaystyle\binom{2}{1} = 2$. The probability depends on both the parameter $p$ and data--the corresponding counts $N_{H}$ and $N_{T}$. Equation~\eqref{eq:general-cointoss-probability} is essentially likelihood--probability to observe data. We are interested how does it depend on $p$ and stress this by writing $p$ in the first position followed by semicolon and data as we care less about the dependency on data: \begin{equation} \label{eq:general-cointoss-likelihood} \likelihood(p; N_{H}, N_{T}) = \binom{N}{N_{H}} \, p^{N_{H}} \, (1 - p)^{N_{T}} \end{equation} Technically, it is easier to work with log-likelihood instead of likelihood (as log is monotonic function, maximum of likelihood and maximum of log-likelihood occur at the same parameter value). We denote log-likelihood by $\loglik$ and write \begin{equation} \label{eq:general-cointoss-loglik} \loglik(p; N_{H}, N_{T}) = \log\likelihood(p; N_{H}, N_{T}) = \log \binom{N}{N_{H}} + N_{H} \log p + N_{T} \log (1 - p). \end{equation} ML estimator of $p$ is the value that maximizes this expression. Fortunately, in this case the binomial coefficient $\displaystyle\binom{N}{N_{H}}$ depends only on data but not on the $p$. Intuitively, $p$ determines the probability of various combinations of heads and tails, but \emph{what kind of combinations are possible} does not depend on $p$. Hence we can ignore the first term on the right hand side of~\eqref{eq:general-cointoss-loglik} when maximizing the log-likelihood. Such approach is very common in practice, many terms that are invariant with respect to parameters are often ignored. Hence, with we can re-define the log-likelihood as \begin{equation} \label{eq:general-cointoss-partial-loglik} \loglik(p; N_{H}, N_{T}) = N_{H} \log p + N_{T} \log (1 - p). \end{equation} It is easy to check that the solution, the value of $p$ that maximizes log-likelihood~\eqref{eq:general-cointoss-partial-loglik} is\footnote{Just differentiate $\loglik(p)$ with respect to $p$, set the result to zero, and isolate $p$.} \begin{equation} \label{eq:general-cointoss-solution} p^{*} = \frac{N_{H}}{N_{H} + N_{T}} = \frac{N_{H}}{N}. \end{equation} This should be surprise to no-one: the intuitive ``fairness'' of the coin is just the average percentage of heads we get. Now it is time to try this out on computer with \texttt{maxLik}. Let's assume we toss a coin and receive $H_{H} = 3$ heads and $H_{T} = 7$ tails: <<>>= NH <- 3 NT <- 7 @ Next, we have to define the log-likelihood function. It has to be a function of the parameter, and the parameter must be its first argument. We can access data in different ways, for instance through the \R workspace environment. So we can write the log-likelihood as <<>>= loglik <- function(p) { NH*log(p) + NT*log(1-p) } @ And finally, we can use \texttt{maxLik} function to compute the likelihood. In its simplest form, \texttt{maxLik} requires two arguments: the log-likelihood function, and the start value for the iterative algorithm (see Section~\ref{sec:non-linear-optimization}, and the documentation and vignette \textsl{Maximum Likelihood Estimation with \maxlik} for more detailed explanations). The start value must be a valid parameter value (the loglik function must not give errors when called with the start value). We can choose $p_{0} = 0.5$ as the initial value, and let the algorithm find the best possible $p$ from there: <<>>= library(maxLik) m <- maxLik(loglik, start=0.5) summary(m) @ As expected, the best bet for $p$ is 0.3. Our intuitive approach--the percentage of heads in the experiment--turns also out to be the ML estimate. Next, we look at an example with continuous outcomes. \section{Continuous case: probability density and likelihood} \label{sec:continuous-outcomes} In the example above we looked at a discrete random process, a case where there were only a small number of distinct possibilities (heads and tails). Discrete cases are easy to understand because we can actually compute the respective probabilities, such as the probability to receive one heads and one tails in our experiment. Now we consider continuous random variables where the outcome can be any number in a certain interval. Unfortunately, in continuous case we cannot compute probability of any particular outcome. Or more precisely--we can do it, but the answer is always 0. This may sound a little counter-intuitive but perhaps the following example helps. If you ask the computer to generate a single random number between 0 and 1, you may receive \Sexpr{x <- runif(1); x}. What is the probability to get the same number again? You can try, you will get close but you won't get exactly the same number.\footnote{As computers operate with finite precision, the actual chances to repeat any particular random number are positive, although small. The exact answer depends on the numeric precision and the quality of random number generator. } But despite the probability to receive this number is zero, we somehow still produced it in the first place. Clearly, zero probability does not mean the number was impossible. However, if we want to receive a negative number from the same random number generator, it will be impossible (because we chose a generator that only produces numbers between 0 and 1). So probability 0-events may be possible and they may also be impossible. And to make matter worse, they may also be more likely and less likely. For instance, in case of standard normal random numbers (these numbers are distributed according to ``bell curve'') the values near $0$ are much more likely than values around $-2$, despite of the probability to receive any particular number still being 0 (see Figure~\ref{fig:standard-normal-intervals}). The solution is to look not at the individual numbers but narrow interval near these numbers. Consider the number of interest $x_{1}$, and compute the probability that the random outcome $X$ falls into the narrow interval of width $\delta$, $[x_{1} - \delta/2,\, x_{1} + \delta/2]$, around this number (Figure~\ref{fig:standard-normal-intervals}). Obviously, the smaller the width $\delta$, the less likely it is that $X$ falls into this narrow interval. But it turns out that when we divide the probability by the width, we get a stable value at the limit which we denote by $f(x_{1})$: \begin{equation} \label{eq:probability-density} f(x_{1}) = \lim_{\delta\to0} \frac{\Pr(X \in [x_{1} - \delta/2,\, x_{1} + \delta/2])}{\delta}. \end{equation} In the example on the Figure the values around $x_{1}$ are less likely than around $x_{2}$ and hence $f(x_{1}) < f(x_{2})$. The result, $f(x)$, is called \emph{probability density function}, often abbreviated as \emph{pdf}. In case of continuous random variables, we have to work with pdf-s instead of probabilities. \begin{figure}[ht] \centering \includegraphics{probability-density.pdf} \caption{Standard normal probability density (thick black curve). While $\Pr(X = x_{1}) = 0$, i.e. the probability to receive a random number exactly equal to $x_{1}$ is 0, the probability to receive a random number in the narrow interval of width $\delta$ around $x_{1}$ is positive. In this example, the probability to get a random number in the interval around $x_{2}$ is four times larger than for the interval around $x_{1}$. } \label{fig:standard-normal-intervals} \end{figure} Consider the following somewhat trivial example: we have sampled two independent datapoints $x_{1}$ and $x_{2}$ from normal distribution with variance 1 and mean (expected value) equal to $\mu$. Say, $x_{1} = \Sexpr{x1 <- rnorm(1); round(x1, 3)}$ and $x_{2} = \Sexpr{x1 <- rnorm(1); round(x1, 3)}$. Assume we do not know $\mu$ and use ML to estimate it. We can proceed in a similar steps as what we did for the discrete case: \begin{enumerate*}[label=\roman*)] \item observe data, in this case $x_{1}$ and $x_{2}$; \item set up the probability model; \item use the model to compute probability to observe the data; \item write the probability as $\loglik(\mu)$, log-likelihood function of the parameter $\mu$; \item and finally, find $\mu^{*}$, the $\mu$ value that maximizes the corresponding log-likelihood. \end{enumerate*} This will be our best estimate for the true mean. As we already have our data points $x_{1}$ and $x_{2}$, our next step is the probability model. The probability density function (pdf) for normal distribution with mean $\mu$ and variance 1 is \begin{equation} \label{eq:standard-normal-pdf} f(x; \mu) = \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x - \mu)^{2} } \end{equation} (This is the thick curve in Figure~\ref{fig:standard-normal-intervals}). We write it as $f(x; \mu)$ as pdf is usually written as a function of data. But as our primary interest is $\mu$, we also add this as an argument. Now we use this pdf and~\eqref{eq:probability-density} to find the probability that we observe a datapoint in the narrow interval around $x$. Here it is just $f(x; \mu)\cdot \delta$. As $x_{1}$ and $x_{2}$ are independent, we can simply multiply the corresponding probabilities to find the combined probability that both random numbers are near their corresponding values: \begin{multline} \label{eq:two-normal-probability-likelihood} \Pr{\Big(X_{1} \in [x_{1} - \delta/2, x_{1} + \delta/2] \quad\text{and}\quad X_{2} \in [x_{2} - \delta/2, x_{2} + \delta/2]\Big)} =\\[2ex]= \underbrace{ \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{1} - \mu)^{2} } \cdot\delta\ }_{ \text{First random value near $x_{1}$} } \times \underbrace{ \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{2} - \mu)^{2} } \cdot\delta }_{ \text{Second random value near $x_{2}$} } \equiv\\[2ex]\equiv \tilde\likelihood(\mu; x_{1}, x_{2}). \end{multline} The interval width $\delta$ must be small for the equation to hold precisely. We denote this probability with $\tilde\likelihood$ to stress that it is essentially the likelihood, just not written in the way it is usually done. As in the coin-toss example above, we write it as a function of the parameter $\mu$, and put data $x_{1}$ and $x_{2}$ after semicolon. Now we can estimate $\mu$ by finding such a value $\mu^{*}$ that maximizes the expression~\eqref{eq:two-normal-probability-likelihood}. But note that $\delta$ plays no role in maximizing the likelihood. It is just a multiplicative factor, and it cannot be negative because it is a width. So for our maximization problem we can just ignore it. This is what is normally done when working with continuous random variables. Hence we write the likelihood as \begin{equation} \label{eq:two-normal-likelihood} \likelihood(\mu; x_{1}, x_{2}) = \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{1} - \mu)^{2} } \times \frac{1}{\sqrt{2\pi}} \, \me^{ \displaystyle -\frac{1}{2} (x_{2} - \mu)^{2} }. \end{equation} We denote this by $\likelihood$ instead of $\tilde\likelihood$ to stress that this is how likelihood function for continuous random variables is usually written. Exactly as in the discrete case, it is better to use log-likelihood instead of likelihood to actually compute the maximum. From~\eqref{eq:two-normal-likelihood} we get log-likelihood as \begin{multline} \label{eq:two-standard-normal-loglik} \loglik(\mu; x_{1}, x_{2}) = -\log{\sqrt{2\pi}} -\frac{1}{2} (x_{1} - \mu)^{2} + (- \log{\sqrt{2\pi}}) -\frac{1}{2} (x_{2} - \mu)^{2} =\\[2ex]= - 2\log{\sqrt{2\pi}} - \frac{1}{2} \sum_{i=1}^{2} (x_{i} - \mu)^{2}. \end{multline} The first term, $- 2\log{\sqrt{2\pi}}$, is just an additive constant and plays no role in the actual maximization but it is typically still included when defining the likelihood function.