kedd/0000755000176200001440000000000012614775365011201 5ustar liggesuserskedd/inst/0000755000176200001440000000000012614740637012150 5ustar liggesuserskedd/inst/CITATION0000644000176200001440000000136512614740637013312 0ustar liggesuserscitHeader("To cite package 'kedd' in publications use:") if(!exists("meta") || is.null(meta)) meta <- packageDescription("kedd") year <- sub(".*(2[[:digit:]]{3})-.*", "\\1", meta$Date, perl = TRUE) vers <- paste("R package version", meta$Version) citEntry(entry="Manual", title = "kedd: Kernel estimator and bandwidth selection for density and its derivatives.", author = personList(as.person("A.C. Guidoum")), year = year, note = vers, url = paste("http://CRAN.R-project.org/package=kedd"), textVersion = paste("A.C. Guidoum (", year, "). kedd: Kernel estimator and bandwidth selection for density and its derivatives.", vers, ".", "http://CRAN.R-project.org/package=kedd" , sep=""))kedd/inst/doc/0000755000176200001440000000000012614740640012707 5ustar liggesuserskedd/inst/doc/kedd.R0000644000176200001440000003276412614740640013755 0ustar liggesusers### R code from vignette source 'kedd.Rnw' ### Encoding: ISO8859-1 ################################################### ### code chunk number 1: kedd.Rnw:132-133 ################################################### library(kedd) ################################################### ### code chunk number 2: kedd.Rnw:135-137 ################################################### kernel.fun(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx kernel.conv(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx ################################################### ### code chunk number 3: kedd.Rnw:139-141 ################################################### plot(kernel.fun(deriv.order = 1, kernel = "gaussian")) plot(kernel.conv(deriv.order = 1, kernel = "gaussian")) ################################################### ### code chunk number 4: kedd.Rnw:146-147 ################################################### plot(kernel.fun(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") ################################################### ### code chunk number 5: kedd.Rnw:149-150 ################################################### plot(kernel.conv(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") ################################################### ### code chunk number 6: kedd.Rnw:176-182 ################################################### fx <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) kernels <- c("gaussian","biweight","triweight","tricube") plot(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[1]),col = 1,ylim=c(-0.6,0.6) ,sub="", main="") for (i in 2:length(kernels))lines(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[i] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),lty = c(8,rep(1,length(kernels))), inset = .015) ################################################### ### code chunk number 7: kedd.Rnw:184-189 ################################################### h <- c(0.14,0.3,0.6,1.2) plot(dkde(x = bimodal, deriv.order = 1, h = h[1], kernel = kernels[1]),col = 1,ylim=c(-0.6,1) ,sub="", main="") for (i in 2:length(h))lines(dkde(x = bimodal, deriv.order = 1, h = h[i], kernel = kernels[1] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c("TRUE",paste("h =",bquote(.(h)))), col = c("black",seq(h)),lty = c(8,rep(1,length(h))), inset = .015) ################################################### ### code chunk number 8: kedd.Rnw:229-233 ################################################### hatf <- dkde(bimodal, deriv.order = 0) hatf1 <- dkde(bimodal, deriv.order = 1) hatf2 <- dkde(bimodal, deriv.order = 2) hatf3 <- dkde(bimodal, deriv.order = 3) ################################################### ### code chunk number 9: kedd.Rnw:237-248 ################################################### fx <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5) fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) fx2 <- function(x) 0.5 * ((-4*x-6)^2 - 4) * dnorm(x,-1.5,0.5) + 0.5 * ((-4*x+6)^2 - 4) * dnorm(x,1.5,0.5) fx3 <- function(x) 0.5 * (-4*x-6) * ((-4*x-6)^2 - 12) * dnorm(x,-1.5,0.5) + 0.5 * (-4*x+6) * ((-4*x+6)^2 - 12) * dnorm(x,1.5,0.5) plot(hatf ,fx = fx) plot(hatf1,fx = fx1) plot(hatf2,fx = fx2) plot(hatf3,fx = fx3) ################################################### ### code chunk number 10: kedd.Rnw:253-254 ################################################### plot(hatf,fx = fx,lwd=2,sub="",main="") ################################################### ### code chunk number 11: kedd.Rnw:256-257 ################################################### plot(hatf1,fx = fx1,lwd=2,sub="",main="") ################################################### ### code chunk number 12: kedd.Rnw:259-260 ################################################### plot(hatf2,fx = fx2,lwd=2,sub="",main="") ################################################### ### code chunk number 13: kedd.Rnw:262-263 ################################################### plot(hatf3,fx = fx3,lwd=2,sub="",main="") ################################################### ### code chunk number 14: kedd.Rnw:312-316 ################################################### h.amise(bimodal, deriv.order = 0) h.amise(bimodal, deriv.order = 1) h.amise(bimodal, deriv.order = 2) h.amise(bimodal, deriv.order = 3) ################################################### ### code chunk number 15: kedd.Rnw:352-356 ################################################### kernels <- eval(formals(h.mlcv.default)$kernel) hmlcv <- numeric() for(i in 1:length(kernels)) hmlcv[i] <- h.mlcv(bimodal, kernel = kernels[i])$h ################################################### ### code chunk number 16: kedd.Rnw:358-359 ################################################### data.frame(kernels,hmlcv) ################################################### ### code chunk number 17: kedd.Rnw:362-364 ################################################### plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50)) plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50)) ################################################### ### code chunk number 18: kedd.Rnw:369-370 ################################################### plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50),sub="",main="") ################################################### ### code chunk number 19: kedd.Rnw:372-373 ################################################### plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50),sub="",main="") ################################################### ### code chunk number 20: kedd.Rnw:412-416 ################################################### h.ucv(bimodal, deriv.order = 0) h.ucv(bimodal, deriv.order = 1) h.ucv(bimodal, deriv.order = 2) h.ucv(bimodal, deriv.order = 3) ################################################### ### code chunk number 21: kedd.Rnw:419-420 ################################################### for (i in 0:3) plot(h.ucv(bimodal, deriv.order = i)) ################################################### ### code chunk number 22: kedd.Rnw:425-426 ################################################### plot(h.ucv(bimodal, deriv.order = 0),sub="",main="") ################################################### ### code chunk number 23: kedd.Rnw:428-429 ################################################### plot(h.ucv(bimodal, deriv.order = 1),sub="",main="") ################################################### ### code chunk number 24: kedd.Rnw:431-432 ################################################### plot(h.ucv(bimodal, deriv.order = 2),sub="",main="") ################################################### ### code chunk number 25: kedd.Rnw:434-435 ################################################### plot(h.ucv(bimodal, deriv.order = 3),sub="",main="") ################################################### ### code chunk number 26: kedd.Rnw:481-485 ################################################### h.bcv(bimodal, whichbcv = 1, deriv.order = 0) h.bcv(bimodal, whichbcv = 2, deriv.order = 0) h.bcv(bimodal, whichbcv = 1, deriv.order = 1, lower=0.1, upper=0.8) h.bcv(bimodal, whichbcv = 2, deriv.order = 1, lower=0.1, upper=0.8) ################################################### ### code chunk number 27: kedd.Rnw:488-499 ################################################### ## deriv.order = 0 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) ## deriv.order = 1 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws = seq(0.1,0.8,length=50)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) ################################################### ### code chunk number 28: kedd.Rnw:504-507 ################################################### plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) ################################################### ### code chunk number 29: kedd.Rnw:509-512 ################################################### plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws=seq(0.1,0.8,length=50),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) ################################################### ### code chunk number 30: kedd.Rnw:553-557 ################################################### h.ccv(bimodal, deriv.order = 0, upper = 0.5) h.ccv(bimodal, deriv.order = 1, upper = 0.5) h.ccv(bimodal, deriv.order = 2, upper = 0.5) h.ccv(bimodal, deriv.order = 3, upper = 0.5) ################################################### ### code chunk number 31: kedd.Rnw:560-562 ################################################### for (i in 0:3) plot(h.ccv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) ################################################### ### code chunk number 32: kedd.Rnw:567-568 ################################################### plot(h.ccv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 33: kedd.Rnw:570-571 ################################################### plot(h.ccv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 34: kedd.Rnw:573-574 ################################################### plot(h.ccv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 35: kedd.Rnw:576-577 ################################################### plot(h.ccv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 36: kedd.Rnw:614-618 ################################################### h.mcv(bimodal, deriv.order = 0, upper = 0.5) h.mcv(bimodal, deriv.order = 1, upper = 0.5) h.mcv(bimodal, deriv.order = 2, upper = 0.5) h.mcv(bimodal, deriv.order = 3, upper = 0.5) ################################################### ### code chunk number 37: kedd.Rnw:621-623 ################################################### for (i in 0:3) plot(h.mcv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) ################################################### ### code chunk number 38: kedd.Rnw:628-629 ################################################### plot(h.mcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 39: kedd.Rnw:631-632 ################################################### plot(h.mcv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 40: kedd.Rnw:634-635 ################################################### plot(h.mcv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 41: kedd.Rnw:637-638 ################################################### plot(h.mcv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") ################################################### ### code chunk number 42: kedd.Rnw:676-680 ################################################### h.tcv(bimodal, deriv.order = 0) h.tcv(bimodal, deriv.order = 1) h.tcv(bimodal, deriv.order = 2) h.tcv(bimodal, deriv.order = 3) ################################################### ### code chunk number 43: kedd.Rnw:683-684 ################################################### for (i in 0:3) plot(h.tcv(bimodal, deriv.order = i)) ################################################### ### code chunk number 44: kedd.Rnw:689-690 ################################################### plot(h.tcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,1.5,length=50),sub="",main="") ################################################### ### code chunk number 45: kedd.Rnw:692-693 ################################################### plot(h.tcv(bimodal, deriv.order = 1),seq.bws=seq(0.3,1.5,length=50),sub="",main="") 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\usepackage{amsmath,amssymb,amsthm,amsopn,array,natbib,titling} \usepackage[left=2.5cm,top=2cm,right=2cm,bottom=2cm]{geometry} \usepackage[utf8]{inputenc} \usepackage{authblk} \usepackage[english]{babel} \usepackage{hyperref} \usepackage{Sweave} \usepackage{color} \definecolor{link}{rgb}{0.45,0.51,0.67} \hypersetup{ colorlinks,% citecolor=link,% filecolor=link,% linkcolor=link,% urlcolor=link } %% load any required packages here \renewcommand{\today}{\begingroup \number \day\space \ifcase \month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space \number \year \endgroup} \newcommand{\argmin}[1]{\underset{#1}{\mathrm{argmin}} \ } \newcommand{\argmax}[1]{\underset{#1}{\mathrm{argmax}} \ } \def\ISE{\mathrm{ISE}} \def\MSE{\mathrm{MSE}} \def\MISE{\mathrm{MISE}} \def\AMSE{\mathrm{AMSE}} \def\AMISE{\mathrm{AMISE}} \def\LSCV{\mathrm{LSCV}} \def\UCV{\mathrm{UCV}} \def\BCV{\mathrm{BCV}} \def\CCV{\mathrm{CCV}} \def\TCV{\mathrm{TCV}} \def\MCV{\mathrm{MCV}} \def\MLCV{\mathrm{MLCV}} \def\rth{r^{th}} \def\Intr{\int_{\boldsymbol{\mathbb{R}}}} \def\Sum2{\sum_{i=1}^{n}\sum_{\substack{j=1 \\ j \neq i}}^{n}} \def\M2{\mu_{2}} \def\RK{\mathrm{R}\left(K^{(r)}\right)} \def\RR{\mathrm{R}} \def\Kr{K^{(r)}} \def\ConvKr{K^{(r)} \ast K^{(r)}} \def\Z{\left(\frac{X_{j}-X_{i}}{h}\right)} \def\z{\left(\frac{x-X_{i}}{h}\right)} \def\hatf{\hat{f}_{h}^{(r)}} \def\N{\mathbb{N}} %%%%%%%%%%%%%%%% \let\code=\texttt \let\proglang=\texttt \let\pkg=\texttt \newcommand{\CRANpkg}[1]{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}}% \newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} \let\cpkg=\CRANpkg \let\cpkg=\CRANpkg \bibliographystyle{plainnat} %\VignetteIndexEntry{Kernel estimator and bandwidth selection for density and its derivatives} %\VignettePackage{kedd} \begin{document} \title{\bfseries{ Kernel Estimator and Bandwidth Selection for Density and its Derivatives}\\ \vskip3pt \large The \CRANpkg{kedd} Package \\Version 1.0.3} \author{by Arsalane Chouaib Guidoum\thanks{Department of Probabilities \& Statistics.\\Faculty of Mathematics. \\University of Science and Technology Houari Boumediene.\\ BP 32 El-Alia, U.S.T.H.B, Algeria.\\ \email{acguidoum@usthb.dz}}} \date{Revised \today} \maketitle \section{Introduction}\label{Sec0} In statistics, the univariate kernel density estimation (KDE) is a non-parametric way to estimate the probability density function $f(x)$ of a random variable $X$, is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. This techniques are widely used in various inference procedures such as signal processing, data mining and econometrics, see e.g., \cite{Silverman,WandandJones,Jeffrey,Wolfgangetall,Alexandre}. The kernel estimator are standard in many books with applications and computer vision, see \cite{Wolfgang,Scott1992,Bowman,VenablesandRipley}, for computational complexity and with implementation in \texttt{S}, for an overview. Estimation of the density derivatives also comes up in various other applications like estimation of modes and inflexion points of densities, a good list of applications which require the estimation of density derivatives can be found in \cite{Singh1977}. There already exist a number of packages that can perform kernel density estimation in \texttt{R} (\code{density} in \texttt{R} base); see for example \pkg{KernSmooth} \citep{KernSmooth}, \pkg{sm} \citep{smarticle}, \pkg{np} \citep{np} and \pkg{feature} \citep{feature}, they exist also of functions for kernel density derivative estimation (KDDE), e.g., \code{kdde} in \pkg{ks} package \citep{ks}. We introduce in this vignette a new \texttt{R} package \CRANpkg{kedd} \citep{kedd} for use with the statistical programming environment \cite{R}, which implements smoothing techniques and computing bandwidth selectors of the $\rth$ derivative of a probability density $f(x)$ for univariate data, using several kernels functions. \section{Convolutions and derivatives in kernels}\label{Sec1} In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables density functions $f(x)$, or in kernel regression to estimate the conditional expectation of a random variable, see e.g., \cite{Silverman,WandandJones}. In general any functions having the following assumptions can be used as a kernel: \begin{itemize} \item[(A1)] $K(x) \geq 0$ and $\Intr K(x) dx = 1$. \item[(A2)] Symmetric about the origin, e.g., $\Intr x K(x) dx = 0$. \item[(A3)] Has finite second moment, e.g., $\M2(K) = \Intr x^{2} K(x) dx < \infty$. We denote $\RR(K) = \Intr \left(K(x)\right)^{2} dx$. \end{itemize} If $K(x)$ is a kernel, then so is the function $\bar{K}(x)$ defined by $\bar{K}(x)=\lambda K(\lambda x)$, where $\lambda > 0$, this can be used to select a scale that is appropriate for the data. The kernel function is very important to spreading a probability mass of $1/n$, the most widely used kernel is the Gaussian of zero mean and unit variance. Some classical of kernel function $K(x;r)$ ($r$ is the maximum derivative of kernel) in \pkg{kedd} package are the following: \begin{table}[!ht] \begin{center} {\renewcommand{\arraystretch}{1.5} \begin{tabular}{rlll} \hline\hline Kernel & $K(x;r)$ & $\RR(K)$ & $\M2(K)$ \\ \hline Gaussian & $K(x;\infty) =\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)1_{]-\infty,+\infty[}$ & $1/\left(2\sqrt{\pi}\right)$ & 1 \\ Epanechnikov & $K(x;2)=\frac{3}{4}\left(1-x^{2}\right)1_{(|x| \leq 1)}$ & 3/5 & 1/5 \\ Uniform & $K(x;0)=\frac{1}{2}1_{(|x| \leq 1)}$ & 1/2 & 1/3 \\ Triangular & $K(x;1)=(1-|x|)1_{(|x| \leq 1)}$ & 2/3 & 1/6 \\ Triweight & $K(x;6)=\frac{35}{32}\left(1-x^{2}\right)^{3} 1_{(|x| \leq 1)}$ & 350/429 & 1/9 \\ Tricube & $K(x;9)=\frac{70}{81}\left(1-|x|^{3}\right)^{3} 1_{(|x| \leq 1)}$ & 175/247 & 35/243 \\ Biweight & $K(x;4)=\frac{15}{16}\left(1-x^{2}\right)^{2} 1_{(|x| \leq 1)}$ & 5/7 & 1/7\\ Cosine & $K(x;\infty)=\frac{\pi}{4}\cos\left(\frac{\pi}{2}x\right) 1_{(|x| \leq 1)}$ & $\pi^{2}/16$ & $\left(-8+\pi^2\right)/\pi^{2}$\\ \hline\hline \end{tabular} } \end{center} \caption{Kernel functions in \pkg{kedd} pakage.}\label{Sec1:Tab1} \end{table}\\ The $\rth$ derivative of the kernel function $K(x)$ is written as: \begin{equation}\label{Sec1:eq1} \Kr(x) = \frac{d^{r}}{dx^{r}} K(x) \end{equation} and convolution of $\Kr(x)$ is: \begin{equation}\label{Sec1:eq2} \ConvKr(x) = \Intr \Kr(x) \Kr(x-y) dy \end{equation} for example the $\rth$ derivative of the Gaussian kernel is given by: $$\Kr(x) = (-1)^{r} H_{r}(x) K(x)$$ and the $\rth$ convolution can be written as: $$\ConvKr(x) = (-1)^{2r} \Intr H_{r}(x) H_{r}(x-y)K(x)K(x-y)dy$$ where $H_{r}(x)$ is the $\rth$ Hermite polynomial, see e.g., \cite{Olver}. We use \code{kernel.fun} for kernel derivative defined by \eqref{Sec1:eq1}, and \code{kernel.conv} for kernel convolution defined by \eqref{Sec1:eq2}.\\ For example the first derivative of the Gaussian kernel displayed on the left in Figure \ref{Sec1:fig1}. On the right is the first convolution of the Gaussian kernel. <>= library(kedd) @ <>= kernel.fun(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx kernel.conv(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx @ <>= plot(kernel.fun(deriv.order = 1, kernel = "gaussian")) plot(kernel.conv(deriv.order = 1, kernel = "gaussian")) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(kernel.fun(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") @ <>= plot(kernel.conv(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") @ \end{center} \caption{(Left) First derivative of the Gaussian kernel. (Right) Convolution of the first derivative Gaussian kernel.}\label{Sec1:fig1} \end{figure} \section{Kernel density derivative estimator}\label{Sec2} Let $(X_{1},X_{2},\dots,X_{n})$ be a data sample, independent and identically distributed of a continuous random variable $X$, with density function $f(x)$. If the kernel $K$ is differentiable $r$ times then a natural estimator of the $\rth$ derivative of $f(x)$ the $\rth $ derivative of the kernel estimate \citep{Bhattacharya,Schuster,Alekseev}: \begin{equation}\label{Sec2:eq1} \hatf(x) = \frac{d^{r}}{dx^{r}} \frac{1}{nh} \sum_{i=1}^{n} K\z = \frac{1}{nh^{r+1}} \sum_{i=1}^{n} \Kr\z \end{equation} where $\Kr$ is $\rth$ derivative of the kernel function $K$, which we take to be a symmetric probability density with at least $r$ non zero derivatives when estimating $f^{(r)}(x)$, and $h$ is the bandwidth, this parameter is very important that controls the degree of smoothing applied to the data. The following assumptions on the density $f^{(r)}(x)$, the bandwidth $h$, and the kernel $K$: \begin{itemize} \item[(A4)] The $(r+2)$ derivatives $f^{(r+2)}(x)$ is continuous, square integrable and ultimately monotone. \item[(A5)] In the asymptotic framework, as $\lim_{n \to \infty} h_{n} = 0$ and $\lim_{n \to \infty} nh_{n}^{2r+1} = \infty$, i.e., as the number of sample $n$ is increased $h$ approaches zero at a rate slower than $1/n^{2r+1}$. \item[(A6)] Assumptions about $K$ are introduced in the previous section. \end{itemize} As seen in Equation \eqref{Sec2:eq1}, when working with a kernel estimator of the $\rth$ derivative function two choices must be made: the kernel function $K$ and the smoothing parameter or bandwidth $h$. The choice of $K$ is a problem of less importance, because $K$ is not very sensitive to the shape of estimator, and different functions that produce good results can be used. In practice, the choice of an efficient method for the computation of $h$, for an observed data sample is a crucial problem, because of the effect of the bandwidth on the shape of the corresponding estimator. If the bandwidth is small, we will obtain an under smoothed estimator, with high variability. On the contrary, if the value of $h$ is big, the resulting estimator will be over smooth and farther from the function that we are trying to estimate.\\ An example is drawn in Figure \ref{Sec2:fig1} where we show in left four different kernel (Gaussian, biweight, triweight and tricube) estimators of the first derivative of a bimodal (separated) Gaussian density (Equation \ref{Sec2:eq3}), and a given value of $h=0.6$. On the right, using the Gaussian kernel and four different values for the bandwidth. \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= fx <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) kernels <- c("gaussian","biweight","triweight","tricube") plot(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[1]),col = 1,ylim=c(-0.6,0.6) ,sub="", main="") for (i in 2:length(kernels))lines(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[i] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),lty = c(8,rep(1,length(kernels))), inset = .015) @ <>= h <- c(0.14,0.3,0.6,1.2) plot(dkde(x = bimodal, deriv.order = 1, h = h[1], kernel = kernels[1]),col = 1,ylim=c(-0.6,1) ,sub="", main="") for (i in 2:length(h))lines(dkde(x = bimodal, deriv.order = 1, h = h[i], kernel = kernels[1] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c("TRUE",paste("h =",bquote(.(h)))), col = c("black",seq(h)),lty = c(8,rep(1,length(h))), inset = .015) @ \end{center} \caption{(Left) Different kernels for estimation, with $h=0.6$. (Right) Effect of the bandwidth on the kernel estimator.}\label{Sec2:fig1} \end{figure} We have implemented in \texttt{R} the function \code{dkde} corresponds to the derivative of kernel density estimator (Equation \ref{Sec2:eq1}). Eight possibilities are allowed for the kernel functions that are summarized in Table \ref{Sec1:Tab1}. We enumerate the arguments and results of this function in Table \ref{Sec2:Tab1}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{y} & The points of the grid at which the density derivative is to be estimated.\\ & The default are $4h$ outside of range($x$).\\ \code{deriv.order} & Derivative order (scalar). \\ \code{h} & The smoothing bandwidth to be used. The default, "ucv" unbiased cross-\\ & validation.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{eval.points} & The coordinates of the points where the density derivative is estimated.\\ \code{est.fx} & The estimated density derivative values (Equation \ref{Sec2:eq1}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{dkde}.}\label{Sec2:Tab1} \end{table}\\ Working with the dataset \code{'bimodal'} correspond to data sample of 200 random numbers of a bi-modality (separated) of a two-component Gaussian mixture density (Equation \ref{Sec2:eq2}), with the following parameters: $-\mu_{1}=\mu_{2} = 3/2$ and $\sigma_{1}=\sigma_{2}=1/2$. The \code{dkde} function enables to compute the $\rth$ derivative of kernel density estimator over a grid of points, with a bandwidth selected by the user, but it also allows to estimate directly this parameter by the unbiased cross-validation method \code{h.ucv} (see following Section). We have chosen this method as the automatic one because it is the fastest in computation time terms. Now we estimate the first three derivatives of $f(x)$, can be written as: \begin{eqnarray} % \nonumber to remove numbering (before each equation) f(x) &=& 0.5\phi(\mu_{1},\sigma_{1}) + 0.5\phi(\mu_{2},\sigma_{2}) \label{Sec2:eq2}\\ f^{(1)}(x) &=& 0.5(-4x-6) \phi(\mu_{1},\sigma_{1}) + 0.5(-4x+6)\phi(\mu_{2},\sigma_{2})\label{Sec2:eq3} \\ f^{(2)}(x) &=& 0.5\left(\left(-4x-6\right)^{2} - 4\right)\phi(\mu_{1},\sigma_{1})+ 0.5 \left(\left(-4x+6\right)^{2} - 4\right) \phi(\mu_{2},\sigma_{2})\label{Sec2:eq4}\\ f^{(3)}(x) &=& 0.5(-4x-6)\left(\left(-4x-6\right)^{2} - 12\right)\phi(\mu_{1},\sigma_{1})+0.5(-4x+6) \notag \\ & & \left(\left(-4x+6\right)^{2} - 12\right)\phi(\mu_{2},\sigma_{2})\label{Sec2:eq5} \end{eqnarray} where $\phi$ is a standard normal density. <>= hatf <- dkde(bimodal, deriv.order = 0) hatf1 <- dkde(bimodal, deriv.order = 1) hatf2 <- dkde(bimodal, deriv.order = 2) hatf3 <- dkde(bimodal, deriv.order = 3) @ By default, the function \code{dkde} selects a grid of 512 points in the data range and used the Gaussian kernel. The output is a list containing the estimated values in the points of the grid, this last sequence and the bandwidth $h$ (by default, using unbiased cross-validation method). In Figure \ref{Sec2:fig3} we show the first three derivatives estimators of $f(x)$ obtained with the code: <>= fx <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5) fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) fx2 <- function(x) 0.5 * ((-4*x-6)^2 - 4) * dnorm(x,-1.5,0.5) + 0.5 * ((-4*x+6)^2 - 4) * dnorm(x,1.5,0.5) fx3 <- function(x) 0.5 * (-4*x-6) * ((-4*x-6)^2 - 12) * dnorm(x,-1.5,0.5) + 0.5 * (-4*x+6) * ((-4*x+6)^2 - 12) * dnorm(x,1.5,0.5) plot(hatf ,fx = fx) plot(hatf1,fx = fx1) plot(hatf2,fx = fx2) plot(hatf3,fx = fx3) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(hatf,fx = fx,lwd=2,sub="",main="") @ <>= plot(hatf1,fx = fx1,lwd=2,sub="",main="") @ <>= plot(hatf2,fx = fx2,lwd=2,sub="",main="") @ <>= plot(hatf3,fx = fx3,lwd=2,sub="",main="") @ \end{center} \caption{Kernel density derivative estimates obtained with the function \code{dkde}. (top left) density estimate $\hat{f}_{h}(x)$. (top right) first derivative $\hat{f}^{(1)}_{h}(x)$. (bottom left) second derivative $\hat{f}^{(2)}_{h}(x)$. (bottom right) third derivative $\hat{f}^{(3)}_{h}(x)$.}\label{Sec2:fig3} \end{figure} \section{Bandwidth selections}\label{Sec3} Despite the great number of bandwidth selection techniques in kernel density estimator or regression estimation, as for example \cite{Rudemo1982,Bowman1984,ScottandGeorge1987,SheatherandJones1991,Chiu1991a,Chiu1991b,Chiu1992,FeluchandKoronacki1992,Stute1992,Jonesetall1996, Sheather2004,DuongandHazelton2003,DuongandHazelton2005,Heidenreichetall2013}, to the best of our knowledge, only few paper have been studied in the context of estimating the $\rth$ derivative of a density $f(x)$, see \citet{PeterandMarron1987,Wolfgangetall1990,JonesandKappenman1991,Stoker1993}. In this section we summarize the techniques of cross-validation methods for bandwidth choice in the kernel estimation of the derivatives of a probability density. The practicality of this methods is demonstrated by an example. \subsection{Optimal bandwidth} We Consider the following $\AMISE$ version of the $\rth$ derivative of a probability density $f(x)$ \cite[p. 131]{Scott1992}: \begin{equation}\label{Sec3:eq1} \AMISE(h,r)= \frac{\RK}{nh^{2r+1}} + \frac{1}{4} h^{4} \M2^{2}(K) \RR\left(f^{(r+2)}\right) \end{equation} The optimal bandwidth minimizing \eqref{Sec3:eq1} is: \begin{equation}\label{Sec3:eq2} h^{\ast} = \left[\frac{(2r+1)\RK}{\M2^{2}(K) \RR\left(f^{(r+2)}\right)}\right]^{1/(2r+5)} n^{-1/(2r+5)} \end{equation} whereof: \begin{equation}\label{Sec3:eq3} \AMISE(h,r) = \frac{2r+5}{4} \RK^{\frac{4}{(2r+5)}} \left[ \frac{\M2^{2}(K) \RR\left(f^{(r+2)}\right)}{2r+1} \right]^{\frac{2r+1}{2r+5}} n^{-\frac{4}{2r+5}} \end{equation} which is the smallest possible $\AMISE$ for estimation of $\hat{f}^{(r)}_{h}$. The function \code{h.amise} provides the optimal bandwidth under $\AMISE$. The same possibilities for the kernel function as in the function \code{dkde} appear here. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab1}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq2}).\\ \code{amise} & The $\AMISE$ value (Equation \ref{Sec3:eq3}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.amise}.}\label{Sec3:Tab1} \end{table}\\ The following example computes this bandwidth for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.amise(bimodal, deriv.order = 0) h.amise(bimodal, deriv.order = 1) h.amise(bimodal, deriv.order = 2) h.amise(bimodal, deriv.order = 3) @ \subsection{Maximum likelihood cross-validation} This method was proposed by \cite{Habbema1974} and \cite{Duin1976}. They proposed to choose $h$ so that the pseudo-likelihood $\prod_{i=1}^{n} \hat{f}_{h}(X_{i})$ is maximized. However this has a trivial maximum at $h = 0$, so the cross-validation principle is invoked by replacing $\hat{f}_{h}(x)$ by the leave-one-out $\hat{f}_{h,i}(x)$, where: $$\hat{f}_{h,i}(X_{i}) = \frac{1}{(n-1) h} \sum_{j \neq i} K\Z$$ Define that $h$ as good which approaches the finite maximum of \begin{equation}\label{Sec3:eq4} h_{mlcv} = \argmax {h > 0} \MLCV(h) \end{equation} \begin{equation}\label{Sec3:eq5} \MLCV(h) = \left(n^{-1} \sum_{i=1}^{n} \log\left[\sum_{j \neq i} K\Z\right]-\log[(n-1)h]\right) \end{equation} The function \code{h.mlcv} computed the maximum likelihood cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab2}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq4}).\\ \code{mlcv} & The $\MLCV$ value (Equation \ref{Sec3:eq5}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.mlcv}.}\label{Sec3:Tab2} \end{table}\\ The following example computes this bandwidth of bimodal Gaussian density (Equation \ref{Sec2:eq2}), by different kernels. <>= kernels <- eval(formals(h.mlcv.default)$kernel) hmlcv <- numeric() for(i in 1:length(kernels)) hmlcv[i] <- h.mlcv(bimodal, kernel = kernels[i])$h @ <>= data.frame(kernels,hmlcv) @ The plot of the maximal likelihood cross validation function $\MLCV$ is shown in Figure \ref{Sec3:fig1} for Gaussian kernel in the left, and Epanechnikov kernel in the right, obtained with the code: <>= plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50)) plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50),sub="",main="") @ <>= plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50),sub="",main="") @ \end{center} \caption{$\MLCV$ function obtained by \code{h.mlcv}, using Gaussian kernel (Left) and Epanechnikov kernel (Right).}\label{Sec3:fig1} \end{figure} \subsection{Unbiased cross-validation} \cite{Rudemo1982} and \cite{Bowman1984} proposed a so-called unbiased (least-squares) cross-validation ($\UCV$) in kernel density estimator, is probably the most popular and best studied one. An adaptation of unbiased cross-validation is proposed by \cite{Wolfgangetall1990} for bandwidth choice in the $\rth$ derivative of kernel density estimator. The essential idea of this methods, it aims to estimate $h$ the minimizer of $\ISE(h)$. The minimization criterion is defined by: \begin{equation}\label{Sec3:eq6} h_{ucv} = \argmin {h > 0 }\UCV(h,r) \end{equation} \begin{equation}\label{Sec3:eq7} \UCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \Sum2 \left(\ConvKr -2 K^{(2r)}\right)\Z \end{equation} In general, cross-validation functions in non-parametric bandwidth selection present several local minima. These minima are more likely to appear at too small values of the bandwidth \citep{PeterandMarron1991}. The function \code{h.ucv} computes the unbiased cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab3}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq6}).\\ \code{min.ucv} & The minimal $\UCV$ value (Equation \ref{Sec3:eq7}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.ucv}.}\label{Sec3:Tab3} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.ucv(bimodal, deriv.order = 0) h.ucv(bimodal, deriv.order = 1) h.ucv(bimodal, deriv.order = 2) h.ucv(bimodal, deriv.order = 3) @ The plot of $\UCV$ function obtained with the code (Figure \ref{Sec3:fig2}): <>= for (i in 0:3) plot(h.ucv(bimodal, deriv.order = i)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.ucv(bimodal, deriv.order = 0),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 1),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 2),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 3),sub="",main="") @ \end{center} \caption{$\UCV$ function obtained by \code{h.ucv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig2} \end{figure} \subsection{Biased cross-validation} Biased cross-validation was proposed by \cite{ScottandGeorge1987}, which has as its immediate target the $\AMISE$ \eqref{Sec3:eq1}. They proposed to estimate $\RR\left(f^{(r+2)}\right)$ by! $$\hat{\RR}\left(f^{(r+2)}\right) = \RR\left(\hat{f}^{(r+2)}_{h}\right) - \frac{\RR\left(K^{(r+2)}\right)}{nh^{2r+5}}$$ There are two versions of $\BCV$, depending on the estimator of $\RR\left(f^{(r+2)}\right)$. We can use \citep{ScottandGeorge1987} $$\hat{\RR}\left(f^{(r+2)}\right) = \frac{(-1)^{r+2}}{n(n-1)h^{2r+5}} \Sum2 K^{(r+2)} \ast K^{(r+2)} \Z$$ or we could use \citep{JonesandKappenman1991} $$\hat{\RR}\left(f^{(r+2)}\right) = \frac{(-1)^{r+2}}{n(n-1) h^{2r+5}} \Sum2 K^{(2r+4)} \Z$$ From this we obtain respectively an adaptation of biased cross-validation for bandwidth choice in the $\rth$ derivative of kernel density estimator, is given by: \begin{equation}\label{Sec3:eq8} \BCV_{1}(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{\M2^{2}(K)}{4} \frac{(-1)^{r+2}}{n(n-1)h^{2r+1}} \Sum2 K^{(r+2)} \ast K^{(r+2)} \Z \end{equation} \begin{equation}\label{Sec3:eq9} \BCV_{2}(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{\M2^{2}(K)}{4} \frac{(-1)^{r+2}}{n(n-1)h^{2r+1}} \Sum2 K^{(2r+4)} \Z \end{equation} The $\BCV$ selectors $h_{bcv_{1}}$ and $h_{bcv_{2}}$ are the minimisers of the appropriate $\BCV$ function. The function \code{h.bcv} computes the biased cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab4}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{whichbcv} & Method selected, \code{1 = BCV1} or \code{2 = BCV2}, by default \code{BCV1}. \\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.bcv} & The minimal $\BCV$ value (Equation \ref{Sec3:eq8} or \ref{Sec3:eq9}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.bcv}.}\label{Sec3:Tab4} \end{table}\\ The following example computes the bandwidth parameter by this method for kernel estimator of Equation \eqref{Sec2:eq2} and its first derivative estimators. <>= h.bcv(bimodal, whichbcv = 1, deriv.order = 0) h.bcv(bimodal, whichbcv = 2, deriv.order = 0) h.bcv(bimodal, whichbcv = 1, deriv.order = 1, lower=0.1, upper=0.8) h.bcv(bimodal, whichbcv = 2, deriv.order = 1, lower=0.1, upper=0.8) @ The plot of $\BCV$ function obtained with the code \code{h.bcv} (Figure \ref{Sec3:fig3}): <>= ## deriv.order = 0 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) ## deriv.order = 1 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws = seq(0.1,0.8,length=50)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) @ <>= plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws=seq(0.1,0.8,length=50),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) @ \end{center} \caption{$\BCV$ function obtained by \code{h.bcv}. (Left) $\BCV_{1}$ vs $\BCV_{2}$ (\code{deriv.order = 0}). (Right) $\BCV_{1}$ vs $\BCV_{2}$ (\code{deriv.order = 1}).}\label{Sec3:fig3} \end{figure} \subsection{Complete cross-validation} \cite{JonesandKappenman1991} proposed a so-called complete cross-validation ($\CCV$) in kernel density estimator. This method can be extended to the estimation of derivative of the density, basing our estimate of integrated squared density derivative \citep{PeterandMarron1987} we get the following. Thus, $h_{ccv}$, say, is the $h$ that minimises: \begin{equation}\label{Sec3:eq10} \CCV(h,r) = \RR\left(\hatf\right) -\bar{\theta}_{r}(h) + \frac{1}{2}\M2(K) h^{2} \bar{\theta}_{r+1}(h)+ \frac{1}{24}\left(6\M2^{2}(K) -\delta(K)\right) h^{4}\bar{\theta}_{r+2}(h) \end{equation} where, $$\RR\left(\hatf\right) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \Sum2 \ConvKr \Z$$ and $$\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 K^{(2r)} \Z$$ with : $\delta(K) = \Intr x^{4} K(x) dx$.\\ The function \code{h.ccv} computes the complete cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab5}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.ccv} & The minimal $\CCV$ value (Equation \ref{Sec3:eq10}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.ccv}.}\label{Sec3:Tab5} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. This time we set Over-smoothing in \code{upper = 0.5}. <>= h.ccv(bimodal, deriv.order = 0, upper = 0.5) h.ccv(bimodal, deriv.order = 1, upper = 0.5) h.ccv(bimodal, deriv.order = 2, upper = 0.5) h.ccv(bimodal, deriv.order = 3, upper = 0.5) @ The plot of $\CCV$ function obtained with the code (Figure \ref{Sec3:fig4}): <>= for (i in 0:3) plot(h.ccv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.ccv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ \end{center} \caption{$\CCV$ function obtained by \code{h.ccv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig4} \end{figure} \subsection{Modified cross-validation} \cite{Stute1992} proposed a so-called modified cross-validation ($\MCV$) in kernel density estimator. This method can be extended to the estimation of derivative of a probability density, the essential idea based on approximated the problematic term by the aid of the Hajek projection. The minimization criterion is defined by: \begin{equation}\label{Sec3:eq11} \MCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 \varphi^{(r)} \Z \end{equation} where$$ \varphi^{(r)} (c) = \left(\ConvKr - K^{(2r)} - \frac{\M2(K)}{2} K^{(2r+2)} \right)(c)$$ The function \code{h.mcv} computes the modified cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab6}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.mcv} & The minimal $\MCV$ value (Equation \ref{Sec3:eq11}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.mcv}.}\label{Sec3:Tab6} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. We set Over-smoothing in \code{upper = 0.5}. <>= h.mcv(bimodal, deriv.order = 0, upper = 0.5) h.mcv(bimodal, deriv.order = 1, upper = 0.5) h.mcv(bimodal, deriv.order = 2, upper = 0.5) h.mcv(bimodal, deriv.order = 3, upper = 0.5) @ The plot of $\MCV$ function obtained with the code (Figure \ref{Sec3:fig5}): <>= for (i in 0:3) plot(h.mcv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.mcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ \end{center} \caption{$\MCV$ function obtained by \code{h.mcv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig5} \end{figure} \subsection{Trimmed cross-validation} \cite{FeluchandKoronacki1992} proposed a so-called trimmed cross-validation ($\TCV$) in kernel density estimator, a simple modification of the unbiased (least-squares) cross-validation criterion \eqref{Sec3:eq7}. We consider the following "trimmed" version of "unbiased", to be minimized with respect to $h$: \begin{equation}\label{Sec3:eq12} \TCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 \varphi^{(r)} \Z \end{equation} where$$ \varphi^{(r)} (c) = \left[\ConvKr - 2 K^{(2r)} 1\left(|c| > \frac{c_{n}}{h^{2r+1}}\right) \right](c)$$ $1(.)$ denotes the indicator function and $c_{n}$ is a sequence of positive constants, as $\lim_{n \to \infty} c_{n}/h \rightarrow 0$, here we take $c_{n} = 1/n$, for assure the convergence. The function \code{h.tcv} computes the trimmed cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab7}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.tcv} & The minimal $\TCV$ value (Equation \ref{Sec3:eq12}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.tcv}.}\label{Sec3:Tab7} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.tcv(bimodal, deriv.order = 0) h.tcv(bimodal, deriv.order = 1) h.tcv(bimodal, deriv.order = 2) h.tcv(bimodal, deriv.order = 3) @ The plot of $\TCV$ function obtained with the code (Figure \ref{Sec3:fig6}): <>= for (i in 0:3) plot(h.tcv(bimodal, deriv.order = i)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.tcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 1),seq.bws=seq(0.3,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 2),seq.bws=seq(0.5,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 3),seq.bws=seq(0.5,1.5,length=50),sub="",main="") @ \end{center} \caption{$\TCV$ function obtained by \code{h.tcv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig6} \end{figure} \vspace{3cm} \section{Summary} We have implemented in \texttt{R} the estimators of the defined functions and the bandwidth selection procedures of the above sections. The package \pkg{kedd} contains seven functions, in Table \ref{Sec4:Tab1} we can find a summary of the contents of the package. The current feature set of the package can be split in four main categories: compute the convolutions and derivatives of a kernel function, compute the kernel estimators for a density of probability and its derivatives, computing the bandwidth selectors with different methods, displaying the kernel estimators and selection functions of the bandwidth. Moreover, the package follows the general \texttt{R} philosophy of working with model objects. This means that instead of merely returning, say, a kernel estimator of $\rth$ derivative of a density, many functions will return an object containing, it's functions are S3 classes (\code{S3method}). The object can then be manipulated at one’s will using various extraction, summary or plotting functions. Whenever possible, we develop a graphical user interface of the various functions of a coherent whole, to facilitate the use of this package. \begin{table}[!ht] \begin{center} {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Function & Description \\ \hline \code{dkde} & Derivatives of kernel density estimator, as defined in Equation \ref{Sec2:eq1}.\\ \code{h.amise} & $\AMISE$ for optimal bandwidth selectors (Equation \ref{Sec3:eq3}). \\ \code{h.mlcv} & Maximum-likelihood cross-validation bandwidth selection (Equation \ref{Sec3:eq5}).\\ \code{h.ucv} & Unbiased cross-validation bandwidth selection (Equation \ref{Sec3:eq7}).\\ \code{h.bcv} & Biased cross-validation bandwidth selection (Equations \ref{Sec3:eq8} and \ref{Sec3:eq9}) .\\ \code{h.ccv} & Complete cross-validation bandwidth selection (Equation \ref{Sec3:eq10}).\\ \code{h.mcv} & Modified cross-validation bandwidth selection (Equation \ref{Sec3:eq11}).\\ \code{h.tcv} & Trimmed cross-validation bandwidth selection (Equation \ref{Sec3:eq12}).\\ \hline\hline \end{tabular} } \end{center} \caption{Summary of contents of the package.}\label{Sec4:Tab1} \end{table} \begin{thebibliography}{1} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{{\tt #1}}\fi \bibitem[Alekseev, 1972]{Alekseev} Alekseev, V. 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(2015). \newblock \pkg{kedd}: Kernel estimator and bandwidth selection for density and its derivatives. \newblock \texttt{R} package version 1.0.3. \newblock \url{http://CRAN.R-project.org/package=kedd} \bibitem[Habbema, Hermans and Van den Broek (1974)]{Habbema1974} Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974). \newblock A stepwise discrimination analysis program using density estimation. \newblock \emph{Compstat 1974: Proceedings in Computational Statistics}. Physica Verlag, Vienna. \bibitem[Heidenreich et all, 2013]{Heidenreichetall2013} Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). \newblock Bandwidth selection for kernel density estimation: a review of fully automatic selectors. \newblock \emph{Advances in Statistical Analysis}. \bibitem[Jeffrey, 1996]{Jeffrey} Jeffrey, S. S. (1996). \newblock \emph{Smoothing Methods in Statistics}. \newblock Springer-Verlag, New York. \bibitem[Jones and Kappenman, 1991]{JonesandKappenman1991} Jones, M. 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(1991). \newblock Local minima in cross-validation functions. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{53}, 245--252. \bibitem[\texttt{R} Development Core Team (2015)]{R} R Development Core Team (2015). \newblock {\em \texttt{R}: A Language and Environment for Statistical Computing}. \newblock Vienna, Austria. \newblock \url{http://www.R-project.org/} \bibitem[Rudemo, 1982]{Rudemo1982} Rudemo, M. (1982). \newblock Empirical choice of histograms and kernel density estimators. \newblock \emph{Scandinavian Journal of Statistics}, \textbf{9}, 65--78. \bibitem[Schuster, 1969]{Schuster} Schuster, E. F. (1969). \newblock Estimation of a probability density function and its derivatives. \newblock \emph{The Annals of Mathematical Statistics}, \textbf{40}(4), 1187--1195. \bibitem[Scott, 1992]{Scott1992} Scott, D. W. (1992). \newblock \emph{Multivariate Density Estimation. 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(1990). \newblock Applications of estimators of a density and its derivatives\textbf{39}(3), 357--363. \bibitem[Stoker, 1993]{Stoker1993} Stoker, T. M. (1993). \newblock Smoothing bias in density derivative estimation. \newblock \emph{Journal of the American Statistical Association}, \textbf{88}, 855--863. \bibitem[Stute, 1992]{Stute1992} Stute, W. (1992). \newblock Modified cross validation in density estimation. \newblock \emph{Journal of Statistical Planning and Inference}, \textbf{30}, 293--305. \bibitem[Tristen and Jeffrey, 2008]{np} Tristen, H. and Jeffrey, S. R. (2008). \newblock Nonparametric Econometrics: The \pkg{np} Package. \newblock {\em Journal of Statistical Software}. \textbf{27 (5)}. \bibitem[Wand and Jones, 1995]{WandandJones} Wand, M. P. and Jones, M. C. (1995). \newblock \emph{Kernel Smoothing}. \newblock Chapman and Hall, London. \bibitem[Wand and Ripley, 2013]{KernSmooth} Wand, M.P. and Ripley, B. D. (2013). \newblock \pkg{KernSmooth}: Functions for kernel smoothing for Wand and Jones (1995). \newblock \texttt{R} package version 2.23-10. \newblock \url{http://CRAN.R-project.org/package=KernSmooth} \bibitem[Wolfgang, 1991]{Wolfgang} Wolfgang, H. (1991). \newblock \emph{Smoothing Techniques, With Implementation in S}. \newblock Springer-Verlag, New York. \bibitem[Wolfgang et all, 1990]{Wolfgangetall1990} Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). \newblock Bandwidth choice for density derivatives. \newblock \emph{Journal of the Royal Statistical Society, Series B}, 223--232. \bibitem[Wolfgang et all, 2004]{Wolfgangetall} Wolfgang, H., Marlene, M., Stefan, S. and Axel, W. (2004). \newblock \emph{Nonparametric and Semiparametric Models}. \newblock Springer-Verlag, Berlin Heidelberg. \bibitem[Venables and Ripley, 2002]{VenablesandRipley} Venables, W. N. and Ripley, B. D. (2002). \newblock \emph{Modern Applied Statistics with S}. \newblock New York: Springer. \end{thebibliography} \end{document}kedd/inst/doc/intro.Rnw0000644000176200001440000001105512614740640014534 0ustar liggesusers\documentclass{article} \usepackage{graphicx} \usepackage{amsmath,amssymb,amsthm,amsopn,array,natbib,titling} \usepackage[utf8]{inputenc} \usepackage{authblk} \usepackage{hyperref} \usepackage[english]{babel} \usepackage[scaled=0.9]{helvet} \usepackage[sc]{mathpazo} \usepackage{Sweave} \usepackage{color} \definecolor{link}{rgb}{0.45,0.51,0.67} \hypersetup{ colorlinks,% citecolor=link,% filecolor=link,% linkcolor=link,% urlcolor=link } %% load any required packages here \renewcommand{\today}{\begingroup \number \day\space \ifcase \month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space \number \year \endgroup} \def\rth{r^{th}} \let\code=\texttt \let\proglang=\texttt \let\pkg=\texttt \newcommand{\CRANpkg}[1]{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}}% \newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} \let\cpkg=\CRANpkg \bibliographystyle{plainnat} %\VignetteIndexEntry{Introduction to kedd} %\VignettePackage{kedd} \begin{document} \title{\bfseries Introduction to \pkg{kedd}} \author{by Arsalane Chouaib Guidoum\thanks{Department of Probabilities \& Statistics.\\Faculty of Mathematics. \\University of Science and Technology Houari Boumediene.\\ BP 32 El-Alia, U.S.T.H.B, Algeria.\\ \email{acguidoum@usthb.dz}}} \date{\today} \maketitle \section{Introduction} \pkg{kedd} \citep{kedd} is a package providing additional smoothing techniques to the \texttt{R} statistical system. Although various packages on the Comprehensive \texttt{R} Archive Network (CRAN) provide functions useful to nonparametric statistics, \pkg{kedd} aims to serve as a central location for more specifically of a nonparametric functions and data sets. The project was officially launched in December 2012 and is under active development. The current feature set of the package can be split in four main categories: compute the convolutions and derivatives of a kernel function, compute the kernel estimators for a density of probability and its derivatives, computing the bandwidth selectors with different methods, displaying the kernel estimators and selection functions of the bandwidth. Moreover, the package follows the general \texttt{R} philosophy of working with model objects. This means that instead of merely returning, say, a kernel estimator of $\rth$ derivative of a density, many functions will return an object containing, it's functions are S3 classes (\code{S3method}). The object can then be manipulated at one’s will using various extraction, summary or plotting functions. Whenever possible, we develop a graphical user interface of the various functions of a coherent whole, to facilitate the use of this package. \section{Documentation} It is a requirement of the \texttt{R} packaging system that every function and data set in a package has a help page. The \pkg{kedd} package follows this requirement strictly. In addition to the help pages, the package includes vignettes and demonstration scripts; running <>= vignette(package = "kedd") @ and <>= demo(package = "kedd") @ at the \texttt{R} prompt will give the list of each. \section{Requirements} \texttt{R} version >= 2.15.0. \section{Licence} This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package. \section{Collaboration and citation} Obviously, the package leaves many other fields of nonparametric statistics untouched. For this situation to change, we hope that experts in their field will join their efforts to ours and contribute code to the \pkg{kedd} project. The project will continue to grow and to improve by and for the community of developers and users. If you use \pkg{kedd} for smoothing techniques and computing bandwidth selectors of the $\rth$ derivative of a probability density, please cite the software in publications. Use <>= citation() @ or <>= citation("kedd") @ for information on how to cite the software. \section*{Note} Please send comments, error reports, etc. to the author via the addresses email. \begin{thebibliography}{1} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{{\tt #1}}\fi \bibitem[Guidoum, 2015]{kedd} Guidoum, A. C. (2015). \newblock \pkg{kedd}: Kernel estimator and bandwidth selection for density and its derivatives. \newblock \texttt{R} package version 1.0.3. \newblock \url{http://CRAN.R-project.org/package=kedd} \end{thebibliography} \end{document} kedd/NAMESPACE0000644000176200001440000000317512614740640012412 0ustar liggesusersimportFrom("graphics", "curve", "legend", "lines.default","plot.default") importFrom("stats", "D", "dnorm", "integrate", "optimize", "sd") export(kernel.fun,kernel.fun.default,kernel.conv,kernel.conv.default,plot.kernel.fun,plot.kernel.conv, dkde,dkde.default,print.dkde,plot.dkde,lines.dkde, h.bcv,h.bcv.default,print.h.bcv,plot.h.bcv,lines.h.bcv, h.ucv,h.ucv.default,print.h.ucv,plot.h.ucv,lines.h.ucv, h.ccv,h.ccv.default,print.h.ccv,plot.h.ccv,lines.h.ccv, h.amise,h.amise.default,print.h.amise,plot.h.amise,lines.h.amise, h.mcv,h.mcv.default,print.h.mcv,plot.h.mcv,lines.h.mcv, h.mlcv,h.mlcv.default,print.h.mlcv,plot.h.mlcv,lines.h.mlcv, h.tcv,h.tcv.default,print.h.tcv,plot.h.tcv,lines.h.tcv) S3method(kernel.fun, default) S3method(kernel.conv, default) S3method(dkde, default) S3method(h.bcv, default) S3method(h.ucv, default) S3method(h.ccv, default) S3method(h.amise, default) S3method(h.mcv, default) S3method(h.mlcv, default) S3method(h.tcv, default) S3method(plot, kernel.fun) S3method(plot, kernel.conv) S3method(plot, dkde) S3method(plot, h.bcv) S3method(plot, h.ucv) S3method(plot, h.ccv) S3method(plot, h.amise) S3method(plot, h.mcv) S3method(plot, h.mlcv) S3method(plot, h.tcv) S3method(lines, dkde) S3method(lines, h.bcv) S3method(lines, h.ucv) S3method(lines, h.ccv) S3method(lines, h.amise) S3method(lines, h.mcv) S3method(lines, h.mlcv) S3method(lines, h.tcv) S3method(print, dkde ) S3method(print, h.bcv) S3method(print, h.ucv) S3method(print, h.ccv) S3method(print, h.amise) S3method(print, h.mcv) S3method(print, h.mlcv) S3method(print, h.tcv) kedd/demo/0000755000176200001440000000000012614740636012116 5ustar liggesuserskedd/demo/kedd.R0000644000176200001440000002152512614740636013155 0ustar liggesusers############################################################################ # EXAMPLE 1: # # Simple example for a Gaussian density derivative # ############################################################################ x <- rnorm(100) SD0 <- dkde(x,deriv.order=0) SD1 <- dkde(x,deriv.order=1) SD2 <- dkde(x,deriv.order=2) SD3 <- dkde(x,deriv.order=3) dev.new() par(mfrow=c(2,2)) plot(SD0) plot(SD1) plot(SD2) plot(SD3) ############################################################################ # EXAMPLE 2 # # Trimodal Gaussian density derivative # # Computing bandwidths with UCV methods # ############################################################################ data(trimodal) h.ucv(trimodal,deriv.order=0,kernel="gaussian") h.ucv(trimodal,deriv.order=1,kernel="gaussian") h.ucv(trimodal,deriv.order=2,kernel="gaussian") h.ucv(trimodal,deriv.order=3,kernel="gaussian") ############################################################################ # Example 3 # # Computing bandwidths with BCV methods different kernels # # derivative order = 0 # ############################################################################ data(outlier) h.bcv(outlier,deriv.order=0,kernel="gaussian") h.bcv(outlier,deriv.order=0,kernel="triweight") h.bcv(outlier,deriv.order=0,kernel="tricube") h.bcv(outlier,deriv.order=0,kernel="biweight") h.bcv(outlier,deriv.order=0,kernel="cosine") ############################################################################ # derivative order = 1 # ############################################################################ h.bcv(outlier,deriv.order=1,kernel="gaussian") h.bcv(outlier,deriv.order=1,kernel="triweight") h.bcv(outlier,deriv.order=1,kernel="tricube") h.bcv(outlier,deriv.order=1,kernel="biweight") h.bcv(outlier,deriv.order=1,kernel="cosine") ############################################################################ # derivative order = 2 # ############################################################################ h.bcv(outlier,deriv.order=2,kernel="gaussian") h.bcv(outlier,deriv.order=2,kernel="triweight") h.bcv(outlier,deriv.order=2,kernel="tricube") h.bcv(outlier,deriv.order=2,kernel="biweight") h.bcv(outlier,deriv.order=2,kernel="cosine") ############################################################################ # Example 4 # # Bimodal Gaussian density derivative # ############################################################################ fx <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5) fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) ############################################################################ # derivative order = 0 # ############################################################################ kernels <- eval(formals(dkde.default)$kernel) dev.new() plot(dkde(bimodal,h=0.3),sub=paste("Derivative order = 0",";", "Bandwidth =0.3 "),ylim=c(0,0.5), main = "Bimodal Gaussian Density") for(i in 2:length(kernels)) lines(dkde(bimodal, h = 0.3, kernel = kernels[i]), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)), lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015) ############################################################################ # derivative order = 1 # ############################################################################ kernels <- eval(formals(dkde.default)$kernel)[-3] dev.new() plot(dkde(bimodal,deriv.order=1,h=0.6),main = "Bimodal Gaussian Density Derivative",sub=paste ("Derivative order = 1",";","Bandwidth =0.6"),ylim=c(-0.6,0.6)) for(i in 2:length(kernels)) lines(dkde(bimodal,deriv.order=1, h = 0.6, kernel = kernels[i]), col = i) curve(fx1,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)), lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015) ############################################################################ # Example 5 # # Show the bandwidth selection # # kernel = "gaussian" ; derivative order = 0 # ############################################################################ ############################################################################ # KDE of f (bimodal gaussian density) # ############################################################################ hbcv1 <- h.bcv(x=bimodal,whichbcv = 1,deriv.order = 0)$h hbcv2 <- h.bcv(x=bimodal,whichbcv = 2,deriv.order = 0)$h hucv <- h.ucv(x=bimodal,deriv.order = 0)$h htcv <- h.tcv(x=bimodal,deriv.order = 0)$h hccv <- h.ccv(x=bimodal,deriv.order = 0)$h hmcv <- h.mcv(x=bimodal,deriv.order = 0)$h h0 <- c(hbcv1,hbcv2,hucv,htcv,hccv,hmcv) h0 dev.new() plot(dkde(x=bimodal,deriv.order = 0,h=h0[1]),ylim=c(0,0.5), sub=paste("Kernel Gaussian",";","Derivative order = 0"), main="Bimodal Gaussian density") for(i in 1:length(h0)) lines(dkde(x=bimodal,deriv.order = 0,h=h0[i]), col = i) curve(fx,lty=8,add=TRUE) legend("topright",title="Bandwidth", c("True",expression(h[bcv1]), expression(h[bcv2]),expression(h[ucv]),expression(h[tcv]), expression(h[ccv]),expression(h[mcv])), lty=c(8,rep(1,length(h0))),col= c("black",seq(h0)),inset = .015) ############################################################################ # KDDE of d/dx f (bimodal gaussian density) # ############################################################################ hbcv1 <- h.bcv(x=bimodal,whichbcv = 1,deriv.order = 1,upper=0.5)$h hbcv2 <- h.bcv(x=bimodal,whichbcv = 2,deriv.order = 1,upper=0.5)$h hucv <- h.ucv(x=bimodal,deriv.order = 1)$h htcv <- h.tcv(x=bimodal,deriv.order = 1)$h hccv <- h.ccv(x=bimodal,deriv.order = 1)$h hmcv <- h.mcv(x=bimodal,deriv.order = 1,upper=0.5)$h h1 <- c(hbcv1,hbcv2,hucv,htcv,hccv,hmcv) h1 dev.new() plot(dkde(x=bimodal,deriv.order = 1,h=h1[1]),ylim=c(-0.7,0.7), sub=paste("Kernel Gaussian",";","Derivative order = 1"), main="Bimodal Gaussian density derivative") for(i in 1:length(h1)) lines(dkde(x=bimodal,deriv.order = 1,h=h1[i]), col = i) curve(fx1,lty=8,add=TRUE) legend("topright",title="Bandwidth", c("True",expression(h[bcv1]), expression(h[bcv2]),expression(h[ucv]),expression(h[tcv]), expression(h[ccv]),expression(h[mcv])), lty=c(8,rep(1,length(h1))),col= c("black",seq(h1)),inset = .015) ############################################################################ # Example 6 # # Bimodal Gaussian density derivative # # CCV and MCV plot # # derivative order = 0 # ############################################################################ data(bimodal) dev.new() plot(h.ccv(bimodal),main="CCV vs MCV",ylab="") lines(h.mcv(bimodal),col="red") legend("topright", c("CCV","MCV"),lty=c(1,1),col=c("black","red"), inset = .015) ############################################################################ # derivative order = 1 # ############################################################################ dev.new() plot(h.ccv(bimodal,deriv.order=1),main="CCV vs UCV",ylab="",ylim=c(-0.7,0.3), seq.bws=seq(0.05,1,length=50)) lines(h.ucv(bimodal,deriv.order=1),col="red") legend("topright", c("CCV","UCV"),lty=c(1,1),col=c("black","red"), inset = .015) ############################################################################ # derivative order = 2 # ############################################################################ dev.new() plot(h.ccv(bimodal,deriv.order=2,upper=0.5),seq.bws=seq(0.1,0.6,length=50), main="CCV vs MCV",ylab="") lines(h.ucv(bimodal,deriv.order=2),col="red") legend("topright", c("CCV","UCV"),lty=c(1,1),col=c("black","red"), inset = .015) kedd/demo/00Index0000644000176200001440000000007112614740636013246 0ustar liggesuserskedd More examples on kedd-package kedd/NEWS0000644000176200001440000000057512614740640011673 0ustar liggesusersVersion 1.0.3: 2015-10-30 o Version 1.0.3 released. o Other minor bug fixes: R (>= 3.2.2). Version 1.0.2: 2015-01-28 o Version 1.0.2 released. o Other minor bug fixes: R (>= 3.1.2). 