\footnote{Additive or multiplicative constants do not play any role for optimization, but they are important when comparing different log-likelihood values. This is often needed for likelihood-based statistical tests. } One can easily check by differentiating the log-likelihood function that the maximum is achieved at $\mu^{*} = \frac{1}{2}(x_{1} + x_{2})$. It is not surprising, our intuitive understanding of mean value carries immediately over to the normal distribution context. Now it is time to demonstrate these results with \texttt{maxLik} package. First, create our ``data'', just two normally distributed random numbers: <<>>= x1 <- rnorm(1) # centered around 0 x2 <- rnorm(1) x1 x2 @ and define the log-likelihood function. We include all the terms as in the final version of~\eqref{eq:two-standard-normal-loglik}: <<>>= loglik <- function(mu) { -2*log(sqrt(2*pi)) - 0.5*((x1 - mu)^2 + (x2 - mu)^2) } @ We also need the parameter start value--we can pick $0$. And we use \texttt{maxLik} to find the best $\mu$: <<>>= m <- maxLik(loglik, start=0) summary(m) @ The answer is the same as sample mean: <<>>= (x1 + x2)/2 @ \section{Vector arguments} \label{sec:vector-arguments} The previous example is instructive but it does have very few practical applications. The problem is that we wrote the probability model as normal density with unknown mean $\mu$ but standard deviation $\sigma$ equal to one. However, in practice we hardly ever know that we are dealing with unit standard deviation. More likely both mean and standard deviation are unknown. So we have to incorporate the unknown $\sigma$ into the model. The more general normal pdf with standard deviation $\sigma$ is \begin{equation} \label{eq:normal-pdf} f(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \, \me^{ -\displaystyle\frac{1}{2} \frac{(x - \mu)^{2}}{\sigma^{2}} }. \end{equation} Similar reasoning as what we did above will give the log-likelihood \begin{equation} \label{eq:two-normal-loglik} \loglik(\mu, \sigma; x_{1}, x_{2}) = - 2\log{\sqrt{2\pi}} - 2\log \sigma - \frac{1}{2} \sum_{i=1}^{2} \frac{(x_{i} - \mu)^{2}}{\sigma^{2}}. \end{equation} We write the log-likelihood as function of both parameters, $\mu$ and $\sigma$; the semicolon that separates data $x_{1}$ and $x_{2}$ shows that though the log-likelihood depends on data too, we are not much interested in that dependency for now. This formula immediately extends to the case of $N$ datapoints as \begin{equation} \label{eq:normal-loglik} \loglik(\mu, \sigma) = - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{(x_{i} - \mu)^{2}}{\sigma^{2}} \end{equation} where we have dropped the dependency on data in the notation. In this case we can actually do the optimization analytically, and derive the well-known intuitive results: the best estimator for mean $\mu$ is the sample average, and the best estimator for $\sigma^{2}$ is the sample variance. However, in general the expression cannot be solved analytically. We have to use numeric optimization to search for the best $\mu$ and $\sigma$ combination. The common multi-dimensional optimizers rely on linear algebra and expect all the parameters submitted as a single vector. So we can write the log-likelihood as \begin{equation} \label{eq:normal-loglik-vector} \loglik(\vec{\theta}) \quad\text{where}\quad \vec{\theta} = (\mu, \sigma). \end{equation} Here we denote both parameters $\mu$ and $\sigma$ as components of a single parameter vector $\vec{\theta}$. (Traditionally vectors are denoted by bold symbols.) We have also dropped dependency on data in notation, but remember that in practical applications log-likelihood always depends on data. This notation can be converted to computer code almost verbatim, just remember to extract the parameters $\mu$ and $\sigma$ from $\vec{\theta}$ in the log-likelihood function. Let us illustrate this using the \emph{CO2} dataset (in package \emph{datasets}). It describes \COii uptake (\si{\micro\mol\per\meter\squared\sec}, variable \emph{uptake}) by different grasses in various conditions. Let us start by plotting the histogram of uptake: <>= data(CO2) hist(CO2$uptake) @ Let us model the uptake as a normal random variable with expected value $\mu$ and standard deviation $\sigma$. We code~\eqref{eq:normal-loglik} while keeping both parameters in a single vector as in~\eqref{eq:normal-loglik-vector}: <<>>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- nrow(CO2) -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } @ The function is similar to the function \texttt{loglik} we used in Section~\ref{sec:continuous-outcomes}. There are just two main differences: \begin{itemize} \item both arguments, $\mu$ and $\sigma$ are passed as components of $\vec{\theta}$, and hence the function starts by unpacking the values. \item instead of using variables \texttt{x1} and \texttt{x2}, we now extract data directly from the data frame. \end{itemize} Besides these two differences, the formula now also includes $\sigma$ and sums over all observations, not just over two observations. As our parameter vector now contains two components, the start vector must also be of length two. Based on the figure we guess that a good starting value might be $\mu=30$ and $\sigma=10$: <<>>= m <- maxLik(loglik, start=c(mu=30, sigma=10)) summary(m) @ Indeed, our guess was close. \section{Final Example: Linear Regression} \label{sec:linear-regression} Now we have the main tools in place to extend the example above to a real statistical model. Let us build the previous example into linear regression. We describe \COii uptake (variable \emph{uptake}) by \COii concentration in air (variable \emph{conc}). We can write the corresponding regression model as \begin{equation} \label{eq:co2-regression} \mathit{uptake}_{i} = \beta_{0} + \beta_{1} \cdot \mathit{conc}_{i} + \epsilon_{i}. \end{equation} In order to turn this regression model into a ML problem, we need a probability model. Assume that the disturbance term $\epsilon$ is normally distributed with mean 0 and (unknown) variance $\sigma^{2}$ (this is a standard assumption in linear regression). Now we can follow~\eqref{eq:two-normal-loglik} and write log of pdf for a single observation as \begin{equation} \label{eq:co2-epsilon-loglik} \loglik(\sigma; \epsilon_{i}) = - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{\epsilon_{i}^{2}}{\sigma^{2}}. \end{equation} Here we have replaced $x_{i}$ by the random outcome $\epsilon_{i}$. As the expected value $\mu=0$ by assumption, we do not include $\mu$ in~\eqref{eq:co2-epsilon-loglik} and hence we drop it also from the argument list of $\loglik$. We do not know $\epsilon_{i}$ but we can express it using linear regression model~\eqref{eq:co2-regression}: \begin{equation} \label{eq:co2-epsilon} \epsilon_{i} = \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i}. \end{equation} This expression depends on two additional unknown parameters, $\beta_{0}$ and $\beta_{1}$. These are the linear regression coefficients we want to find. Now we plug this into~\eqref{eq:co2-epsilon-loglik}: \begin{multline} \label{eq:co2-single-loglik} \loglik(\beta_{0}, \beta_{1}, \sigma; \mathit{uptake}_{i}, \mathit{conc}_{i}) =\\= - \log{\sqrt{2\pi}} - \log \sigma - \frac{1}{2} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i} )^{2}}{\sigma^{2}}. \end{multline} We have designed log-likelihood formula for a single linear regression observation. It depends on three parameters, $\beta_{0}$, $\beta_{1}$ and $\sigma$. For $N$ observations we have \begin{multline} \label{eq:co2-loglik} \loglik(\beta_{0}, \beta_{1}, \sigma; \vec{\mathit{uptake}}, \vec{\mathit{conc}}) =\\= - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i})^{2}}{\sigma^{2}} \end{multline} where vectors $\vec{\mathit{uptake}}$ and $\vec{\mathit{conc}}$ contain the data values for all the observations. This is a fully specified log-likelihood function that we can use for optimization. Let us repeat what we have done: \begin{itemize} \item We wrote log-likelihood as a function of parameters $\beta_{0}$, $\beta_{1}$ and $\sigma$. Note that in case of linear regression we typically do not call $\sigma$ a parameter. But it is still a parameter, although one we usually do not care much about (sometimes called ``nuisance parameter''). \item The likelihood function also depends on data, here the vectors $\vec{\mathit{uptake}}$ and $\vec{\mathit{conc}}$. \item The function definition itself is just sum of log-likelihood contributions of individual normal disturbance terms, but as we do not observe the disturbance terms, we express those through the regression equation in~\eqref{eq:co2-single-loglik}. \end{itemize} Finally, we combine the three parameters into a single vector $\vec{\theta}$, suppress dependency on data in the notation, and write \begin{equation} \label{eq:co2-loglik-simplified} \loglik(\vec{\theta}) = - N\log{\sqrt{2\pi}} - N\log \sigma - \frac{1}{2} \sum_{i=1}^{N} \frac{( \mathit{uptake}_{i} - \beta_{0} - \beta_{1} \cdot \mathit{conc}_{i})^{2}}{\sigma^{2}}. \end{equation} This is the definition we can easily code and estimate. We guess start values $\beta_{0} = 30$ (close to the mean), $\beta_{1} = 0$ (uptake does not depend on concentration) and $\sigma=10$ (close to sample standard deviation). We can convert~\eqref{eq:co2-loglik-simplified} into code almost verbatim, below we choose to compute the expected uptake $\mu$ as an auxiliary variable: <<>>= loglik <- function(theta) { beta0 <- theta[1] beta1 <- theta[2] sigma <- theta[3] N <- nrow(CO2) ## compute new mu based on beta1, beta2 mu <- beta0 + beta1*CO2$conc ## use this mu in a similar fashion as previously -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } m <- maxLik(loglik, start=c(beta0=30, beta1=0, sigma=10)) summary(m) @ These are the linear regression estimates: $\beta_{0} = \Sexpr{round(coef(m)["beta0"], 3)}$ and $\beta_{1} = \Sexpr{round(coef(m)["beta1"], 3)}$. Note that \maxlik output also provides standard errors, $z$-values and $p$-values, hence we see that the results are highly statistically significant. One can check that a linear regression model will give similar results: <<>>= summary(lm(uptake ~ conc, data=CO2)) @ Indeed, the results are close although not identical. \section{Non-linear optimization} \label{sec:non-linear-optimization} Finally, we discuss the magic inside \texttt{maxLik} that finds the optimal parameter values. Although not necessary in everyday work, this knowledge helps to understand the issues and potential solutions when doing non-linear optimization. So how does the optimization work? Consider the example in Section~\ref{sec:vector-arguments} where we computed the normal distribution parameters for \COii intake. There are two parameters, $\mu$ and $\sigma$, and \maxlik returns the combination that gives the largest possible log-likelihood value. We can visualize the task by plotting the log-likelihood value for different combinations of $\mu$, $\sigma$ (Figure~\ref{fig:mu-sigma-plot}). \begin{figure}[ht] \centering <>= loglik <- function(theta) { mu <- theta[1] sigma <- theta[2] N <- nrow(CO2) -N*log(sqrt(2*pi)) - N*log(sigma) - 0.5*sum((CO2$uptake - mu)^2/sigma^2) } m <- maxLik(loglik, start=c(mu=30, sigma=10)) params <- coef(m) np <- 33 # number of points mu <- seq(6, 36, length.out=np) sigma <- seq(5, 50, length.out=np) X <- as.matrix(expand.grid(mu=mu, sigma=sigma)) ll <- matrix(apply(X, 1, loglik), nrow=np) levels <- quantile(ll, c(0.05, 0.4, 0.6, 0.8, 0.9, 0.97)) # where to draw the contours colors <- colorRampPalette(c("Blue", "White"))(30) par(mar=c(0,0,0,0), mgp=2:0) ## Perspective plot if(require(plot3D)) { persp3D(mu, sigma, ll, xlab=expression(mu), ylab=expression(sigma), zlab=expression(log-likelihood), theta=40, phi=30, colkey=FALSE, col=colors, alpha=0.5, facets=TRUE, shade=1, lighting="ambient", lphi=60, ltheta=0, image=TRUE, bty="b2", contour=list(col="gray", side=c("z"), levels=levels) ) ## add the dot for maximum scatter3D(rep(coef(m)[1], 2), rep(coef(m)[2], 2), c(maxValue(m), min(ll)), col="red", pch=16, facets=FALSE, bty="n", add=TRUE) ## line from max on persp to max at bottom surface segments3D(coef(m)[1], coef(m)[2], maxValue(m), coef(m)[1], coef(m)[2], min(ll), col="red", lty=2, bty="n", add=TRUE) ## contours for the bottom image contour3D(mu, sigma, z=min(ll) + 0.1, colvar=ll, col="black", levels=levels, add=TRUE) } else { plot(1:2, type="n") text(1.5, 1.5, "This figure requires 'plot3D' package", cex=1.5) } @ \caption{Log-likelihood surface as a function of $\mu$ and $\sigma$. The optimum, denoted as the red dot, is at $\mu=\Sexpr{round(coef(m)[1], 3)}$ and $\sigma=\Sexpr{round(coef(m)[2], 3)}$. The corresponding countour plot is shown at the bottom of the figure box. } \label{fig:mu-sigma-plot} \end{figure} So how does the algorithm find the optimal parameter value $\vec{\theta}^*$, the red dot on the figure? All the common methods are iterative, i.e. they start with a given start value (that's why we need the start value), and repeatedly find a new and better parameter that gives a larger log-likelihood value. While humans can look at the figure and immediately see where is its maximum, computers cannot perceive the image in this way. And more importantly--even humans cannot visualize the function in more than three dimensions. This visualization is so helpful for us because we can intuitively understand the 3-dimensional structure of the surface. It is 3-D because we have two parameters, $\mu$ and $\sigma$, and a single log-likelihood value. Add one more parameter as we did in Section~\ref{sec:linear-regression}, and visualization options are very limited. In case of 5 parameters, it is essentially impossible to solve the problem by just visualizations. Non-linear optimization is like climbing uphill in whiteout conditions where you cannot distinguish any details around you--sky is just a white fog and the ground is covered with similar white snow. But you can still feel which way the ground goes up and so you can still go uphill. This is what the popular algorithms do. They rely on the slope of the function, the gradient, and follow the direction suggested by gradient. Most optimizers included in the \texttt{maxLik} package need gradients, including the default Newton-Raphson method. But how do we know the gradient if the log-likelihood function only returns a single value? There are two ways: \begin{enumerate*}[label=\roman*)] \item provide a separate function that computes gradient; \item compute the log-likelihood value in multiple points nearby and deduce the gradient from that information. \end{enumerate*} The first option is superior, in high dimensions it is much faster and much less error prone. But computing and coding gradient can easily be days of work. The second approach, numeric gradient, forces the computer to do more work and hence it is slower. Unfortunately importantly, it may also unreliable for more complex cases. In practice you may notice how the algorithm refuses to converge for thousands of iterations. But numeric gradient works very well in simple cases we demonstrated here. This also hints why it is useful to choose good start values. The closer we start to our final destination, the less work the computer has to do. And while we may not care too much about a few seconds of computer's work, we also help the algorithm to find the correct maximum. The less the algorithm has to work, the less likely it is that it gets stuck in a wrong place or just keeps wandering around in a clueless manner. 