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WW+lRˑ yT8HQ`i/ މ;(1qxjh긑|y[`dMSA4.o|VWP3!ΧJlEړ4i v,epʱǴ8|pq8šiUkfk lӰ86F84f=MsJ kfFMq_]\^#d(iti]J Rp!v߬oJ,N⺺KپDn2~䆽rǢ@ܞQ9n)c3A.-^ ÑGd܋]BYm>BgeYh Df|QɎu5AX70!j׵TK="?IJ?"[rKp VD+NJJN[^|JhAmӕņN🹗Kq`v}by#ʅMÓaN0~t֢JK2Z?yLV}-X%ElkB [$r]r(gs}t-*zHK 'ݪ vX^rtc'6=/ C~cQ.j\bidVm- #ri5݊mQ bq= >*V; l!٬/0#-R5a I]xj;,"9AX_ AֻzN-5&yfC'·laϙwkedd/data/outlier.rda0000644000176200001440000000316212614740636014260 0ustar liggesusers ;ǥYڑ6<[uX'8LaZK͎K%ƌˌ62.1!%ŰXkUyOz8hhuAP#hvnGqm6y:Rb8{zpcTr0fQD匭%o| ƿ<4W\+ckʙ٬ZdiGGW߻CG{{t/ħe5ɖ x˯`JCd|% fY4% DlckWLfz:@Y=@}\A=9!2o?w6^%2y[lNes$Cj߿B`"K6M{ , c<4H`T춺Pa4]xPighdD7S~pN/!k~y-) bpv:z6Cuz-L5Q^ Yvnmn%>w?qB$6W{SFA 2R6O |lf%Xg&).Aڜ':]5_klS"xאy,^E]%&(XYT.\45t{ݖf`MS MkoiYG\aB|C?U(QXz.R&%D1e3uQRMêڽy5tQzl$a|# J~n<Xscj/p"0wf yq%L >U%cs[hO,L[H9Yh9P௠"+ ±v,J.vJsF)ƾCrYXZPx| ^rW黫!^ot'vYq77~vkedd/R/0000755000176200001440000000000012614740640011366 5ustar liggesuserskedd/R/utils.R0000644000176200001440000013342112614740640012655 0ustar liggesusers## Sat Jul 20 02:11:02 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or ## (at your option) any later version. ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## A copy of the GNU General Public License is available at ## http://www.r-project.org/Licenses/ ## Unlimited use and distribution (see LICENCE). ## kedd : Kernel estimator and bandwidth selection for density and its derivatives. ################################################################################################### #### #### r(th) derivative of Kernel functions K^r(x) kernel_fun_der <- function(kernel,u,deriv.order=0) { if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (kernel=="gaussian") {Kr <- expression( dnorm(X) )} else if (kernel=="epanechnikov") {Kr <- expression( (3/4)*(1-X^2)) } else if (kernel=="uniform") {Kr <- expression( 0.5 ) } else if (kernel=="triweight") {Kr <- expression( (35/32)*(1-X^2)^3 )} else if (kernel=="biweight") {Kr <- expression( (15/16)*(1-X^2)^2 )} else if (kernel=="cosine") {Kr <- expression( (pi/4)*cos((pi*X)/2) )} else if (kernel=="triangular") {Kr <- expression(1-abs(X));Kr1 <- expression(1-X); Kr2 <- expression(1+X)} else if (kernel=="tricube") {Kr <- expression((70/81)*(1-abs(X)^3)^3);Kr1 <- expression((70/81)*(1-X^3)^3); Kr2 <- expression((70/81)*(1+X^3)^3)} else if (kernel=="silverman") {r <- r%%8;Kr <- expression(0.5*exp(-abs(X)/sqrt(2))*sin((abs(X)/sqrt(2))+0.25*pi));Kr1 <- expression(0.5*exp(-X/sqrt(2)) * sin((X/sqrt(2)) + 0.25*pi)); Kr2 <- expression(0.5*exp(X/sqrt(2)) * sin((-X/sqrt(2)) + 0.25*pi))} if (kernel=="epanechnikov" && r >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order >= 3' ") if (kernel=="uniform" && r >= 1) stop(" 'uniform kernel derivative = 0' for 'order >= 1' ") if (kernel=="triweight" && r >= 7) stop(" 'triweight kernel derivative = 0' for 'order >= 7' ") if (kernel=="biweight" && r >= 5) stop(" 'biweight kernel derivative = 0' for 'order >= 5' ") if (kernel=="triangular" && r >= 2) stop(" 'triangular kernel derivative = 0' for 'order >= 2' ") if (kernel=="tricube" && r >= 10) stop(" 'tricube kernel derivative = 0' for 'order >= 10' ") if (r == 0) {DKr <- Kr if (kernel=="gaussian" || kernel =="silverman"){ K <- function(X) eval(DKr);fx <- K(u)}else{ K <- function(X) eval(DKr)* (X >= -1 & X <= 1);fx <- K(u)} } else { if (kernel=="gaussian"){ if (r == 1){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)/sqrt(pi)} else if (r == 2){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^2-1)/sqrt(pi) } else if (r == 3){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^2-3)/sqrt(pi)} else if (r == 4){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^4-6*u^2+3)/sqrt(pi)} else if (r == 5){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^4-10*u^2+15)/sqrt(pi)} else if (r == 6){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^6-15*u^4+45*u^2-15)/sqrt(pi)} else if (r == 7){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^6-21*u^4+105*u^2-105)/sqrt(pi)} else if (r == 8){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^8-28*u^6+210*u^4-420*u^2+105)/sqrt(pi)} else if (r == 9){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^8-36*u^6+378*u^4-1260*u^2+945)/sqrt(pi)} else if (r == 10){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^10-45*u^8+630*u^6-3150*u^4+4725*u^2-945)/sqrt(pi)} else if (r == 11){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^10-55*u^8+990*u^6-6930*u^4+17325*u^2-10395)/sqrt(pi)} else if (r == 12){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^12-66*u^10+1485*u^8-13860*u^6+51975*u^4-62370*u^2+10395)/sqrt(pi)} else if (r == 13){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^12-78*u^10+2145*u^8-25740*u^6+135135*u^4-270270*u^2+135135)/sqrt(pi)} else if (r == 14){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^14-91*u^12+3003*u^10-45045*u^8+315315*u^6-945945*u^4+945945*u^2-135135)/sqrt(pi)} else if (r == 15){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^14-105*u^12+4095*u^10-75075*u^8+675675*u^6-2837835*u^4+4729725*u^2-2027025)/sqrt(pi)} else if (r == 16){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^16-120*u^14+5460*u^12-120120*u^10+1351350*u^8-7567560*u^6+18918900*u^4-16216200*u^2+2027025)/sqrt(pi)} else if (r == 17){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^16-136*u^14+7140*u^12-185640*u^10+2552550*u^8-18378360*u^6+64324260*u^4-91891800*u^2+34459425)/sqrt(pi)} else if (r == 18){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^18-153*u^16+9180*u^14-278460*u^12+4594590*u^10-41351310*u^8+192972780*u^6-413513100*u^4+310134825*u^2-34459425)/sqrt(pi)} else if (r == 19){fx <- -(1/2)*u*exp(-(1/2)*u^2)*sqrt(2)*(u^18-171*u^16+11628*u^14-406980*u^12+7936110*u^10-87297210*u^8+523783260*u^6-1571349780*u^4+1964187225*u^2-654729075)/sqrt(pi)} else if (r == 20){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^20-190*u^18+14535*u^16-581400*u^14+13226850*u^12-174594420*u^10+1309458150*u^8-5237832600*u^6+9820936125*u^4-6547290750*u^2+654729075)/sqrt(pi)} else if (r == 21){fx <- -(1/2)*exp(-(1/2)*u^2)*sqrt(2)*u*(u^20-210*u^18+17955*u^16-813960*u^14+21366450*u^12-333316620*u^10+3055402350*u^8-15713497800*u^6+41247931725*u^4-45831035250*u^2+13749310575)/sqrt(pi)} else if (r == 22){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^22-231*u^20+21945*u^18-1119195*u^16+33575850*u^14-611080470*u^12+6721885170*u^10-43212118950*u^8+151242416325*u^6-252070693875*u^4+151242416325*u^2-13749310575)/sqrt(pi)} else if (r == 23){fx <- -(1/2)*exp(-(1/2)*u^2)*sqrt(2)*u*(u^22-253*u^20+26565*u^18-1514205*u^16+51482970*u^14-1081142370*u^12+14054850810*u^10-110430970650*u^8+496939367925*u^6-1159525191825*u^4+1159525191825*u^2-316234143225)/sqrt(pi)} else if (r == 24){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^24-276*u^22+31878*u^20-2018940*u^18+77224455*u^16-1853386920*u^14+28109701620*u^12-265034329560*u^10+1490818103775*u^8-4638100767300*u^6+6957151150950*u^4-3794809718700*u^2+316234143225)/sqrt(pi)} else if (r == 25){fx <- -(1/2)*exp(-(1/2)*u^2)*sqrt(2)*u*(u^24-300*u^22+37950*u^20-2656500*u^18+113565375*u^16-3088978200*u^14+54057118500*u^12-602350749000*u^10+4141161399375*u^8-16564645597500*u^6+34785755754750*u^4-31623414322500*u^2+7905853580625)/sqrt(pi)} else if (r == 26){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^26-325*u^24+44850*u^22-3453450*u^20+164038875*u^18-5019589575*u^16+100391791500*u^14-1305093289500*u^12+10767019638375*u^10-53835098191875*u^8+150738274937250*u^6-205552193096250*u^4+102776096548125*u^2-7905853580625)/sqrt(pi)} else if (r == 27){fx <- -(1/2)*exp(-(1/2)*u^2)*sqrt(2)*u*(u^26-351*u^24+52650*u^22-4440150*u^20+233107875*u^18-7972289325*u^16+180705224700*u^14-2710578370500*u^12+26428139112375*u^10-161505294575625*u^8+581419060472250*u^6-1109981842719750*u^4+924984868933125*u^2-213458046676875)/sqrt(pi)} else if (r == 28){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^28-378*u^26+61425*u^24-5651100*u^22+326351025*u^20-12401338950*u^18+316234143225*u^16-5421156741000*u^14+61665657928875*u^12-452214824811750*u^10+2034966711652875*u^8-5179915266025500*u^6+6474894082531875*u^4-2988412653476250*u^2+213458046676875)/sqrt(pi)} else if (r == 29){fx <- -(1/2)*exp(-(1/2)*u^2)*sqrt(2)*u*(u^28-406*u^26+71253*u^24-7125300*u^22+450675225*u^20-18928359450*u^18+539458244325*u^16-10480903032600*u^14+137561852302875*u^12-1192202719958250*u^10+6557114959770375*u^8-21459648959248500*u^6+37554385678684875*u^4-28887988983603750*u^2+6190283353629375)/sqrt(pi)} else if (r == 30){fx <- (1/2)*exp(-(1/2)*u^2)*sqrt(2)*(u^30-435*u^28+82215*u^26-8906625*u^24+614557125*u^22-28392539175*u^20+899097073875*u^18-19651693186125*u^16+294775397791875*u^14-2980506799895625*u^12+19671344879311125*u^10-80473683597181875*u^8+187771928393424375*u^6-216659917377028125*u^4+92854250304440625*u^2-6190283353629375)/sqrt(pi)} else {K <- function(X) (-1)^r * Hermite(X,r) * eval(Kr); fx <- K(u)}} else if (kernel=="cosine"){ if (r%%2==0){ if ((r%/%2)%%2==0){ K <- function(X)( ((-1)^r/2^(r+2))*pi^(r+1)*cos(0.5*pi*X))* (X >= -1 & X <= 1)}else{ K <- function(X)(-((-1)^r/2^(r+2))*pi^(r+1)*cos(0.5*pi*X))* (X >= -1 & X <= 1)}}else{ if ((r%/%2)%%2==0){ K <- function(X)( ((-1)^r/2^(r+2))*pi^(r+1)*sin(0.5*pi*X))* (X >= -1 & X <= 1)}else{ K <- function(X)(-((-1)^r/2^(r+2))*pi^(r+1)*sin(0.5*pi*X))* (X >= -1 & X <= 1)}} fx <- K(u)} else if (kernel=="tricube"){ if (r == 1){fx <- (u < 0 & u >= -1) *((70/9)*(u^3+1)^2*u^2)+ (u >= 0 & u <= 1)*(-(70/9)*(u^3-1)^2*u^2)} else if (r == 2){fx <- (u < 0 & u >= -1) *((560/9)*u^7+(700/9)*u^4+(140/9)*u)+ (u >= 0 & u <= 1)*(-(560/9)*u^7+(700/9)*u^4-(140/9)*u)} else if (r == 3){fx <- (u < 0 & u >= -1) *((3920/9)*u^6+(2800/9)*u^3+(140/9))+ (u >= 0 & u <= 1)*(-(3920/9)*u^6+(2800/9)*u^3-(140/9))} else if (r == 4){fx <- (u < 0 & u >= -1) *((7840/3)*u^5+(2800/3)*u^2)+ (u >= 0 & u <= 1)*(-(7840/3)*u^5+(2800/3)*u^2)} else if (r == 5){fx <- (u < 0 & u >= -1) *((39200/3)*u^4+(5600/3)*u)+ (u >= 0 & u <= 1)*(-(39200/3)*u^4+(5600/3)*u)} else if (r == 6){fx <- (u < 0 & u >= -1) *((156800/3)*u^3+(5600/3))+ (u >= 0 & u <= 1)*(-(156800/3)*u^3+(5600/3))} else if (r == 7){fx <- (u <0 & u >= -1) *(156800*u^2)+ (u >= 0 & u <= 1)*(-156800*u^2)} else if (r == 8){fx <- (u < 0 & u >= -1) *(313600*u)+ (u >= 0 & u <= 1)*( -313600*u)} else if (r == 9){fx <- (u < 0 & u >= -1) *(313600)+ (u >= 0 & u <= 1)*( -313600)}} else if (kernel=="triweight"){ if (r == 1){fx <- (-(105/16)*(u^2-1)^2*u)*(u >= -1 & u <= 1)} else if (r == 2){fx <- (-(525/16)*u^4+(315/8)*u^2-(105/16))*(u >= -1 & u <= 1)} else if (r == 3){fx <- (-(525/4)*u^3+(315/4)*u)*(u >= -1 & u <= 1)} else if (r == 4){fx <- (-(1575/4)*u^2+(315/4))*(u >= -1 & u <= 1)} else if (r == 5){fx <- (-(1575/2)*u)*(u >= -1 & u <= 1)} else if (r == 6){fx <- (-1575/2)*(u >= -1 & u <= 1)}} else if (kernel=="biweight"){ if (r == 1){fx <- (((15/4)*(u^2-1))*u)*(u >= -1 & u <= 1)} else if (r == 2){fx <- ((45/4)*u^2-(15/4))*(u >= -1 & u <= 1)} else if (r == 3){fx <- ((45/2)*u)*(u >= -1 & u <= 1)} else if (r == 4){fx <- (45/2)*(u >= -1 & u <= 1)}} else if (kernel =="triangular"){ if (r == 1){fx <- (u <= 0 & u >= -1) - (u >= 0 & u <= 1)}} else if (kernel=="epanechnikov"){ if (r == 1){fx <- (-(3/2)*u)*(u >= -1 & u <= 1)} else if (r == 2){fx <- (-3/2)*(u >= -1 & u <= 1)}} else if (kernel=="silverman"){ if (r == 1){fx <- (u < 0)*((1/4)*sqrt(2)*exp((1/2)*u*sqrt(2))*(sin(-(1/2)*u*sqrt(2)+(1/4)*pi)-cos(-(1/2)*u*sqrt(2)+(1/4)*pi)))+ (u >= 0)* (-(1/4)*sqrt(2)*exp(-(1/2)*u*sqrt(2))*(sin((1/2)*u*sqrt(2)+(1/4)*pi)-cos((1/2)*u*sqrt(2)+(1/4)*pi)))} else if (r == 2){fx <- (u < 0)*(-(1/2)*exp((1/2)*u*sqrt(2))*cos(-(1/2)*u*sqrt(2)+(1/4)*pi))+ (u >= 0)* (-(1/2)*exp(-(1/2)*u*sqrt(2))*cos((1/2)*u*sqrt(2)+(1/4)*pi))} else if (r == 3){fx <- (u < 0)*(-(1/4)*exp((1/2)*u*sqrt(2))*sqrt(2)*(cos(-(1/2)*u*sqrt(2)+(1/4)*pi)+sin(-(1/2)*u*sqrt(2)+(1/4)*pi)))+ (u >= 0)* ((1/4)*exp(-(1/2)*u*sqrt(2))*sqrt(2)*(cos((1/2)*u*sqrt(2)+(1/4)*pi)+sin((1/2)*u*sqrt(2)+(1/4)*pi)))} else if (r == 4){fx <- (u < 0)*(-(1/2)*exp((1/2)*u*sqrt(2))*sin(-(1/2)*u*sqrt(2)+(1/4)*pi))+ (u >= 0)* (-(1/2)*exp(-(1/2)*u*sqrt(2))*sin((1/2)*u*sqrt(2)+(1/4)*pi))} else if (r == 5){fx <- (u < 0)*(-(1/4)*sqrt(2)*exp((1/2)*u*sqrt(2))*(sin(-(1/2)*u*sqrt(2)+(1/4)*pi)-cos(-(1/2)*u*sqrt(2)+(1/4)*pi)))+ (u >= 0)* ((1/4)*sqrt(2)*exp(-(1/2)*u*sqrt(2))*(sin((1/2)*u*sqrt(2)+(1/4)*pi)-cos((1/2)*u*sqrt(2)+(1/4)*pi)))} else if (r == 6){fx <- (u < 0)*((1/2)*exp((1/2)*u*sqrt(2))*cos(-(1/2)*u*sqrt(2)+(1/4)*pi))+ (u >= 0)* ((1/2)*exp(-(1/2)*u*sqrt(2))*cos((1/2)*u*sqrt(2)+(1/4)*pi))} else if (r == 7){fx <- (u < 0)*((1/4)*exp((1/2)*u*sqrt(2))*sqrt(2)*(cos(-(1/2)*u*sqrt(2)+(1/4)*pi)+sin(-(1/2)*u*sqrt(2)+(1/4)*pi)))+ (u >= 0)* (-(1/4)*exp(-(1/2)*u*sqrt(2))*sqrt(2)*(cos((1/2)*u*sqrt(2)+(1/4)*pi)+sin((1/2)*u*sqrt(2)+(1/4)*pi)))}} } return(fx) } #### #### Kernels density convolution kernel_fun_conv <- function(kernel,u,deriv.order=0) { if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (kernel=="epanechnikov" && r >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order >= 3' ") if (kernel=="uniform" && r >= 1) stop(" 'uniform kernel derivative = 0' for 'order >= 1' ") if (kernel=="triweight" && r >= 7) stop(" 'triweight kernel derivative = 0' for 'order >= 7' ") if (kernel=="biweight" && r >= 5) stop(" 'biweight kernel derivative = 0' for 'order >= 5' ") if (kernel=="triangular" && r >= 2) stop(" 'triangular kernel derivative = 0' for 'order >= 2' ") if (kernel=="tricube" && r >= 10) stop(" 'tricube kernel derivative = 0' for 'order >= 10' ") if (kernel=="uniform"){ if (r==0){fx <- (0.5-0.25*abs(u))*(abs(u)<=2)} } else if(kernel=="epanechnikov"){ if (r==0){fx <- ((3/5)-(3/160)*abs(u)^5+(3/8)*abs(u)^3-(3/4)*u^2)*(abs(u)<=2)} else if (r==1){fx <- (-(3/2)-(3/8)*abs(u)^3+(9/4)*abs(u))*(abs(u)<=2)} else if (r==2){fx <- ((9/2)-(9/4)*abs(u))*(abs(u)<=2)} } else if (kernel=="triweight"){ if (r==0) {fx <- (-(35/22)*u^2+(350/429)+(35/11264)*abs(u)^11-(35/768)*abs(u)^9+ (35/64)*abs(u)^7-(35/32)*u^6+(35/24)*u^4-(175/1757184)*abs(u)^13)* (abs(u)<=2)} else if (r==1) {fx <- (-(35/11)+(735/32)*abs(u)^5+(35/2)*u^2-(175/11264)*abs(u)^11+ (175/512)*abs(u)^9-(105/32)*abs(u)^7-(525/16)*u^4)*(abs(u)<=2)} else if (r==2) {fx <- (35-(2205/16)*abs(u)^5+(3675/8)*abs(u)^3-(1575/4)*u^2-(875/512)* abs(u)^9+(1575/64)*abs(u)^7)*(abs(u)<=2)} else if (r==3) {fx <- ((11025/4)*abs(u)+(33075/32)*abs(u)^5-(11025/4)*abs(u)^3-(7875/64)* abs(u)^7-(1575/2))*(abs(u)<=2)} else if (r==4) {fx <- (33075-(165375/32)*abs(u)^5+(165375/8)*abs(u)^3+(165375/4)*u^2- 99225*abs(u))*(abs(u)<=2)} else if (r==5) {fx <- (-(826875/2)-(826875/8)*abs(u)^3+(2480625/4)*abs(u))*(abs(u)<=2)} else if (r==6) {fx <- ((2480625/2)-(2480625/4)*abs(u))*(abs(u)<=2)} } else if (kernel=="biweight"){ if (r==0) {fx <- (-(15/32)*abs(u)^5-(15/14)*u^2-(5/3584)*abs(u)^9+(15/448)*abs(u)^7+ (15/16)*u^4+(5/7))*(abs(u)<=2)} else if (r==1) {fx <- ((45/32)*abs(u)^5-(75/8)*abs(u)^3+(45/4)*u^2-(45/448)*abs(u)^7- (15/7)) *(abs(u)<=2)} else if (r==2) {fx <- ((45/2)-(135/32)*abs(u)^5+(225/8)*abs(u)^3-(225/4)*abs(u)) * (abs(u)<=2)} else if (r==3) {fx <- (-(675/2)-(675/8)*abs(u)^3+(2025/4)*abs(u)) *(abs(u)<=2)} else if (r==4) {fx <- ((2025/2)-(2025/4)*abs(u)) *(abs(u)<=2)} } else if (kernel=="cosine"){ fx <- ((-1)^(r+1) * (pi^(2*r+2)/2^(2*r+5)) * cos(0.5*pi*abs(u))*abs(u) + (-1)^r * (pi^(2*r+2)/ 2^(2*r+4))* cos(0.5*pi*abs(u))+(pi^(2*r+1)/2^(2*r+4))*sin(0.5*pi*abs(u)))*(abs(u)<=2) } else if (kernel=="triangular"){ if (r==0) {fx <- ((1/6)*(u+2)^3 *(u <=- 1 & u > -2 ) + (-(1/2)*u^3-u^2+(2/3)) * (u <= 0 & u > -1 )+((2/3)+(1/2)*u^3-u^2)*(u <= 1 & u > 0 )-(1/6)*(u-2)^3 * (u <=2 & u >1) )} else if (r==1) {fx <- (u+2)*(u <=-1 & u >= -2 )+(-3*u-2)*(u <=0 & u > -1 )+ (3*u-2)*(u <= 1 & u > 0 )+(2-u)*(u <= 2 & u > 1)} } else if (kernel=="tricube"){ if (r==0) {fx <- (u <= -1 & u > -2)*((2870/351)*u^6+(21560/2187)*u^4+(7840/1683)*u^2+ (22400/20007)+(70/81)*u^9+(15680/6561)*u+(11305/2187)*u^7+(980/99)*u^5+ (2800/351)*u^3+(35/625482)*u^16+(1085/6561)*u^10+(245/303046029)*u^19+ (665/312741)*u^13+(245/99)*u^8)+ (u <= 0 & u > -1)*(-(350/117)*u^6+(980/729)*u^4- (210/187)*u^2+(175/247)-(70/81)*u^9-(2905/729)*u^7-(35/625482)*u^16- (1085/6561)*u^10-(245/101015343)*u^19-(1295/312741)*u^13-(245/99)*u^8)+ (u > 0 & u <= 1)*(-(350/117)*u^6+(980/729)*u^4-(210/187)*u^2+(175/247)+ (70/81)*u^9+(2905/729)*u^7-(35/625482)*u^16-(1085/6561)*u^10+(245/101015343)* u^19+(1295/312741)*u^13-(245/99)*u^8)+(u > 1 & u < 2)*((2870/351)*u^6+(21560/2187)* u^4+(7840/1683)*u^2+(22400/20007)-(15680/6561)*u-(70/81)*u^9+(245/99)*u^8- (11305/2187)*u^7-(980/99)*u^5-(2800/351)*u^3-(245/303046029)*u^19- (665/312741)*u^13+(35/625482)*u^16+(1085/6561)*u^10)} else if (r==1) {fx <- (u <= -1 & u > -2)*((86240/729)*u^2+(28700/117)*u^4+(15680/1683)+ (560/9)*u^7+(10850/729)*u^8+(1400/104247)*u^14+(13720/99)*u^6+(490/1772199)* u^17+(2660/8019)*u^11+(158270/729)*u^5+(19600/99)*u^3+(5600/117)*u)+ (u <= 0 & u > -1)* ((3920/243)*u^2-(3500/39)*u^4-(420/187)-(560/9)*u^7- (10850/729)*u^8-(1400/104247)*u^14-(13720/99)*u^6-(490/590733)*u^17- (5180/8019)*u^11-(40670/243)*u^5)+(u > 0 & u <= 1)*((3920/243)*u^2- (3500/39)*u^4-(420/187)+(560/9)*u^7-(10850/729)*u^8-(1400/104247)*u^14- (13720/99)*u^6+(490/590733)*u^17+(5180/8019)*u^11+(40670/243)*u^5)+ (u > 1 & u < 2)*((86240/729)*u^2+(28700/117)*u^4+(15680/1683)-(560/9)*u^7+ (10850/729)*u^8-(158270/729)*u^5+(13720/99)*u^6-(19600/99)*u^3-(5600/117)*u+ (1400/104247)*u^14-(2660/8019)*u^11-(490/1772199)*u^17)} else if (r==2) {fx <- (u <= -1 & u > -2)*((114800/39)*u^2+(172480/729)+(3165400/729)*u^3+ (607600/729)*u^6+(26600/729)*u^9+(7840/104247)*u^15+(7840/3)*u^5+ (137200/33)*u^4+(39200/33)*u+(19600/8019)*u^12)+ (u <= 0 & u > -1)* (-(14000/13)*u^2+(7840/243)-(813400/243)*u^3-(607600/729)*u^6-(51800/729)* u^9-(7840/34749)*u^15-(7840/3)*u^5-(137200/33)*u^4-(19600/8019)*u^12)+ (u > 0 & u <= 1)*(-(14000/13)*u^2+(7840/243)+(813400/243)*u^3+(51800/729)* u^9-(607600/729)*u^6+(7840/34749)*u^15+(7840/3)*u^5-(137200/33)*u^4- (19600/8019)*u^12)+ (u > 1 & u < 2)*((114800/39)*u^2+(172480/729)- (3165400/729)*u^3+(19600/8019)*u^12-(26600/729)*u^9+(607600/729)*u^6- (7840/3)*u^5-(39200/33)*u+(137200/33)*u^4-(7840/104247)*u^15)} else if (r==3) {fx <- (u <= -1 & u > -2)*( (229600/39)+(212800/81)*u^7+(78400/243)*u^10+ (548800/34749)*u^13+(6076000/243)*u^4+(6330800/243)*u+(156800/3)*u^3+ (548800/11)*u^2)+ (u <= 0 & u > -1)* ( -(28000/13)-(414400/81)*u^7- (548800/11583)*u^13-(6076000/243)*u^4-(1626800/81)*u-(78400/243)*u^10- (156800/3)*u^3-(548800/11)*u^2) +(u > 0 & u <= 1)*( (-28000/13)+ (414400/81)*u^7+(548800/11583)*u^13+(1626800/81)*u-(6076000/243)*u^4- (78400/243)*u^10+(156800/3)*u^3-(548800/11)*u^2)+ (u > 1 & u < 2)* ((229600/39)+(78400/243)*u^10-(6330800/243)*u+(6076000/243)*u^4- (548800/34749)*u^13-(212800/81)*u^7-(156800/3)*u^3+(548800/11)*u^2)} else if (r==4) {fx <- (u <= -1 & u > -2)*( -(589568000/81)*u^2-(29478400/33)-(16777600/27)* u^5-(11603200/3)*u+(2195200/891)*u^11+(784000/27)*u^8-(21952000/3)*u^3- (10976000/3)*u^4)+(u <= 0 & u > -1)* ( (551936000/81)*u^2+(4076800/11)+ (2038400/3)*u^5+2822400*u-(2195200/297)*u^11-(784000/27)*u^8+(21952000/3)* u^3+(10976000/3)*u^4)+(u > 0 & u <= 1)*((551936000/81)*u^2+(4076800/11)- (2038400/3)*u^5-2822400*u-(784000/27)*u^8+(2195200/297)*u^11-(21952000/3)* u^3+(10976000/3)*u^4)+ (u > 1 & u < 2)*(-(589568000/81)*u^2-(29478400/33)+ (16777600/27)*u^5+(11603200/3)*u+(784000/27)*u^8-(2195200/891)*u^11- (10976000/3)*u^4+(21952000/3)*u^3)} else if (r==5) {fx <- (u <= -1 & u > -2)*((2885120000/81)+(4406080000/27)*u^3+219520000*u^2+ (43904000/27)*u^6+43904000*u^4+137984000*u+(21952000/81)*u^9)+ (u <= 0 & u > -1)* ( -(1944320000/81)-(486080000/3)*u^3-219520000*u^2- (43904000/27)*u^6-43904000*u^4-125440000*u-(21952000/27)*u^9)+ (u > 0 & u <= 1)*( -(1944320000/81)+(486080000/3)*u^3-219520000*u^2- (43904000/27)*u^6-43904000*u^4+125440000*u+(21952000/27)*u^9)+ (u > 1 & u < 2)*( (2885120000/81)+43904000*u^4+219520000*u^2-137984000*u- (4406080000/27)*u^3-(21952000/81)*u^9+(43904000/27)*u^6)} else if (r==6) {fx <- (u <= -1 & u > -2)*( (-2320640000/3)-(3512320000/3)*u^3-(22798720000/9)*u+ (175616000/9)*u^7-2985472000*u^2+(439040000/9)*u^4)+ (u <= 0 & u > -1)* ( 689920000+(3512320000/3)*u^3+2540160000*u-(175616000/3)*u^7+2985472000* u^2-(439040000/9)*u^4)+ (u > 0 & u <= 1)*( 689920000-(3512320000/3)*u^3- 2540160000*u+(175616000/3)*u^7+2985472000*u^2-(439040000/9)*u^4)+ (u > 1 & u < 2)*((-2320640000/3)+(3512320000/3)*u^3+(22798720000/9)*u- (175616000/9)*u^7-2985472000*u^2+(439040000/9)*u^4)} else if (r==7) {fx <- (u <= -1 & u > -2)*( 9834496000+24586240000*u+(2458624000/3)*u^5+ (49172480000/3)*u^2)+(u <= 0 & u > -1)* ( -9834496000-24586240000*u- 2458624000*u^5-(49172480000/3)*u^2)+(u > 0 & u <= 1)*( -9834496000+ 24586240000*u+2458624000*u^5-(49172480000/3)*u^2)+ (u > 1 & u < 2)* (9834496000-24586240000*u-(2458624000/3)*u^5+(49172480000/3)*u^2)} else if (r==8) {fx <- (u <= -1 & u > -2)*(-98344960000*u+(49172480000/3)*u^3-(196689920000/3))+ (u <= 0 & u > -1)*(98344960000*u-49172480000*u^3+(196689920000/3))+ (u > 0 & u <= 1)* (49172480000*u^3-98344960000*u+(196689920000/3))+ (u > 1 & u < 2)*( 98344960000*u-(49172480000/3)*u^3-(196689920000/3));return(fx)} else if (r==9) {fx <- (u <= -1 & u > -2)*( 98344960000*u+196689920000)+ (u <= 0 & u > -1)* (-295034880000*u-196689920000)+(u > 0 & u <= 1)* ( 295034880000*u- 196689920000)+ (u > 1 & u < 2)*( -98344960000*u+196689920000)} } else if (kernel=="silverman"){ r <- r%%4 if (r==0) {fx <- (u < 0)*((1/16)*exp((1/2)*u*sqrt(2))*(3*sqrt(2)*cos((1/2)*u*sqrt(2))- 3*sqrt(2)*sin((1/2)*u*sqrt(2))+2*sin((1/2)*u*sqrt(2))*u))+ (u >= 0)*((1/16)* exp(-(1/2)*u*sqrt(2))*(3*sqrt(2)*cos((1/2)*u*sqrt(2))+3*sqrt(2)*sin((1/2)*u* sqrt(2))+2*sin((1/2)*u*sqrt(2))*u))} else if (r==1) {fx <- (u < 0)*(-(1/16)*exp((1/2)*u*sqrt(2))*(sqrt(2)*cos((1/2)*u*sqrt(2))+ sqrt(2)*sin((1/2)*u*sqrt(2))-2*cos((1/2)*u*sqrt(2))*u))+ (u >= 0)* (-(1/16)*exp(-(1/2)*u*sqrt(2))*(sqrt(2)*cos((1/2)*u*sqrt(2))+2*cos((1/2)* u*sqrt(2))*u-sqrt(2)*sin((1/2)*u*sqrt(2))))} else if (r==2) {fx <- (u < 0)*((1/16)*exp((1/2)*u*sqrt(2))*(sqrt(2)*cos((1/2)*u*sqrt(2))- sqrt(2)*sin((1/2)*u*sqrt(2))-2*sin((1/2)*u*sqrt(2))*u))+ (u >= 0)* ((1/16)*exp(-(1/2)*u*sqrt(2))*(sqrt(2)*cos((1/2)*u*sqrt(2))+sqrt(2)* sin((1/2)*u*sqrt(2))-2*sin((1/2)*u*sqrt(2))*u))} else if (r==3) {fx <- (u < 0)*(-(1/16)*exp((1/2)*u*sqrt(2))*(3*sqrt(2)*cos((1/2)*u*sqrt(2))+ 3*sqrt(2)*sin((1/2)*u*sqrt(2))+2*cos((1/2)*u*sqrt(2))*u))+ (u >= 0)* (-(1/16)*exp(-(1/2)*u*sqrt(2))*(3*sqrt(2)*cos((1/2)*u*sqrt(2))-3*sqrt(2)* sin((1/2)*u*sqrt(2))-2*cos((1/2)*u*sqrt(2))*u))} } else if (kernel=="gaussian"){ if (r==0) {fx <- dnorm(u,mean=0,sd=sqrt(2))} else if (r==1) {fx <- (1/8)*exp(-(1/4)*u^2)*(u^2-2)/sqrt(pi)} else if (r==2) {fx <- (1/32)*exp(-(1/4)*u^2)*(12-12*u^2+u^4)/sqrt(pi)} else if (r==3) {fx <- (1/128)*exp(-(1/4)*u^2)*(u^6-30*u^4+180*u^2-120)/sqrt(pi)} else if (r==4) {fx <- (1/512)*exp(-(1/4)*u^2)*(u^8-56*u^6+840*u^4-3360*u^2+1680)/sqrt(pi)} else if (r==5) {fx <- (1/2048)*exp(-(1/4)*u^2)*(u^10-90*u^8+2520*u^6-25200*u^4+75600*u^2-30240)/sqrt(pi)} else if (r==6) {fx <- (1/8192)*exp(-(1/4)*u^2)*(u^12-132*u^10+5940*u^8-110880*u^6+831600*u^4-1995840*u^2+665280)/sqrt(pi)} else if (r==7) {fx <- (1/32768)*exp(-(1/4)*u^2)*(u^14-182*u^12+12012*u^10-360360*u^8+5045040*u^6-30270240*u^4+60540480*u^2-17297280)/sqrt(pi)} else if (r==8) {fx <- (1/131072)*exp(-(1/4)*u^2)*(u^16-240*u^14+21840*u^12-960960*u^10+21621600*u^8-242161920*u^6+1210809600*u^4-2075673600*u^2+518918400)/sqrt(pi)} else if (r==9) {fx <- (1/524288)*exp(-(1/4)*u^2)*(u^18-306*u^16+36720*u^14-2227680*u^12+73513440*u^10-1323241920*u^8+12350257920*u^6-52929676800*u^4+79394515200*u^2-17643225600)/sqrt(pi)} else if (r==10) {fx <- (1/2097152)*exp(-(1/4)*u^2)*(u^20-380*u^18+58140*u^16-4651200*u^14+211629600*u^12-5587021440*u^10+83805321600*u^8-670442572800*u^6+2514159648000*u^4-3352212864000*u^2+670442572800)/sqrt(pi)} else if (r==11) {fx <- (1/8388608)*exp(-(1/4)*u^2)*(u^22-462*u^20+87780*u^18-8953560*u^16+537213600*u^14-19554575040*u^12+430200650880*u^10-5531151225600*u^8+38718058579200*u^6-129060195264000*u^4+154872234316800*u^2-28158588057600)/sqrt(pi)} else if (r==12) {fx <- (1/33554432)*exp(-(1/4)*u^2)*(u^24-552*u^22+127512*u^20-16151520*u^18+1235591280*u^16-59308381440*u^14+1799020903680*u^12-33924394183680*u^10+381649434566400*u^8-2374707592857600*u^6+7124122778572800*u^4-7771770303897600*u^2+1295295050649600)/sqrt(pi)} else if (r==13) {fx <- (1/134217728)*exp(-(1/4)*u^2)*(u^26-650*u^24+179400*u^22-27627600*u^20+2624622000*u^18-160626866400*u^16+6425074656000*u^14-167051941056000*u^12+2756357027424000*u^10-27563570274240000*u^8+154355993535744000*u^6-420970891461120000*u^4+420970891461120000*u^2-64764752532480000)/sqrt(pi)} else if (r==14) {fx <- (1/536870912)*exp(-(1/4)*u^2)*(u^28-756*u^26+245700*u^24-45208800*u^22+5221616400*u^20-396842846400*u^18+20238985166400*u^16-693908062848000*u^14+15786408429792000*u^12-231533990303616000*u^10+2083805912732544000*u^8-10608466464820224000*u^6+26521166162050560000*u^4-24481076457277440000*u^2+3497296636753920000)/sqrt(pi)} else if (r==15) {fx <- (1/2147483648)*exp(-(1/4)*u^2)*(u^30-870*u^28+328860*u^26-71253000*u^24+9832914000*u^22-908561253600*u^20+57542212728000*u^18-2515416727824000*u^16+75462501834720000*u^14-1526019481546560000*u^12+20143457156414592000*u^10-164810104007028480000*u^8+769113818699466240000*u^6-1774878043152614400000*u^4+1521324036987955200000*u^2-202843204931727360000)/sqrt(pi)} else if (r==16) {fx <- (1/8589934592)*exp(-(1/4)*u^2)*(u^32-992*u^30+431520*u^28-108743040*u^26+17670744000*u^24-1950850137600*u^22+150215460595200*u^20-8154553575168000*u^18+311911674250176000*u^16-8317644646671360000*u^14+151381132569418752000*u^12-1816573590833025024000*u^10+13624301931247687680000*u^8-58689300626913116160000*u^6+125762787057670963200000*u^4-100610229646136770560000*u^2+12576278705767096320000)/sqrt(pi)} else if (r==17) {fx <- (1/34359738368)*exp(-(1/4)*u^2)*(u^34-1122*u^32+556512*u^30-161388480*u^28+30502422720*u^26-3965314953600*u^24+364808975731200*u^22-24077392398259200*u^20+1143676138917312000*u^18-38884988723188608000*u^16+933239729356526592000*u^14-15440875522080712704000*u^12+169849630742887839744000*u^10-1175882058989223505920000*u^8+4703528235956894023680000*u^6-9407056471913788047360000*u^4+7055292353935341035520000*u^2-830034394580628357120000)/sqrt(pi)} else if (r==18) {fx <- (1/137438953472)*exp(-(1/4)*u^2)*(u^36-1260*u^34+706860*u^32-233735040*u^30+50837371200*u^28-7686610525440*u^26+832716140256000*u^24-65665615631616000*u^22+3792189302725824000*u^20-160114659448423680000*u^18+4899508579121764608000*u^16-106898368999020318720000*u^14+1621291929818474833920000*u^12-16462348825849129082880000*u^10+105829385309030115532800000*u^8-395096371820379097989120000*u^6+740805697163210808729600000*u^4-522921668585795864985600000*u^2+58102407620643984998400000)/sqrt(pi)} else if (r==19) {fx <- (1/549755813888)*exp(-(1/4)*u^2)*(u^38-1406*u^36+885780*u^34-331281720*u^32+82157866560*u^30-14295468781440*u^28+1801229066461440*u^26-167256984742848000*u^24+11540731947256512000*u^22-592424239959167616000*u^20+22512121118448369408000*u^18-626246278385927367168000*u^16+12524925567718547343360000*u^14-175348957948059662807040000*u^12+1653290174938848249323520000*u^10-9919741049633089495941120000*u^8+34719093673715813235793920000*u^6-61268988835969082180812800000*u^4+40845992557312721453875200000*u^2-4299578163927654889881600000)/sqrt(pi)} else if (r==20) {fx <- (1/2199023255552)*exp(-(1/4)*u^2)*(u^40-1560*u^38+1096680*u^36-460605600*u^34+129199870800*u^32-25633254366720*u^30+3716821883174400*u^28-401416763382835200*u^26+32615112024855360000*u^24-2000393537524462080000*u^22+92418181433630148096000*u^20-3192628085889041479680000*u^18+81412016190170557731840000*u^16-1502991068126225681203200000*u^14+19538883885640933855641600000*u^12-171942178193640217929646080000*u^10+967174752339226225854259200000*u^8-3185987419470392273402265600000*u^6+5309979032450653789003776000000*u^4-3353670967863570814107648000000*u^2+335367096786357081410764800000)/sqrt(pi)} else if (r==21) {fx <- (1/8796093022208)*exp(-(1/4)*u^2)*(u^42-1722*u^40+1343160*u^38-629494320*u^36+198290710800*u^34-44496435503520*u^32+7356744003248640*u^30-914338183260902400*u^28+86404958318155276800*u^26-6240358100755658880000*u^24+344467767161712370176000*u^22-14467646220791919547392000*u^20+458142130325077452334080000*u^18-10783960913805669262632960000*u^16+184867901379525758787993600000*u^14-2243063870071579206627655680000*u^12+18505276928090528454678159360000*u^10-97969113148714562407119667200000*u^8+304792796462667527488816742400000*u^6-481251783888422411824447488000000*u^4+288751070333053447094668492800000*u^2-27500101936481280675682713600000)/sqrt(pi)} else if (r==22) {fx <- (1/35184372088832)*exp(-(1/4)*u^2)*(u^44-1892*u^42+1629012*u^40-847086240*u^38+297750813360*u^36-75033204966720*u^34+14031209328776640*u^32-1988422807735203840*u^30+216240980341203417600*u^28-18164242348661087078400*u^26+1180675752662970660096000*u^24-59248455951814527670272000*u^22+2281065554144859315305472000*u^20-66677300813465118447390720000*u^18+1457375289208594731778682880000*u^16-23318004627337515708458926080000*u^14+265242302635964241183720284160000*u^12-2059528467526310578603004559360000*u^10+10297642337631552893015022796800000*u^8-30350945837229840105728488243200000*u^6+45526418755844760158592732364800000*u^4-26015096431911291519195847065600000*u^2+2365008766537390138108713369600000)/sqrt(pi)} else if (r==23) {fx <- (1/140737488355328)*exp(-(1/4)*u^2)*(u^46-2070*u^44+1958220*u^42-1124018280*u^40+438367129200*u^38-123268836731040*u^36+25886455713518400*u^34-4149229044366806400*u^32+514504401501483993600*u^30-49735425478476786048000*u^28+3759998166172845025228800*u^26-222181709819304478763520000*u^24+10220358651688006023121920000*u^22-363215822852296829437102080000*u^20+9858715191705199656149913600000*u^18-201117789910786072985458237440000*u^16+3016766848661791094781873561600000*u^14-32297150968026234073547116953600000*u^12+236845773765525716539345524326400000*u^10-1121901033626174446765320904704000000*u^8+3141322894153288450942898533171200000*u^6-4487604134504697787061283618816000000*u^4+2447784073366198792942518337536000000*u^2-212850788988365112429784203264000000)/sqrt(pi)} else if (r==24) {fx <- (1/562949953421312)*exp(-(1/4)*u^2)*(u^48-2256*u^46+2334960*u^44-1472581440*u^42+633946309920*u^40-197791248695040*u^38+46349082610871040*u^36-8342834869956787200*u^34+1170082590511439404800*u^32-128969103309705321062400*u^30+11220311987944362932428800*u^28-771141442080539852446924800*u^26+41770161446029242007541760000*u^24-1773625316785241660627927040000*u^22+58529635453912974800721592320000*u^20-1482750764832462028284947005440000*u^18+28357608377420836290949611479040000*u^16-400342706504764747636935691468800000*u^14+4047909587992621337217905324851200000*u^12-28122319242896106132250710677913600000*u^10+126550436593032477595128198050611200000*u^8-337467830914753273587008528134963200000*u^6+460183405792845373073193447456768000000*u^4-240095689978875846820796581281792000000*u^2+20007974164906320568399715106816000000)/sqrt(pi)} else if (r==25) {fx <- (1/2251799813685248)*exp(-(1/4)*u^2)*(u^50-2450*u^48+2763600*u^46-1906884000*u^44+901956132000*u^42-310633691860800*u^40+80764759883808000*u^38-16222178913804864000*u^36+2554993178924266080000*u^34-318522482972558504640000*u^32+31597430310877803660288000*u^30-2499069488223971744040960000*u^28+157441377758110219874580480000*u^26-7872068887905510993729024000000*u^24+310384430437417290609887232000000*u^22-9559840457472452550784526745600000*u^20+227046210864970748081132510208000000*u^18-4086831795569473465460385183744000000*u^16+54491090607592979539471802449920000000*u^14-521967288977995909272835160309760000000*u^12+3444984107254773001200712058044416000000*u^10-14764217602520455719431623105904640000000*u^8+37581644806415705467644131542302720000000*u^6-49019536704020485392579302011699200000000*u^4+24509768352010242696289651005849600000000*u^2-1960781468160819415703172080467968000000)/sqrt(pi)} else if (r==26) {fx <- (1/9007199254740992)*exp(-(1/4)*u^2)*(u^52-2652*u^50+3248700*u^48-2443022400*u^46+1264264092000*u^44-478397532412800*u^42+137300091802473600*u^40-30598306173122688000*u^38+5377652309926312416000*u^36-752871323389683738240000*u^34+84472162484322515430528000*u^32-7617853198586175937007616000*u^30+552294356897497755433052160000*u^28-32118041062654484854414417920000*u^26+1491194763623243939669240832000000*u^24-54875967301335376979828062617600000*u^22+1584543555826059010292535308083200000*u^20-35419208894935436700656671592448000000*u^18+602126551213902423911163417071616000000*u^16-7605809067965083249404169478799360000000*u^14+69212862518482257569577942257074176000000*u^12-435052278687602761865918494187323392000000*u^10+1779759321903829480360575658039050240000000*u^8-4333327044635410908704010297834209280000000*u^6+5416658805794263635880012872292761600000000*u^4-2599996226781246545222406178700525568000000*u^2+199999709752403580401723552207732736000000)/sqrt(pi)} else if (r==27) {fx <- (1/36028797018963968)*exp(-(1/4)*u^2)*(u^54-2862*u^52+3795012*u^50-3099259800*u^48+1747982527200*u^46-723664766260800*u^44+228195622960905600*u^42-56136123248382777600*u^40+10946544033434641632000*u^38-1710093434556567348288000*u^36+215471772754127485884288000*u^34-21978120820921003560197376000*u^32+1816857987862802960976316416000*u^30-121589726880049121234568867840000*u^28+6565845251522652546666718863360000*u^26-284519960899314943688891150745600000*u^24+9815938651026365557266744700723200000*u^22-266762568045540052203366826572595200000*u^20+5631654214294734435404410783199232000000*u^18-90699273135483617749144721034682368000000*u^16+1088391277625803412989736652416188416000000*u^14-9432724406090296245911050987606966272000000*u^12+56596346436541777475466305925641797632000000*u^10-221463964316902607512694240578598338560000000*u^8+516749250072772750862953228016729456640000000*u^6-620099100087327301035543873620075347968000000*u^4+286199584655689523554866403209265545216000000*u^2-21199969233754779522582696534019670016000000)/sqrt(pi)} else if (r==28) {fx <- (1/144115188075855872)*exp(-(1/4)*u^2)*(u^56-3080*u^54+4407480*u^52-3896212320*u^50+2386430046000*u^48-1076757236755200*u^46+371481246680544000*u^44-100406074102798464000*u^42+21612407450627369376000*u^40-3746150624775410691840000*u^38+526708777843422743272704000*u^36-60332096371155696047600640000*u^34+5641051010703057580450659840000*u^32-430455584816725624600542658560000*u^30+26749739913610806671605150924800000*u^28-1348186891645984656248899606609920000*u^26+54770092473118126660111546518528000000*u^24-1778417120303600348022445510483968000000*u^22+45646039421125742265909434769088512000000*u^20-912920788422514845318188695381770240000000*u^18+13967688062864477133368287039341084672000000*u^16-159630720718451167238494709021040967680000000*u^14+1320581416852641474427547138264975278080000000*u^12-7578989001067333679323314010912032030720000000*u^10+28421208754002501297462427540920120115200000000*u^8-63663507608965602906315837691661069058048000000*u^6+73457893394960311045749043490378156605440000000*u^4-32647952619982360464777352662390291824640000000*u^2+2331996615713025747484096618742163701760000000)/sqrt(pi)} else if (r==29) {fx <- (1/576460752303423488)*exp(-(1/4)*u^2)*(u^58-3306*u^56+5091240*u^54-4857042960*u^52+3220219482480*u^50-1577907546415200*u^48+593293237452115200*u^46-175445285932268352000*u^44+41492810122981465248000*u^42-7938957670197120350784000*u^40+1238477396550750774722304000*u^38-158299929050032326296323584000*u^36+16621492550253394261113976320000*u^34-1434562664721869873920760110080000*u^32+101649011671721065352099573514240000*u^30-5895642676959821790421775263825920000*u^28+278569116486351579597428881215774720000*u^26-10651172100948736866960516046485504000000*u^24+326635944429094597253455825425555456000000*u^22-7942410859275879154268241649821401088000000*u^20+150905806326241703931096591346606620672000000*u^18-2198913177896664828710264616764839329792000000*u^16+23988143758872707222293795819252792688640000000*u^14-189819224526731857150324819091478620405760000000*u^12+1044005734897025214326786505003132412231680000000*u^10-3758420645629290771576431418011276684034048000000*u^8+8095059852124626277241544592639672857919488000000*u^6-8994510946805140308046160658488525397688320000000*u^4+3854790405773631560591211710780796599009280000000*u^2-265847614191284935213187014536606662000640000000)/sqrt(pi)} else if (r==30) {fx <- (1/2305843009213693952)*exp(-(1/4)*u^2)*(u^60-3540*u^58+5851620*u^56-6007663200*u^54+4298483019600*u^52-2279915393595840*u^50+930965452384968000*u^48-300036865797212544000*u^46+77634539025028745760000*u^44-16320505315039376330880000*u^42+2810391015249780604177536000*u^40-398564543980877976592450560000*u^38+46698479069759536257415457280000*u^36-4526160279069001206487959705600000*u^34+362739416651101382405677913548800000*u^32-23989166754526171423095499349360640000*u^30+1304410942277360571130817777121484800000*u^28-58007921903628505398523425853167206400000*u^26+2094730513186584917168901489142149120000000*u^24-60857433856789203909328085368761384960000000*u^22+1405806722091830610305478772018387992576000000*u^20-25438407352137887234099139684142258913280000000*u^18+353825120443372431528833488333978692157440000000*u^16-3692088213322147111605219008702386352947200000000*u^14+27998335617692948929672910815993096509849600000000*u^12-147831212061418770348672969108443549572005888000000*u^10+511723426366449589668483354606150748518481920000000*u^8-1061352291723006556349446957701645996927221760000000*u^6+1137163169703221310374407454680334996707737600000000*u^4-470550277118574335327341015729793791741132800000000*u^2+31370018474571622355156067715319586116075520000000)/sqrt(pi)} else if (r>=31) {fx <- NA} } return(fx) } #### #### Kernel distribution H(x) # kernel_fun_dist <- function(kernel,u) # { # if (kernel=="epanechnikov") {H <- function(X) (3/4)*X-(1/4)*X^3+0.5 } # else if (kernel=="gaussian") {H <- pnorm} # else if (kernel=="silverman") {H <- function(X) (0.25*sqrt(2)*exp(0.5*sqrt(2)*X)*(cos(0.25*pi-0.5*sqrt(2)*X)+sin(0.25*pi-0.5*sqrt(2)*X))) *(X < 0)+ # (-0.25*sqrt(2)*exp(-0.5*sqrt(2)*X)*cos(0.25*pi+0.5*sqrt(2)*X)-0.25*sqrt(2)*exp(-0.5*sqrt(2)*X)*sin(0.25*pi+0.5*sqrt(2)*X)+1)*(X >= 0)} # else if (kernel=="uniform") {H <- function(X) 0.5*X+0.5 } # else if (kernel=="triangular") {H <- function(X) (0.5+X+0.5*X^2) *(X <0) + (0.5+X+0.5*X^2-X^2) *(X >=0)} # else if (kernel=="tricube") {H <- function(X) (0.5+(70/81)*X+(7/81)*X^10 + (10/27)*X^7 + (35/54)*X^4 )*(X < 0)+(0.5+(70/81)*X-(7/81)*X^10 + (10/27)*X^7 - (35/54)*X^4 )*(X >= 0)} # else if (kernel=="triweight") {H <- function(X) ((35/32) * X -(35/32)* X^3 +(21/32)* X^5 -(5/32)* X^7 + 0.5)} # else if (kernel=="biweight") {H <- function(X) ((15/16) * X -(5/8)* X^3+ (3/16)* X^5 + 0.5) } # else if (kernel=="cosine") {H <- function(X) (1/2)*sin((1/2)*pi)+(1/2)*sin((1/2)*pi*X) } # if (kernel=="gaussian"){Fx <- H(u)}else{ # Fx <- u # I0 <- (u <= -1) # I1 <- (u >= 1) # I2 <- (u > -1 & u < 1) # Fx[I0] <- 0;Fx[I1] <- 1 # U <- Fx[I2] # Fx[I2] <- H(U) # } # return(Fx) # } #### #### 1/(n-1) * sum[H((x-x[j])/h)] ; j != i #### Author: Alejandro Quintela del Rio and Graciela Estevez Perez #### Alejandro Quintela del Rio # kernel_dist_wit_i <-function(kernel,y,x,bw) # { # n <- length(x) # AUX <- matrix(0, n, n) # result <- matrix(0,n,length(y)) # for(j in 1:length(y)){ # AUX <- matrix(rep.int(outer(y[j],x,"-"),n),nrow=n,byrow=TRUE) # aux <- kernel_fun_dist(kernel, AUX/bw) # diag(aux) <- 0 # result[,j] <- (1/(n-1))*apply(aux,1,sum) # } # return(result) # } #### #### integral function #### Author: Alejandro Quintela del Rio and Graciela Estevez Perez #### Alejandro Quintela del Rio # simp_int <- function (x, fx, n.pts = 256, ret = FALSE) # { # if (class(fx) == "function") # fx = fx(x) # n.x = length(x) # if (n.x != length(fx)) # stop("Unequal input vector lengths") # if (n.pts < 64) # n.pts = 64 # ap = approx(x, fx, n = 2 * n.pts + 1) # h = diff(ap$x)[1] # integral = h * (ap$y[2 * (1:n.pts) - 1] + 4 * ap$y[2 * (1:n.pts)] + ap$y[2 * (1:n.pts) + 1])/3 # invisible(list(value = sum(integral), cdf = list(x = ap$x[2 * # (1:n.pts)], y = cumsum(integral)))) # } #### #### \int x K(x) H(x) dx A1_kM <- function(kernel) { if (kernel=="gaussian") {xKr <- 1/(2*sqrt(pi))} else if (kernel=="silverman") {xKr <- (5/32)*sqrt(2)} else if (kernel=="epanechnikov") {xKr <- 9/70 } else if (kernel=="uniform") {xKr <- 1/6 } else if (kernel=="triangular") {xKr <- 7/60 } else if (kernel=="triweight") {xKr <- 245/2574} else if (kernel=="tricube") {xKr <- 4291/39366} else if (kernel=="biweight") {xKr <- 25/231} else if (kernel=="cosine") {xKr <- 1/8} return(xKr) } #### #### \int x^2 K(x) dx A2_kM <-function(kernel) { if (kernel=="gaussian") {xKr <- 1} else if (kernel=="silverman") {xKr <- 0} else if (kernel=="epanechnikov") {xKr <- 1/5 } else if (kernel=="uniform") {xKr <- 1/3 } else if (kernel=="triangular") {xKr <- 1/6 } else if (kernel=="triweight") {xKr <- 1/9} else if (kernel=="tricube") {xKr <- 35/243} else if (kernel=="biweight") {xKr <- 1/7} else if (kernel=="cosine") {xKr <- (-8+pi^2)/pi^2} return(xKr) } #### #### \int K(x)^2 dx A3_kM <-function(kernel) { if (kernel=="gaussian") {xKr <- 1/(2*sqrt(pi))} else if (kernel=="silverman") {xKr <- (3/16)*sqrt(2)} else if (kernel=="epanechnikov") {xKr <- 3/5} else if (kernel=="uniform") {xKr <- 1/2} else if (kernel=="triangular") {xKr <- 2/3} else if (kernel=="triweight") {xKr <- 350/429} else if (kernel=="tricube") {xKr <- 175/247} else if (kernel=="biweight") {xKr <- 5/7} else if (kernel=="cosine") {xKr <- (1/16)*pi^2} return(xKr) } #### #### \int k(x,r)^2 dx A3_kMr <- function(kernel,r) { if (kernel=="gaussian"){ xKr <- integrate(function(x) kernel_fun_der(kernel,x,deriv.order=r)^2, -Inf, Inf)$value} else if (kernel =="silverman"){ xKr <- integrate(function(x) kernel_fun_der(kernel,x,deriv.order=r)^2, -100, +100)$value}else{ xKr <- integrate(function(x) kernel_fun_der(kernel,x,deriv.order=r)^2, -1, +1)$value} return(xKr) } #### #### \int K''(x)^2 dx A4_kM <-function(kernel) { if (kernel=="gaussian") {xKr <- 3/(8*sqrt(pi))} else if (kernel=="silverman") {xKr <- (1/16)*sqrt(2)} else if (kernel=="epanechnikov") {xKr <- 9/2} else if (kernel=="uniform") {xKr <- 0} else if (kernel=="triangular") {xKr <- 0} else if (kernel=="triweight") {xKr <- 35} else if (kernel=="tricube") {xKr <- 7840/243} else if (kernel=="biweight") {xKr <- 22.5} else if (kernel=="cosine") {xKr <- (1/256)*pi^6} return(xKr) } #### #### \int x^4 * K(x) dx A5_kM <-function(kernel) { if (kernel=="gaussian") {xKr <- 3} else if (kernel=="silverman") {xKr <- -24} else if (kernel=="epanechnikov") {xKr <- 3/35} else if (kernel=="uniform") {xKr <- 1/5} else if (kernel=="triangular") {xKr <- 1/15} else if (kernel=="triweight") {xKr <- 1/33} else if (kernel=="tricube") {xKr <- 1/22} else if (kernel=="biweight") {xKr <- 1/21} else if (kernel=="cosine") {xKr <- (pi^4-48*pi^2+384)/pi^4} return(xKr) } #### #### Derivatives DD <- function(expr,name, order = 1) { if(order < 1) stop("'order' must be >= 1") if(order == 1) D(expr,name) else DD(D(expr, name), name, order - 1) } #### #### Hermite Polynomial Hermite <-function (x, n, prob = TRUE) { if (any(n < 0 || n != round(n))) stop("Argument 'n' must be a vector of non-negative integers") if ((length(n) != 1) && (length(x) != length(n)) && (length(x) != 1)) stop(paste("Argument 'n' must be either a vector of same length", "as argument 'x',\n a single integer or 'x' must be a ", "single value!", sep = "")) H <- function(x, n) { if (n <= 1) { return(switch(n + 1, 1, x)) } else { return(x * Recall(x, n - 1) - (n - 1) * Recall(x, n - 2)) } } scale <- 1 if (!prob) { x <- sqrt(2) * x scale <- 2^(n/2) } scale * mapply(H, x, n) } kedd/R/MCV.R0000644000176200001440000001770412614740640012147 0ustar liggesusers## Wed Jun 19 14:21:28 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Modified Cross-Validation (MCV) h.mcv <- function(x, ...) UseMethod("h.mcv") h.mcv.default <- function(x,deriv.order=0,lower=0.1*hos,upper=2*hos,tol=0.1 * lower, kernel=c("gaussian","epanechnikov","triweight","tricube", "biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") if (missing(kernel)) kernel <- "gaussian" r <- deriv.order name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if (kernel=="epanechnikov" && (2*r) + 2 >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.mcv=NA),class="h.mcv")) else if (kernel=="triweight" && (2*r) + 2 >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.mcv=NA),class="h.mcv")) else if (kernel=="biweight" && (2*r) + 2 >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.mcv=NA),class="h.mcv")) else if (kernel=="tricube" && (2*r) + 2 >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.mcv=NA),class="h.mcv")) hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= 2*hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=2*hos lower=0.1*hos } R_Kr1 <- A3_kMr(kernel,r) fmcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r+2) diag(D3) <- 0 D4 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D3) Q3 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 - 0.5 * A2_kM(kernel) * Q3 } obj <- optimize(fmcv , c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,h = obj$minimum , min.mcv=obj$objective),class="h.mcv") } ###### print.h.mcv <- function(x, digits=NULL, ...) { class(x) <- "h.mcv" cat("\nCall:\t","\tModified Cross-Validation","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin MCV = ",format(x$min.mcv,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.mcv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.mcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && (2*r) + 2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for '(2 * order) + 2 >= 3' ") else if (kernel=="triweight" && (2*r) + 2 >= 7) stop(" 'triweight kernel derivative = 0' for '(2 * order) + 2 >= 7' ") else if (kernel=="biweight" && (2*r) + 2 >= 5) stop(" 'biweight kernel derivative = 0' for '(2 * order) + 2 >= 5' ") else if (kernel=="tricube" && (2*r) + 2 >= 10) stop(" 'tricube kernel derivative = 0' for '(2 * order) + 2 >= 10' ") if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(MCV~(h[(.(r))])) if(is.null(main)){ if(r !=0) {main <- "Modified Cross-Validation function for \nBandwidth Choice for Density Derivative"}else{ main <- "Modified Cross-Validation function for \nBandwidth Choice for Density Function"} } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fmcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r+2) diag(D3) <- 0 D4 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D3) Q3 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 - 0.5 * A2_kM(kernel) * Q3 } D <- lapply(1:length(seq.bws), function(i) fmcv(seq.bws[i])) Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, mcv=Minf)) } plot.h.mcv <- function(x,seq.bws=NULL,...) plot.mcv(x,seq.bws,...) ##### lines.mcv <- function(f,seq.bws=NULL,...) { class(f) <- "h.mcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && (2*r) + 2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for '(2 * order) + 2 >= 3' ") else if (kernel=="triweight" && (2*r) + 2 >= 7) stop(" 'triweight kernel derivative = 0' for '(2 * order) + 2 >= 7' ") else if (kernel=="biweight" && (2*r) + 2 >= 5) stop(" 'biweight kernel derivative = 0' for '(2 * order) + 2 >= 5' ") else if (kernel=="tricube" && (2*r) + 2 >= 10) stop(" 'tricube kernel derivative = 0' for '(2 * order) + 2 >= 10' ") if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fmcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r+2) diag(D3) <- 0 D4 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D3) Q3 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 - 0.5 * A2_kM(kernel) * Q3 } D <- lapply(1:length(seq.bws), function(i) fmcv(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.mcv <- function(x,seq.bws=NULL,...) lines.mcv(x,seq.bws,...) kedd/R/CCV.R0000644000176200001440000002044012614740640012124 0ustar liggesusers## Mon Jun 10 02:18:40 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Complete Cross-Validation (CCV) h.ccv <- function(x, ...) UseMethod("h.ccv") h.ccv.default <- function(x,deriv.order=0,lower=0.1*hos,upper=hos,tol=0.1 * lower, kernel=c("gaussian","triweight","tricube", "biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") if (missing(kernel)) kernel <- "gaussian" r <- deriv.order name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if (kernel=="triweight" && 2*(r+2) >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ccv=NA),class="h.ccv")) else if (kernel=="biweight" && 2*(r+2) >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ccv=NA),class="h.ccv")) else if (kernel=="tricube" && 2*(r+2) >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ccv=NA),class="h.ccv")) hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=hos lower=0.1*hos } R_Kr1 <- A3_kMr(kernel,r) fccv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D2 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*r) diag(D2) <- 0 D2 <- ((-1)^r /((n-1)*h^(2*r+1)))* colSums(D2) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+1)) diag(D3) <- 0 D3 <- ((-1)^(r+1) /((n-1)*h^(2*r+3)))* colSums(D3) Q3 <- mean(D3) D4 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D4) <- 0 D4 <- ((-1)^(r+2) /((n-1)*h^(2*r+5)))* colSums(D4) Q4 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 + 0.5 * h^2 *A2_kM(kernel) * Q3 + (h^4 / 24) *(6*(A2_kM(kernel))^2 - A5_kM(kernel)) * Q4 } obj <- optimize(fccv ,c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,h = obj$minimum , min.ccv=obj$objective),class="h.ccv") } ###### print.h.ccv <- function(x, digits=NULL, ...) { class(x) <- "h.ccv" cat("\nCall:\t","\tComplete Cross-Validation","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin CCV = ",format(x$min.ccv,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.ccv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.ccv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="triweight" && 2*(r+2) >= 7) stop(" 'triweight kernel derivative = 0' for '2 * (order + 2) >= 7' ") else if (kernel=="biweight" && 2*(r+2) >= 5) stop(" 'biweight kernel derivative = 0' for '2 * (order + 2) >= 5' ") else if (kernel=="tricube" && 2*(r+2) >= 10) stop(" 'tricube kernel derivative = 0' for '2 * (order + 2) >= 10' ") if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(CCV~(h[(.(r))])) if(is.null(main)){ if(r !=0) {main <- "Complete Cross-Validation function for \nBandwidth Choice for Density Derivative"}else{ main <- "Complete Cross-Validation function for \nBandwidth Choice for Density Function"} } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fccv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D2 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*r) diag(D2) <- 0 D2 <- ((-1)^r /((n-1)*h^(2*r+1)))* colSums(D2) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+1)) diag(D3) <- 0 D3 <- ((-1)^(r+1) /((n-1)*h^(2*r+3)))* colSums(D3) Q3 <- mean(D3) D4 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D4) <- 0 D4 <- ((-1)^(r+2) /((n-1)*h^(2*r+5)))* colSums(D4) Q4 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 + 0.5 * h^2 *A2_kM(kernel) * Q3 + (h^4 / 24) *(6*(A2_kM(kernel))^2 - A5_kM(kernel)) * Q4 } D <- lapply(1:length(seq.bws), function(i) fccv(seq.bws[i])) Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, ccv=Minf)) } plot.h.ccv <- function(x,seq.bws=NULL,...) plot.ccv(x,seq.bws,...) ##### lines.ccv <- function(f,seq.bws=NULL,...) { class(f) <- "h.ccv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="triweight" && 2*(r+2) >= 7) stop(" 'triweight kernel derivative = 0' for '2 * (order + 2) >= 7' ") else if (kernel=="biweight" && 2*(r+2) >= 5) stop(" 'biweight kernel derivative = 0' for '2 * (order + 2) >= 5' ") else if (kernel=="tricube" && 2*(r+2) >= 10) stop(" 'tricube kernel derivative = 0' for '2 * (order + 2) >= 10' ") if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fccv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D2 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*r) diag(D2) <- 0 D2 <- ((-1)^r /((n-1)*h^(2*r+1)))* colSums(D2) Q2 <- mean(D2) D3 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+1)) diag(D3) <- 0 D3 <- ((-1)^(r+1) /((n-1)*h^(2*r+3)))* colSums(D3) Q3 <- mean(D3) D4 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D4) <- 0 D4 <- ((-1)^(r+2) /((n-1)*h^(2*r+5)))* colSums(D4) Q4 <- mean(D4) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - Q2 + 0.5 * h^2 *A2_kM(kernel) * Q3 + (h^4 / 24) *(6*(A2_kM(kernel))^2 - A5_kM(kernel)) * Q4 } D <- lapply(1:length(seq.bws), function(i) fccv(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.ccv <- function(x,seq.bws=NULL,...) lines.ccv(x,seq.bws,...) kedd/R/dkde.R0000644000176200001440000001177012614740640012426 0ustar liggesusers## Tue Feb 26 00:39:10 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Derivatives of Kernel Density Estimator (dkde) dkde <- function(x, ...) UseMethod("dkde") dkde.default <- function(x,y=NULL,deriv.order=0,h, kernel=c("gaussian","epanechnikov","uniform","triangular", "triweight","tricube", "biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (kernel=="epanechnikov" && r >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order >= 3' ") if (kernel=="uniform" && r >= 1) stop(" 'uniform kernel derivative = 0' for 'order >= 1' ") if (kernel=="triweight" && r >= 7) stop(" 'triweight kernel derivative = 0' for 'order >= 7' ") if (kernel=="biweight" && r >= 5) stop(" 'biweight kernel derivative = 0' for 'order >= 5' ") if (kernel=="triangular" && r >= 2) stop(" 'triangular kernel derivative = 0' for 'order >= 2' ") if (kernel=="tricube" && r >= 10) stop(" 'tricube kernel derivative = 0' for 'order >= 10' ") name <- deparse(substitute(x)) if (missing(r)) r <- 0 if (missing(kernel)) kernel <- "gaussian" if (missing(h)) h <- h.ucv(x,deriv.order=r,kernel=kernel)$h x <- x[!is.na(x)] if (is.null(y)){ #from <- min(x) - 3 * h #to <- max(x) + 3 * h #y <- seq(from - 4 * h, to + 4 * h,length.out=512L) #tau <- if (kernel == "gaussian") 4 else 1 tau <- 4 range.x <- c(min(x) - tau * h, max(x) + tau * h) y <- seq(range.x[1L], range.x[2L],length.out=512L) } aux <- outer(y,x,"-")/h fn <- kernel_fun_der(kernel, aux,r)/h^(r+1) fn <- apply(fn,1,mean) structure(list(x=x, data.name=name, n=length(x), kernel=kernel, deriv.order=r, h = h, eval.points=y, est.fx = fn),class="dkde") } ###### print.dkde <- function(x, digits=NULL, ...) { class(x) <- "dkde" cat("\nData: ",x$data.name," (",x$n," obs.);", "\tKernel: ",x$kernel,"\n", "\nDerivative order: ",x$deriv.order,";","\tBandwidth 'h' = ",formatC(x$h,digits=digits), "\n\n",sep="") print(summary(as.data.frame(x[c("eval.points","est.fx")])), digits=digits, ...) invisible(x) } ###### plot.dkde1d <- function(f,fx=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,col="red",lty=1,...) { class(f) <- "dkde" #if(!is.function(fx)) stop("fx must be a function.") if(is.null(xlab)) xlab <- "x" if(is.null(ylab)){ if(f$deriv.order !=0){ylab <- "density derivative function"}else{ ylab <- "density function"} } if(is.null(main)){ if(f$deriv.order !=0) {main <- "Kernel density derivative estimate"}else{ main <- "Kernel density estimation"} } if(is.null(sub)){ if(f$deriv.order !=0) {sub <- paste("Kernel",f$kernel,";","Derivative order = ",f$deriv.order,";", "Bandwidth = ", formatC(f$h))}else{ sub <- paste("Kernel",f$kernel,";", "Bandwidth = ", formatC(f$h))} } gn <- if (!is.null(fx)) function(par) fx(par, ...) if (is.null(fx)){ plot.default(f$eval.points,f$est.fx, main=main,sub=sub, xlab=xlab, ylab=ylab, type=type,font.lab=2,cex.lab=0.9, font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,las=las,lwd=lwd,...)}else{ plot.default(f$eval.points,f$est.fx, ylim=c(min(f$est.fx,fx(f$eval.points),na.rm = TRUE),max(f$est.fx,fx(f$eval.points),na.rm = TRUE)), main=main,sub=sub, xlab=xlab, ylab=ylab, type=type,col=col,lty=lty, font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,las=las,lwd=lwd,...) curve(fx,xlim = c(min(f$eval.points,na.rm = TRUE), max(f$eval.points,na.rm = TRUE)), n = length(f$eval.points), lty = 8,lwd=lwd,add=TRUE) legend("topright", c("Estimate","True"),lty=c(lty,8),col=c(col,"black"),lwd=c(lwd,lwd), inset = .015,cex=1) } invisible(NULL) } plot.dkde <- function(x,fx=NULL,...) plot.dkde1d(x,fx,...) ###### lines.dkde1d <- function(f,...) { class(f) <- "dkde" lines.default(f$eval.points,f$est.fx,...) invisible(NULL) } lines.dkde <- function(x,...) lines.dkde1d(x,...) kedd/R/TCV.R0000644000176200001440000002030312614740640012143 0ustar liggesusers## Fri Jun 21 01:27:08 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Trimmed Cross-Validation (TCV) h.tcv <- function(x, ...) UseMethod("h.tcv") h.tcv.default <- function(x,deriv.order=0,lower=0.1*hos,upper=2*hos,tol=0.1 * lower, kernel=c("gaussian","epanechnikov", "uniform", "triangular","triweight","tricube", "biweight", "cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") if (missing(kernel)) kernel <- "gaussian" r <- deriv.order name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if (kernel=="epanechnikov" && 2*r >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) else if (kernel=="uniform" && 2*r >= 1) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) else if (kernel=="triweight" && 2*r >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) else if (kernel=="biweight" && 2*r >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) else if (kernel=="triangular" && 2*r >= 2) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) else if (kernel=="tricube" && 2*r >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.tcv=NA),class="h.tcv")) hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= 2*hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=2*hos lower=0.1*hos } R_Kr1 <- A3_kMr(kernel,r) ftcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r)*(abs(outer(x,x,"-")/h)> 1/(n*h^(2*r+1))) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - 2 * Q2 } obj <- optimize(ftcv , c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,h = obj$minimum , min.tcv=obj$objective),class="h.tcv") } ###### print.h.tcv <- function(x, digits=NULL, ...) { class(x) <- "h.tcv" cat("\nCall:\t","\tTrimmed Cross-Validation","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin TCV = ",format(x$min.tcv,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.tcv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.tcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(TCV~(h[(.