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It includes several optimizers and associated tools for a typical Maximum Likelihood workflow. However, as predictive modeling and complex (deep) models have gained popularity in the recend decade, \texttt{maxLik} also includes a few popular algorithms for stochastic gradient ascent, the mirror image for the more widely known stochastic gradient descent. This vignette gives a brief overview of these methods, and their usage in \texttt{maxLik}. \section{Stochastic Gradient Ascent} \label{sec:stochastic-gradient-ascent} In machine learning literature, it is more common to describe the optimization problems as minimization and hence to talk about gradient descent. As \texttt{maxLik} is primarily focused on maximizing likelihood, it implements the maximization version of the method, stochastic gradient ascent (SGA). The basic method is simple and intuitive, it is essentially just a careful climb in the gradient's direction. Given and objective function $f(\vec{\theta})$, and the initial parameter vector $\vec{\theta}_{0}$, the algorithm will compute the gradient $\vec{g}(\vec{\theta}_{0}) = \nabla_{\vec{\theta}} f(\vec{\theta})\big|_{\vec{\theta} = \vec{\theta}_{0}}$, and update the parameter vector as $\vec{\theta}_{1} = \vec{\theta}_{0} + \rho \vec{g}(\vec{\theta}_{0})$. Here $\rho$, the \emph{learning rate}, is a small positive constant to ensure we do not overshoot the optimum. Depending on the task it is typically of order $0.1 \dots 0.001$. In common tasks, the objective function $f(\vec{\theta})$ depends on data, ``predictors'' $\mat{X}$ and ``outcome'' $\vec{y}$ in an additive form $f(\vec{\theta}; \mat{X}, \vec{y}) = \sum_{i} f(\vec{\theta}; \vec{x}_{i}, y_{i})$ where $i$ denotes ``observations'', typically arranged as the rows of the design matrix $\mat{X}$. Observations are often considered to be independent of each other. The overview above does not specify how to compute the gradient $\vec{g}(\vec{\theta}_{0})$ in a sense of which observations $i$ to include. A natural approach is to include the complete data and compute \begin{equation} \label{eq:full-batch-gradient} \vec{g}_{N}(\vec{\theta}_{0}) = \frac{1}{N}\sum_{i=1}^{N} \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}}. \end{equation} In SGA context, this approach is called ``full batch'' and it has a number of advantages. In particular, it is deterministic (given data $\mat{X}$ and $\vec{y}$), and computing of the sum can be done in parallel. However, there are also a number of reasons why full-batch approach may not be desirable \citep[see][]{bottou2018SIAM}: \begin{itemize} \item Data over different observations is often more or less redundant. If we use all the observations to compute the update then we spend a substantial effort on redundant calculations. \item Full-batch gradient is deterministic and hence there is no stochastic noise. While advantageous in the latter steps of optimization, the noise helps the optimizer to avoid local optima and overcome flat areas in the objective function early in the process. \item SGA achieves much more rapid initial convergence compared to the full batch method (although full-batch methods may achieve better final result). \item Cost of computing the full-batch gradient grows with the sample size but that of minibatch gradient does not grow. \item It is empirically known that large-batch optimization tend to find sharp optima \citep[see][]{keskar+2016ArXiv} that do not generalize well to validation data. Small batch approach leads to a better validation performance. \end{itemize} In contrast, SGA is an approach where the gradient is computed on just a single observation as \begin{equation} \label{eq:stochastic-gradient} \vec{g}_{1}(\vec{\theta}_{0}) = \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i}, y_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}} \end{equation} where $i$ is chosen randomly. In applications, all the observations are usually walked through in a random order, to ensure that each observation is included once, and only once, in an \emph{epoch}. Epoch is a full walk-through of the data, and in many ways similar to iteration in a full-batch approach. As SGA only accesses a single observation at time, it suffers from other kind of performance issues. In particular, one cannot parallelize the gradient function \eqref{eq:stochastic-gradient}, operating on individual data vectors may be inefficient compared to larger matrices, and while we gain in terms of gradient computation speed, we lose by running the optimizer for many more loops. \emph{Minibatch} approach offers a balance between the full-batch and SGA. In case of minibatch, we compute gradient not on individual observations but on \emph{batches} \begin{equation} \label{eq:minibatch-gradient} \vec{g}_{m}(\vec{\theta}_{0}) = \frac{1}{|\mathcal{B}|}\sum_{i\in\mathcal{B}} \nabla_{\vec{\theta}} f(\vec{\theta}; \vec{x}_{i}, y_{i})\big|_{\vec{\theta} = \vec{\theta}_{0}} \end{equation} where $\mathcal{B}$ is the batch, a set of observations that are included in the gradient computation. Normally the full data is partitioned into a series of minibatches and walked through sequentially in one epoch. \section{SGA in \texttt{maxLik} package} \label{sec:sga-in-maxlik} \maxlik implements two different optimizers: \texttt{maxSGA} for simple SGA (including momentum), and \texttt{maxAdam} for the Adaptive Moments method \citep[see][p. 301]{goodfellow+2016DL}. The usage of both methods mostly follows that of the package's main workhorse, \texttt{maxNR} \citep[see][]{henningsen+toomet2011}, but their API has some important differences due to the different nature of SGA. The basic usage of the maxSGA is as follows: <>= maxSGA(fn, grad, start, nObs, control) @ where \texttt{fn} is the objective function, \texttt{grad} is the gradient function, \texttt{nObs} is number of observations, and \texttt{control} is a list of control parameters. From the user's perspective, \texttt{grad} is typically the most important (and the most complex) argument. Next, we describe the API and explain the differences between the \texttt{maxSGA} API and \texttt{maxNR} API, and thereafter give a few toy examples that demonstrate how to use \texttt{maxSGA} in practice. \subsection{The objective function} Unlike in \texttt{maxNR} and the related optimizers, SGA does not directly need the objective function \texttt{fn}. The function can still be provided (and perhaps will in most cases), but one can run the optimizer without it. If provided, the function can be used for printing the value at each epoch (by setting a suitable \texttt{printLevel} control option), and for stopping through \emph{patience} stopping condition. If \texttt{fn} is not provided, do not forget to add the argument name for the gradient, \texttt{grad=}, as otherwise the gradient will be treated as the objective function with unexpected results! If provided, the function should accept two (or more) arguments: the first must be the numeric parameter vector, and another one, named \texttt{index}, is the list of indices in the current minibatch. As the function is not needed by the optimizer itself, it is up to the user to decide what it does. An obvious option is to compute the objective function value on the same minibatch as used for the gradient computation. But one can also opt for something else, for instance to compute the value on the validation data instead (and ignore the provided \emph{index}). The latter may be a useful option if one wants to employ the patience-based stopping criteria. \subsection{Gradient function} \label{sec:gradient-function} Gradient is the work-horse of the SGA methods. Although \maxlik can also compute numeric gradient using the finite difference method (this will be automatically done if the objective function is provided but the gradient isn't), this is not advisable, and may be very slow in high-dimensional problems. \texttt{maxLik} uses the numerator layout, i.e. the gradient should be a $1\times K$ matrix where columns correspond to the components of the parameter vector $\vec{\theta}$. For compatibility with other optimizers in \texttt{maxLik} it also accepts a observation-wise matrix where rows correspond to the individual observations and columns to the parameter vector components. The requirements for the gradient function arguments are the same as for \texttt{fn}: the first formal argument must be the parameter vector, and it must also have an argument \texttt{index}, a numeric index for the observations to be included in the minibatch. \subsection{Stopping Conditions} \label{sec:stopping-conditions} \texttt{maxSGA} uses three stopping criteria: \begin{itemize} \item Number of epochs (control option \texttt{iterlim}): number of times all data is iterated through using the minibatches. \item Gradient norm. However, in case of stochastic approach one cannot expect the gradient at optimum to be close to zero, and hence the corresponding criterion (control option \texttt{gradtol}) is set to zero by default. If interested, one may make it positive. \item Patience. Normally, each new iteration has better (higher) value of the objective function. However, in certain situations this may not be the case. In such cases the algorithm does not stop immediately, but continues up to \emph{patience} more epochs. It also returns the best parameters, not necessarily the last parameters. Patience can be controlled with the options \texttt{SG\_patience} and \texttt{SG\_patienceStep}. The former controls the patience itself--how many times the algorithm is allowed to produce an inferior result (default value \texttt{NULL} means patience criterion is not used). The latter controls how often the patience criterion is checked. If computing the objective function is costly, it may be useful to increase the patience step and decrease the patience. \end{itemize} \subsection{Optimizers} \label{sec:optimizers} \texttt{maxLik} currently implements two optimizers: \emph{SGA}, the stock gradient ascent (including momentum), and \emph{Adam}. Here we give some insight into the momentum, and into the Adam method, the basic gradient-only based optimization technique was explained in Section~\ref{sec:stochastic-gradient-ascent}. It is easy and intuitive to extend the SGA method with momentum. As implemented in \texttt{maxSGA}, the momentum $\mu$ ($0 < \mu < 1$) is incorporated into the the gradient update as \begin{equation} \label{eq:gradient-update-momentum} \vec{\theta}_{t+1} = \vec{\theta}_{t} + \vec{v}_{t} \quad\text{where}\quad \vec{v}_{t} = \mu \vec{v}_{t-1} + \rho \vec{g}(\vec{\theta}_{t}). \end{equation} See \citet[p. 288]{goodfellow+2016DL}. The algorithm takes the initial ``velocity'' $\vec{v}_{0} = \vec{0}$. It is easy to see that $\mu=0$ is equivalent to no-momentum case, and if $\vec{g}(\vec{\theta})$ is constant, $\vec{v}_{t} \to \rho \vec{g}(\vec{\theta})/(1 - \mu)$. So the movement speeds up in a region with stable gradient. As a downside, it is also easier overshoot a maximum. But this behavior makes momentum-equipped SGA less prone of getting stuck in a local optimum. Momentum can be set by the control option \texttt{SG\_momentum}, the default value is 0. Adaptive Moments method, usually referred to as \emph{Adam}, \citep[p. 301]{goodfellow+2016DL} adapts the learning rate by variance of the gradient--if gradient components are unstable, it slows down, and if they are stable, it speeds up. The adaptation is proportional to the weighted average of the gradient divided by the square root of the weighted average of the gradient squared, all operations done component-wise. In this way a stable gradient component (where moving average is similar to the gradient value) will have higher speed than a fluctuating gradient (where the components frequently