(r))])) if(is.null(main)){ if(r !=0) {main <- "Trimmed Cross-Validation function for \nBandwidth Choice for Density Derivative"}else{ main <- "Trimmed Cross-Validation function for \nBandwidth Choice for Density Function"} } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } if (kernel=="epanechnikov" && 2*r >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2*order >= 3' ") else if (kernel=="uniform" && 2*r >= 1) stop(" 'uniform kernel derivative = 0' for '2*order >= 1' ") else if (kernel=="triweight" && 2*r >= 7) stop(" 'triweight kernel derivative = 0' for '2*order >= 7' ") else if (kernel=="biweight" && 2*r >= 5) stop(" 'biweight kernel derivative = 0' for '2*order >= 5' ") else if (kernel=="triangular" && 2*r >= 2) stop(" 'triangular kernel derivative = 0' for '2*order >= 2' ") else if (kernel=="tricube" && 2*r >= 10) stop(" 'tricube kernel derivative = 0' for '2*order >= 10' ") R_Kr1 <- A3_kMr(kernel,r) ftcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r)*(abs(outer(x,x,"-")/h)> 1/(n*h^(2*r+1))) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - 2 * Q2 } D <- lapply(1:length(seq.bws), function(i) ftcv(seq.bws[i])) Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, tcv=Minf)) } plot.h.tcv <- function(x,seq.bws=NULL,...) plot.tcv(x,seq.bws,...) ##### lines.tcv <- function(f,seq.bws=NULL,...) { class(f) <- "h.tcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && 2*r >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2*order >= 3' ") else if (kernel=="uniform" && 2*r >= 1) stop(" 'uniform kernel derivative = 0' for '2*order >= 1' ") else if (kernel=="triweight" && 2*r >= 7) stop(" 'triweight kernel derivative = 0' for '2*order >= 7' ") else if (kernel=="biweight" && 2*r >= 5) stop(" 'biweight kernel derivative = 0' for '2*order >= 5' ") else if (kernel=="triangular" && 2*r >= 2) stop(" 'triangular kernel derivative = 0' for '2*order >= 2' ") else if (kernel=="tricube" && 2*r >= 10) stop(" 'tricube kernel derivative = 0' for '2*order >= 10' ") if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) ftcv <- function(h) { L1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(L1) <- 0 L2 <- ((-1)^(r)/((n-1)*h^(2*r+1)))* colSums(L1) Q1 <- mean(L2) D1 <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r)*(abs(outer(x,x,"-")/h)> 1/(n*h^(2*r+1))) diag(D1) <- 0 D2 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D1) Q2 <- mean(D2) (1/(n*h^(2*r+1)))* R_Kr1 + Q1 - 2 * Q2 } D <- lapply(1:length(seq.bws), function(i) ftcv(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.tcv <- function(x,seq.bws=NULL,...) lines.tcv(x,seq.bws,...) kedd/R/UCV.R0000644000176200001440000001777212614740640012164 0ustar liggesusers## Tue Apr 16 03:33:05 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Unbiased Cross-Validation (UCV) h.ucv <- function(x, ...) UseMethod("h.ucv") h.ucv.default <- function(x,deriv.order=0,lower=0.1*hos,upper=2*hos,tol=0.1 * lower, kernel=c("gaussian","epanechnikov", "uniform", "triangular","triweight","tricube", "biweight", "cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (missing(kernel)) kernel <- "gaussian" name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if (kernel=="epanechnikov" && 2*r >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) else if (kernel=="uniform" && 2*r >= 1) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) else if (kernel=="triweight" && 2*r >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) else if (kernel=="biweight" && 2*r >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) else if (kernel=="triangular" && 2*r >= 2) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) else if (kernel=="tricube" && 2*r >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,min.ucv=NA),class="h.ucv")) hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= 2*hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=2*hos lower=0.1*hos } R_Kr1 <- A3_kMr(kernel,r) fucv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D) <- 0 D <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D) D1 <- mean(D) D2 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(D2) <- 0 D3 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D2) D4 <- mean(D3) (1/(n*h^(2*r+1)))* R_Kr1 + D4 - 2*D1 } obj <- optimize(fucv , c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = obj$minimum , min.ucv=obj$objective),class="h.ucv") } ###### print.h.ucv <- function(x, digits=NULL, ...) { class(x) <- "h.ucv" cat("\nCall:\t","\tUnbiased Cross-Validation","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin UCV = ",format(x$min.ucv,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.ucv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.ucv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && 2*r >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2*order >= 3' ") else if (kernel=="uniform" && 2*r >= 1) stop(" 'uniform kernel derivative = 0' for '2*order >= 1' ") else if (kernel=="triweight" && 2*r >= 7) stop(" 'triweight kernel derivative = 0' for '2*order >= 7' ") else if (kernel=="biweight" && 2*r >= 5) stop(" 'biweight kernel derivative = 0' for '2*order >= 5' ") else if (kernel=="triangular" && 2*r >= 2) stop(" 'triangular kernel derivative = 0' for '2*order >= 2' ") else if (kernel=="tricube" && 2*r >= 10) stop(" 'tricube kernel derivative = 0' for '2*order >= 10' ") if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(UCV~(h[(.(r))])) if(is.null(main)){ if(r !=0) {main <- "Unbiased Cross-Validation function for \nBandwidth Choice for Density Derivative"}else{ main <- "Unbiased Cross-Validation function for \nBandwidth Choice for Density Function"} } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fucv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D) <- 0 D <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D) D1 <- mean(D) D2 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(D2) <- 0 D3 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D2) D4 <- mean(D3) (1/(n*h^(2*r+1)))* R_Kr1 + D4 - 2*D1 } D <- lapply(1:length(seq.bws), function(i) fucv(seq.bws[i])) Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, ucv=Minf)) } plot.h.ucv <- function(x,seq.bws=NULL,...) plot.ucv(x,seq.bws,...) ##### lines.ucv <- function(f,seq.bws=NULL,...) { class(f) <- "h.ucv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && 2*r >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2*order >= 3' ") else if (kernel=="uniform" && 2*r >= 1) stop(" 'uniform kernel derivative = 0' for '2*order >= 1' ") else if (kernel=="triweight" && 2*r >= 7) stop(" 'triweight kernel derivative = 0' for '2*order >= 7' ") else if (kernel=="biweight" && 2*r >= 5) stop(" 'biweight kernel derivative = 0' for '2*order >= 5' ") else if (kernel=="triangular" && 2*r >= 2) stop(" 'triangular kernel derivative = 0' for '2*order >= 2' ") else if (kernel=="tricube" && 2*r >= 10) stop(" 'tricube kernel derivative = 0' for '2*order >= 10' ") if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) fucv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=2*r) diag(D) <- 0 D <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D) D1 <- mean(D) D2 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r) diag(D2) <- 0 D3 <- ((-1)^r / ((n-1)*h^(2*r+1)))* colSums(D2) D4 <- mean(D3) (1/(n*h^(2*r+1)))* R_Kr1 + D4 - 2*D1 } D <- lapply(1:length(seq.bws), function(i) fucv(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.ucv <- function(x,seq.bws=NULL,...) lines.ucv(x,seq.bws,...) kedd/R/MLCV.R0000644000176200001440000001067212614740640012260 0ustar liggesusers## Mon Apr 15 04:04:20 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Maximum-Likelihood (Kullback-Leibler information) Cross-Validation (MLCV) h.mlcv <- function(x, ...) UseMethod("h.mlcv") h.mlcv.default <- function(x,lower=0.1,upper=5,tol=0.1 * lower,kernel=c("gaussian", "epanechnikov","uniform","triangular","triweight", "tricube","biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 2L) stop("argument 'x' must be numeric and need at least 3 data points") if (!is.numeric(lower) || lower < 0) stop("invalid 'lower'") if (!is.numeric(upper)) stop("invalid 'upper'") if (!is.numeric(tol) || tol < 0) stop("invalid 'tol'") if (lower >= upper ) stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") if (missing(kernel)) kernel <- "gaussian" name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) fmlcv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=0) diag(D) <- 0 D <- (1/((n-1)*h))* colSums(D) mean(log(D)) } obj <- optimize(fmlcv ,c(lower,upper),tol=tol,maximum = TRUE) structure(list(x=x, data.name=name,n=n, kernel=kernel, h = obj$maximum, mlcv=obj$objective),class="h.mlcv") } ###### print.h.mlcv <- function(x, digits=NULL, ...) { class(x) <- "h.mlcv" cat("\nCall:\t","\tMaximum-Likelihood Cross-Validation","\n", "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMax CV = ",formatC(x$mlcv,digits=digits),";","\tBandwidth 'h' = ",formatC(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.mlcv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.mlcv" n <- f$n r <- 0 kernel <- f$kernel x <- sort(f$x) if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(MLCV~(h)) if(is.null(main)) main <- "Maximum-Likelihood Cross-Validation function for \nBandwidth Choice for Density Function" if(is.null(sub)) sub <- paste("Kernel",kernel) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } fmlcv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=0) diag(D) <- 0 D <- (1/((n-1)*h))* colSums(D) mean(log(D)) } D <- lapply(1:length(seq.bws), function(i) fmlcv(seq.bws[i])) Maxf <- c(do.call("rbind",D)) plot.default(seq.bws,Maxf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,seq.bws=seq.bws, mlcv=Maxf)) } plot.h.mlcv <- function(x,seq.bws=NULL,...) plot.mlcv(x,seq.bws,...) lines.mlcv <- function(f,seq.bws=NULL,...) { class(f) <- "h.mlcv" r <- 0 n <- f$n kernel <- f$kernel x <- sort(f$x) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } fmlcv <- function(h) { D <- kernel_fun_der(kernel, outer(x,x,"-")/h,deriv.order=0) diag(D) <- 0 D <- (1/((n-1)*h))* colSums(D) mean(log(D)) } D <- lapply(1:length(seq.bws), function(i) fmlcv(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.mlcv <- function(x,seq.bws=NULL,...) lines.mlcv(x,seq.bws,...) kedd/R/AMISE.R0000644000176200001440000001555312614740640012360 0ustar liggesusers## Mon May 20 13:31:03 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Aymptotic Mean Integrated Squared Error (AMISE) h.amise <- function(x, ...) UseMethod("h.amise") h.amise.default <- function(x,deriv.order=0,lower=0.1*hos,upper=2*hos,tol=0.1 * lower, kernel=c("gaussian","epanechnikov","triweight", "tricube", "biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") if (missing(kernel)) kernel <- "gaussian" r <- deriv.order name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if (kernel=="epanechnikov" && r+2 >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,amise=NA),class="h.amise")) else if (kernel=="triweight" && r+2 >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,amise=NA),class="h.amise")) else if (kernel=="biweight" && r+2 >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,amise=NA),class="h.amise")) else if (kernel=="tricube" && r+2 >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = NA ,amise=NA),class="h.amise")) hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=hos lower=0.1*hos } R_Kr <- A3_kMr(kernel,r) famise <- function(h) { D <- ((-1)^(r+2)/h^(2*r+5)) * mean(kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2)) 0.25*(2*r+5)* R_Kr^(4/(2*r+5)) * ((A2_kM(kernel)^2 * D) / (2*r+1))^((2*r+1)/(2*r+5)) * n^(-4/(2*r+5)) } obj <- optimize(famise , c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r, h = obj$minimum , amise=obj$objective),class="h.amise") } ###### print.h.amise <- function(x, digits=NULL, ...) { class(x) <- "h.amise" cat("\nCall:\t","\tAymptotic Mean Integrated Squared Error","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nAMISE = ",format(x$amise,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") invisible(x) } ###### plot.amise <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.amise" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && r+2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order + 2 >= 3' ") else if (kernel=="triweight" && r+2 >= 7) stop(" 'triweight kernel derivative = 0' for 'order + 2 >= 7' ") else if (kernel=="biweight" && r+2 >= 5) stop(" 'biweight kernel derivative = 0' for 'order + 2 >= 5' ") else if (kernel=="tricube" && r+2 >= 10) stop(" 'tricube kernel derivative = 0' for 'order + 2 >= 10' ") if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(AMISE~(h[(.(r))])) if(is.null(main)){ if(r !=0) {main <- "Aymptotic Mean Integrated Squared Error for \nBandwidth Choice for Density Derivative"}else{ main <- "Aymptotic Mean Integrated Squared Error for \nBandwidth Choice for Density Function"} } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr <- A3_kMr(kernel,r) famise <- function(h) { D <- ((-1)^(r+2)/h^(2*r+5)) * mean(kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2)) 0.25*(2*r+5)* R_Kr^(4/(2*r+5)) * ((A2_kM(kernel)^2 * D) / (2*r+1))^((2*r+1)/(2*r+5)) * n^(-4/(2*r+5)) } D <- lapply(1:length(seq.bws), function(i) famise(seq.bws[i])) Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, amise=Minf)) } plot.h.amise <- function(x,seq.bws=NULL,...) plot.amise(x,seq.bws,...) ##### lines.amise <- function(f,seq.bws=NULL,...) { class(f) <- "h.amise" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if (kernel=="epanechnikov" && r+2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order + 2 >= 3' ") else if (kernel=="triweight" && r+2 >= 7) stop(" 'triweight kernel derivative = 0' for 'order + 2 >= 7' ") else if (kernel=="biweight" && r+2 >= 5) stop(" 'biweight kernel derivative = 0' for 'order + 2 >= 5' ") else if (kernel=="tricube" && r+2 >= 10) stop(" 'tricube kernel derivative = 0' for 'order + 2 >= 10' ") if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr <- A3_kMr(kernel,r) famise <- function(h) { D <- ((-1)^(r+2)/h^(2*r+5)) * mean(kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2)) 0.25*(2*r+5)* R_Kr^(4/(2*r+5)) * ((A2_kM(kernel)^2 * D) / (2*r+1))^((2*r+1)/(2*r+5)) * n^(-4/(2*r+5)) } D <- lapply(1:length(seq.bws), function(i) famise(seq.bws[i])) Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.amise <- function(x,seq.bws=NULL,...) lines.amise(x,seq.bws,...) kedd/R/kernel.R0000644000176200001440000001000112614740640012761 0ustar liggesusers## Mon Jun 03 00:13:35 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Kernels functions kernel.fun <- function(x,...) UseMethod("kernel.fun") kernel.fun.default <- function(x=NULL,deriv.order=0,kernel=c("gaussian","epanechnikov", "uniform","triangular","triweight","tricube", "biweight","cosine","silverman"),...) { if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (missing(kernel)) kernel <- "gaussian" if (is.null(x)){ if (kernel == "gaussian"){x <- seq(-5,5,length=1000)} else if (kernel == "silverman"){x <- seq(-10,10,length=1000)} else {x <- seq(-1.5,1.5,length=1000)} } kx <- kernel_fun_der(kernel,u=x,deriv.order=r) structure(list(kernel = kernel,deriv.order=r,x=x,kx=kx),class="kernel.fun") } ############## ############## kernel.conv <- function(x,...) UseMethod("kernel.conv") kernel.conv.default <- function(x=NULL,deriv.order=0,kernel=c("gaussian","epanechnikov", "uniform","triangular","triweight","tricube", "biweight","cosine","silverman"),...) { if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") r <- deriv.order if (missing(kernel)) kernel <- "gaussian" if (is.null(x)){ if (kernel == "gaussian"){x <- seq(-8,8,length=1000)} else if (kernel == "silverman"){x <- seq(-10,10,length=1000)} else {x <- seq(-2.5,2.5,length=1000)} } kx <- kernel_fun_conv(kernel,u=x,deriv.order=r) structure(list(kernel = kernel,deriv.order=r,x=x,kx=kx),class="kernel.conv") } ############# ############# plot.kernel.fun1d <- function(f,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "kernel.fun" r <- f$deriv.order kernel <- f$kernel if(is.null(xlab)) xlab <- "x" if(is.null(ylab)) ylab <- "" if(is.null(main)){ if(r != 0) {main <- paste("Derivative of ",kernel,"kernel")}else{ main <- paste(kernel,"kernel")} } if(is.null(sub)){ if(r != 0) {sub <- paste("Derivative order = ",r)} } plot.default(f$x,f$kx,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) } plot.kernel.fun <- function(x,...) plot.kernel.fun1d (x,...) ################################ ################################ plot.kernel.conv1d <- function(f,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "kernel.conv" r <- f$deriv.order kernel <- f$kernel if(is.null(xlab)) xlab <- "x" if(is.null(ylab)) ylab <- "" if(is.null(main)){ if(r != 0) {main <- paste("Convolution of derivative",kernel,"kernel")}else{ main <- paste("Convolution of",kernel,"kernel")} } if(is.null(sub)){ if(r !=0) {sub <- paste("Derivative order = ",r)} } plot.default(f$x,f$kx,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) } plot.kernel.conv <- function(x,...) plot.kernel.conv1d (x,...) kedd/R/BCV.R0000644000176200001440000002642312614740640012132 0ustar liggesusers## Tue May 21 02:04:46 2013 ## Original file Copyright 2013 A.C. Guidoum ## This file is part of the R package kedd. ## Arsalane Chouaib GUIDOUM and ## Department of Probabilities-Statistics ## Faculty of Mathematics ## University of Science and Technology Houari Boumediene ## BP 32 El-Alia, U.S.T.H.B, Algeris ## Algeria ############################################################################## ## Biased Cross-Validation (BCV) h.bcv <- function(x, ...) UseMethod("h.bcv") h.bcv.default <- function(x,whichbcv = 1,deriv.order=0,lower=0.1*hos,upper=2*hos,tol=0.1 * lower, kernel=c("gaussian","epanechnikov","triweight","tricube", "biweight","cosine"),...) { if (!is.numeric(x) || length(dim(x)) >=1 || length(x) < 3L) stop("argument 'x' must be numeric and need at least 3 data points") if (any(deriv.order < 0 || deriv.order != round(deriv.order))) stop("argument 'deriv.order' is non-negative integers") if (missing(kernel)) kernel <- "gaussian" r <- deriv.order name <- deparse(substitute(x)) x <- x[!is.na(x)] x <- sort(x) n <- length(x) if(whichbcv == 1){ if (kernel=="epanechnikov" && r+2 >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv1=NA),class="h.bcv")) else if (kernel=="triweight" && r+2 >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv1=NA),class="h.bcv")) else if (kernel=="biweight" && r+2 >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv1=NA),class="h.bcv")) else if (kernel=="tricube" && r+2 >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv1=NA),class="h.bcv")) } else if (whichbcv == 2) { if (kernel=="epanechnikov" && 2*(r+2) >= 3) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv2=NA),class="h.bcv")) else if (kernel=="triweight" && 2*(r+2) >= 7) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv2=NA),class="h.bcv")) else if (kernel=="biweight" && 2*(r+2) >= 5) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv2=NA),class="h.bcv")) else if (kernel=="tricube" && 2*(r+2) >= 10) return(structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = NA ,min.bcv2=NA),class="h.bcv")) } hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) if (!is.numeric(upper)){ stop("argument 'upper' must be numeric. Default 2*hos (Oversmoothing) boundary was used") upper= hos } if (!is.numeric(lower)){ stop("argument 'lower' must be numeric. Default 0.1*hos boundary was used") lower=0.1*hos } if (lower < 0 | lower >= upper){ stop("the boundaries must be positive and 'lower' must be smaller than 'upper'. Default boundaries were used") upper=hos lower=0.1*hos } R_Kr1 <- A3_kMr(kernel,r) R_Kr2 <- A3_kMr(kernel,r+2) fbcv1 <- function(h) { D1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) ##(1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * (D - (1/((n-1)*h^(2*r+5))) * R_Kr2 ) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } fbcv2 <- function(h) { D1 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } if(whichbcv == 1) {obj <- optimize(fbcv1 ,c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = obj$minimum , min.bcv1=obj$objective),class="h.bcv")}else{ obj <- optimize(fbcv2 ,c(lower, upper),tol=tol) structure(list(x=x, data.name=name,n=n, kernel=kernel, deriv.order=r,whichbcv=whichbcv, h = obj$minimum , min.bcv2=obj$objective),class="h.bcv")} } ###### print.h.bcv <- function(x, digits=NULL, ...) { class(x) <- "h.bcv" if (x$whichbcv == 1){ cat("\nCall:\t","\tBiased Cross-Validation 1","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin BCV = ",format(x$min.bcv1,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="")}else{ cat("\nCall:\t","\tBiased Cross-Validation 2","\n", "\nDerivative order = ",x$deriv.order, "\nData: ",x$data.name," (",x$n," obs.);","\tKernel: ",x$kernel, "\nMin BCV = ",format(x$min.bcv2,digits=digits),";","\tBandwidth 'h' = ",format(x$h,digits=digits), "\n\n",sep="") } invisible(x) } ###### plot.bcv <- function(f,seq.bws=NULL,main=NULL,sub = NULL, xlab=NULL, ylab=NULL, type="l",las=1,lwd=1,...) { class(f) <- "h.bcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if(f$whichbcv == 1){ if (kernel=="epanechnikov" && r+2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order + 2 >= 3' ") else if (kernel=="triweight" && r+2 >= 7) stop(" 'triweight kernel derivative = 0' for 'order + 2 >= 7' ") else if (kernel=="biweight" && r+2 >= 5) stop(" 'biweight kernel derivative = 0' for 'order + 2 >= 5' ") else if (kernel=="tricube" && r+2 >= 10) stop(" 'tricube kernel derivative = 0' for 'order + 2 >= 10' ") } else if (f$whichbcv == 2){ if (kernel=="epanechnikov" && 2*(r+2) >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2 * (order + 2) >= 3' ") else if (kernel=="triweight" && 2*(r+2) >= 7) stop(" 'triweight kernel derivative = 0' for '2 * (order + 2) >= 7' ") else if (kernel=="biweight" && 2*(r+2) >= 5) stop(" 'biweight kernel derivative = 0' for '2 * (order + 2) >= 5' ") else if (kernel=="tricube" && 2*(r+2) >= 10) stop(" 'tricube kernel derivative = 0' for '2 * (order + 2) >= 10' ") } if(is.null(xlab)) xlab <- "Bandwidths" if(is.null(ylab)) ylab <- bquote(BCV~(h[(.(r))])) if(is.null(main)){ if(f$whichbcv == 1){ if(r !=0) {main <- "Biased Cross-Validation (1) function for \nBandwidth Choice for Density Derivative"}else{ main <- "Biased Cross-Validation (1) function for \nBandwidth Choice for Density Function"}}else{ if(r !=0) {main <- "Biased Cross-Validation (2) function for \nBandwidth Choice for Density Derivative"}else{ main <- "Biased Cross-Validation (2) function for \nBandwidth Choice for Density Function"} } } if(is.null(sub)) sub <- paste("Kernel",kernel,";","Derivative order = ",r) if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) R_Kr2 <- A3_kMr(kernel,r+2) fbcv1 <- function(h) { D1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) ##(1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * (D - (1/((n-1)*h^(2*r+5))) * R_Kr2 ) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } fbcv2 <- function(h) { D1 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } if(f$whichbcv == 1){ D <- lapply(1:length(seq.bws), function(i) fbcv1(seq.bws[i]))}else{ D <- lapply(1:length(seq.bws), function(i) fbcv2(seq.bws[i]))} Minf <- c(do.call("rbind",D)) plot.default(seq.bws,Minf,type=type,las=las,lwd=lwd,xlab=xlab,ylab=ylab, main=main,sub=sub,font.main=2,cex.main=0.9,font.sub=2,cex.sub=0.7,...) return(list(kernel=kernel,deriv.order=r,seq.bws=seq.bws, bcv=Minf)) } plot.h.bcv <- function(x,seq.bws=NULL,...) plot.bcv(x,seq.bws,...) ##### lines.bcv <- function(f,seq.bws=NULL,...) { class(f) <- "h.bcv" r <- f$deriv.order n <- f$n kernel <- f$kernel x <- sort(f$x) if(f$whichbcv == 1){ if (kernel=="epanechnikov" && r+2 >= 3) stop(" 'epanechnikov kernel derivative = 0' for 'order + 2 >= 3' ") else if (kernel=="triweight" && r+2 >= 7) stop(" 'triweight kernel derivative = 0' for 'order + 2 >= 7' ") else if (kernel=="biweight" && r+2 >= 5) stop(" 'biweight kernel derivative = 0' for 'order + 2 >= 5' ") else if (kernel=="tricube" && r+2 >= 10) stop(" 'tricube kernel derivative = 0' for 'order + 2 >= 10' ") } else if (f$whichbcv == 2){ if (kernel=="epanechnikov" && 2*(r+2) >= 3) stop(" 'epanechnikov kernel derivative = 0' for '2 * (order + 2) >= 3' ") else if (kernel=="triweight" && 2*(r+2) >= 7) stop(" 'triweight kernel derivative = 0' for '2 * (order + 2) >= 7' ") else if (kernel=="biweight" && 2*(r+2) >= 5) stop(" 'biweight kernel derivative = 0' for '2 * (order + 2) >= 5' ") else if (kernel=="tricube" && 2*(r+2) >= 10) stop(" 'tricube kernel derivative = 0' for '2 * (order + 2) >= 10' ") } if(is.null(seq.bws)){ hos <- ((243 *(2*r+1)*A3_kMr(kernel,r))/(35* A2_kM(kernel)^2))^(1/(2*r+5)) * sd(x,na.rm = TRUE) * n^(-1/(2*r+5)) seq.bws <- seq(0.15*hos,2*hos,length=50) } R_Kr1 <- A3_kMr(kernel,r) R_Kr2 <- A3_kMr(kernel,r+2) fbcv1 <- function(h) { D1 <- kernel_fun_conv(kernel,outer(x,x,"-")/h,deriv.order=r+2) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) ##(1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * (D - (1/((n-1)*h^(2*r+5))) * R_Kr2 ) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } fbcv2 <- function(h) { D1 <- kernel_fun_der(kernel,outer(x,x,"-")/h,deriv.order=2*(r+2)) diag(D1) <- 0 D1 <- ((-1)^(r+2)/((n-1)*h^(2*r+5)))* colSums(D1) D <- mean(D1) (1/(n*h^(2*r+1)))* R_Kr1 + (0.25*h^4)*(A2_kM(kernel))^2 * D } if(f$whichbcv == 1){ D <- lapply(1:length(seq.bws), function(i) fbcv1(seq.bws[i]))}else{ D <- lapply(1:length(seq.bws), function(i) fbcv2(seq.bws[i]))} Minf <- c(do.call("rbind",D)) lines.default(seq.bws,Minf,...) invisible(NULL) } lines.h.bcv <- function(x,seq.bws=NULL,...) lines.bcv(x,seq.bws,...) kedd/vignettes/0000755000176200001440000000000012614740771013202 5ustar liggesuserskedd/vignettes/kedd.Rnw0000644000176200001440000014626412614740640014611 0ustar liggesusers\documentclass[a4paper,11pt]{article} \usepackage{a4wide} \usepackage{graphicx} \usepackage{amsmath,amssymb,amsthm,amsopn,array,natbib,titling} \usepackage{amsmath,amssymb,amsthm,amsopn,array,natbib,titling} \usepackage[left=2.5cm,top=2cm,right=2cm,bottom=2cm]{geometry} \usepackage[utf8]{inputenc} \usepackage{authblk} \usepackage[english]{babel} \usepackage{hyperref} \usepackage{Sweave} \usepackage{color} \definecolor{link}{rgb}{0.45,0.51,0.67} \hypersetup{ colorlinks,% citecolor=link,% filecolor=link,% linkcolor=link,% urlcolor=link } %% load any required packages here \renewcommand{\today}{\begingroup \number \day\space \ifcase \month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space \number \year \endgroup} \newcommand{\argmin}[1]{\underset{#1}{\mathrm{argmin}} \ } \newcommand{\argmax}[1]{\underset{#1}{\mathrm{argmax}} \ } \def\ISE{\mathrm{ISE}} \def\MSE{\mathrm{MSE}} \def\MISE{\mathrm{MISE}} \def\AMSE{\mathrm{AMSE}} \def\AMISE{\mathrm{AMISE}} \def\LSCV{\mathrm{LSCV}} \def\UCV{\mathrm{UCV}} \def\BCV{\mathrm{BCV}} \def\CCV{\mathrm{CCV}} \def\TCV{\mathrm{TCV}} \def\MCV{\mathrm{MCV}} \def\MLCV{\mathrm{MLCV}} \def\rth{r^{th}} \def\Intr{\int_{\boldsymbol{\mathbb{R}}}} \def\Sum2{\sum_{i=1}^{n}\sum_{\substack{j=1 \\ j \neq i}}^{n}} \def\M2{\mu_{2}} \def\RK{\mathrm{R}\left(K^{(r)}\right)} \def\RR{\mathrm{R}} \def\Kr{K^{(r)}} \def\ConvKr{K^{(r)} \ast K^{(r)}} \def\Z{\left(\frac{X_{j}-X_{i}}{h}\right)} \def\z{\left(\frac{x-X_{i}}{h}\right)} \def\hatf{\hat{f}_{h}^{(r)}} \def\N{\mathbb{N}} %%%%%%%%%%%%%%%% \let\code=\texttt \let\proglang=\texttt \let\pkg=\texttt \newcommand{\CRANpkg}[1]{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}}% \newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} \let\cpkg=\CRANpkg \let\cpkg=\CRANpkg \bibliographystyle{plainnat} %\VignetteIndexEntry{Kernel estimator and bandwidth selection for density and its derivatives} %\VignettePackage{kedd} \begin{document} \title{\bfseries{ Kernel Estimator and Bandwidth Selection for Density and its Derivatives}\\ \vskip3pt \large The \CRANpkg{kedd} Package \\Version 1.0.3} \author{by Arsalane Chouaib Guidoum\thanks{Department of Probabilities \& Statistics.\\Faculty of Mathematics. \\University of Science and Technology Houari Boumediene.\\ BP 32 El-Alia, U.S.T.H.B, Algeria.\\ \email{acguidoum@usthb.dz}}} \date{Revised \today} \maketitle \section{Introduction}\label{Sec0} In statistics, the univariate kernel density estimation (KDE) is a non-parametric way to estimate the probability density function $f(x)$ of a random variable $X$, is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. This techniques are widely used in various inference procedures such as signal processing, data mining and econometrics, see e.g., \cite{Silverman,WandandJones,Jeffrey,Wolfgangetall,Alexandre}. The kernel estimator are standard in many books with applications and computer vision, see \cite{Wolfgang,Scott1992,Bowman,VenablesandRipley}, for computational complexity and with implementation in \texttt{S}, for an overview. Estimation of the density derivatives also comes up in various other applications like estimation of modes and inflexion points of densities, a good list of applications which require the estimation of density derivatives can be found in \cite{Singh1977}. There already exist a number of packages that can perform kernel density estimation in \texttt{R} (\code{density} in \texttt{R} base); see for example \pkg{KernSmooth} \citep{KernSmooth}, \pkg{sm} \citep{smarticle}, \pkg{np} \citep{np} and \pkg{feature} \citep{feature}, they exist also of functions for kernel density derivative estimation (KDDE), e.g., \code{kdde} in \pkg{ks} package \citep{ks}. We introduce in this vignette a new \texttt{R} package \CRANpkg{kedd} \citep{kedd} for use with the statistical programming environment \cite{R}, which implements smoothing techniques and computing bandwidth selectors of the $\rth$ derivative of a probability density $f(x)$ for univariate data, using several kernels functions. \section{Convolutions and derivatives in kernels}\label{Sec1} In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables density functions $f(x)$, or in kernel regression to estimate the conditional expectation of a random variable, see e.g., \cite{Silverman,WandandJones}. In general any functions having the following assumptions can be used as a kernel: \begin{itemize} \item[(A1)] $K(x) \geq 0$ and $\Intr K(x) dx = 1$. \item[(A2)] Symmetric about the origin, e.g., $\Intr x K(x) dx = 0$. \item[(A3)] Has finite second moment, e.g., $\M2(K) = \Intr x^{2} K(x) dx < \infty$. We denote $\RR(K) = \Intr \left(K(x)\right)^{2} dx$. \end{itemize} If $K(x)$ is a kernel, then so is the function $\bar{K}(x)$ defined by $\bar{K}(x)=\lambda K(\lambda x)$, where $\lambda > 0$, this can be used to select a scale that is appropriate for the data. The kernel function is very important to spreading a probability mass of $1/n$, the most widely used kernel is the Gaussian of zero mean and unit variance. Some classical of kernel function $K(x;r)$ ($r$ is the maximum derivative of kernel) in \pkg{kedd} package are the following: \begin{table}[!ht] \begin{center} {\renewcommand{\arraystretch}{1.5} \begin{tabular}{rlll} \hline\hline Kernel & $K(x;r)$ & $\RR(K)$ & $\M2(K)$ \\ \hline Gaussian & $K(x;\infty) =\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)1_{]-\infty,+\infty[}$ & $1/\left(2\sqrt{\pi}\right)$ & 1 \\ Epanechnikov & $K(x;2)=\frac{3}{4}\left(1-x^{2}\right)1_{(|x| \leq 1)}$ & 3/5 & 1/5 \\ Uniform & $K(x;0)=\frac{1}{2}1_{(|x| \leq 1)}$ & 1/2 & 1/3 \\ Triangular & $K(x;1)=(1-|x|)1_{(|x| \leq 1)}$ & 2/3 & 1/6 \\ Triweight & $K(x;6)=\frac{35}{32}\left(1-x^{2}\right)^{3} 1_{(|x| \leq 1)}$ & 350/429 & 1/9 \\ Tricube & $K(x;9)=\frac{70}{81}\left(1-|x|^{3}\right)^{3} 1_{(|x| \leq 1)}$ & 175/247 & 35/243 \\ Biweight & $K(x;4)=\frac{15}{16}\left(1-x^{2}\right)^{2} 1_{(|x| \leq 1)}$ & 5/7 & 1/7\\ Cosine & $K(x;\infty)=\frac{\pi}{4}\cos\left(\frac{\pi}{2}x\right) 1_{(|x| \leq 1)}$ & $\pi^{2}/16$ & $\left(-8+\pi^2\right)/\pi^{2}$\\ \hline\hline \end{tabular} } \end{center} \caption{Kernel functions in \pkg{kedd} pakage.}\label{Sec1:Tab1} \end{table}\\ The $\rth$ derivative of the kernel function $K(x)$ is written as: \begin{equation}\label{Sec1:eq1} \Kr(x) = \frac{d^{r}}{dx^{r}} K(x) \end{equation} and convolution of $\Kr(x)$ is: \begin{equation}\label{Sec1:eq2} \ConvKr(x) = \Intr \Kr(x) \Kr(x-y) dy \end{equation} for example the $\rth$ derivative of the Gaussian kernel is given by: $$\Kr(x) = (-1)^{r} H_{r}(x) K(x)$$ and the $\rth$ convolution can be written as: $$\ConvKr(x) = (-1)^{2r} \Intr H_{r}(x) H_{r}(x-y)K(x)K(x-y)dy$$ where $H_{r}(x)$ is the $\rth$ Hermite polynomial, see e.g., \cite{Olver}. We use \code{kernel.fun} for kernel derivative defined by \eqref{Sec1:eq1}, and \code{kernel.conv} for kernel convolution defined by \eqref{Sec1:eq2}.\\ For example the first derivative of the Gaussian kernel displayed on the left in Figure \ref{Sec1:fig1}. On the right is the first convolution of the Gaussian kernel. <>= library(kedd) @ <>= kernel.fun(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx kernel.conv(x = seq(-0.02,0.02,by=0.01), deriv.order = 1, kernel = "gaussian")$kx @ <>= plot(kernel.fun(deriv.order = 1, kernel = "gaussian")) plot(kernel.conv(deriv.order = 1, kernel = "gaussian")) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(kernel.fun(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") @ <>= plot(kernel.conv(deriv.order = 1, kernel = "gaussian"),main = "", sub = "") @ \end{center} \caption{(Left) First derivative of the Gaussian kernel. (Right) Convolution of the first derivative Gaussian kernel.}\label{Sec1:fig1} \end{figure} \section{Kernel density derivative estimator}\label{Sec2} Let $(X_{1},X_{2},\dots,X_{n})$ be a data sample, independent and identically distributed of a continuous random variable $X$, with density function $f(x)$. If the kernel $K$ is differentiable $r$ times then a natural estimator of the $\rth$ derivative of $f(x)$ the $\rth $ derivative of the kernel estimate \citep{Bhattacharya,Schuster,Alekseev}: \begin{equation}\label{Sec2:eq1} \hatf(x) = \frac{d^{r}}{dx^{r}} \frac{1}{nh} \sum_{i=1}^{n} K\z = \frac{1}{nh^{r+1}} \sum_{i=1}^{n} \Kr\z \end{equation} where $\Kr$ is $\rth$ derivative of the kernel function $K$, which we take to be a symmetric probability density with at least $r$ non zero derivatives when estimating $f^{(r)}(x)$, and $h$ is the bandwidth, this parameter is very important that controls the degree of smoothing applied to the data. The following assumptions on the density $f^{(r)}(x)$, the bandwidth $h$, and the kernel $K$: \begin{itemize} \item[(A4)] The $(r+2)$ derivatives $f^{(r+2)}(x)$ is continuous, square integrable and ultimately monotone. \item[(A5)] In the asymptotic framework, as $\lim_{n \to \infty} h_{n} = 0$ and $\lim_{n \to \infty} nh_{n}^{2r+1} = \infty$, i.e., as the number of sample $n$ is increased $h$ approaches zero at a rate slower than $1/n^{2r+1}$. \item[(A6)] Assumptions about $K$ are introduced in the previous section. \end{itemize} As seen in Equation \eqref{Sec2:eq1}, when working with a kernel estimator of the $\rth$ derivative function two choices must be made: the kernel function $K$ and the smoothing parameter or bandwidth $h$. The choice of $K$ is a problem of less importance, because $K$ is not very sensitive to the shape of estimator, and different functions that produce good results can be used. In practice, the choice of an efficient method for the computation of $h$, for an observed data sample is a crucial problem, because of the effect of the bandwidth on the shape of the corresponding estimator. If the bandwidth is small, we will obtain an under smoothed estimator, with high variability. On the contrary, if the value of $h$ is big, the resulting estimator will be over smooth and farther from the function that we are trying to estimate.\\ An example is drawn in Figure \ref{Sec2:fig1} where we show in left four different kernel (Gaussian, biweight, triweight and tricube) estimators of the first derivative of a bimodal (separated) Gaussian density (Equation \ref{Sec2:eq3}), and a given value of $h=0.6$. On the right, using the Gaussian kernel and four different values for the bandwidth. \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= fx <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) kernels <- c("gaussian","biweight","triweight","tricube") plot(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[1]),col = 1,ylim=c(-0.6,0.6) ,sub="", main="") for (i in 2:length(kernels))lines(dkde(x = bimodal, deriv.order = 1, h = 0.6, kernel = kernels[i] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)),lty = c(8,rep(1,length(kernels))), inset = .015) @ <>= h <- c(0.14,0.3,0.6,1.2) plot(dkde(x = bimodal, deriv.order = 1, h = h[1], kernel = kernels[1]),col = 1,ylim=c(-0.6,1) ,sub="", main="") for (i in 2:length(h))lines(dkde(x = bimodal, deriv.order = 1, h = h[i], kernel = kernels[1] ), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c("TRUE",paste("h =",bquote(.(h)))), col = c("black",seq(h)),lty = c(8,rep(1,length(h))), inset = .015) @ \end{center} \caption{(Left) Different kernels for estimation, with $h=0.6$. (Right) Effect of the bandwidth on the kernel estimator.}\label{Sec2:fig1} \end{figure} We have implemented in \texttt{R} the function \code{dkde} corresponds to the derivative of kernel density estimator (Equation \ref{Sec2:eq1}). Eight possibilities are allowed for the kernel functions that are summarized in Table \ref{Sec1:Tab1}. We enumerate the arguments and results of this function in Table \ref{Sec2:Tab1}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{y} & The points of the grid at which the density derivative is to be estimated.\\ & The default are $4h$ outside of range($x$).\\ \code{deriv.order} & Derivative order (scalar). \\ \code{h} & The smoothing bandwidth to be used. The default, "ucv" unbiased cross-\\ & validation.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{eval.points} & The coordinates of the points where the density derivative is estimated.\\ \code{est.fx} & The estimated density derivative values (Equation \ref{Sec2:eq1}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{dkde}.}\label{Sec2:Tab1} \end{table}\\ Working with the dataset \code{'bimodal'} correspond to data sample of 200 random numbers of a bi-modality (separated) of a two-component Gaussian mixture density (Equation \ref{Sec2:eq2}), with the following parameters: $-\mu_{1}=\mu_{2} = 3/2$ and $\sigma_{1}=\sigma_{2}=1/2$. The \code{dkde} function enables to compute the $\rth$ derivative of kernel density estimator over a grid of points, with a bandwidth selected by the user, but it also allows to estimate directly this parameter by the unbiased cross-validation method \code{h.ucv} (see following Section). We have chosen this method as the automatic one because it is the fastest in computation time terms. Now we estimate the first three derivatives of $f(x)$, can be written as: \begin{eqnarray} % \nonumber to remove numbering (before each equation) f(x) &=& 0.5\phi(\mu_{1},\sigma_{1}) + 0.5\phi(\mu_{2},\sigma_{2}) \label{Sec2:eq2}\\ f^{(1)}(x) &=& 0.5(-4x-6) \phi(\mu_{1},\sigma_{1}) + 0.5(-4x+6)\phi(\mu_{2},\sigma_{2})\label{Sec2:eq3} \\ f^{(2)}(x) &=& 0.5\left(\left(-4x-6\right)^{2} - 4\right)\phi(\mu_{1},\sigma_{1})+ 0.5 \left(\left(-4x+6\right)^{2} - 4\right) \phi(\mu_{2},\sigma_{2})\label{Sec2:eq4}\\ f^{(3)}(x) &=& 0.5(-4x-6)\left(\left(-4x-6\right)^{2} - 12\right)\phi(\mu_{1},\sigma_{1})+0.5(-4x+6) \notag \\ & & \left(\left(-4x+6\right)^{2} - 12\right)\phi(\mu_{2},\sigma_{2})\label{Sec2:eq5} \end{eqnarray} where $\phi$ is a standard normal density. <>= hatf <- dkde(bimodal, deriv.order = 0) hatf1 <- dkde(bimodal, deriv.order = 1) hatf2 <- dkde(bimodal, deriv.order = 2) hatf3 <- dkde(bimodal, deriv.order = 3) @ By default, the function \code{dkde} selects a grid of 512 points in the data range and used the Gaussian kernel. The output is a list containing the estimated values in the points of the grid, this last sequence and the bandwidth $h$ (by default, using unbiased cross-validation method). In Figure \ref{Sec2:fig3} we show the first three derivatives estimators of $f(x)$ obtained with the code: <>= fx <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5) fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) fx2 <- function(x) 0.5 * ((-4*x-6)^2 - 4) * dnorm(x,-1.5,0.5) + 0.5 * ((-4*x+6)^2 - 4) * dnorm(x,1.5,0.5) fx3 <- function(x) 0.5 * (-4*x-6) * ((-4*x-6)^2 - 12) * dnorm(x,-1.5,0.5) + 0.5 * (-4*x+6) * ((-4*x+6)^2 - 12) * dnorm(x,1.5,0.5) plot(hatf ,fx = fx) plot(hatf1,fx = fx1) plot(hatf2,fx = fx2) plot(hatf3,fx = fx3) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(hatf,fx = fx,lwd=2,sub="",main="") @ <>= plot(hatf1,fx = fx1,lwd=2,sub="",main="") @ <>= plot(hatf2,fx = fx2,lwd=2,sub="",main="") @ <>= plot(hatf3,fx = fx3,lwd=2,sub="",main="") @ \end{center} \caption{Kernel density derivative estimates obtained with the function \code{dkde}. (top left) density estimate $\hat{f}_{h}(x)$. (top right) first derivative $\hat{f}^{(1)}_{h}(x)$. (bottom left) second derivative $\hat{f}^{(2)}_{h}(x)$. (bottom right) third derivative $\hat{f}^{(3)}_{h}(x)$.}\label{Sec2:fig3} \end{figure} \section{Bandwidth selections}\label{Sec3} Despite the great number of bandwidth selection techniques in kernel density estimator or regression estimation, as for example \cite{Rudemo1982,Bowman1984,ScottandGeorge1987,SheatherandJones1991,Chiu1991a,Chiu1991b,Chiu1992,FeluchandKoronacki1992,Stute1992,Jonesetall1996, Sheather2004,DuongandHazelton2003,DuongandHazelton2005,Heidenreichetall2013}, to the best of our knowledge, only few paper have been studied in the context of estimating the $\rth$ derivative of a density $f(x)$, see \citet{PeterandMarron1987,Wolfgangetall1990,JonesandKappenman1991,Stoker1993}. In this section we summarize the techniques of cross-validation methods for bandwidth choice in the kernel estimation of the derivatives of a probability density. The practicality of this methods is demonstrated by an example. \subsection{Optimal bandwidth} We Consider the following $\AMISE$ version of the $\rth$ derivative of a probability density $f(x)$ \cite[p. 131]{Scott1992}: \begin{equation}\label{Sec3:eq1} \AMISE(h,r)= \frac{\RK}{nh^{2r+1}} + \frac{1}{4} h^{4} \M2^{2}(K) \RR\left(f^{(r+2)}\right) \end{equation} The optimal bandwidth minimizing \eqref{Sec3:eq1} is: \begin{equation}\label{Sec3:eq2} h^{\ast} = \left[\frac{(2r+1)\RK}{\M2^{2}(K) \RR\left(f^{(r+2)}\right)}\right]^{1/(2r+5)} n^{-1/(2r+5)} \end{equation} whereof: \begin{equation}\label{Sec3:eq3} \AMISE(h,r) = \frac{2r+5}{4} \RK^{\frac{4}{(2r+5)}} \left[ \frac{\M2^{2}(K) \RR\left(f^{(r+2)}\right)}{2r+1} \right]^{\frac{2r+1}{2r+5}} n^{-\frac{4}{2r+5}} \end{equation} which is the smallest possible $\AMISE$ for estimation of $\hat{f}^{(r)}_{h}$. The function \code{h.amise} provides the optimal bandwidth under $\AMISE$. The same possibilities for the kernel function as in the function \code{dkde} appear here. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab1}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq2}).\\ \code{amise} & The $\AMISE$ value (Equation \ref{Sec3:eq3}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.amise}.}\label{Sec3:Tab1} \end{table}\\ The following example computes this bandwidth for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.amise(bimodal, deriv.order = 0) h.amise(bimodal, deriv.order = 1) h.amise(bimodal, deriv.order = 2) h.amise(bimodal, deriv.order = 3) @ \subsection{Maximum likelihood cross-validation} This method was proposed by \cite{Habbema1974} and \cite{Duin1976}. They proposed to choose $h$ so that the pseudo-likelihood $\prod_{i=1}^{n} \hat{f}_{h}(X_{i})$ is maximized. However this has a trivial maximum at $h = 0$, so the cross-validation principle is invoked by replacing $\hat{f}_{h}(x)$ by the leave-one-out $\hat{f}_{h,i}(x)$, where: $$\hat{f}_{h,i}(X_{i}) = \frac{1}{(n-1) h} \sum_{j \neq i} K\Z$$ Define that $h$ as good which approaches the finite maximum of \begin{equation}\label{Sec3:eq4} h_{mlcv} = \argmax {h > 0} \MLCV(h) \end{equation} \begin{equation}\label{Sec3:eq5} \MLCV(h) = \left(n^{-1} \sum_{i=1}^{n} \log\left[\sum_{j \neq i} K\Z\right]-\log[(n-1)h]\right) \end{equation} The function \code{h.mlcv} computed the maximum likelihood cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab2}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq4}).\\ \code{mlcv} & The $\MLCV$ value (Equation \ref{Sec3:eq5}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.mlcv}.}\label{Sec3:Tab2} \end{table}\\ The following example computes this bandwidth of bimodal Gaussian density (Equation \ref{Sec2:eq2}), by different kernels. <>= kernels <- eval(formals(h.mlcv.default)$kernel) hmlcv <- numeric() for(i in 1:length(kernels)) hmlcv[i] <- h.mlcv(bimodal, kernel = kernels[i])$h @ <>= data.frame(kernels,hmlcv) @ The plot of the maximal likelihood cross validation function $\MLCV$ is shown in Figure \ref{Sec3:fig1} for Gaussian kernel in the left, and Epanechnikov kernel in the right, obtained with the code: <>= plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50)) plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.mlcv(bimodal, kernel = kernels[1]), seq.bws = seq(0.1,1,length=50),sub="",main="") @ <>= plot(h.mlcv(bimodal, kernel = kernels[2]), seq.bws = seq(0.1,1,length=50),sub="",main="") @ \end{center} \caption{$\MLCV$ function obtained by \code{h.mlcv}, using Gaussian kernel (Left) and Epanechnikov kernel (Right).}\label{Sec3:fig1} \end{figure} \subsection{Unbiased cross-validation} \cite{Rudemo1982} and \cite{Bowman1984} proposed a so-called unbiased (least-squares) cross-validation ($\UCV$) in kernel density estimator, is probably the most popular and best studied one. An adaptation of unbiased cross-validation is proposed by \cite{Wolfgangetall1990} for bandwidth choice in the $\rth$ derivative of kernel density estimator. The essential idea of this methods, it aims to estimate $h$ the minimizer of $\ISE(h)$. The minimization criterion is defined by: \begin{equation}\label{Sec3:eq6} h_{ucv} = \argmin {h > 0 }\UCV(h,r) \end{equation} \begin{equation}\label{Sec3:eq7} \UCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \Sum2 \left(\ConvKr -2 K^{(2r)}\right)\Z \end{equation} In general, cross-validation functions in non-parametric bandwidth selection present several local minima. These minima are more likely to appear at too small values of the bandwidth \citep{PeterandMarron1991}. The function \code{h.ucv} computes the unbiased cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab3}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth (Equation \ref{Sec3:eq6}).\\ \code{min.ucv} & The minimal $\UCV$ value (Equation \ref{Sec3:eq7}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.ucv}.}\label{Sec3:Tab3} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.ucv(bimodal, deriv.order = 0) h.ucv(bimodal, deriv.order = 1) h.ucv(bimodal, deriv.order = 2) h.ucv(bimodal, deriv.order = 3) @ The plot of $\UCV$ function obtained with the code (Figure \ref{Sec3:fig2}): <>= for (i in 0:3) plot(h.ucv(bimodal, deriv.order = i)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.ucv(bimodal, deriv.order = 0),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 1),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 2),sub="",main="") @ <>= plot(h.ucv(bimodal, deriv.order = 3),sub="",main="") @ \end{center} \caption{$\UCV$ function obtained by \code{h.ucv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig2} \end{figure} \subsection{Biased cross-validation} Biased cross-validation was proposed by \cite{ScottandGeorge1987}, which has as its immediate target the $\AMISE$ \eqref{Sec3:eq1}. They proposed to estimate $\RR\left(f^{(r+2)}\right)$ by! $$\hat{\RR}\left(f^{(r+2)}\right) = \RR\left(\hat{f}^{(r+2)}_{h}\right) - \frac{\RR\left(K^{(r+2)}\right)}{nh^{2r+5}}$$ There are two versions of $\BCV$, depending on the estimator of $\RR\left(f^{(r+2)}\right)$. We can use \citep{ScottandGeorge1987} $$\hat{\RR}\left(f^{(r+2)}\right) = \frac{(-1)^{r+2}}{n(n-1)h^{2r+5}} \Sum2 K^{(r+2)} \ast K^{(r+2)} \Z$$ or we could use \citep{JonesandKappenman1991} $$\hat{\RR}\left(f^{(r+2)}\right) = \frac{(-1)^{r+2}}{n(n-1) h^{2r+5}} \Sum2 K^{(2r+4)} \Z$$ From this we obtain respectively an adaptation of biased cross-validation for bandwidth choice in the $\rth$ derivative of kernel density estimator, is given by: \begin{equation}\label{Sec3:eq8} \BCV_{1}(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{\M2^{2}(K)}{4} \frac{(-1)^{r+2}}{n(n-1)h^{2r+1}} \Sum2 K^{(r+2)} \ast K^{(r+2)} \Z \end{equation} \begin{equation}\label{Sec3:eq9} \BCV_{2}(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{\M2^{2}(K)}{4} \frac{(-1)^{r+2}}{n(n-1)h^{2r+1}} \Sum2 K^{(2r+4)} \Z \end{equation} The $\BCV$ selectors $h_{bcv_{1}}$ and $h_{bcv_{2}}$ are the minimisers of the appropriate $\BCV$ function. The function \code{h.bcv} computes the biased cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab4}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{whichbcv} & Method selected, \code{1 = BCV1} or \code{2 = BCV2}, by default \code{BCV1}. \\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.bcv} & The minimal $\BCV$ value (Equation \ref{Sec3:eq8} or \ref{Sec3:eq9}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.bcv}.}\label{Sec3:Tab4} \end{table}\\ The following example computes the bandwidth parameter by this method for kernel estimator of Equation \eqref{Sec2:eq2} and its first derivative estimators. <>= h.bcv(bimodal, whichbcv = 1, deriv.order = 0) h.bcv(bimodal, whichbcv = 2, deriv.order = 0) h.bcv(bimodal, whichbcv = 1, deriv.order = 1, lower=0.1, upper=0.8) h.bcv(bimodal, whichbcv = 2, deriv.order = 1, lower=0.1, upper=0.8) @ The plot of $\BCV$ function obtained with the code \code{h.bcv} (Figure \ref{Sec3:fig3}): <>= ## deriv.order = 0 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) ## deriv.order = 1 plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws = seq(0.1,0.8,length=50)) lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"), inset = .015) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 0),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) @ <>= plot(h.bcv(bimodal, whichbcv = 2, deriv.order = 1),seq.bws=seq(0.1,0.8,length=50),sub="",main="") lines(h.bcv(bimodal, whichbcv = 1, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("red","black"),inset = .015) @ \end{center} \caption{$\BCV$ function obtained by \code{h.bcv}. (Left) $\BCV_{1}$ vs $\BCV_{2}$ (\code{deriv.order = 0}). (Right) $\BCV_{1}$ vs $\BCV_{2}$ (\code{deriv.order = 1}).}\label{Sec3:fig3} \end{figure} \subsection{Complete cross-validation} \cite{JonesandKappenman1991} proposed a so-called complete cross-validation ($\CCV$) in kernel density estimator. This method can be extended to the estimation of derivative of the density, basing our estimate of integrated squared density derivative \citep{PeterandMarron1987} we get the following. Thus, $h_{ccv}$, say, is the $h$ that minimises: \begin{equation}\label{Sec3:eq10} \CCV(h,r) = \RR\left(\hatf\right) -\bar{\theta}_{r}(h) + \frac{1}{2}\M2(K) h^{2} \bar{\theta}_{r+1}(h)+ \frac{1}{24}\left(6\M2^{2}(K) -\delta(K)\right) h^{4}\bar{\theta}_{r+2}(h) \end{equation} where, $$\RR\left(\hatf\right) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \Sum2 \ConvKr \Z$$ and $$\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 K^{(2r)} \Z$$ with : $\delta(K) = \Intr x^{4} K(x) dx$.\\ The function \code{h.ccv} computes the complete cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab5}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.ccv} & The minimal $\CCV$ value (Equation \ref{Sec3:eq10}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.ccv}.}\label{Sec3:Tab5} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. This time we set Over-smoothing in \code{upper = 0.5}. <>= h.ccv(bimodal, deriv.order = 0, upper = 0.5) h.ccv(bimodal, deriv.order = 1, upper = 0.5) h.ccv(bimodal, deriv.order = 2, upper = 0.5) h.ccv(bimodal, deriv.order = 3, upper = 0.5) @ The plot of $\CCV$ function obtained with the code (Figure \ref{Sec3:fig4}): <>= for (i in 0:3) plot(h.ccv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.ccv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.ccv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ \end{center} \caption{$\CCV$ function obtained by \code{h.ccv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig4} \end{figure} \subsection{Modified cross-validation} \cite{Stute1992} proposed a so-called modified cross-validation ($\MCV$) in kernel density estimator. This method can be extended to the estimation of derivative of a probability density, the essential idea based on approximated the problematic term by the aid of the Hajek projection. The minimization criterion is defined by: \begin{equation}\label{Sec3:eq11} \MCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 \varphi^{(r)} \Z \end{equation} where$$ \varphi^{(r)} (c) = \left(\ConvKr - K^{(2r)} - \frac{\M2(K)}{2} K^{(2r+2)} \right)(c)$$ The function \code{h.mcv} computes the modified cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab6}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.mcv} & The minimal $\MCV$ value (Equation \ref{Sec3:eq11}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.mcv}.}\label{Sec3:Tab6} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. We set Over-smoothing in \code{upper = 0.5}. <>= h.mcv(bimodal, deriv.order = 0, upper = 0.5) h.mcv(bimodal, deriv.order = 1, upper = 0.5) h.mcv(bimodal, deriv.order = 2, upper = 0.5) h.mcv(bimodal, deriv.order = 3, upper = 0.5) @ The plot of $\MCV$ function obtained with the code (Figure \ref{Sec3:fig5}): <>= for (i in 0:3) plot(h.mcv(bimodal, deriv.order = i), seq.bws=seq(0.1,0.5,length=50)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.mcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 1),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 2),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ <>= plot(h.mcv(bimodal, deriv.order = 3),seq.bws=seq(0.1,0.5,length=50),sub="",main="") @ \end{center} \caption{$\MCV$ function obtained by \code{h.mcv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig5} \end{figure} \subsection{Trimmed cross-validation} \cite{FeluchandKoronacki1992} proposed a so-called trimmed cross-validation ($\TCV$) in kernel density estimator, a simple modification of the unbiased (least-squares) cross-validation criterion \eqref{Sec3:eq7}. We consider the following "trimmed" version of "unbiased", to be minimized with respect to $h$: \begin{equation}\label{Sec3:eq12} \TCV(h,r) = \frac{\RK}{nh^{2r+1}} + \frac{(-1)^r}{n(n-1) h^{2r+1}} \Sum2 \varphi^{(r)} \Z \end{equation} where$$ \varphi^{(r)} (c) = \left[\ConvKr - 2 K^{(2r)} 1\left(|c| > \frac{c_{n}}{h^{2r+1}}\right) \right](c)$$ $1(.)$ denotes the indicator function and $c_{n}$ is a sequence of positive constants, as $\lim_{n \to \infty} c_{n}/h \rightarrow 0$, here we take $c_{n} = 1/n$, for assure the convergence. The function \code{h.tcv} computes the trimmed cross-validation for bandwidth selection. We enumerate the arguments and results of this function in Table \ref{Sec3:Tab7}. \begin{table}[!ht] {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Arguments & Description \\ \hline \code{x} & The data sample.\\ \code{deriv.order} & Derivative order (scalar). \\ \code{lower,upper} & Range over which to minimize. The default is almost always satisfactory,\\ & \code{hos} (Over-smoothing) is calculated internally from an kernel.\\ \code{tol} & The convergence tolerance for optimize.\\ \code{kernel} & The kernel function (see Table \ref{Sec1:Tab1}), by default \code{"gaussian"}. \\ \hline Results & Description \\ \hline \code{h} & Value of bandwidth.\\ \code{min.tcv} & The minimal $\TCV$ value (Equation \ref{Sec3:eq12}).\\ \hline\hline \end{tabular} } \caption{Summary of arguments and results of \code{h.tcv}.}\label{Sec3:Tab7} \end{table}\\ The following example computes the bandwidth $h$ by this method for a first three derivatives estimators of \eqref{Sec2:eq2}. <>= h.tcv(bimodal, deriv.order = 0) h.tcv(bimodal, deriv.order = 1) h.tcv(bimodal, deriv.order = 2) h.tcv(bimodal, deriv.order = 3) @ The plot of $\TCV$ function obtained with the code (Figure \ref{Sec3:fig6}): <>= for (i in 0:3) plot(h.tcv(bimodal, deriv.order = i)) @ \setkeys{Gin}{width=0.45\textwidth} \begin{figure}[!ht] \begin{center} <>= plot(h.tcv(bimodal, deriv.order = 0),seq.bws=seq(0.1,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 1),seq.bws=seq(0.3,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 2),seq.bws=seq(0.5,1.5,length=50),sub="",main="") @ <>= plot(h.tcv(bimodal, deriv.order = 3),seq.bws=seq(0.5,1.5,length=50),sub="",main="") @ \end{center} \caption{$\TCV$ function obtained by \code{h.tcv}. (top left) \code{deriv.order = 0}. (top right) \code{deriv.order = 1}. (bottom left) \code{deriv.order = 2}. (bottom right) \code{deriv.order = 3}.}\label{Sec3:fig6} \end{figure} \vspace{3cm} \section{Summary} We have implemented in \texttt{R} the estimators of the defined functions and the bandwidth selection procedures of the above sections. The package \pkg{kedd} contains seven functions, in Table \ref{Sec4:Tab1} we can find a summary of the contents of the package. The current feature set of the package can be split in four main categories: compute the convolutions and derivatives of a kernel function, compute the kernel estimators for a density of probability and its derivatives, computing the bandwidth selectors with different methods, displaying the kernel estimators and selection functions of the bandwidth. Moreover, the package follows the general \texttt{R} philosophy of working with model objects. This means that instead of merely returning, say, a kernel estimator of $\rth$ derivative of a density, many functions will return an object containing, it's functions are S3 classes (\code{S3method}). The object can then be manipulated at one’s will using various extraction, summary or plotting functions. Whenever possible, we develop a graphical user interface of the various functions of a coherent whole, to facilitate the use of this package. \begin{table}[!ht] \begin{center} {\renewcommand{\arraystretch}{1} \begin{tabular}{ll} \hline\hline Function & Description \\ \hline \code{dkde} & Derivatives of kernel density estimator, as defined in Equation \ref{Sec2:eq1}.\\ \code{h.amise} & $\AMISE$ for optimal bandwidth selectors (Equation \ref{Sec3:eq3}). \\ \code{h.mlcv} & Maximum-likelihood cross-validation bandwidth selection (Equation \ref{Sec3:eq5}).\\ \code{h.ucv} & Unbiased cross-validation bandwidth selection (Equation \ref{Sec3:eq7}).\\ \code{h.bcv} & Biased cross-validation bandwidth selection (Equations \ref{Sec3:eq8} and \ref{Sec3:eq9}) .