shift the sign and the average is much smaller). More specifically, the algorithm is as follows: \begin{enumerate} \item Initialize the first and second moment averages $\vec{s} = \vec{0}$ and $\vec{r} = \vec{0}$. \item Compute the gradient $\vec{g}_{t} = \vec{g}(\vec{\theta}_{t})$. \item Update the average first moment: $\vec{s}_{t+1} = \mu_{1} \vec{s}_{t} + (1 - \mu_{1}) \vec{g}_{t}$. $\mu_{1}$ is the decay parameter, the larger it is, the longer memory does the method have. It can be adjusted with the control parameter \texttt{Adam\_momentum1}, the default value is 0.9. \item Update the average second moment: $\vec{r}_{t+1} = \mu_{2} \vec{r}_{t} + (1 - \mu_{2}) \vec{g}_{t} \elemProd \vec{g}_{t}$ where $\elemProd$ denotes element-wise multiplication. The control parameter for the $\mu_{2}$ is \texttt{Adam\_momentum2}, the default value is 0.999. \item As the algorithm starts with the averages $\vec{s}_{0} = \vec{r}_{0}= 0$, we also correct the resulting bias: $\hat{\vec{s}} = \vec{s}/(1 - \mu_{1}^{t})$ and $\hat{\vec{r}} = \vec{r}/(1 - \mu_{2}^{t})$. \item Finally, update the estimate: $\vec{\theta}_{t+1} = \vec{\theta}_{t} + \rho \hat{\vec{s}}/(\delta + \sqrt{\hat{\vec{r}}})$ where division and square root are done element-wise and $\delta=10^{-8}$ takes care of numerical stabilization. \end{enumerate} Adam optimizer can be used with \texttt{maxAdam}. \subsection{Controlling Optimizers} \label{sec:control-options} Both \texttt{maxSGA} and \texttt{maxAdam} are designed to be similar to \texttt{maxNR}, and mostly expect similar arguments. In particular, both functions expect the objective function \texttt{fn}, gradient \texttt{grad} and Hessian function \texttt{hess}, and the initial parameter start values \texttt{start}. As these optimizers only need gradient, one can leave out both \texttt{fn} and \texttt{hess}. The Hessian is mainly included for compatibility reasons and only used to compute the final Hessian, if requested by the user. As SGA methods are typically used in contexts where Hessian is not needed, by default the algorithms do not return Hessian matrix and hence do not use the \texttt{hess} function even if provided. Check out the argument \texttt{finalHessian} if interested. An important SGA-specific control options is \texttt{SG\_batchSize}. This determines the batch size, or \texttt{NULL} for the full-batch approach. Finally, unlike the traditional optimizers, stochastic optimizers need to know the size of data (argument \texttt{nObs}) in order to calculate the batches. \section{Example usage: Linear regression} \label{sec:example-usage-cases} \subsection{Setting Up} \label{sec:setting-up} We demonstrate the usage of \texttt{maxSGA} and \texttt{maxAdam} to solve a linear regression (OLS) problem. Although OLS is not a task where one commonly relies on stochastic optimization, it is a simple and easy-to understand model. We use the Boston housing data, a popular dataset where one traditionally attempts to predict the median house price across 500 neighborhoods using a number of neighborhood descriptors, such as mean house size, age, and proximity to Charles river. All variables in the dataset are numeric, and there are no missing values. The data is provided in \emph{MASS} package. First, we create the design matrix $\mat{X}$ and extract the house price $y$: <<>>= i <- which(names(MASS::Boston) == "medv") X <- as.matrix(MASS::Boston[,-i]) X <- cbind("const"=1, X) # add constant y <- MASS::Boston[,i] @ Although the model and data are simple, it is not an easy task for stock gradient ascent. The problem lies in different scaling of variables, the means are <<>>= colMeans(X) @ One can see that \emph{chas} has an average value \Sexpr{round(mean(X[,"chas"]), 3)} while that of \emph{tax} is \Sexpr{round(mean(X[,"tax"]), 3)}. This leads to extremely elongated contours of the loss function: <>= eigenvals <- eigen(crossprod(X))$values @ One can see that the ratio of the largest and the smallest eigenvalue is $\mat{X}^{\transpose} \mat{X} = \Sexpr{round(eigenvals[1]/eigenvals[14], -5)}$. Solely gradient-based methods, such as SGA, have trouble working in the resulting narrow valleys. For reference, let's also compute the analytic solution to this linear regression model (reminder: $\hat{\vec{\beta}} = (\mat{X}^{\transpose}\,\mat{X})^{-1}\,\mat{X}^{\transpose}\,\vec{y}$): <<>>= betaX <- solve(crossprod(X)) %*% crossprod(X, y) betaX <- drop(betaX) # matrix to vector betaX @ Next, we provide the gradient function. As a reminder, OLS gradient in numerator layout can be expressed as \begin{equation} \label{eq:ols-gradient} \vec{g}_{m}(\vec{\theta}) = -\frac{2}{|\mathcal{B}|} \sum_{i\in\mathcal{B}} \left(y_{i} - \vec{x}_{i}^{\transpose} \cdot \vec{\theta} \right) \vec{x}_{i}^{\transpose} = -\frac{2}{|\mathcal{B}|} \left(y_{\mathcal{B}} - \mat{X}_{\mathcal{B}} \cdot \vec{\theta} \right)^{\transpose} \mat{X}_{\mathcal{B}} \end{equation} where $y_{\mathcal{B}}$ and $\mat{X}_{\mathcal{B}}$ denote the elements of the outcome vector and the slice of the design matrix that correspond to the minibatch $\mathcal{B}$. We choose to divide the value by batch size $|\mathcal{B}|$ in order to have gradient values of roughly similar size, independent of the batch size. We implement it as: <<>>= gradloss <- function(theta, index) { e <- y[index] - X[index,,drop=FALSE] %*% theta g <- t(e) %*% X[index,,drop=FALSE] 2*g/length(index) } @ The \texttt{gradloss} function has two arguments: \texttt{theta} is the parameter vector, and \texttt{index} tells which observations belong to the current minibatch. The actual argument will be an integer vector, and hence we can use \texttt{length(index)} to find the size of the minibatch. Finally, we return the negative of~\eqref{eq:ols-gradient} as \texttt{maxSGA} performs maximization, not minimization. First, we demonstrate how the models works without the objective function. We have to supply the gradient function, initial parameter values (we use random normals below), and also \texttt{nObs}, number of observations to select the batches from. The latter is needed as the optimizer itself does not have access to data but still has to partition it into batches. Finally, we may also provide various control parameters, such as number of iterations, stopping conditions, and batch size. We start with only specifying the iteration limit, the only stopping condition we use here: <>= library(maxLik) set.seed(3) start <- setNames(rnorm(ncol(X), sd=0.1), colnames(X)) # add names for better reference res <- try(maxSGA(grad=gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000) ) ) @ This run was a failure. We encountered a run-away growth of the gradient because the default learning rate $\rho=0.1$ is too big for such strongly curved objective function. But before we repeat the exercise with a smaller learning rate, let's incorporate gradient clipping. Gradient clipping, performed with \texttt{SG\_clip} control option, caps the $L_{2}$-norm of the gradient while keeping it's direction. We clip the squared norm at 10,000, i.e. the gradient norm cannot exceed 100: <<>>= res <- maxSGA(grad=gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000, SG_clip=1e4) # limit ||g|| <= 100 ) summary(res) @ This time the gradient did not explode and we were able to get a result. But the estimates are rather far from the analytic solution shown above, e.g. the constant estimate \Sexpr{round(coef(res)[1], 3)} is very different from the corresponding analytic value \Sexpr{round(betaX[1], 3)}. Let's analyze what is happening inside the optimizer. We can ask for both the parameter values and the objective function value to be stored for each epoch. But before we can store its value, in this case mean squared error (MSE), we have to supply an objective function to maxSGA. We compute MSE on the same minibatch as <<>>= loss <- function(theta, index) { e <- y[index] - X[index,] %*% theta -crossprod(e)/length(index) } @ Now we can store the values with the control options \texttt{storeParameters} and \texttt{storeValues}. The corresponding numbers can be retrieved with \texttt{storedParameters} and \texttt{storedValues} methods. For \texttt{iterlim=R}, the former returns a $(R+1) \times K$ matrix, one row for each epoch and one column for each parameter component, and the latter returns a numeric vector of length $R+1$ where $R$ is the number of epochs. The first value in both cases is the initial value, so we have $R+1$ values in total. Let's retrieve the values and plot both. We decrease the learning rate to $0.001$ using the \texttt{SG\_learningRate} control. Note that although we maximize negative loss, we plot positive loss. \setkeys{Gin}{width=\textwidth, height=80mm} <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(X), control=list(iterlim=1000, # will misbehave with larger numbers SG_clip=1e4, SG_learningRate=0.001, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") ## add some arrows to see which way the parameters move iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 10 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) ## plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We can see how the parameters (the first and the second components, ``const'' and ``crim'' in this figure) evolve through the iterations while the loss is rapidly falling. One can see an initial jump where the loss is falling very fast, followed but subsequent slow movement. It is possible the initial jump be limited by gradient clipping. \subsection{Training and Validation Sets} \label{sec:training-validation} However, as we did not specify the batch size, \texttt{maxSGA} will automatically pick the full batch (equivalent to control option \texttt{SG\_batchSize = NULL}). So there was nothing stochastic in what we did above. Let us pick a small batch size--a single observation at time. However, as smaller batch sizes introduce more noise to the gradient, we also make the learning rate smaller and choose \texttt{SG\_learningRate = 1e-5}. But now the existing loss function, calculated just at the single observation, carries little meaning. Instead, we split the data into training and validation sets and feed batches of training data to gradient descent while calculating the loss on the complete validation set. This can be achieved with small modifications in the \texttt{gradloss} and \texttt{loss} function. But as the first step, we split the data: <<>>= i <- sample(nrow(X), 0.8*nrow(X)) # training indices, 80% of data Xt <- X[i,] # training data yt <- y[i] Xv <- X[-i,] # validation data yv <- y[-i] @ Thereafter we modify \texttt{gradloss} to only use the batches of training data while \texttt{loss} will use the complete validation data and just ignore \texttt{index}: <<>>= gradloss <- function(theta, index) { e <- yt[index] - Xt[index,,drop=FALSE] %*% theta g <- -2*t(e) %*% Xt[index,,drop=FALSE] -g/length(index) } loss <- function(theta, index) { e <- yv - Xv %*% theta -crossprod(e)/length(yv) } @ Note that because the optimizer only uses training data, the \texttt{nObs} argument now must equal to the size of training data in this case. Another thing to discuss is the computation speed. \texttt{maxLik} implements SGA in a fairly complex loop that does printing, storing, and complex function calls, computes stopping conditions and does many other checks. Hence a smaller batch size leads to many more such auxiliary computations per epoch and the algorithm gets considerably slower. This is less of a problem for complex objective functions or larger batch sizes, but for linear regression, the slow-down is very large. For demonstration purposes we lower the number of epochs from 1000 to 100. How do the convergence properties look now with the updated approach? <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(Xt), # note: only training data now control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-5, SG_clip=1e4, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We can see the parameters evolving and loss decreasing over epochs. The convergence seems to be smooth and not ruptured by gradient clipping. Next, we try to improve the convergence by introducing momentum. We add momentum $\mu = 0.95$ to the gradient and decrease the learning rate down to $1\cdot10^{-6}$: <>= res <- maxSGA(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, SGA_momentum = 0.99, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ We achieved a lower loss but we are still far from the correct solution. As the next step, we use Adam optimizer. Adam has two momentum parameters but we leave those untouched at the initial values. \texttt{SGA\_momentum} is not used, so we remove that argument. <>= res <- maxAdam(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=100, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, storeParameters=TRUE, storeValues=TRUE ) ) par <- storedParameters(res) val <- storedValues(res) par(mfrow=c(1,2)) plot(par[,1], par[,2], type="b", pch=".", xlab=names(start)[1], ylab=names(start)[2], main="Parameters") iB <- c(40, nrow(par)/2, nrow(par)) iA <- iB - 1 arrows(par[iA,1], par[iA,2], par[iB,1], par[iB,2], length=0.1) plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") @ As visible from the figure, Adam was marching toward the solution without any stability issues. \subsection{Sequence of Batch Sizes } \label{sec:sequence-batch-sizes} The OLS' loss function is globally convex and hence there is no danger to get stuck in a local maximum. However, when the objective function is more complex, the noise that is generated by the stochastic sampling helps the algorithm to leave local maxima. A suggested strategy is to increase the batch size over time to achieve good exploratory properties early in the process and stable convergence later \citep[see][for more information]{smith+2018arXiv}. This approach is in some ways similar to Simulated Annealing. Here we introduce such an approach by using batch sizes $B=1$, $B=10$ and $B=100$ in succession. We also introduce patience stopping condition. If the objective function value is worse than the best value so far for more than \emph{patience} times then the algorithm stops. Here we use patience value 5. We also store the loss values from all the batch sizes into a single vector \texttt{val}. If the algorithm stops early, some of the stored values are left uninitialized (\texttt{NA}-s), hence we use \texttt{na.omit} to include only the actual values in the final \texttt{val}-vector. We allow the algorithm to run for 200 epochs, but as we now have introduced early stopping through patience, the actual number of epochs may be less than that. \setkeys{Gin}{width=\textwidth, height=110mm} <>= val <- NULL # loop over batch sizes for(B in c(1,10,100)) { res <- maxAdam(loss, gradloss, start=start, nObs=nrow(Xt), control=list(iterlim=200, SG_batchSize=1, SG_learningRate=1e-6, SG_clip=1e4, SG_patience=5, # worse value allowed only 5 times storeValues=TRUE ) ) cat("Batch size", B, ",", nIter(res), "epochs, function value", maxValue(res), "\n") val <- c(val, na.omit(storedValues(res))) start <- coef(res) } plot(seq(length=length(val))-1, -val, type="l", xlab="epoch", ylab="MSE", main="Loss", log="y") summary(res) @ Two first batch sizes run through all 200 epochs, but the last run stopped early after 7 epochs only. The figure shows that Adam works well for approximately 170 epochs, thereafter the steady pace becomes uneven. It may be advantageous to slow down the movement further. As explained above, this dataset is not an easy task for methods that are solely gradient-based, and so we did not achieve a result that is close to the analytic solution. But our task here is to demonstrate the usage of the package, not to solve a linear regression exercise. We believe every \emph{R}-savy user can adapt the method to their needs. \bibliographystyle{apecon} \bibliography{maxlik} \end{document} maxLik/inst/CITATION0000644000176200001440000000150214077525067013624 0ustar liggesuserscitHeader("To cite package 'maxLik' in publications use:") citEntry( entry = "Article", title = "maxLik: A package for maximum likelihood estimation in {R}", author = personList( as.person( "Arne Henningsen" ), as.person( "Ott Toomet" ) ), journal = "Computational Statistics", year = "2011", volume = "26", number = "3", pages = "443-458", doi = "10.1007/s00180-010-0217-1", url = "http://dx.doi.org/10.1007/s00180-010-0217-1", textVersion = paste( "Arne Henningsen and Ott Toomet (2011).", "maxLik: A package for maximum likelihood estimation in R.", "Computational Statistics 26(3), 443-458.", "DOI 10.1007/s00180-010-0217-1." ) ) maxLik/inst/tinytest/0000755000176200001440000000000014077525067014354 5ustar liggesusersmaxLik/inst/tinytest/test-maxSG.R0000644000176200001440000001635114077525067016501 0ustar liggesusers### tests for stochastic gradient ascent ### ### do not run unless 'NOT_CRAN' explicitly defined ### (Suggested by Sebastian Meyer and others) if(!identical(Sys.getenv("NOT_CRAN"), "true")) { message("We are on CRAN: skipping slow optimizer tests") q("no") } if(!requireNamespace("tinytest", quietly = TRUE)) { message("These tests require 'tinytest' package\n") q("no") } library(maxLik) ### Test the following things: ### ### 1. basic 2-D SGA ### SGA without function, only gradient ### SGA neither function nor gradient ### SGA in 1-D case ### 2. SGA w/momentum ### 3. SGA full batch ### 4. SGA, no gradient supplied ### SGA, return numeric hessian, gradient provided ### SGA, return numeric hessian, no gradient provided ### SGA, printlevel 1, storeValues ### SGA, NA as iterlim: should give informative error ### SGA, storeValues but no fn (should fail) ### ### using highly unequally scaled data ### SGA without gradient clipping (fails) ### SGA with gradient clipping (works, although does not converge) ## ---------- OLS ## log-likelihood function(s): ## return log-likelihood on validation data loglik <- function(beta, index) { e <- yValid - XValid %*% beta -crossprod(e)/length(y) } ## gradlik: work on training data gradlik <- function(beta, index) { e <- yTrain[index] - XTrain[index,,drop=FALSE] %*% beta g <- t(-2*t(XTrain[index,,drop=FALSE]) %*% e) -g/length(index) } ### create random data set.seed(1) N <- 1000 x <- rnorm(N) X <- cbind(1, x) y <- 100 + 100*x + rnorm(N) ## training-validation iTrain <- sample(N, 0.8*N) XTrain <- X[iTrain,,drop=FALSE] XValid <- X[-iTrain,,drop=FALSE] yTrain <- y[iTrain] yValid <- y[-iTrain] ## Analytic solution (training data): start <- c(const=10, x=10) b0 <- drop(solve(crossprod(XTrain)) %*% crossprod(XTrain, yTrain)) names(b0) <- names(start) tol <- 1e-3 # coefficient tolerance ## ---------- 1. working example res <- maxSGA(loglik, gradlik, start=start, control=list(printLevel=0, iterlim=200, SG_batchSize=100, SG_learningRate=0.1, storeValues=TRUE), nObs=length(yTrain)) expect_equal(coef(res), b0, tolerance=tol) # SGA usually ends with gradient not equal to 0 so we don't test that ## ---------- store parameters res <- maxSGA(loglik, gradlik, start=start, control=list(printLevel=0, iterlim=20, SG_batchSize=100, SG_learningRate=0.1, storeParameters=TRUE), nObs=length(yTrain)) expect_equal(dim(storedParameters(res)), c(1 + nIter(res), 2)) ## ---------- no function, only gradient expect_silent( res <- maxSGA(grad=gradlik, start=start, control=list(printLevel=0, iterlim=10, SG_batchSize=100), nObs=length(yTrain)) ) ## ---------- neither function nor gradient expect_error( res <- maxSGA(start=start, control=list(printLevel=0, iterlim=10, SG_batchSize=100), nObs=length(yTrain)) ) ## ---------- 1D case N1 <- 1000 t <- rexp(N1, 2) loglik1 <- function(theta, index) sum(log(theta) - theta*t[index]) gradlik1 <- function(theta, index) sum(1/theta - t[index]) expect_silent( res <- maxSGA(loglik1, gradlik1, start=1, control=list(iterlim=300, SG_batchSize=20), nObs=length(t)) ) expect_equal(coef(res), 1/mean(t), tolerance=0.2) expect_null(hessian(res)) ## ---------- 2. SGA with momentum expect_silent( res <- maxSGA(loglik, gradlik, start=start, control=list(printLevel=0, iterlim=200, SG_batchSize=100, SG_learningRate=0.1, SGA_momentum=0.9), nObs=length(yTrain)) ) expect_equal(coef(res), b0, tolerance=tol) ## ---------- 3. full batch expect_silent( res <- maxSGA(loglik, gradlik, start=start, control=list(printLevel=0, iterlim=200, SG_batchSize=NULL, SG_learningRate=0.1), nObs=length(yTrain)) ) expect_equal(coef(res), b0, tolerance=tol) ## ---------- 4. no gradient expect_silent( res <- maxSGA(loglik, start=start, control=list(iterlim=1000, SG_learningRate=0.02), nObs=length(yTrain)) ) expect_equal(coef(res), b0, tolerance=tol) ## ---------- return Hessian, gradient provided expect_silent( res <- maxSGA(loglik, gradlik, start=start, control=list(iterlim=1000, SG_learningRate=0.02), nObs=length(yTrain), finalHessian=TRUE) ) expect_equal(coef(res), b0, tolerance=tol) expect_equal(dim(hessian(res)), c(2,2)) ## ---------- return Hessian, no gradient expect_silent( res <- maxSGA(loglik, start=start, control=list(iterlim=1000, SG_learningRate=0.02), nObs=length(yTrain), finalHessian=TRUE) ) expect_equal(coef(res), b0, tolerance=tol) expect_equal(dim(hessian(res)), c(2,2)) ### ---------- SGA, printlevel 1, storeValues ---------- ### it should just work expect_silent( res <- maxSGA(loglik, gradlik, start=start, control=list(iterlim=2, storeValues=TRUE, printLevel=1), nObs=length(yTrain), finalHessian=TRUE) ) ### ---------- SGA, NA as iterlim ---------- ### should give informative error expect_error( res <- maxSGA(loglik, gradlik, start=start, control=list(iterlim=NA), nObs=length(yTrain), finalHessian=TRUE), pattern = "invalid class \"MaxControl\" object: NA in 'iterlim'" ) ### ---------- SGA, fn missing but storeValues=TRUE ### should give informative error expect_error( res <- maxSGA(grad=gradlik, start=start, control=list(iterlim=10, storeValues=TRUE), nObs=length(yTrain)), pattern = "Cannot compute the objective function value: no objective function supplied" ) ## ---------- gradient by observations gradlikO <- function(beta, index) { e <- yTrain[index] - XTrain[index,,drop=FALSE] %*% beta g <- -2*drop(e)*XTrain[index,,drop=FALSE] -g/length(index) } expect_silent( res <- maxSGA(grad=gradlikO, start=start, control=list(printLevel=0, iterlim=100, SG_batchSize=100), nObs=length(yTrain)) ) expect_equal(coef(res), b0, tolerance=tol) ## ---------- 0 iterations expect_silent( res <- maxSGA(grad=gradlik, start=start, control=list(iterlim=0), nObs=length(yTrain)) ) expect_equal(coef(res), start) # should return start values exactly ### -------------------- create unequally scaled data set.seed(1) N <- 1000 x <- rnorm(N, sd=100) XTrain <- cbind(1, x) yTrain <- 1 + x + rnorm(N) start <- c(const=10, x=10) ## ---------- no gradient clipping: ## should fail with informative "NA/Inf in gradient" message expect_error( res <- maxSGA(loglik, gradlik, start=start, control=list(iterlim=100, SG_learningRate=0.5), nObs=length(yTrain)), pattern = "NA/Inf in gradient" ) ## ---------- gradient clipping: should not fail expect_silent( res <- maxSGA(loglik, gradlik, start=start, control=list(iterlim=100, SG_learningRate=0.5, SG_clip=1e6), nObs=length(yTrain) ) ) maxLik/inst/tinytest/test-maxControl.R0000644000176200001440000000465714077525067017616 0ustar liggesusers### Does maxControl stuff behave? ### ### do not run unless 'NOT_CRAN' explicitly defined ### (Suggested by Sebastian Meyer and others) if (!identical(Sys.getenv("NOT_CRAN"), "true")) { message("skipping slow optimizer tests") q("no") } ### test for: ### 1. create maxControl object ### 2. SGA_batchSize NULL ### 3. negative batch size ### 4. more than 1 batch size ### SG_clip: NULL, negative, more than one ### ### printing: ### * #of cols, rows library(maxLik) set.seed(3) ### ---------- create maxControl object maxControl(tol=1e-4, lambdatol=1e-5, qrtol=1e-6, qac="marquardt", marquardt_lambda0=0.1, marquardt_lambdaStep=3, marquardt_maxLambda=1e10, nm_alpha=2, nm_beta=1, nm_gamma=4, sann_temp=5, sann_tmax=100, sann_randomSeed=1, SGA_momentum=0.9, Adam_momentum1=0.5, Adam_momentum2=0.55, SG_learningRate=0.5, SG_batchSize=10, SG_clip=1000, SG_patience=7, SG_patienceStep=10, iterlim=10, printLevel=3) ### ---------- SG_batchSize expect_silent(maxControl(SG_batchSize=NULL)) expect_error(maxControl(SG_batchSize=-1)) # should fail expect_error(maxControl(SG_batchSize=2:3)) # should fail expect_silent(maxControl(SG_clip=NULL)) expect_error(maxControl(SG_clip=-1)) # fails expect_error(maxControl(SG_clip=2:3)) # fails expect_error(maxControl(Adam_momentum1=NA)) # should fail w/'NA in Adam_momentum' ### ---------- printing ---------- ### ---------- max.columns, max.rows ---------- loglik <- function(beta) { e <- y - X %*% beta -crossprod(e) } gradlik <- function(beta) { e <- y - X %*% beta l <- crossprod(e) g <- t(-2*t(X) %*% e) -g } ## linear regression with many columns X <- matrix(rnorm(20*15), 20, 15) beta <- rep(1, ncol(X)) y <- X %*% beta + rnorm(20, sd=0.3) m <- maxNR(loglik, gradlik, start=rep(1, ncol(X)), iterlim=1) ## print estimates + gradient, and hessian ## should print only 4 rows for estimates, 4 rows + 2 cols for Hessia ## should give message "reached getOption("max.cols") -- omitted 13 columns" etc expect_stdout(print(summary(m, hessian=TRUE), max.rows=4, max.cols=2, digits=3), pattern=paste0('reached getOption\\("max.rows"\\) -- omitted 11 rows', '.*', 'reached getOption\\("max.cols"\\) -- omitted 13 columns', '.