\\ \code{h.ccv} & Complete cross-validation bandwidth selection (Equation \ref{Sec3:eq10}).\\ \code{h.mcv} & Modified cross-validation bandwidth selection (Equation \ref{Sec3:eq11}).\\ \code{h.tcv} & Trimmed cross-validation bandwidth selection (Equation \ref{Sec3:eq12}).\\ \hline\hline \end{tabular} } \end{center} \caption{Summary of contents of the package.}\label{Sec4:Tab1} \end{table} \begin{thebibliography}{1} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{{\tt #1}}\fi \bibitem[Alekseev, 1972]{Alekseev} Alekseev, V. 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(2015). \newblock \pkg{kedd}: Kernel estimator and bandwidth selection for density and its derivatives. \newblock \texttt{R} package version 1.0.3. \newblock \url{http://CRAN.R-project.org/package=kedd} \bibitem[Habbema, Hermans and Van den Broek (1974)]{Habbema1974} Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974). \newblock A stepwise discrimination analysis program using density estimation. \newblock \emph{Compstat 1974: Proceedings in Computational Statistics}. Physica Verlag, Vienna. \bibitem[Heidenreich et all, 2013]{Heidenreichetall2013} Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). \newblock Bandwidth selection for kernel density estimation: a review of fully automatic selectors. \newblock \emph{Advances in Statistical Analysis}. \bibitem[Jeffrey, 1996]{Jeffrey} Jeffrey, S. S. (1996). \newblock \emph{Smoothing Methods in Statistics}. \newblock Springer-Verlag, New York. \bibitem[Jones and Kappenman, 1991]{JonesandKappenman1991} Jones, M. C. and Kappenman, R. F. (1991). \newblock On a class of kernel density estimate bandwidth selectors. \newblock \emph{Scandinavian Journal of Statistics}, \textbf{19}, 337--349. \bibitem[Jones et all, 1996]{Jonesetall1996} Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). \newblock A brief survey of bandwidth selection for density estimation. \newblock \emph{Journal of the American Statistical Association}, \textbf{91}, 401--407. \bibitem[Olver et all, 2010]{Olver} Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). \newblock \emph{NIST Handbook of Mathematical Functions}. \newblock Cambridge University Press, New York, USA. \bibitem[Peter and Marron, 1987]{PeterandMarron1987} Peter, H. and Marron, J.S. (1987). \newblock Estimation of integrated squared density derivatives. \newblock \emph{Statistics and Probability Letters}, \textbf{6}, 109--115. \bibitem[Peter and Marron, 1991]{PeterandMarron1991} Peter, H. and Marron, J.S. (1991). \newblock Local minima in cross-validation functions. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{53}, 245--252. \bibitem[\texttt{R} Development Core Team (2015)]{R} R Development Core Team (2015). \newblock {\em \texttt{R}: A Language and Environment for Statistical Computing}. \newblock Vienna, Austria. \newblock \url{http://www.R-project.org/} \bibitem[Rudemo, 1982]{Rudemo1982} Rudemo, M. (1982). \newblock Empirical choice of histograms and kernel density estimators. \newblock \emph{Scandinavian Journal of Statistics}, \textbf{9}, 65--78. \bibitem[Schuster, 1969]{Schuster} Schuster, E. F. (1969). \newblock Estimation of a probability density function and its derivatives. \newblock \emph{The Annals of Mathematical Statistics}, \textbf{40}(4), 1187--1195. \bibitem[Scott, 1992]{Scott1992} Scott, D. W. (1992). \newblock \emph{Multivariate Density Estimation. Theory, Practice and Visualization}. \newblock New York: Wiley. \bibitem[Scott and George, 1987]{ScottandGeorge1987} Scott, D.W. and George, R. T. (1987). \newblock Biased and unbiased cross-validation in density estimation. \newblock \emph{Journal of the American Statistical Association}, \textbf{82}, 1131--1146. \bibitem[Sheather, 2004]{Sheather2004} Sheather, S. J. (2004). \newblock Density estimation. \newblock \emph{Statistical Science}, \textbf{19}, 588--597. \bibitem[Sheather and Jones, 1991]{SheatherandJones1991} Sheather, S. J. and Jones, M. C. (1991). \newblock A reliable data-based bandwidth selection method for kernel density estimation. \newblock \emph{Journal of the Royal Statistical Society, Series B}, \textbf{53}, 683--690. \bibitem[Silverman, 1986]{Silverman} Silverman, B. W. (1986). \newblock \emph{Density Estimation for Statistics and Data Analysis}. \newblock Chapman \& Hall/CRC. London. \bibitem[Singh, 1977]{Singh1977} Singh, R. S. (1990). \newblock Applications of estimators of a density and its derivatives\textbf{39}(3), 357--363. \bibitem[Stoker, 1993]{Stoker1993} Stoker, T. M. (1993). \newblock Smoothing bias in density derivative estimation. \newblock \emph{Journal of the American Statistical Association}, \textbf{88}, 855--863. \bibitem[Stute, 1992]{Stute1992} Stute, W. (1992). \newblock Modified cross validation in density estimation. \newblock \emph{Journal of Statistical Planning and Inference}, \textbf{30}, 293--305. \bibitem[Tristen and Jeffrey, 2008]{np} Tristen, H. and Jeffrey, S. R. (2008). \newblock Nonparametric Econometrics: The \pkg{np} Package. \newblock {\em Journal of Statistical Software}. \textbf{27 (5)}. \bibitem[Wand and Jones, 1995]{WandandJones} Wand, M. P. and Jones, M. C. (1995). \newblock \emph{Kernel Smoothing}. \newblock Chapman and Hall, London. \bibitem[Wand and Ripley, 2013]{KernSmooth} Wand, M.P. and Ripley, B. D. (2013). \newblock \pkg{KernSmooth}: Functions for kernel smoothing for Wand and Jones (1995). \newblock \texttt{R} package version 2.23-10. \newblock \url{http://CRAN.R-project.org/package=KernSmooth} \bibitem[Wolfgang, 1991]{Wolfgang} Wolfgang, H. (1991). \newblock \emph{Smoothing Techniques, With Implementation in S}. \newblock Springer-Verlag, New York. \bibitem[Wolfgang et all, 1990]{Wolfgangetall1990} Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). \newblock Bandwidth choice for density derivatives. \newblock \emph{Journal of the Royal Statistical Society, Series B}, 223--232. \bibitem[Wolfgang et all, 2004]{Wolfgangetall} Wolfgang, H., Marlene, M., Stefan, S. and Axel, W. (2004). \newblock \emph{Nonparametric and Semiparametric Models}. \newblock Springer-Verlag, Berlin Heidelberg. \bibitem[Venables and Ripley, 2002]{VenablesandRipley} Venables, W. N. and Ripley, B. D. (2002). \newblock \emph{Modern Applied Statistics with S}. \newblock New York: Springer. \end{thebibliography} \end{document}kedd/vignettes/intro.Rnw0000644000176200001440000001105512614740640015022 0ustar liggesusers\documentclass{article} \usepackage{graphicx} \usepackage{amsmath,amssymb,amsthm,amsopn,array,natbib,titling} \usepackage[utf8]{inputenc} \usepackage{authblk} \usepackage{hyperref} \usepackage[english]{babel} \usepackage[scaled=0.9]{helvet} \usepackage[sc]{mathpazo} \usepackage{Sweave} \usepackage{color} \definecolor{link}{rgb}{0.45,0.51,0.67} \hypersetup{ colorlinks,% citecolor=link,% filecolor=link,% linkcolor=link,% urlcolor=link } %% load any required packages here \renewcommand{\today}{\begingroup \number \day\space \ifcase \month \or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space \number \year \endgroup} \def\rth{r^{th}} \let\code=\texttt \let\proglang=\texttt \let\pkg=\texttt \newcommand{\CRANpkg}[1]{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}}% \newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} \let\cpkg=\CRANpkg \bibliographystyle{plainnat} %\VignetteIndexEntry{Introduction to kedd} %\VignettePackage{kedd} \begin{document} \title{\bfseries Introduction to \pkg{kedd}} \author{by Arsalane Chouaib Guidoum\thanks{Department of Probabilities \& Statistics.\\Faculty of Mathematics. \\University of Science and Technology Houari Boumediene.\\ BP 32 El-Alia, U.S.T.H.B, Algeria.\\ \email{acguidoum@usthb.dz}}} \date{\today} \maketitle \section{Introduction} \pkg{kedd} \citep{kedd} is a package providing additional smoothing techniques to the \texttt{R} statistical system. Although various packages on the Comprehensive \texttt{R} Archive Network (CRAN) provide functions useful to nonparametric statistics, \pkg{kedd} aims to serve as a central location for more specifically of a nonparametric functions and data sets. The project was officially launched in December 2012 and is under active development. The current feature set of the package can be split in four main categories: compute the convolutions and derivatives of a kernel function, compute the kernel estimators for a density of probability and its derivatives, computing the bandwidth selectors with different methods, displaying the kernel estimators and selection functions of the bandwidth. Moreover, the package follows the general \texttt{R} philosophy of working with model objects. This means that instead of merely returning, say, a kernel estimator of $\rth$ derivative of a density, many functions will return an object containing, it's functions are S3 classes (\code{S3method}). The object can then be manipulated at one’s will using various extraction, summary or plotting functions. Whenever possible, we develop a graphical user interface of the various functions of a coherent whole, to facilitate the use of this package. \section{Documentation} It is a requirement of the \texttt{R} packaging system that every function and data set in a package has a help page. The \pkg{kedd} package follows this requirement strictly. In addition to the help pages, the package includes vignettes and demonstration scripts; running <>= vignette(package = "kedd") @ and <>= demo(package = "kedd") @ at the \texttt{R} prompt will give the list of each. \section{Requirements} \texttt{R} version >= 2.15.0. \section{Licence} This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package. \section{Collaboration and citation} Obviously, the package leaves many other fields of nonparametric statistics untouched. For this situation to change, we hope that experts in their field will join their efforts to ours and contribute code to the \pkg{kedd} project. The project will continue to grow and to improve by and for the community of developers and users. If you use \pkg{kedd} for smoothing techniques and computing bandwidth selectors of the $\rth$ derivative of a probability density, please cite the software in publications. Use <>= citation() @ or <>= citation("kedd") @ for information on how to cite the software. \section*{Note} Please send comments, error reports, etc. to the author via the addresses email. \begin{thebibliography}{1} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{{\tt #1}}\fi \bibitem[Guidoum, 2015]{kedd} Guidoum, A. C. (2015). \newblock \pkg{kedd}: Kernel estimator and bandwidth selection for density and its derivatives. \newblock \texttt{R} package version 1.0.3. \newblock \url{http://CRAN.R-project.org/package=kedd} \end{thebibliography} \end{document} kedd/MD50000644000176200001440000000504112614775365011511 0ustar liggesusers22d10973a19c5b47a56063a0f966973b *DESCRIPTION ff478c2d745c0fcaccb5c5512bb0edec *NAMESPACE 33fa2fdfa9575a278aa42fd11ec8fde0 *NEWS edb8b8e833de7114bf2c78a5fe2815c5 *R/AMISE.R f6d9200a93faaf97f5a3522912e13892 *R/BCV.R 44f186462327282597f8c40adccb3173 *R/CCV.R 578c8fe528d47d3f919ac8a312e025f5 *R/MCV.R 3c8d7bf4b231daa5fc46809d3c93484e *R/MLCV.R 5e1badb27d2e5bb120905506c4a7ba90 *R/TCV.R 533ed3698acd96173d58a309d5036a2d *R/UCV.R f2d49390f5fc00a97928f2e85b063563 *R/dkde.R 998bfdecf2f42e959ed389a5540e8fa5 *R/kernel.R b85646e7b187a331a297741571bae434 *R/utils.R 8a0b1f0c54e028f585449c8cffe24828 *build/vignette.rds 8e74a4e4b226a3d61ca1cf47e853be4a *data/bimodal.rda 33bd82497da86fe085ba2fdff451104c *data/claw.rda 1cb22e7d41066b1abba52da624fccfc9 *data/kurtotic.rda 455ae8d4b056e17386e7dcc215fef4b9 *data/outlier.rda 342ed9ed0633fc87c1b6b8e93ba93c1c *data/trimodal.rda b6a6fc610e4d5c50e139584844f6025a *demo/00Index 1f51ee57b46aa20ed84bba39774258c1 *demo/kedd.R 50d6c2080dddb8f307f51a654debd708 *inst/CITATION ff7451022eebc094eadacd4f846bef82 *inst/doc/Thumbs.db 822473d992d6b1a3ba6c9e75f88a9b0b *inst/doc/intro.R ba35671e7e3a63971e2a24f5a5a65755 *inst/doc/intro.Rnw fa02c70b68e65b96e4fa3d306ef08760 *inst/doc/intro.pdf 71e0ab5f0322d13359f7692893803db9 *inst/doc/kedd.R 5642242c2b8aa11389da53db7785d73e *inst/doc/kedd.Rnw 5ca6f399017af6df95ac575110e57f46 *inst/doc/kedd.pdf 7c51660c39f0e7cc834a3b600142a3f7 *man/claw.Rd 6a791ad20deb6ab3223d699c565b7abb *man/dkde.Rd 152cc7e51441b5b421760a1c264d13b8 *man/h.amise.Rd 8b65d02a1f6411a77cd36d3d58ce3a1a *man/h.bcv.Rd 2ecd9d35b9016bfc23419a60ad99ceb7 *man/h.ccv.Rd 18daaa43c9498a8470dbb28b53e6e1d3 *man/h.mcv.Rd 4b6199f72c22f201db5cc9d94c305000 *man/h.mlcv.Rd 930c3fff7f0cbfaec406c470e0dc68f4 *man/h.tcv.Rd df201d1546437e099d1013feddf9ea15 *man/h.ucv.Rd 326225e603f7b24069293b1e349615e5 *man/kedd-package.Rd c92866ad5bcd629bbd9dcf35aa15130f *man/kernel.conv.Rd 40f972aa03d24ca26f017941034e6d8f *man/kernel.fun.Rd fcaed1a1f6f45bc4ea1f8cf287696f15 *man/plot.dkde.Rd c7850224bcd4cfdc5217cb06c3713a91 *man/plot.h.amise.Rd ad3cb53e45290f69f087397823ec71b4 *man/plot.h.bcv.Rd e7b6b6f53992cd6405d0ab2737d4332c *man/plot.h.ccv.Rd aab6ac3f50e2fb8f7f4c186380772e9b *man/plot.h.mcv.Rd 158081da82fb371d31b8e807fa0eb1aa *man/plot.h.mlcv.Rd ca9b72cf8d313e52f2a6e38527f9680d *man/plot.h.tcv.Rd f85203a2c5780749307966df3deeed9a *man/plot.h.ucv.Rd b1494c916d5e5002148fcf62d293b22b *man/plot.kernel.conv.Rd d729c28befd325f760323aedc3aa5f0d *man/plot.kernel.fun.Rd ba35671e7e3a63971e2a24f5a5a65755 *vignettes/intro.Rnw 5642242c2b8aa11389da53db7785d73e *vignettes/kedd.Rnw kedd/build/0000755000176200001440000000000012614740771012271 5ustar liggesuserskedd/build/vignette.rds0000644000176200001440000000040412614740771014626 0ustar liggesusersuON0 IHHiڴ B,dJ̪M&9~~[!2gr.%\!gET B'FA{8OxTdTFwj:H-s 4t4Hó" Depends: R (>= 2.15.0) Description: Smoothing techniques and computing bandwidth selectors of the nth derivative of a probability density for one-dimensional data. License: GPL (>= 2) LazyLoad: yes LazyData: yes Classification/MSC: 62G05, 62G07, 65D10, 68N15 NeedsCompilation: no Packaged: 2015-10-30 19:16:09 UTC; A.C Repository: CRAN Date/Publication: 2015-10-31 00:19:16 kedd/man/0000755000176200001440000000000012614740640011740 5ustar liggesuserskedd/man/plot.h.mlcv.Rd0000644000176200001440000000255312614740640014400 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.mlcv} \alias{plot.h.mlcv} \alias{lines.h.mlcv} \title{ Plot for Maximum-Likelihood Cross-validation } \description{ The \code{\link{plot.h.mlcv}} function loops through calls to the \code{\link{h.mlcv}} function. Plot for maximum-likelihood cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.mlcv}(x, seq.bws=NULL, \dots) \method{lines}{h.mlcv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.mlcv} (output from \code{\link{h.mlcv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the maximum- likelihood cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d maximum-likelihood cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{seq.bws}{the sequence of bandwidths.} \item{mlcv}{the values of the maximum-likelihood cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.mlcv}}. } \examples{ plot(h.mlcv(bimodal)) } \keyword{plot} kedd/man/h.amise.Rd0000644000176200001440000001076712614740640013566 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{h.amise} \alias{h.amise} \alias{h.amise.default} \alias{print.h.amise} \title{ AMISE for Optimal Bandwidth Selectors } \description{ The (S3) generic function \code{h.amise} evaluates the asymptotic mean integrated squared error \bold{AMISE} for optimal smoothing parameters \eqn{h} of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.amise(x, \dots) \method{h.amise}{default}(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "triweight", "tricube", "biweight", "cosine"), \dots) } \arguments{ \item{x}{vector of data values.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.amise} asymptotic mean integrated squared error implements for choosing the optimal bandwidth \eqn{h} of a r'th derivative kernel density estimator.\cr We Consider the following AMISE version of the r'th derivative of \eqn{f} the r'th derivative of the kernel estimate (see Scott 1992, pp 131): \deqn{AMISE(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} h^{4} \mu_{2}^{2}(K) R\left(f^{(r+2)}\right)}{AMISE(h;r)= R(K(x;r))/ n h^(2r+1) + 0.25 mu(K(x))^2 h^4 R(f(x;r+2))} The optimal bandwidth minimizing this function is: \deqn{h_{(r)}^{\ast} = \left[\frac{(2r+1)R\left(K^{(r)}\right)}{\mu_{2}^{2}(K) R\left(f^{(r+2)}\right)}\right]^{1/(2r+5)} n^{-1/(2r+5)}}{h(r) = ( ((2r+1)R(K(x;r)))/(mu(K(x))^2 R(f(x;r+2))) )^(1/(2r+5)) n^-1/(2r+5)} whereof \deqn{\inf_{h > 0} AMISE(h;r) = \frac{2r+5}{4} R\left(K^{(r)}\right)^{\frac{4}{(2r+5)}} \left[ \frac{\mu_{2}^{2}(K)R\left(f^{(r+2)}\right)}{2r+1} \right]^{\frac{2r+1}{2r+5}} n^{-\frac{4}{2r+5}}}{inf AMISE(h;r) = 2r+5/4 R(K(x;r))^(4/2r+5) ((mu(K(x))^2 R(f(x;r+2)))/(2r+1))^((2r+1)/(2r+5)) n^-4/2r+5} which is the smallest possible AMISE for estimation of \eqn{f^{(r)}(x)}{f(x;r)} using the kernel \eqn{K(x)}{K(x)}, where \eqn{R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx}{R(K(x;r)) = int K(x;r)^2 dx} and \eqn{\mu_{2}(K) = \int_{R}x^{2} K(x) dx}{mu(K(x)) = int x^2 K(x) dx}.\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{h}{value of bandwidth parameter.} \item{amise}{the AMISE value.} } \references{ Bowman, A. W. and Azzalini, A. (1997). \emph{Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations}. Oxford University Press, Oxford. Radhey, S. S. (1987). MISE of kernel estimates of a density and its derivatives. \emph{Statistics and Probability Letters}, \bold{5}, 153--159. Scott, D. W. (1992). \emph{Multivariate Density Estimation. Theory, Practice and Visualization}. New York: Wiley. Sheather, S. J. (2004). Density estimation. \emph{Statistical Science}, \bold{19}, 588--597. Silverman, B. W. (1986). \emph{Density Estimation for Statistics and Data Analysis}. Chapman & Hall/CRC. London. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing}. Chapman and Hall, London. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.amise}}, see \code{\link[sm]{nmise}} in package \CRANpkg{sm} this function evaluates the mean integrated squared error of a density estimate (\code{deriv.order = 0}) which is constructed from data which follow a normal distribution. } \examples{ ## Derivative order = 0 h.amise(kurtotic,deriv.order = 0) ## Derivative order = 1 h.amise(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/plot.kernel.fun.Rd0000644000176200001440000000252612614740640015260 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.kernel.fun} \alias{plot.kernel.fun} \title{ Plot of r'th Derivative Kernel Function } \description{ The \code{\link{plot.kernel.fun}} function loops through calls to the \code{\link{kernel.fun}} function. Plot for r'th derivative kernel function one-dimensional. } \usage{ \method{plot}{kernel.fun}(x, \dots) } \arguments{ \item{x}{object of class \code{kernel.fun} (output from \code{\link{kernel.fun}}).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d for r'th derivative kernel function are sent to graphics window. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{kernel.fun}}. } \examples{ ## Gaussian kernel dev.new() par(mfrow=c(2,2)) plot(kernel.fun(kernel="gaussian",deriv.order=0)) plot(kernel.fun(kernel="gaussian",deriv.order=1)) plot(kernel.fun(kernel="gaussian",deriv.order=2)) plot(kernel.fun(kernel="gaussian",deriv.order=3)) ## Silverman kernel dev.new() par(mfrow=c(2,2)) plot(kernel.fun(kernel="silverman",deriv.order=0)) plot(kernel.fun(kernel="silverman",deriv.order=1)) plot(kernel.fun(kernel="silverman",deriv.order=2)) plot(kernel.fun(kernel="silverman",deriv.order=3)) } \keyword{plot} kedd/man/h.bcv.Rd0000644000176200001440000001441312614740640013232 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{h.bcv} \alias{h.bcv} \alias{h.bcv.default} \alias{print.h.bcv} \title{ Biased Cross-Validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.bcv} computes the biased cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.bcv(x, \dots) \method{h.bcv}{default}(x, whichbcv = 1, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, tol = 0.1 * lower, kernel = c("gaussian","epanechnikov", "triweight","tricube","biweight","cosine"), \dots) } \arguments{ \item{x}{vector of data values.} \item{whichbcv}{method selected, \code{1 = BCV1} or \code{2 = BCV2}, see details.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.bcv} biased cross-validation implements for choosing the bandwidth \eqn{h} of a r'th derivative kernel density estimator. if \code{whichbcv = 1} then \bold{BCV1} is selected (Scott and George 1987), and if \code{whichbcv = 2} used \bold{BCV2} (Jones and Kappenman 1991).\cr Scott and George (1987) suggest a method which has as its immediate target the \bold{AMISE} (e.g. Silverman 1986, section 3.3). We denote \eqn{\hat{\theta}_{r}(h)}{hat(theta)(h;r)} and \eqn{\bar{\theta}_{r}(h)}{bar(theta)(h;r)} (Peter and Marron 1987, Jones and Kappenman 1991) by: %%\deqn{\hat{\theta}_{r}(h) = (n-1)^{-1} n R\left(\hat{f}_{h}^{(r)}\right) - (n-1)^{-1} h^{-2r-1} R\left(K^{(r)}\right)}{} \deqn{\hat{\theta}_{r}(h)= \frac{(-1)^{r}}{n(n-1)h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{hat(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)} and %%\deqn{\bar{\theta}_{r}(h) = (-1)^{r} n^{-1} \sum_{i=1}^{n} \hat{f}_{h,i}^{(2r)}(X_{i})}{} \deqn{\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{bar(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K((x(j)-x(i)/h);2r)), i=1...n, j=1...n, j != i)} Scott and George (1987) proposed using \eqn{\hat{\theta}_{r}(h)}{hat(theta)(h;r)} to estimate \eqn{f^{(r)}(x)}{f(x;r)}. Thus, \eqn{\hat{h}^{(r)}_{BCV1}}{h(r)_(BCV1)}, say, is the \eqn{h} that minimises: \deqn{BCV1(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} \mu_{2}^{2}(K) h^{4} \hat{\theta}_{r+2}(h)}{BCV1(h;r) = R(K(x;r))/ n h^(2r+1) + 0.25 mu(K(x))^2 h^4 hat(theta)(h;r+2)} and we define \eqn{\hat{h}^{(r)}_{BCV2}}{h(r)_(BCV2)} as the minimiser of (Jones and Kappenman 1991): \deqn{BCV2(h;r)= \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{1}{4} \mu_{2}^{2}(K) h^{4} \bar{\theta}_{r+2}(h)}{BCV2(h;r) = R(K(x;r))/ n h^(2r+1) + 0.25 mu(K(x))^2 h^4 bar(theta)(h;r+2)} where \eqn{K^{(r)} \ast K^{(r)} (x)}{K(x;r)*K(x;r)} is the convolution of the r'th derivative kernel function \eqn{K^{(r)}(x)}{K(x;r)} (see \code{\link{kernel.conv}} and \code{\link{kernel.fun}}); \eqn{R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx}{R(K(x;r)) = int K(x;r)^2 dx} and \eqn{\mu_{2}(K) = \int_{R}x^{2} K(x) dx}{mu(K(x)) = int x^2 K(x) dx}.\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{whichbcv}{method selected.} \item{h}{value of bandwidth parameter.} \item{min.bcv}{the minimal BCV value.} } \references{ Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. \emph{Scandinavian Journal of Statistics}, \bold{19}, 337--349. Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). A brief survey of bandwidth selection for density estimation. \emph{Journal of the American Statistical Association}, \bold{91}, 401--407. Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. \emph{Statistics and Probability Letters}, \bold{6}, 109--115. Scott, D.W. and George, R. T. (1987). Biased and unbiased cross-validation in density estimation. \emph{Journal of the American Statistical Association}, \bold{82}, 1131--1146. Sheather,S. J. (2004). Density estimation. \emph{Statistical Science}, \bold{19}, 588--597. Tarn, D. (2007). \CRANpkg{ks}: Kernel density estimation and kernel discriminant analysis for multivariate data in \R. \emph{Journal of Statistical Software}, \bold{21}(7), 1--16. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing}. Chapman and Hall, London. Wolfgang, H. (1991). \emph{Smoothing Techniques}, \emph{With Implementation in S}. Springer-Verlag, New York. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.bcv}}, see \code{\link[stats]{bw.bcv}} in package "stats" and \code{\link[MASS]{bcv}} in package \CRANpkg{MASS} for Gaussian kernel only if \code{deriv.order = 0}, \code{\link[ks]{Hbcv}} for bivariate data in package \CRANpkg{ks} for Gaussian kernel only if \code{deriv.order = 0}, \code{\link[locfit]{kdeb}} in package \CRANpkg{locfit} if \code{deriv.order = 0}. } \examples{ ## EXAMPLE 1: x <- rnorm(100) h.bcv(x,whichbcv = 1, deriv.order = 0) h.bcv(x,whichbcv = 2, deriv.order = 0) ## EXAMPLE 2: ## Derivative order = 0 h.bcv(kurtotic,deriv.order = 0) ## Derivative order = 1 h.bcv(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/dkde.Rd0000644000176200001440000002372112614740640013143 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{dkde} \alias{dkde} \alias{dkde.default} \alias{print.dkde} \title{ Derivatives of Kernel Density Estimator } \description{ The (S3) generic function \code{dkde} computes the r'th derivative of kernel density estimator for one-dimensional data. Its default method does so with the given kernel and bandwidth \eqn{h} for one-dimensional observations. } \usage{ dkde(x, \dots) \method{dkde}{default}(x, y = NULL, deriv.order = 0, h, kernel = c("gaussian", "epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine"), \dots) } \arguments{ \item{x}{the data from which the estimate is to be computed.} \item{y}{the points of the grid at which the density derivative is to be estimated; the defaults are \eqn{\tau * h} outside of range(\eqn{x}), where \eqn{\tau = 4}.} \item{deriv.order}{derivative order (scalar).} \item{h}{the smoothing bandwidth to be used, can also be a character string giving a rule to choose the bandwidth, see \code{\link{h.bcv}}. The default \code{\link{h.ucv}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ A simple estimator for the density derivative can be obtained by taking the derivative of the kernel density estimate. If the kernel \eqn{K(x)} is differentiable \eqn{r} times then the r'th density derivative estimate can be written as: \deqn{\hat{f}^{(r)}_{h}(x)=\frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_{i}}{h}\right)}{hat(f)(x;r) = n^-1 h^-(r+1) sum(K(x-X(i)/h),i=1...n)} where, \deqn{K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)}{K(x;r) = d^r /d x^r K(x)} for \eqn{r = 0, 1, 2, \dots} The following assumptions on the density \eqn{f^{(r)}(x)}{f(x;r)}, the bandwidth \eqn{h}, and the kernel \eqn{K(x)}: \enumerate{ \item{}{The \eqn{(r+2)} derivative \eqn{f^{(r+2)}(x)}{f(x;r+2)} is continuous, square integrable and ultimately monotone.} \item{}{\eqn{\lim_{n \to \infty} h = 0}{lim_(n -- > Inf) h = 0} and \eqn{\lim_{n \to \infty}n h^{2r+1} = \infty}{lim_(n --> Inf) nh^(2r+1) = Inf} i.e., as the number of samples \eqn{n} is increased \eqn{h} approaches zero at a rate slower than \eqn{1/n^{2r+1}}.} \item{}{\eqn{K(x) \geq 0}{K(x) >= 0} and \eqn{\int_{R} K(x) dx = 1}{int K(x) dx = 1}. The kernel function is assumed to be symmetric about the origin i.e., \eqn{\int_{R} xK^{(r)}(x) dx = 0}{int x k(x;r) dx = 0} for even \eqn{r} and has finite second moment i.e., \eqn{\mu_{2}(K)=\int_{R}x^{2} K(x) dx < \infty}{mu(K(x)) = int x^2 K(x) dx < Inf}.} } Some theoretical properties of the estimator \eqn{\hat{f}^{(r)}_{h}}{hat(f)(x;r)} have been investigated, among others, by Bhattacharya (1967), Schuster (1969). Let us now turn to the statistical properties of estimator. We are interested in the mean squared error since it combines squared bias and variance. The \bold{bias} can be written as: \deqn{E\left[\hat{f}^{(r)}_{h}(x)\right]- f^{(r)}(x) = \frac{1}{2}h^{2}\mu_{2}(K) f^{(r+2)}(x)+o(h^{2})}{E[hat(f)(x;r)] - f(x;r) = 0.5 h^2 mu(K(x)) f(x;r+2) + o(h^2)} The \bold{variance} of the estimator can be written as: \deqn{VAR\left[\hat{f}^{(r)}_{h}(x)\right]=\frac{f(x) R\left(K^{(r)}\right)}{nh^{2r+1}} + o(1/nh^{2r+1})}{VAR(hat(f)(x;r)) = f(x) R(K(x;r)) / n h^(2r+1) + o(1/nh^(2r+1)) } with, \eqn{R\left(K^{(r)}\right) = \int_{R} \left(K^{(r)}(x)\right)^{2}dx.}{R(K(x;r)) = int K(x;r)^2 dx.} The \bold{MSE} (Mean Squared Error) for kernel density derivative estimators can be written as: \deqn{MSE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{f(x)R\left(K^{(r)}\right)}{nh^{2r+1}}+\frac{1}{4}h^{4}\mu_{2}^{2}(K) f^{(r+1)}(x)^{2}+o(h^{4}+1/nh^{2r+1})}{MSE(hat(f)(x;r),f(x;r)) = f(x) R(K(x;r)) / nh^(2r+1) + 1/4 h^4 mu(K(x))^2 f(x;r+1)^2 + o(h^4 + 1/ nh^(2r+1))} It follows that the MSE-optimal bandwidth for estimating \eqn{\hat{f}^{(r)}_{h}S(x)}{hat(f)(x;r)}, is of order \eqn{n^{-1/(2r+5)}}{n^(-1/2r+5)}. Therefore, the estimation of \eqn{\hat{f}^{(1)}_{h}(x)}{hat(f)(x;1)} requires a bandwidth of order \eqn{n^{-1/7}}{n^-1/7} compared to the optimal \eqn{n^{-1/5}}{n^-1/5} for estimating \eqn{f(x)}{f(x)} itself. It reveals the increasing difficulty in problems of estimating higher derivatives.\cr The \bold{MISE} (Mean Integrated Squared Error) can be written as: \deqn{MISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)+o(h^{4}+1/nh^{2r+1})}{MISE(hat(f)(x;r),f(x;r))=AMISE(hat(f)(x;r),f(x;r)) + o(h^4 + 1/nh^(2r+1))} where, \deqn{AMISE\left(\hat{f}^{(r)}_{h}(x),f^{(r)}(x)\right)=\frac{1}{nh^{2r+1}}R\left(K^{(r)}\right)+\frac{1}{4}h^{4}\mu_{2}^{2}(K)R\left(f^{(r+2)}\right)}{AMISE(hat(f)(x;r),f(x;r)) = R(K(x;r))/n h^(2r+1) + 1/4 h^2 mu(K(x))^2 R(f(x;r+2))} with: \eqn{R\left(f^{(r)}(x)\right) = \int_{R} \left(f^{(r)}(x)\right)^{2}dx.}{R(f(x;r)) = int f(x;r)^2 dx.}\cr The performance of kernel is measured by \bold{MISE} or \bold{AMISE} (Asymptotic MISE).\cr If the bandwidth \code{h} is missing from \code{dkde}, then the default bandwidth is \code{h.ucv(x,deriv.order,kernel)} (Unbiased cross-validation, see \code{\link{h.ucv}}).\cr For more details see references. } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{h}{the bandwidth value to use.} \item{eval.points}{the coordinates of the points where the density derivative is estimated.} \item{est.fx}{the estimated density derivative values.