*', 'reached getOption\\("max.rows"\\) -- omitted 11 rows') ) maxLik/inst/tinytest/test-optimizers.R0000644000176200001440000006220314077525067017664 0ustar liggesusers### This code tests all the methods and main parameters. It includes: ### * analytic gradients/Hessian ### * fixed parameters ### * inequality constraints ### * equality constraints ## do not run unless 'NOT_CRAN' explicitly defined ## (Suggested by Sebastian Meyer and others) if (!identical(Sys.getenv("NOT_CRAN"), "true")) { message("skipping slow optimizer tests") q("no") } if(!requireNamespace("tinytest", quietly = TRUE)) { message("These tests require 'tinytest' package\n") q("no") } library(maxLik) ## data to fit a normal distribution # set seed for pseudo random numbers set.seed( 123 ) tol <- .Machine$double.eps^0.25 ## generate a variable from normally distributed random numbers truePar <- c(mu=1, sigma=2) NOBS <- 100 x <- rnorm(NOBS, truePar[1], truePar[2] ) xSaved <- x ## log likelihood function llf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output sum(dnorm(x, mu, sigma, log=TRUE)) } ## log likelihood function (individual observations) llfInd <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - 0.5 * ( x - mu )^2 / sigma^2 return( llValues ) } ## function to calculate analytical gradients gf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] N <- length( x ) llGrad <- c( sum( ( x - mu ) / sigma^2 ), - N / sigma + sum( ( x - mu )^2 / sigma^3 ) ) return( llGrad ) } ## function to calculate analytical gradients (individual observations) gfInd <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] llGrads <- cbind( ( x - mu ) / sigma^2, - 1 / sigma + ( x - mu )^2 / sigma^3 ) return( llGrads ) } ## log likelihood function with gradients as attributes llfGrad <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output N <- length( x ) llValue <- -0.5 * N * log( 2 * pi ) - N * log( sigma ) - 0.5 * sum( ( x - mu )^2 / sigma^2 ) attributes( llValue )$gradient <- c( sum( ( x - mu ) / sigma^2 ), - N / sigma + sum( ( x - mu )^2 / sigma^3 ) ) return( llValue ) } ## log likelihood function with gradients as attributes (individual observations) llfGradInd <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - 0.5 * ( x - mu )^2 / sigma^2 attributes( llValues )$gradient <- cbind( ( x - mu ) / sigma^2, - 1 / sigma + ( x - mu )^2 / sigma^3 ) return( llValues ) } ## function to calculate analytical Hessians hf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] N <- length( x ) llHess <- matrix( c( N * ( - 1 / sigma^2 ), sum( - 2 * ( x - mu ) / sigma^3 ), sum( - 2 * ( x - mu ) / sigma^3 ), N / sigma^2 + sum( - 3 * ( x - mu )^2 / sigma^4 ) ), nrow = 2, ncol = 2 ) return( llHess ) } ## log likelihood function with gradients and Hessian as attributes llfGradHess <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output N <- length( x ) llValue <- -0.5 * N * log( 2 * pi ) - N * log( sigma ) - 0.5 * sum( ( x - mu )^2 / sigma^2 ) attributes( llValue )$gradient <- c( sum( ( x - mu ) / sigma^2 ), - N / sigma + sum( ( x - mu )^2 / sigma^3 ) ) attributes( llValue )$hessian <- matrix( c( N * ( - 1 / sigma^2 ), sum( - 2 * ( x - mu ) / sigma^3 ), sum( - 2 * ( x - mu ) / sigma^3 ), N / sigma^2 + sum( - 3 * ( x - mu )^2 / sigma^4 ) ), nrow = 2, ncol = 2 ) return( llValue ) } ## log likelihood function with gradients as attributes (individual observations) llfGradHessInd <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output N <- length( x ) llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - 0.5 * ( x - mu )^2 / sigma^2 attributes( llValues )$gradient <- cbind( ( x - mu ) / sigma^2, - 1 / sigma + ( x - mu )^2 / sigma^3 ) attributes( llValues )$hessian <- matrix( c( N * ( - 1 / sigma^2 ), sum( - 2 * ( x - mu ) / sigma^3 ), sum( - 2 * ( x - mu ) / sigma^3 ), N / sigma^2 + sum( - 3 * ( x - mu )^2 / sigma^4 ) ), nrow = 2, ncol = 2 ) return( llValues ) } # start values startVal <- c( mu = 0, sigma = 1 ) ## basic NR: test if all methods work ml <- maxLik( llf, start = startVal ) expect_equal( coef(ml), truePar, tol=2*max(stdEr(ml)) ) expect_stdout( print( ml ), pattern = "Estimate\\(s\\): 1.18.*1.81" ) expect_stdout( print( summary( ml )), pattern = "Estimates:" ) expect_equal( activePar( ml ), c(mu=TRUE, sigma=TRUE) ) expect_equal( AIC( ml ), 407.167892384587, tol = 0.1, check.attributes=FALSE ) expect_equal( coef( ml ), c(mu=1.181, sigma=1.816), tol = 0.001 ) expect_stdout( condiNumber( ml, digits = 3), "mu[[:space:]]+1[[:space:]\n]+sigma[[:space:]]+1\\." ) expect_equal( hessian( ml), matrix(c(-30.3, 0, 0, -60.6), 2, 2), tol = 0.01, check.attributes = FALSE ) expect_equal( logLik( ml ), -201.583946192294, tol = tol, check.attributes = FALSE ) expect_equal( maximType( ml ), "Newton-Raphson maximisation" ) expect_equal( nIter( ml ) > 5, TRUE ) expect_error( nObs( ml ), "cannot return the number of observations" ) expect_equal( nParam( ml ), 2 ) expect_equal( returnCode( ml ), 1 ) expect_equal( returnMessage( ml ), "gradient close to zero (gradtol)" ) expect_equal( vcov( ml ), matrix(c(0.032975, 0, 0, 0.0165), 2, 2), tol=0.01, check.attributes = FALSE ) expect_equal( logLik( summary( ml ) ), logLik(ml) ) mlInd <- maxLik( llfInd, start = startVal ) expect_stdout( print( summary( mlInd ), digits = 2 ), "mu +1\\.18" ) expect_equal( nObs( mlInd ), length(x) ) ## Marquardt (1963) correction mlM <- maxLik( llf, start = startVal, qac="marquardt") expect_equal( coef(mlM), coef(ml), # coefficients should be the same as above tol=tol ) expect_equal( returnMessage(mlM), returnMessage(ml) ) ## test plain results with analytical gradients ## compare coefficients, Hessian mlg <- maxLik(llf, gf, start = startVal ) expect_equal(coef(ml), coef(mlg), tol=tol) expect_equal(hessian(ml), hessian(mlg), tolerance = 1e-2) ## gradient with individual components mlgInd <- maxLik( llfInd, gfInd, start = startVal ) expect_equal(coef(mlInd), coef(mlgInd), tolerance = 1e-3) expect_equal(hessian(mlg), hessian(mlgInd), tolerance = 1e-3) ## with analytical gradients as attribute mlG <- maxLik( llfGrad, start = startVal ) expect_equal(coef(mlG), coef(mlg), tolerance = tol) expect_equivalent(gradient(mlG), gf( coef( mlG ) ), tolerance = tol) mlGInd <- maxLik( llfGradInd, start = startVal ) expect_equal(coef(mlGInd), coef(mlgInd), tolerance = tol) expect_equivalent(gradient(mlGInd), colSums( gfInd( coef( mlGInd ) ) ), tolerance = tol) expect_equivalent(estfun(mlGInd), gfInd( coef( mlGInd ) ), tolerance=tol) ## with analytical gradients as argument and attribute expect_warning(mlgG <- maxLik( llfGrad, gf, start = startVal)) expect_equal(coef(mlgG), coef(mlg), tolerance = tol) ## with analytical gradients and Hessians mlgh <- maxLik( llf, gf, hf, start = startVal ) expect_equal(coef(mlg), coef(mlgh), tolerance = tol) ## with analytical gradients and Hessian as attribute mlGH <- maxLik( llfGradHess, start = startVal ) expect_equal(coef(mlGH), coef(mlgh), tolerance = tol) ## with analytical gradients and Hessian as argument and attribute expect_warning(mlgGhH <- maxLik( llfGradHess, gf, hf, start = startVal )) expect_equal(coef(mlgGhH), coef(mlgh), tolerance = tol) ## ---------- BHHH method ---------- ## cannot do BHHH if llf not provided by individual x <- xSaved[1] expect_error( maxLik( llfInd, start = startVal, method = "BHHH" ) ) ## 2 observations: can do BHHH x <- xSaved[1:2] expect_silent( maxLik( llfInd, start = startVal, method = "BHHH" ) ) ## x <- xSaved mlBHHH <- maxLik( llfInd, start = startVal, method = "BHHH" ) expect_stdout(print( mlBHHH ), pattern = "Estimate\\(s\\): 1\\.18.* 1\\.81") expect_stdout(print(summary( mlBHHH)), pattern = "mu *1.18") expect_equivalent(activePar( mlBHHH ), c(TRUE, TRUE)) expect_equivalent(AIC( mlBHHH ), 407.168, tolerance=0.01) expect_equal(coef( mlBHHH ), setNames(c(1.180808, 1.816485), c("mu", "sigma")), tolerance=tol) expect_equal(condiNumber( mlBHHH, printLevel=0), setNames(c(1, 1.72), c("mu", "sigma")), tol=0.01) expect_equivalent(hessian( mlBHHH ), matrix(c(-30.306411, -1.833632, -1.833632, -55.731646), 2, 2), tolerance=0.01) expect_equivalent(logLik( mlBHHH ), -201.583946192983, tolerance=tol) expect_equal(maximType( mlBHHH ), "BHHH maximisation") expect_equal(nIter(mlBHHH) > 3, TRUE) # here 12 iterations expect_equal(nParam( mlBHHH ), 2) expect_equal(returnCode( mlBHHH ), 8) expect_equal(returnMessage( mlBHHH ), "successive function values within relative tolerance limit (reltol)") expect_equivalent(vcov( mlBHHH ), matrix(c(0.03306213, -0.00108778, -0.00108778, 0.01797892), 2, 2), tol=0.001) expect_equivalent(logLik(summary(mlBHHH)), -201.583946192983, tolerance=tol) expect_equal(coef(ml), coef(mlBHHH), tol=tol) expect_equal(stdEr(ml), stdEr(mlBHHH), tol=0.1) expect_equal(nObs( mlBHHH ), length(x)) # final Hessian = usual Hessian expect_silent(mlBhhhH <- maxLik( llfInd, start = startVal, method = "BHHH", finalHessian = TRUE ) ) # do not test Hessian equality--BHHH may be imprecise, at least # for diagonal elements expect_stdout(print(hessian( mlBhhhH )), pattern="mu.*\nsigma.+") ## Marquardt (1963) correction expect_silent(mlBHHHM <- maxLik( llfInd, start = startVal, method = "BHHH", qac="marquardt")) expect_equal(coef(mlBHHHM), coef(mlBHHH), tolerance=tol) expect_equal(returnMessage(mlBHHHM), "successive function values within relative tolerance limit (reltol)") ## BHHH with analytical gradients expect_error( maxLik( llf, gf, start = startVal, method = "BHHH" ) ) # need individual log-likelihood expect_error( maxLik( llfInd, gf, start = startVal, method = "BHHH" ) ) # need individual gradient x <- xSaved[1] # test with a single observation expect_error(maxLik( llf, gfInd, start = startVal, method = "BHHH" )) # gradient must have >= 2 rows expect_error( maxLik( llfInd, gfInd, start = startVal, method = "BHHH" ) ) # ditto even if individual likelihood components x <- xSaved[1:2] # test with 2 observations expect_silent(maxLik( llf, gfInd, start = startVal, method = "BHHH", iterlim=1)) # should work with 2 obs expect_silent( maxLik( llfInd, gfInd, start = startVal, method = "BHHH", iterlim=1) ) # should work with 2 obs x <- xSaved expect_silent(mlgBHHH <- maxLik( llfInd, gfInd, start = startVal, method = "BHHH" )) # individual log-likelihood, gradient expect_equal(coef(mlBHHH), coef(mlgBHHH), tolerance = tol) expect_equal(coef(mlg), coef(mlgBHHH), tolerance = tol) expect_silent(mlgBHHH2 <- maxLik( llf, gfInd, start = startVal, method = "BHHH" )) # aggregated log-likelihood, individual gradient expect_equal(coef(mlgBHHH), coef(mlgBHHH2), tolerance=tol) # final Hessian = usual Hessian expect_silent( mlgBhhhH <- maxLik( llf, gfInd, start = startVal, method = "BHHH", finalHessian = TRUE ) ) expect_equal(hessian(mlgBhhhH), hessian(mlBhhhH), tolerance = 1e-2) ## with analytical gradients as attribute expect_error( maxLik( llfGrad, start = startVal, method = "BHHH" ) ) # no individual gradients provided x <- xSaved[1] expect_error( maxLik( llfGrad, start = startVal, method = "BHHH" ), pattern = "gradient is not a matrix") # get an error about need a matrix expect_error( maxLik( llfGradInd, start = startVal, method = "BHHH" ), pattern = "at least as many rows") # need at least two obs x <- xSaved[1:2] expect_error( maxLik( llfGrad, start = startVal, method = "BHHH" ), pattern = "gradient is not a matrix") # enough obs but no individual grad x <- xSaved expect_silent(mlGBHHH <- maxLik( llfGradInd, start = startVal, method = "BHHH" )) expect_equal(coef(mlGBHHH), coef(mlgBHHH), tolerance = tol) # final Hessian = usual Hessian expect_silent(mlGBhhhH <- maxLik( llfGradInd, start = startVal, method = "BHHH", finalHessian = TRUE )) expect_equal(hessian(mlGBhhhH), hessian(mlgBhhhH), tolerance = tol) ## with analytical gradients as argument and attribute expect_warning(mlgGBHHH <- maxLik( llfGradInd, gfInd, start = startVal, method = "BHHH" ), pattern = "both as attribute 'gradient' and as argument 'grad'") # warn about double gradient expect_equal(coef(mlgGBHHH), coef(mlgBHHH), tolerance = tol) ## with unused Hessian expect_silent(mlghBHHH <- maxLik( llfInd, gfInd, hf, start = startVal, method = "BHHH" )) expect_equal(coef(mlgBHHH), coef(mlghBHHH), tolerance = tol) ## final Hessian = usual Hessian expect_silent( mlghBhhhH <- maxLik( llfInd, gfInd, hf, start = startVal, method = "BHHH", finalHessian = TRUE ) ) expect_equivalent(hessian(mlghBhhhH), hessian(mlghBHHH), tolerance = 0.2) # BHHH and ordinary hessian differ quite a bit ## with unused Hessian as attribute expect_silent(mlGHBHHH <- maxLik( llfGradHessInd, start = startVal, method = "BHHH" )) expect_equal(coef(mlGHBHHH), coef(mlghBHHH), tolerance = tol) ## final Hessian = usual Hessian expect_silent(mlGHBhhhH <- maxLik( llfGradHessInd, start = startVal, method = "BHHH", finalHessian = TRUE )) expect_equal(hessian(mlGHBhhhH), hessian(mlghBhhhH), tolerance = tol) ## with analytical gradients and Hessian as argument and attribute expect_warning( mlgGhHBHHH <- maxLik( llfGradHessInd, gfInd, hf, start = startVal, method = "BHHH" ), pattern = "both as attribute 'gradient' and as argument 'grad': ignoring" ) expect_equal(coef(mlgGhHBHHH), coef(mlghBHHH), tolerance = tol) expect_equal(hessian(mlgGhHBHHH), hessian(mlGHBHHH), tolerance = tol) ## ---------- Test BFGS methods ---------- optimizerNames <- c(bfgsr = "BFGSR", bfgs = "BFGS", nm = "Nelder-Mead", sann = "SANN", cg = "CG") successCodes <- list(bfgsr = 1:4, bfgs = 0, nm = 0, sann = 0, cg = 0) successMsgs <- list(bfgsr = c("successive function values within tolerance limit (tol)"), bfgs = c("successful convergence "), # includes space at end... nm = c("successful convergence "), sann = c("successful convergence "), cg = c("successful convergence ") ) for(optimizer in c("bfgsr", "bfgs", "nm", "sann", "cg")) { expect_silent(mlResult <- maxLik( llf, start = startVal, method = optimizer )) expect_stdout(print( mlResult ), pattern = paste0(optimizerNames[optimizer], " maximization") ) expect_stdout(print( summary( mlResult )), pattern = paste0(optimizerNames[optimizer], " maximization,.*Estimates:") ) expect_equal(coef(ml), coef(mlResult), tolerance=0.001) expect_equal(stdEr(ml), stdEr(mlResult), tolerance=0.01) expect_equal(activePar( mlResult ), c(mu=TRUE, sigma=TRUE)) expect_equivalent(AIC( mlResult ), 407.167893392749, tolerance=tol) expect_equivalent( hessian( mlResult ), matrix(c(-30.32596, 0.00000, 0.00000, -60.59508), 2, 2), tolerance = 0.01) expect_equivalent(logLik( mlResult ), -201.5839, tolerance = 0.01) expect_equal(maximType( mlResult ), paste0(optimizerNames[optimizer], " maximization") ) expect_true(nIter( mlResult ) > 1 & is.integer(nIter(mlResult))) expect_error( nObs( mlResult ), pattern = "cannot return the number of observations") expect_equal(nParam( mlResult ), 2) expect_true(returnCode( mlResult ) %in% successCodes[[optimizer]]) expect_equal(returnMessage( mlResult), successMsgs[[optimizer]]) expect_equal(logLik( summary( mlResult ) ), logLik(mlResult)) ## individual observations expect_silent(mlIndResult <- maxLik( llfInd, start = startVal, method = optimizer)) expect_stdout(print( summary( mlIndResult )), pattern = paste0(optimizerNames[optimizer], " maximization,.*Estimates:") ) expect_equal(coef(mlResult), coef(mlIndResult), tolerance = tol) expect_equal(stdEr(mlResult), stdEr(mlIndResult), tolerance = 0.01) expect_equal(nObs( mlIndResult ), length(x)) ## with analytic gradients expect_silent(mlgResult <- maxLik( llf, gf, start = startVal, method = optimizer)) expect_equal(coef(mlgResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlgResult), stdEr(mlResult), tolerance = 0.01) expect_silent(mlgIndResult <- maxLik( llfInd, gfInd, start = startVal, method = optimizer )) expect_equal(coef(mlgIndResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlgIndResult), stdEr(mlResult), tolerance = 0.01) ## with analytical gradients as attribute expect_silent(mlGResult <- maxLik( llfGrad, start = startVal, method = optimizer)) expect_equal(coef(mlGResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlGResult), stdEr(mlResult), tolerance = 0.01) expect_silent(mlGIndResult <- maxLik( llfGradInd, start = startVal, method = optimizer )) expect_equal(coef(mlGIndResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlGIndResult), stdEr(mlResult), tolerance = 0.01) ## with analytical gradients as argument and attribute expect_warning(mlgGResult <- maxLik( llfGrad, gf, start = startVal, method = optimizer )) expect_equal(coef(mlgGResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlgGResult), stdEr(mlResult), tolerance = 0.01) ## with analytical gradients and Hessians expect_silent(mlghResult <- maxLik( llf, gf, hf, start = startVal, method = optimizer )) expect_equal(coef(mlghResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlghResult), stdEr(mlResult), tolerance = 0.01) ## with analytical gradients and Hessian as attribute expect_silent(mlGHResult <- maxLik( llfGradHess, start = startVal, method = optimizer )) expect_equal(coef(mlGHResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlGHResult), stdEr(mlResult), tolerance = 0.01) ## with analytical gradients and Hessian as argument and attribute expect_warning(mlgGhHResult <- maxLik( llfGradHess, gf, hf, start = startVal, method = optimizer )) expect_equal(coef(mlgGhHResult), coef(mlResult), tolerance = tol) expect_equal(stdEr(mlgGhHResult), stdEr(mlResult), tolerance = 0.01) } ### ---------- with fixed parameters ---------- ## start values startValFix <- c( mu = 1, sigma = 1 ) ## fix mu (the mean ) at its start value isFixed <- c( TRUE, FALSE ) successMsgs <- list(bfgsr = c("successive function values within tolerance limit (tol)"), bfgs = c("successful convergence "), # includes space at end... nm = c("successful convergence "), sann = c("successful convergence "), cg = c("successful convergence ") ) ## NR method with fixed parameters for(optimizer in c("nr", "bfgsr", "bfgs", "sann", "cg")) { expect_silent( mlFix <- maxLik( llf, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equivalent(coef(mlFix)[1], 1) expect_equivalent(stdEr(mlFix)[1], 0) expect_silent( mlFix3 <- maxLik(llf, start = startValFix, fixed = "mu", method=optimizer) ) expect_equal(coef(mlFix), coef(mlFix3)) mlFix4 <- maxLik( llf, start = startValFix, fixed = which(isFixed), method=optimizer) expect_equal(coef(mlFix), coef(mlFix4), tolerance=tol) expect_equivalent(activePar( mlFix ), !isFixed) expect_equal(nParam( mlFix ), 2) ## with analytical gradients mlgFix <- maxLik( llf, gf, start = startValFix, fixed = isFixed, method=optimizer) expect_equal(coef(mlgFix), coef(mlFix), tolerance=tol) ## with analytical gradients and Hessians mlghFix <- maxLik( llf, gf, hf, start = startValFix, fixed = isFixed, method=optimizer) expect_equal(coef(mlghFix), coef(mlFix), tolerance=tol) } ## Repeat the previous for NM as that one does not like 1-D optimization for(optimizer in c("nm")) { expect_warning( mlFix <- maxLik( llf, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equivalent(coef(mlFix)[1], 1) expect_equivalent(stdEr(mlFix)[1], 0) expect_warning( mlFix3 <- maxLik(llf, start = startValFix, fixed = "mu", method=optimizer) ) expect_equal(coef(mlFix), coef(mlFix3)) expect_warning( mlFix4 <- maxLik( llf, start = startValFix, fixed = which(isFixed), method=optimizer) ) expect_equal(coef(mlFix), coef(mlFix4), tolerance=tol) expect_equivalent(activePar( mlFix ), !isFixed) expect_equal(nParam( mlFix ), 2) ## with analytical gradients expect_warning( mlgFix <- maxLik( llf, gf, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equal(coef(mlgFix), coef(mlFix), tolerance=tol) ## with analytical gradients and Hessians expect_warning( mlghFix <- maxLik( llf, gf, hf, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equal(coef(mlghFix), coef(mlFix), tolerance=tol) } ## Repeat for BHHH as that one need a different log-likelihood function for(optimizer in c("bhhh")) { expect_silent( mlFix <- maxLik( llfInd, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equivalent(coef(mlFix)[1], 1) expect_equivalent(stdEr(mlFix)[1], 0) expect_silent( mlFix3 <- maxLik(llfInd, start = startValFix, fixed = "mu", method=optimizer) ) expect_equal(coef(mlFix), coef(mlFix3)) expect_silent( mlFix4 <- maxLik( llfInd, start = startValFix, fixed = which(isFixed), method=optimizer) ) expect_equal(coef(mlFix), coef(mlFix4), tolerance=tol) expect_equivalent(activePar( mlFix ), !isFixed) expect_equal(nParam( mlFix ), 2) ## with analytical gradients expect_silent( mlgFix <- maxLik( llf, gfInd, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equal(coef(mlgFix), coef(mlFix), tolerance=tol) ## with analytical gradients and Hessians expect_silent( mlghFix <- maxLik( llf, gfInd, hf, start = startValFix, fixed = isFixed, method=optimizer) ) expect_equal(coef(mlghFix), coef(mlFix), tolerance=tol) } ### ---------- inequality constraints ---------- A <- matrix( -1, nrow = 1, ncol = 2 ) inEq <- list( ineqA = A, ineqB = 2.5 ) # A theta + B > 0 i.e. # mu + sigma < 2.5 for(optimizer in c("bfgs", "nm", "sann")) { expect_silent( mlInEq <- maxLik( llf, start = startVal, constraints = inEq, method = optimizer ) ) expect_stdout( print( summary( mlInEq)), pattern = "constrained likelihood estimation. Inference is probably wrong.*outer iterations, barrier value" ) expect_true(sum(coef( mlInEq )) < 2.5) } ### ---------- equality constraints ---------- eqCon <- list(eqA = A, eqB = 2.5) # A theta + B = 0 i.e. # mu + sigma = 2.5 for(optimizer in c("nr", "bhhh", "bfgs", "nm", "sann")) { expect_silent( mlEq <- maxLik(llfInd, start = startVal, constraints = eqCon, method = optimizer, SUMTTol = 0) ) expect_stdout( print( summary( mlEq)), pattern = "constrained likelihood estimation. Inference is probably wrong.*outer iterations, barrier value" ) expect_equal(sum(coef( mlEq )), 2.5, tolerance=1e-4) } ### ---------- convergence tolerance parameters ---------- a <- maxNR(llf, gf, hf, start=startVal, tol=1e-3, reltol=0, gradtol=0, iterlim=10) expect_equal(returnCode(a), 2) # should stop with code 2: tolerance a <- maxNR(llf, gf, hf, start=startVal, tol=0, reltol=1e-3, gradtol=0, iterlim=10) expect_equal(returnCode(a), 8) # 8: relative tolerance a <- maxNR(llf, gf, hf, start=startVal, tol=0, reltol=0, gradtol=1e-3, iterlim=10) expect_equal(returnCode(a), 1) # 1: gradient a <- maxNR(llf, gf, hf, start=startVal, tol=0, reltol=0, gradtol=0, iterlim=10) expect_equal(returnCode(a), 4) # 4: iteration limit maxLik/inst/tinytest/test-methods.R0000644000176200001440000001040614077525067017120 0ustar liggesusers## Test methods. Note: only test if methods work in terms of dim, length, etc, ## not in terms of values here ## ## ... ## * printing summary with max.columns, max.rows ## if(!requireNamespace("tinytest", quietly = TRUE)) { message("These tests require 'tinytest' package\n") q("no") } require(sandwich) library(maxLik) set.seed(0) compareTolerance = 0.001 # tolerance when comparing different optimizers ## Test standard methods for "lm" x <- runif(20) y <- x + rnorm(20) m <- lm(y ~ x) expect_equal( nObs(m), length(y), info = "nObs.lm must be correct" ) expect_equal( stdEr(m), c(`(Intercept)` = 0.357862322670879, x = 0.568707094458801) ) ## Test maxControl methods: set.seed(9) x <- rnorm(20, sd=2) ll1 <- function(par) dnorm(x, mean=par, sd=1, log=TRUE) ll2 <- function(par) dnorm(x, mean=par[1], sd=par[2], log=TRUE) for(method in c("NR", "BFGS", "BFGSR")) { m <- maxLik(ll2, start=c(0, 2), method=method, control=list(iterlim=1)) expect_equal(maxValue(m), -41.35, tolerance=0.01) expect_true(is.vector(gradient(m)), info=paste0("'gradient' returns a vector for ", method)) expect_equal(length(gradient(m)), 2, info="'gradient(m)' is of length 2") expect_true(is.matrix(estfun(m)), info="'estfun' returns a matrix") expect_equal(dim(estfun(m)), c(20,2), info="'estfun(m)' is 20x2 matrix") expect_stdout( show(maxControl(m)), pattern = "Adam_momentum2 = 0\\.999" ) } ## Test methods for non-likelihood optimization hatf <- function(theta) exp(- theta %*% theta) for(optimizer in c(maxNR, maxBFGSR, maxBFGS, maxNM, maxSANN, maxCG)) { name <- as.character(quote(optimizer)) res <- optimizer(hatf, start=c(1,1)) if(name %in% c("maxNR", "maxBFGS", "maxNM", "maxCG")) { expect_equal(coef(res), c(0,0), tol=1e-5, info=paste0(name, ": result (0,0)")) } expect_equal(objectiveFn(res), hatf, info=paste0(name, ": objectiveFn correct")) } ## Test maxLik vcov related methods set.seed( 15 ) t <- rexp(20, 2) loglik <- function(theta) log(theta) - theta*t gradlik <- function(theta) 1/theta - t hesslik <- function(theta) -100/theta^2 a <- maxLik(loglik, start=1) expect_equal(dim(vcov(a)), c(1,1), info="vcov 1D numeric correct") expect_equal(length(stdEr(a)), 1, info="stdEr 1D numeric correct") a <- maxLik(loglik, gradlik, hesslik, start=1) expect_equal(dim(vcov(a)), c(1,1), info="vcov 1D analytic correct") expect_equal(length(stdEr(a)), 1, info="stdEr 1D analytic correct") ## ---------- both individual and aggregated likelihood ---------- NOBS <- 100 x <- rnorm(NOBS, 2, 1) ## log likelihood function llf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output sum(dnorm(x, mu, sigma, log=TRUE)) } ## log likelihood function (individual observations) llfInd <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output llValues <- -0.5 * log( 2 * pi ) - log( sigma ) - 0.5 * ( x - mu )^2 / sigma^2 return( llValues ) } startVal <- c(mu=2, sigma=1) ml <- maxLik( llf, start = startVal) mlInd <- maxLik( llfInd, start = startVal) ## ---------- Various summary methods ---------- ## These should work and produce consistent results expect_stdout( show(confint(ml)), pattern = "2.5 % +97.5 %\nmu +[[:digit:] .]+\n" ) expect_stdout( show(glance(ml)), pattern = "df logLik AIC +nobs.*1 2 -140. 284. NA" ) expect_stdout( show(glance(mlInd)), pattern = "df logLik AIC nobs.*1 2 -140. 284. 100" ) expect_stdout( show(tidy(ml)), pattern = "term.*estimate std.error statistic.*p.value" ) ### ---------- estfun, bread, sandwich ---------- expect_error( estfun( ml ) ) expect_equal(dim(estfun( mlInd )), c(NOBS, 2)) expect_equal(colnames(estfun( mlInd )), names(startVal)) expect_error(bread( ml ) ) expect_equal(dim(bread( mlInd )), c(2, 2)) expect_equal(colnames(bread( mlInd )), names(startVal)) expect_equal(rownames(bread( mlInd )), names(startVal)) expect_error(sandwich( ml ) ) expect_equal(dim(sandwich( mlInd )), c(2, 2)) expect_equal(colnames(sandwich( mlInd )), names(startVal)) expect_equal(rownames(sandwich( mlInd )), names(startVal)) maxLik/inst/tinytest/test-parameters.R0000644000176200001440000002451414077525067017625 0ustar liggesusers ### Test battery for various optimization parameters for different optimizers. ### ### ... ### library(maxLik) library(tinytest) tol <- .Machine$double.eps^(0.25) set.seed( 123 ) # generate a variable from normally distributed random numbers N <- 50 x <- rnorm(N, 1, 2 ) ## log likelihood function llf <- function( param ) { mu <- param[ 1 ] sigma <- param[ 2 ] if(!(sigma > 0)) return(NA) # to avoid warnings in the output N <- length( x ) llValue <- -0.5 * N * log( 2 * pi ) - N * log( sigma ) - 0.5 * sum( ( x - mu )^2 / sigma^2 ) return( llValue ) } # start values startVal <- c( mu = 0, sigma = 1 ) # expect_silent(ml <- maxLik( llf, start = startVal )) expect_equivalent(coef(ml), c(1.069, 1.833), tolerance=tol) ## tol expect_silent(mlTol <- maxLik( llf, start = startVal, tol=1)) expect_equal(returnCode(mlTol), 2) # tolerance limit expect_silent(mlTolC <- maxLik(llf, start=startVal, control=list(tol=1))) expect_equal(coef(mlTol), coef(mlTolC)) expect_equal(hessian(mlTol), hessian(mlTolC)) expect_equal(returnCode(mlTol), returnCode(mlTolC)) expect_silent(ml <- maxLik( llf, start = startVal, tol=-1)) # negative tol switches tol off expect_silent(ml <- maxLik( llf, start = startVal, control=list(tol=-1))) expect_false(returnCode(ml) == 2) # should not be w/in tolerance limit expect_error(ml <- maxLik( llf, start = startVal, tol=c(1,2)), pattern="'tol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, control=list(tol=c(1,2))), pattern="'tol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, tol=TRUE), pattern="object of class \"logical\" is not valid for slot 'tol'") expect_error(ml <- maxLik( llf, start = startVal, control=list(tol=TRUE)), pattern="object of class \"logical\" is not valid for slot 'tol'") ## ----- reltol: play w/reltol, leave other tolerances at default value ----- expect_silent(mlRelTol <- maxLik( llf, start = startVal, reltol=1)) expect_equal(returnCode(mlRelTol), 8) mlRelTolC <- maxLik(llf, start=startVal, control=list(reltol=1)) expect_equal(coef(mlRelTol), coef(mlRelTolC)) expect_silent(ml0 <- maxLik( llf, start = startVal, reltol=0)) expect_true(nIter(ml0) > nIter(mlRelTol)) # switching off reltol makes more iterations expect_silent(ml1 <- maxLik( llf, start = startVal, reltol=-1)) expect_equal(nIter(ml0), nIter(ml1)) expect_error(ml <- maxLik( llf, start = startVal, reltol=c(1,2)), pattern="invalid class \"MaxControl\" object: 'reltol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, control=list(reltol=c(1,2))), pattern="invalid class \"MaxControl\" object: 'reltol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, reltol=TRUE), pattern="assignment of an object of class \"logical\" is not valid for slot 'reltol'") expect_error(ml <- maxLik( llf, start = startVal, control=list(reltol=TRUE)), pattern="assignment of an object of class \"logical\" is not valid for slot 'reltol'") ## gradtol expect_silent(mlGradtol <- maxLik( llf, start = startVal, gradtol=0.1)) expect_equal(returnCode(mlGradtol), 1) mlGradtolC <- maxLik(llf, start=startVal, control=list(gradtol=0.1)) expect_equal(coef(mlGradtol), coef(mlGradtolC)) expect_silent(ml <- maxLik( llf, start = startVal, gradtol=-1)) expect_true(nIter(ml) > nIter(mlGradtol)) # switching off gradtol makes more iterations expect_error(ml <- maxLik( llf, start = startVal, gradtol=c(1,2)), pattern="object: 'gradtol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, control=list(gradtol=c(1,2))), pattern="object: 'gradtol' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, gradtol=TRUE), pattern="assignment of an object of class \"logical\" is not valid for slot 'gradtol' ") expect_error(ml <- maxLik( llf, start = startVal, control=list(gradtol=TRUE)), pattern="assignment of an object of class \"logical\" is not valid for slot 'gradtol' ") ## examples with steptol, lambdatol ## qac expect_silent(mlMarq <- maxLik( llf, start = startVal, qac="marquardt")) expect_equal(maximType(mlMarq), "Newton-Raphson maximisation with Marquardt (1963) Hessian correction") expect_silent(mlMarqC <- maxLik(llf, start=startVal, control=list(qac="marquardt"))) expect_equal(coef(mlMarq), coef(mlMarqC)) expect_error(ml <- maxLik( llf, start = startVal, qac=-1), pattern = "assignment of an object of class \"numeric\" is not valid for slot 'qac'") # qac should be "stephalving" or "marquardt" expect_error(ml <- maxLik( llf, start = startVal, qac=c("a", "b")), pattern = "invalid class \"MaxControl\" object: 'qac' must be of length 1, not 2") expect_error(ml <- maxLik( llf, start = startVal, qac=TRUE), pattern = "assignment of an object of class \"logical\" is not valid for slot 'qac'") mlMarqCl <- maxLik(llf, start = startVal, control=list(qac="marquardt", lambda0=1000, lambdaStep=4)) expect_equal(coef(mlMarqCl), coef(mlMarq)) ## NM: alpha, beta, gamma expect_silent(mlNMAlpha <- maxLik(llf, start=startVal, method="nm", beta=0.8)) expect_silent(mlNMAlphaC <- maxLik(llf, start=startVal, method="nm", control=list(beta=0.8))) expect_equal(coef(mlNMAlpha), coef(mlNMAlphaC)) ## likelihood function with additional parameter llf1 <- function( param, sigma ) { mu <- param N <- length( x ) ll <- -0.5*N*log( 2 * pi ) - N*log( sigma ) - 0.5*sum( ( x - mu )^2/sigma^2 ) ll } ## log-lik mixture logLikMix <- function(param) { rho <- param[1] if(rho < 0 || rho > 1) return(NA) mu1 <- param[2] mu2 <- param[3] ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) ll } ## loglik mixture with additional parameter logLikMixA <- function(param, rho) { mu1 <- param[1] mu2 <- param[2] ll <- log(rho*dnorm(x - mu1) + (1 - rho)*dnorm(x - mu2)) ll } ## Test the following with all the main optimizers: pl2Patterns <- c(NR = "----- Initial parameters: -----\n.*-----Iteration 1 -----", BFGS = "initial value.*final value", BFGSR = "-------- Initial parameters: -------\n.*Iteration 1") for(method in c("NR", "BFGS", "BFGSR")) { ## create data in loop, we need to mess with 'x' for constraints N <- 100 x <- rnorm(N, 1, 2 ) startVal <- c(1,2) ## two parameters at the same time ## iterlim, printLevel expect_stdout(ml2 <- maxLik(llf, start=startVal, method=method, iterlim=1, printLevel=2), pattern = pl2Patterns[method]) expect_stdout(ml2C <- maxLik(llf, start=startVal, method=method, control=list(iterlim=1, printLevel=2)), pattern = pl2Patterns[method]) expect_equal(coef(ml2), coef(ml2C)) ## what about additional parameters for the loglik function? expect_silent(mlsM <- maxLik(llf1, start=0, method=method, tol=1, sigma=1)) expect_silent(mlsCM <- maxLik(llf1, start=0, method=method, control=list(tol=1), sigma=1)) expect_equal(coef(mlsM), coef(mlsCM)) ## And what about unused parameters? expect_error(maxLik(llf1, start=0, method=method, control=list(tol=1), sigma=1, unusedPar=2), pattern = "unused argument") N <- 100 ## Does this work with constraints? x <- c(rnorm(N, mean=-1), rnorm(N, mean=1)) ## First test inequality constraints ## Inequality constraints: x + y + z < 0.5 A <- matrix(c(-1, 0, 0, 0, -1, 0, 0, 0, 1), 3, 3, byrow=TRUE) B <- rep(0.5, 3) start <- c(0.4, 0, 0.9) ## analytic gradient if(!(method %in% c("NR", "BFGSR"))) { expect_silent(mix <- maxLik(logLikMix, start=start, method=method, constraints=list(ineqA=A, ineqB=B))) expect_silent(mixGT <- try(maxLik(logLikMix, start=start, method=method, constraints=list(ineqA=A, ineqB=B), tol=1))) expect_silent( mixGTC <- try(maxLik(logLikMix, start=start, method=method, constraints=list(ineqA=A, ineqB=B), control=list(tol=1))) ) ## 2d inequality constraints: x + y < 0.5 A2 <- matrix(c(-1, -1), 1, 2, byrow=TRUE) B2 <- 0.5 start2 <- c(-0.5, 0.5) expect_silent( mixA <- maxLik(logLikMixA, start=start2, method=method, constraints=list(ineqA=A2, ineqB=B2), tol=1, rho=0.5) ) expect_silent( mixAC <- maxLik(logLikMixA, start=start2, method=method, constraints=list(ineqA=A2, ineqB=B2), control=list(tol=1), rho=0.5) ) expect_equal(coef(mixA), coef(mixAC)) expect_equal(hessian(mixA), hessian(mixAC)) } } ### Test adding both default and user-specified parameters through control list estimate <- function(control=NULL, ...) { maxLik(llf, start=c(1,1), control=c(list(iterlim=100), control), ...) } expect_silent(m <- estimate(control=list(iterlim=1), fixed=2)) expect_stdout(show(maxControl(m)), pattern = "iterlim = 1") # iterlim should be 1 expect_equal(coef(m)[2], 1) # sigma should be 1.000 ## Does print.level overwrite 'printLevel'? expect_silent(m <- estimate(control=list(printLevel=2, print.level=1))) expect_stdout(show(maxControl(m)), pattern = "printLevel = 1") ## Does open parameters override everything? expect_silent(m <- estimate(control=list(printLevel=2, print.level=1), print.level=0)) expect_stdout(show(maxControl(m)), pattern = "printLevel = 0") ### does both printLevel, print.level work for condiNumber? expect_silent(condiNumber(hessian(m), print.level=0)) expect_silent(condiNumber(hessian(m), printLevel=0)) expect_silent(condiNumber(hessian(m), printLevel=0, print.level=1)) maxLik/inst/tinytest/test-basic.R0000644000176200001440000001027514077525067016542 0ustar liggesusers### general optimization tests for the functions of various forms ### test for: ### 1. numeric gradient, Hessian ### 2. analytic gradient, numeric Hessian ### 3. analytic gradient, Hessian ### ### a) maxLik(, method="NR") ### c) maxLik(, method="BFGS") ### b) maxLik(, method="BHHH") ### ### i) maxNR() ### ii) maxBFGS() if(!requireNamespace("tinytest", quietly = TRUE)) { cat("These tests require 'tinytest' package\n") q("no") } library(maxLik) ## ---------- define log-likelihood functions ---------- ## log-likelihood function(s) logLL <- function(x, X) # per observation for maxLik dgamma(x = X, shape = x[1], scale = x[2], log = TRUE) logLLSum <- function(x, X) sum(logLL(x, X)) # gradient of log-likelihood function d.logLL <- function(x, X){ # analytic 1. derivatives shape <- x[1] scale <- x[2] cbind(shape= log(X) - log(scale) - psigamma(shape, 0), scale= (X/scale - shape)/scale ) } d.logLLSum <- function(x, X) { ## analytic 1. derivatives, summed colSums(d.logLL(x, X)) } ## Hessian of log-likelihood function dd.logLL <- function(x, X){ # analytic 2. derivatives shape <- x[1] scale <- x[2] hessian <- matrix(0, 2, 2) hessian[1,1] <- -psigamma(shape, 1)*length(X) hessian[2,2] <- (shape*length(X) - 2*sum(X)/scale)/scale^2 hessian[cbind(c(2,1), c(1,2))] <- -length(X)/scale return(hessian) } ## ---------- create data ---------- ## sample size 1000 should give precision 0.1 or better param <- c(1.5, 2) set.seed(100) testData <- rgamma(1000, shape=param[1], scale=param[2]) start <- c(1,1) mTol <- .Machine$double.eps^0.25 ## estimation with maxLik() / NR doTests <- function(method="NR") { suppressWarnings(rLLSum <- maxLik( logLLSum, start=start, method=method, X=testData )) stdDev <- stdEr(rLLSum) tol <- 2*max(stdDev) expect_equal(coef(rLLSum), param, tolerance=tol, info=paste("coefficient values should be close to the true values", paste(param, collapse=", "))) # should equal to param, but as N is small, it may be way off ## rLL <- suppressWarnings(maxLik( logLL, start = start, method=method, X=testData )) expect_equal(coef(rLL), coef(rLLSum), tolerance=mTol) ## rLLSumGSum <- suppressWarnings(maxLik( logLLSum, grad=d.logLLSum, start = start, method=method, X=testData )) expect_equal(coef(rLLSumGSum), coef(rLLSum), tolerance=mTol) rLLG <- suppressWarnings(maxLik( logLL, grad=d.logLL, start = start, method=method, X=testData )) expect_equal(coef(rLLG), coef(rLLSum), tolerance=mTol) rLLGH <- suppressWarnings(maxLik( logLL, grad=d.logLL, hess=dd.logLL, start = start, method=method, X=testData )) expect_equal(coef(rLLGH), coef(rLLSum), tolerance=mTol) } doTests("NR") doTests("BFGS") ## maxBHHH: cannot run the same tests method <- "BHHH" expect_error( maxLik( logLLSum, start=start, method=method, X=testData), pattern = "not provided by .* returns a numeric vector" ) rLL <- suppressWarnings(maxLik( logLL, start = start, method=method, X=testData )) stdDev <- stdEr(rLL) tol <- 2*max(stdDev) expect_equal(coef(rLL), param, tolerance=tol, info=paste("coefficient values should be close to the true values", paste(param, collapse=", "))) # should equal to param, but as N is small, it may be way off ## rLLG <- suppressWarnings(maxLik( logLL, grad=d.logLL, start = start, method=method, X=testData )) expect_equal(coef(rLLG), coef(rLL), tolerance=mTol) ## Do the other basic functions work? expect_equal(class(logLik(rLL)), "numeric") expect_equal(class(gradient(rLL)), "numeric") expect_true(inherits(hessian(rLL), "matrix"), info="Hessian must inherit from matrix class") ## test maxNR with gradient and hessian as attributes W <- matrix(-c(4,1,2,4), 2, 2) c <- c(1,2) start <- c(0,0) f <- function(x) { hess <- 2*W grad <- 2*W %*% (x - c) val <- t(x - c) %*% W %*% (x - c) attr(val, "gradient") <- as.vector(grad) # gradient matrices only work for BHHH-type problems attr(val, "hessian") <- hess val } res <- maxNR(f, start=start) expect_equal(coef(res), c, tolerance=mTol) expect_equal(sqrt(sum(gradient(res)^2)), 0, tolerance=mTol) expect_equal(maxValue(res), 0, tolerance=mTol)