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \references{ Alekseev, V. G. (1972). Estimation of a probability density function and its derivatives. \emph{Mathematical notes of the Academy of Sciences of the USSR}. \bold{12} (5), 808--811. Alexandre, B. T. (2009). \emph{Introduction to Nonparametric Estimation}. Springer-Verlag, New York. Bowman, A. W. and Azzalini, A. (1997). \emph{Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations}. Oxford University Press, Oxford. Bhattacharya, P. K. (1967). Estimation of a probability density function and Its derivatives. \emph{Sankhya: The Indian Journal of Statistics, Series A}, \bold{29}, 373--382. Jeffrey, S. S. (1996). \emph{Smoothing Methods in Statistics}. Springer-Verlag, New York. Radhey, S. S. (1987). MISE of kernel estimates of a density and its derivatives. \emph{Statistics and Probability Letters}, \bold{5}, 153--159. Scott, D. W. (1992). \emph{Multivariate Density Estimation. Theory, Practice and Visualization}. New York: Wiley. Schuster, E. F. (1969) Estimation of a probability density function and its derivatives. \emph{The Annals of Mathematical Statistics}, \bold{40} (4), 1187--1195. Silverman, B. W. (1986). \emph{Density Estimation for Statistics and Data Analysis}. Chapman & Hall/CRC. London. Stoker, T. M. (1993). Smoothing bias in density derivative estimation. \emph{Journal of the American Statistical Association}, \bold{88}, 855--863. Venables, W. N. and Ripley, B. D. (2002). \emph{Modern Applied Statistics with S}. New York: Springer. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing}. Chapman and Hall, London. Wolfgang, H. (1991). \emph{Smoothing Techniques}, \emph{With Implementation in S}. Springer-Verlag, New York. } \note{This function are available in other packages such as \CRANpkg{KernSmooth}, \CRANpkg{sm}, \CRANpkg{np}, \CRANpkg{GenKern} and \CRANpkg{locfit} if \code{deriv.order=0}, and in \CRANpkg{ks} package for Gaussian kernel only if \code{0 <= deriv.order <= 10}. } \seealso{ \code{\link{plot.dkde}}, see \code{\link[stats]{density}} in package "stats" if \code{deriv.order = 0}, and \code{\link[ks]{kdde}} in package \CRANpkg{ks}. } \examples{ ## EXAMPLE 1: Simple example of a Gaussian density derivative x <- rnorm(100) dkde(x,deriv.order=0) ## KDE of f dkde(x,deriv.order=1) ## KDDE of d/dx f dkde(x,deriv.order=2) ## KDDE of d^2/x^2 f dkde(x,deriv.order=3) ## KDDE of d^3/x^3 f dev.new() par(mfrow=c(2,2)) plot(dkde(x,deriv.order=0)) plot(dkde(x,deriv.order=1)) plot(dkde(x,deriv.order=2)) plot(dkde(x,deriv.order=3)) ## EXAMPLE 2: Bimodal Gaussian density derivative ## show the kernels in the dkde parametrization fx <- function(x) 0.5 * dnorm(x,-1.5,0.5) + 0.5 * dnorm(x,1.5,0.5) fx1 <- function(x) 0.5 *(-4*x-6)* dnorm(x,-1.5,0.5) + 0.5 *(-4*x+6) * dnorm(x,1.5,0.5) ## 'h = 0.3' ; 'Derivative order = 0' kernels <- eval(formals(dkde.default)$kernel) dev.new() plot(dkde(bimodal,h=0.3),sub=paste("Derivative order = 0",";", "Bandwidth =0.3 "),ylim=c(0,0.5), main = "Bimodal Gaussian Density") for(i in 2:length(kernels)) lines(dkde(bimodal, h = 0.3, kernel = kernels[i]), col = i) curve(fx,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)), lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015) ## 'h = 0.6' ; 'Derivative order = 1' kernels <- eval(formals(dkde.default)$kernel)[-3] dev.new() plot(dkde(bimodal,deriv.order=1,h=0.6),main = "Bimodal Gaussian Density Derivative",sub=paste ("Derivative order = 1",";","Bandwidth =0.6"),ylim=c(-0.6,0.6)) for(i in 2:length(kernels)) lines(dkde(bimodal,deriv.order=1, h = 0.6, kernel = kernels[i]), col = i) curve(fx1,add=TRUE,lty=8) legend("topright", legend = c(TRUE,kernels), col = c("black",seq(kernels)), lty = c(8,rep(1,length(kernels))),cex=0.7, inset = .015) } \keyword{smooth} \keyword{nonparametric} \keyword{density derivative} kedd/man/plot.h.ucv.Rd0000644000176200001440000000275012614740640014233 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.ucv} \alias{plot.h.ucv} \alias{lines.h.ucv} \title{ Plot for Unbiased Cross-Validation } \description{ The \code{\link{plot.h.ucv}} function loops through calls to the \code{\link{h.ucv}} function. Plot for unbiased cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.ucv}(x, seq.bws=NULL, \dots) \method{lines}{h.ucv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.ucv} (output from \code{\link{h.ucv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the unbiased cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d unbiased cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{ucv}{the values of the unbiased cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.ucv}}. } \examples{ par(mfrow=c(2,1)) plot(h.ucv(trimodal,deriv.order=0),seq.bws=seq(0.06,0.2,length=50)) plot(h.ucv(trimodal,deriv.order=1),seq.bws=seq(0.06,0.2,length=50)) } \keyword{plot} kedd/man/h.ucv.Rd0000644000176200001440000001547712614740640013270 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{h.ucv} \alias{h.ucv} \alias{h.ucv.default} \alias{print.h.ucv} \title{ Unbiased (Least-Squares) Cross-Validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.ucv} computes the unbiased (least-squares) cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.ucv(x, \dots) \method{h.ucv}{default}(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine"), \dots) } \arguments{ \item{x}{vector of data values.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.ucv} unbiased (least-squares) cross-validation implements for choosing the bandwidth \eqn{h} of a r'th derivative kernel density estimator.\cr Rudemo (1982) and Bowman (1984) proposed a so-called unbiased (least-squares) cross-validation (UCV) in kernel density estimator. An adaptation of unbiased cross-validation is proposed by Wolfgang et al. (1990) for bandwidth choice in the r'th derivative of kernel density estimator. The essential idea of this methods, for the estimation of \eqn{f^{(r)}(x)}{f(x;r)} (\eqn{r} is derivative order), is to use the bandwidth \eqn{h} which minimizes the function: \deqn{UCV(h;r) = \int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} - 2n^{-1}(-1)^{r}\sum_{i=1}^{n} \hat{f}_{h,i}^{(2r)}(X_{i})}{UCV(h;r)= int (hat(f)(x;r))^2 - 2 n^(-1) (-1)^r sum(hat(f)(X_i;2r),i=1,...n)} The bandwidth minimizing this function is: \deqn{\hat{h}^{(r)}_{ucv} = \arg \min_{h^{(r)}} UCV(h;r)}{h(r) = argmin UCV(h;r)} for \eqn{r = 0, 1, 2, \dots}\cr where \deqn{\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{int (hat(f)(x;r))^2 dx = R(k(x;r))/n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)} and \eqn{K^{(r)} \ast K^{(r)} (x)}{K(x;r)*K(x;r)} is the convolution of the r'th derivative kernel function \eqn{K^{(r)}(x)}{K(x;r)} (see \code{\link{kernel.conv}} and \code{\link{kernel.fun}}).\cr The estimate \eqn{\hat{f}_{h,i}^{(2r)}(x)}{hat(f)(x;2r)} on the subset \eqn{\{X_{j}\}_{j \neq i}}{(X_j)_(j != i)} denoting the leave-one-out estimator, can be written: \deqn{\hat{f}_{h,i}^{(2r)}(X_{i}) = \frac{1}{(n-1) h^{2r+1}} \sum_{j \neq i} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{hat(f)(X_i;2r) = 1 /(n-1) h^(2r+1) sum(K(x(j)-x(i)/h;2r), j != i)} The function \eqn{UCV(h;r)} is unbiased cross-validation in the sense that \eqn{E[UCV]=MISE[\hat{f}_{h}^{(r)}(x)]-R(f^{(r)}(x))}{E[UCV(h;r)] = MISE[hat(f)(x;r)]-R(f(x;r))} (see, Scott and George 1987). Can be simplified to give the computationally: \deqn{UCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1 ;j \neq i}^{n} \left(K^{(r)} \ast K^{(r)} -2K^{(2r)}\right) \left(\frac{X_{j}-X_{i}}{h}\right)}{UCV(h;r) = R(k(x;r))/n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r) - 2 K(.;r))(x(j)-x(i)/h), i=1...n, j=1...n, j != i)} where \eqn{R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx}{R(K(x;r)) = int K(x;r)^2 dx}.\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{h}{value of bandwidth parameter.} \item{min.ucv}{the minimal UCV value.} } \references{ Bowman, A. (1984). An alternative method of cross-validation for the smoothing of kernel density estimates. \emph{Biometrika}, \bold{71}, 353--360. Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. \emph{Scandinavian Journal of Statistics}, \bold{19}, 337--349. Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). A brief survey of bandwidth selection for density estimation. \emph{Journal of the American Statistical Association}, \bold{91}, 401--407. Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. \emph{Statistics and Probability Letters}, \bold{6}, 109--115. Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. \emph{Scandinavian Journal of Statistics}, \bold{9}, 65--78. Scott, D.W. and George, R. T. (1987). Biased and unbiased cross-validation in density estimation. \emph{Journal of the American Statistical Association}, \bold{82}, 1131--1146. Sheather, S. J. (2004). Density estimation. \emph{Statistical Science}, \bold{19}, 588--597. Tarn, D. (2007). \CRANpkg{ks}: Kernel density estimation and kernel discriminant analysis for multivariate data in \R. \emph{Journal of Statistical Software}, \bold{21}(7), 1--16. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing}. Chapman and Hall, London. Wolfgang, H. (1991). \emph{Smoothing Techniques}, \emph{With Implementation in S}. Springer-Verlag, New York. Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. \emph{Journal of the Royal Statistical Society, Series B}, 223--232. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.ucv}}, see \code{\link[stats]{bw.ucv}} in package "stats" and \code{\link[MASS]{ucv}} in package \CRANpkg{MASS} for Gaussian kernel only if \code{deriv.order = 0}, \code{\link[ks]{hlscv}} in package \CRANpkg{ks} for Gaussian kernel only if \code{0 <= deriv.order <= 5}, \code{\link[locfit]{kdeb}} in package \CRANpkg{locfit} if \code{deriv.order = 0}. } \examples{ ## Derivative order = 0 h.ucv(kurtotic,deriv.order = 0) ## Derivative order = 1 h.ucv(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/h.tcv.Rd0000644000176200001440000001054112614740640013252 0ustar liggesusers\name{h.tcv} \alias{h.tcv} \alias{h.tcv.default} \alias{print.h.tcv} \title{ Trimmed Cross-Validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.tcv} computes the trimmed cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.tcv(x, \dots) \method{h.tcv}{default}(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine"), \dots) } \arguments{ \item{x}{vector of data values.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.tcv} trimmed cross-validation implements for choosing the bandwidth \eqn{h} of a r'th derivative kernel density estimator.\cr Feluch and Koronacki (1992) proposed a so-called trimmed cross-validation (TCV) in kernel density estimator, a simple modification of the unbiased (least-squares) cross-validation criterion. We consider the following "trimmed" version of "unbiased", to be minimized with respect to \eqn{h}: \deqn{\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} - 2 \frac{(-1)^{r}}{n(n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1; j \neq i} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)\chi\left(|X_{i}-X_{j}| > c_{n}\right)}{int (hat(f)(x;r))^2 - 2 (-1)^r / n (n-1)h^(2r+1) sum(sum(K(x(j)-x(i)/h;2r)),i=1...n,j=1...n, j!=i) chi(|X(i)-X(j)| > c(n))} where \eqn{\chi(.)}{chi(.)} denotes the indicator function and \eqn{c_{n}}{c(n)} is a sequence of positive constants, \eqn{c_{n}/ h^{2r+1} \rightarrow 0}{c(n)/h^(2r+1) --> 0} as \eqn{n \rightarrow \infty}{n --> Inf}, and \deqn{\int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{int (hat(f)(x;r))^2 dx = R(k(x;r))/n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)} the trimmed cross-validation function is defined by: \deqn{TCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n(n-1)h^{2r+1}}\sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} \varphi^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{TCV(h;r) = R(K(x;r))/ n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum( sum(varphi(x(j)-x(i)/h;r) ),i=1...n,j=1...n,j != i) } whit \deqn{\varphi^{(r)}(c) = \left(K^{(r)} \ast K^{(r)} - 2 K^{(2r)} \chi\left(|c| > c_{n}/h^{2r+1}\right) \right)(c)}{varphi(c;r)= K(c;r)*K(c;r) - 2 K(c;2r) chi( |c| > c(n)/h^(2r+1) )} here we take \eqn{c_{n} = 1/n}{c(n) = 1/n}, for assure the convergence. Where \eqn{K^{(r)} \ast K^{(r)} (x)}{K(x;r)*K(x;r)} is the convolution of the r'th derivative kernel function \eqn{K^{(r)}(x)}{K(x;r)} (see \code{\link{kernel.conv}} and \code{\link{kernel.fun}}).\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{h}{value of bandwidth parameter.} \item{min.tcv}{the minimal TCV value.} } \references{ Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. \emph{Computational Statistics and Data Analysis}, \bold{13}, 143--151. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.tcv}}. } \examples{ ## Derivative order = 0 h.tcv(kurtotic,deriv.order = 0) ## Derivative order = 1 h.tcv(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/plot.h.ccv.Rd0000644000176200001440000000267012614740640014212 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.ccv} \alias{plot.h.ccv} \alias{lines.h.ccv} \title{ Plot for Complete Cross-Validation } \description{ The \code{\link{plot.h.ccv}} function loops through calls to the \code{\link{h.ccv}} function. Plot for complete cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.ccv}(x, seq.bws=NULL, \dots) \method{lines}{h.ccv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.ccv} (output from \code{\link{h.ccv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the complete cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d complete cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{ccv}{the values of the complete cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.ccv}}. } \examples{ par(mfrow=c(2,1)) plot(h.ccv(trimodal,deriv.order=0),main="") plot(h.ccv(trimodal,deriv.order=1),main="") } \keyword{plot} kedd/man/plot.dkde.Rd0000644000176200001440000000252312614740640014115 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.dkde} \alias{plot.dkde} \alias{lines.dkde} \title{ Plot for Kernel Density Derivative Estimate } \description{ The \code{\link{plot.dkde}} function loops through calls to the \code{\link{dkde}} function. Plot for kernel density derivative estimate for 1-dimensional data. } \usage{ \method{plot}{dkde}(x, fx = NULL, \dots) \method{lines}{dkde}(x, \dots) } \arguments{ \item{x}{object of class \code{dkde} (output from \code{\link{dkde}}).} \item{fx}{add to graphics the true density derivative (class :\code{\link{function}}), to compare it by the density derivative to estimate.} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \details{ The 1-d plot is a standard plot of a 1-d curve. If \code{!is.null(fx)} then a true density derivative is added. } \value{ Plot of 1-d kernel density derivative estimates are sent to graphics window. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{dkde}}, \code{\link[stats]{plot.density}} in package "stats" if \code{deriv.order = 0}. } \examples{ plot(dkde(kurtotic,deriv.order=0,kernel="gaussian"),sub="") lines(dkde(kurtotic,deriv.order=0,kernel="biweight"),col="red") } \keyword{plot} kedd/man/plot.h.mcv.Rd0000644000176200001440000000267012614740640014224 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.mcv} \alias{plot.h.mcv} \alias{lines.h.mcv} \title{ Plot for Modified Cross-Validation } \description{ The \code{\link{plot.h.mcv}} function loops through calls to the \code{\link{h.mcv}} function. Plot for modified cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.mcv}(x, seq.bws=NULL, \dots) \method{lines}{h.mcv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.mcv} (output from \code{\link{h.mcv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the modified cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d modified cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{mcv}{the values of the modified cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.mcv}}. } \examples{ par(mfrow=c(2,1)) plot(h.mcv(trimodal,deriv.order=0),main="") plot(h.mcv(trimodal,deriv.order=1),main="") } \keyword{plot} kedd/man/h.mlcv.Rd0000644000176200001440000000644612614740640013430 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{h.mlcv} \alias{h.mlcv} \alias{h.mlcv.default} \alias{print.h.mlcv} \title{ Maximum-Likelihood Cross-validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.mlcv} computes the maximum likelihood cross-validation (Kullback-Leibler information) bandwidth selector of a one-dimensional kernel density estimate. } \usage{ h.mlcv(x, \dots) \method{h.mlcv}{default}(x, lower = 0.1, upper = 5, tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine"), \dots) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{x}{vector of data values.} \item{lower, upper}{range over which to maximize. The default is almost always satisfactory.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.mlcv} maximum-likelihood cross-validation implements for choosing the optimal bandwidth \eqn{h} of kernel density estimator.\cr This method was proposed by Habbema, Hermans, and Van den Broeck (1971) and by Duin (1976). The maximum-likelihood cross-validation (MLCV) function is defined by: \deqn{MLCV(h) = n^{-1} \sum_{i=1}^{n} \log\left[\hat{f}_{h,i}(x)\right]}{MLCV(h) = n^-1 sum( log(hat(f(h))),i=1...n)} the estimate \eqn{\hat{f}_{h,i}(x)}{hat(f)(x)} on the subset \eqn{\{X_{j}\}_{j \neq i}}{(X_j)_(j != i)} denoting the leave-one-out estimator, can be written: \deqn{\hat{f}_{h,i}(X_{i}) = \frac{1}{(n-1) h} \sum_{j \neq i} K \left(\frac{X_{j}-X_{i}}{h}\right)}{hat(f)(X_i) = 1/(n-1) h sum(K(x(j)-x(i)/h), j != i)} Define that \eqn{h_{mlcv}}{h(mlcv)} as good which approaches the finite maximum of \eqn{MLCV(h)}{MLCV(h)}: \deqn{h_{mlcv} = \arg \max_{h} MLCV(h) = \arg \max_{h} \left(n^{-1} \sum_{i=1}^{n} \log\left[\sum_{j \neq i} K \left(\frac{X_{j}-X_{i}}{h}\right)\right]-\log[(n-1)h]\right)}{h(mlcv)= argmax MLCV(h) = argmax n^-1 sum(log(sum(K(x(j)-x(i)/h),j != i)),i=1...n) - log((n-1)h)} } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{h}{value of bandwidth parameter.} \item{mlcv}{the maximal likelihood CV value.} } \references{ Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974) A stepwise discrimination analysis program using density estimation. \emph{Compstat 1974: Proceedings in Computational Statistics}. Physica Verlag, Vienna. Duin, R. P. W. (1976). On the choice of smoothing parameters of Parzen estimators of probability density functions. \emph{IEEE Transactions on Computers}, \bold{C-25}, 1175--1179. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.mlcv}}, see \code{\link[locfit]{lcv}} in package \CRANpkg{locfit}. } \examples{ h.mlcv(bimodal) h.mlcv(bimodal, kernel ="epanechnikov") } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/plot.h.tcv.Rd0000644000176200001440000000272612614740640014235 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.tcv} \alias{plot.h.tcv} \alias{lines.h.tcv} \title{ Plot for Trimmed Cross-Validation } \description{ The \code{\link{plot.h.tcv}} function loops through calls to the \code{\link{h.tcv}} function. Plot for trimmed cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.tcv}(x, seq.bws=NULL, \dots) \method{lines}{h.tcv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.tcv} (output from \code{\link{h.tcv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the trimmed cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d trimmed cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{tcv}{the values of the trimmed cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.tcv}}. } \examples{ par(mfrow=c(2,1)) plot(h.tcv(trimodal,deriv.order=0),main="") plot(h.tcv(trimodal,deriv.order=1),seq.bws=seq(0.1,0.5,length.out=50),main="") } \keyword{plot} kedd/man/kernel.conv.Rd0000644000176200001440000000606512614740640014462 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{kernel.conv} \alias{kernel.conv} \alias{kernel.conv.default} \title{ Convolutions of r'th Derivative for Kernel Function } \description{ The (S3) generic function \code{kernel.conv} computes the convolution of r'th derivative for kernel function. } \usage{ kernel.conv(x, \dots) \method{kernel.conv}{default}(x = NULL, deriv.order = 0,kernel = c("gaussian","epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine", "silverman"), \dots) } \arguments{ \item{x}{points at which the convolution of kernel derivative is to be evaluated.} \item{deriv.order}{derivative order (scalar).} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ The convolution of r'th derivative for kernel function is written \eqn{K^{(r)}\ast K^{(r)}}{K(x;r)*K(x;r)}. It is defined as the integral of the product of the derivative for kernel. As such, it is a particular kind of integral transform: \deqn{K^{(r)} \ast K^{(r)}(x) = \int_{-\infty}^{+\infty} K^{(r)}(y)K^{(r)}(x-y)dy}{K(x;r)*k(x;r) = int K(y;r) K(x-y;r) dy} where: \deqn{K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)}{K(x;r) = d^r / dx^r K(x)} for \eqn{r = 0, 1, 2, \dots} } \value{ \item{kernel }{name of kernel to use.} \item{deriv.order }{the derivative order to use.} \item{x }{the n coordinates of the points where the convolution of kernel derivative is evaluated.} \item{kx }{the convolution of kernel derivative values.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \references{ Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). \emph{NIST Handbook of Mathematical Functions}. Cambridge University Press, New York, USA. Scott, D. W. (1992). \emph{Multivariate Density Estimation. Theory, Practice and Visualization}. New York: Wiley. Silverman, B. W. (1986). \emph{Density Estimation for Statistics and Data Analysis}. Chapman & Hall/CRC. London. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing.} Chapman and Hall, London. Wolfgang, H. (1991). \emph{Smoothing Techniques, With Implementation in S.} Springer-Verlag, New York. } \seealso{ \code{\link{plot.kernel.conv}}, \code{\link[stats]{kernapply}} in package "stats" for computes the convolution between an input sequence, and \code{\link[stats]{convolve}} use the Fast Fourier Transform (\code{\link[stats]{fft}}) to compute the several kinds of convolutions of two sequences. } \examples{ kernels <- eval(formals(kernel.conv.default)$kernel) kernels ## gaussian kernel.conv(x = 0,kernel=kernels[1],deriv.order=0) kernel.conv(x = 0,kernel=kernels[1],deriv.order=1) ## silverman kernel.conv(x = 0,kernel=kernels[9],deriv.order=0) kernel.conv(x = 0,kernel=kernels[9],deriv.order=1) } \keyword{nonparametric} \keyword{kernel} kedd/man/plot.kernel.conv.Rd0000644000176200001440000000262612614740640015436 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.kernel.conv} \alias{plot.kernel.conv} \title{ Plot for Convolutions of r'th Derivative Kernel Function } \description{ The \code{\link{plot.kernel.conv}} function loops through calls to the \code{\link{kernel.conv}} function. Plot for convolutions of r'th derivative kernel function one-dimensional. } \usage{ \method{plot}{kernel.conv}(x, \dots) } \arguments{ \item{x}{object of class \code{kernel.conv} (output from \code{\link{kernel.conv}}).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d for convolution of r'th derivative kernel function are sent to graphics window. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{kernel.conv}}. } \examples{ ## Gaussian kernel dev.new() par(mfrow=c(2,2)) plot(kernel.conv(kernel="gaussian",deriv.order=0)) plot(kernel.conv(kernel="gaussian",deriv.order=1)) plot(kernel.conv(kernel="gaussian",deriv.order=2)) plot(kernel.conv(kernel="gaussian",deriv.order=3)) ## Silverman kernel dev.new() par(mfrow=c(2,2)) plot(kernel.conv(kernel="silverman",deriv.order=0)) plot(kernel.conv(kernel="silverman",deriv.order=1)) plot(kernel.conv(kernel="silverman",deriv.order=2)) plot(kernel.conv(kernel="silverman",deriv.order=3)) } \keyword{plot} kedd/man/h.mcv.Rd0000644000176200001440000000723612614740640013252 0ustar liggesusers\name{h.mcv} \alias{h.mcv} \alias{h.mcv.default} \alias{print.h.mcv} \title{ Modified Cross-Validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.mcv} computes the modified cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.mcv(x, \dots) \method{h.mcv}{default}(x, deriv.order = 0, lower = 0.1 * hos, upper = 2 * hos, tol = 0.1 * lower, kernel = c("gaussian", "epanechnikov", "triweight", "tricube", "biweight", "cosine"), \dots) } \arguments{ \item{x}{vector of data values.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.mcv} modified cross-validation implements for choosing the bandwidth \eqn{h} of a r'th derivative kernel density estimator.\cr Stute (1992) proposed a so-called modified cross-validation (MCV) in kernel density estimator. This method can be extended to the estimation of derivative of a density, the essential idea based on approximated the problematic term by the aid of the Hajek projection (see Stute 1992). The minimization criterion is defined by: \deqn{MCV(h;r) = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n(n-1)h^{2r+1}}\sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} \varphi^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{MCV(h;r) = R(K(x;r))/ n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum( sum(varphi(x(j)-x(i)/h;r) ),i=1...n,j=1...n,j != i) } whit \deqn{\varphi^{(r)}(c) = \left(K^{(r)} \ast K^{(r)} - K^{(2r)} - \frac{\mu_{2}(K)}{2}K^{(2r+2)} \right)(c)}{varphi(c;r)= K(c;r)*K(c;r) - K(c;2r) - 0.5 mu(K(c)) K(c;2r+2)} and \eqn{K^{(r)} \ast K^{(r)} (x)}{K(x;r)*K(x;r)} is the convolution of the r'th derivative kernel function \eqn{K^{(r)}(x)}{K(x;r)} (see \code{\link{kernel.conv}} and \code{\link{kernel.fun}}); \eqn{R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx}{R(K(x;r)) = int K(x;r)^2 dx} and \eqn{\mu_{2}(K) = \int_{R}x^{2} K(x) dx}{mu(K(x)) = int x^2 K(x) dx}.\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{h}{value of bandwidth parameter.} \item{min.mcv}{the minimal MCV value.} } \references{ Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). Bandwidth selection for kernel density estimation: a review of fully automatic selectors. \emph{Advances in Statistical Analysis}. Stute, W. (1992). Modified cross validation in density estimation. \emph{Journal of Statistical Planning and Inference}, \bold{30}, 293--305. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.mcv}}. } \examples{ ## Derivative order = 0 h.mcv(kurtotic,deriv.order = 0) ## Derivative order = 1 h.mcv(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/kernel.fun.Rd0000644000176200001440000001005312614740640014275 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{kernel.fun} \alias{kernel.fun} \alias{kernel.fun.default} \title{ Derivatives of Kernel Function } \description{ The (S3) generic function \code{kernel.fun} computes the r'th derivative for kernel density.} \usage{ kernel.fun(x, \dots) \method{kernel.fun}{default}(x = NULL, deriv.order = 0, kernel = c("gaussian","epanechnikov", "uniform", "triangular", "triweight", "tricube", "biweight", "cosine", "silverman"), \dots) } \arguments{ \item{x}{points at which the derivative of kernel function is to be evaluated.} \item{deriv.order}{derivative order (scalar).} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ We give a short survey of some kernels functions \eqn{K(x;r)}{K(x;r)}; where \eqn{r} is derivative order, \itemize{ \item Gaussian: \eqn{K(x;\infty) =\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)1_{]-\infty,+\infty[}}{K(x;Inf) = 1/sqrt(2pi) exp(-0.5 x^2)} \item Epanechnikov: \eqn{K(x;2)=\frac{3}{4}(1-x^{2})1_{(|x| \leq 1)}}{K(x;2) = 0.75 (1-x^2) (abs(x) <= 1)} \item uniform (rectangular): \eqn{K(x;0)=\frac{1}{2}1_{(|x| \leq 1)}}{K(x;0) = 0.5 (abs(x) <= 1)} \item triangular: \eqn{K(x;1)=(1-|x|)1_{(|x| \leq 1)}}{K(x;1) = (1-abs(x)) (abs(x) <= 1)} \item triweight: \eqn{K(x;6)=\frac{35}{32}(1-x^{2})^{3} 1_{(|x| \leq 1)}}{K(x;6) = 35/36 (1-x^2)^3 (abs(x) <= 1)} \item tricube: \eqn{K(x;9)=\frac{70}{81}(1-|x|^{3})^{3} 1_{(|x| \leq 1)}}{K(x;9) = 70/81 (1-abs(x)^3)^3 (abs(x) <= 1)} \item biweight: \eqn{K(x;4)=\frac{15}{16}(1-x^{2})^{2} 1_{(|x| \leq 1)}}{K(x;4) = 15/16 (1-x^2)^2 (abs(x) <= 1)} \item cosine: \eqn{K(x;\infty)=\frac{\pi}{4}\cos\left(\frac{\pi}{2}x\right) 1_{(|x| \leq 1)}}{0.25 pi cos(0.5 pi x) (abs(x) <= 1)} \item Silverman: \eqn{K(x;r \%\% 8)\footnote{\%\% indicates x mod y}=\frac{1}{2}\exp\left(-\frac{|x|}{\sqrt{2}}\right)\sin\left(\frac{|x|}{\sqrt{2}}+\frac{\pi}{4}\right)1_{]-\infty,+\infty[}}{K(x;r mod 8)= 0.5 exp(-abs(x)/sqrt(2)) sin(abs(x)/sqrt(2) + 0.25 pi)} } The r'th derivative for kernel function \eqn{K(x)} is written: \deqn{K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)}{K(x;r) = d^r / dx^r K(x)} for \eqn{r = 0, 1, 2, \dots}\cr The r'th derivative of the \bold{Gaussian kernel} \eqn{K(x)} is given by: \deqn{K^{(r)}(x) = (-1)^{r} H_{r}(x) K(x)}{K(x;r) = (-1)^r H(x;r) K(x)} where \eqn{H_{r}(x)}{H(x;r)} is the r'th \bold{Hermite polynomial}. This polynomials are set of orthogonal polynomials, for more details see, \code{\link[orthopolynom]{hermite.h.polynomials}} in package \CRANpkg{orthopolynom}. } \value{ \item{kernel }{name of kernel to use.} \item{deriv.order }{the derivative order to use.} \item{x }{the n coordinates of the points where the derivative of kernel function is evaluated.} \item{kx }{the kernel derivative values.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \references{ Jones, M. C. (1992). Differences and derivatives in kernel estimation. \emph{Metrika}, \bold{39}, 335--340. Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). \emph{NIST Handbook of Mathematical Functions}. Cambridge University Press, New York, USA. Silverman, B. W. (1986). \emph{Density Estimation for Statistics and Data Analysis}. Chapman & Hall/CRC. London. } \seealso{ \code{\link{plot.kernel.fun}}, \code{\link[stats]{deriv}} and \code{\link[stats]{D}} in package "stats" for symbolic and algorithmic derivatives of simple expressions. } \examples{ kernels <- eval(formals(kernel.fun.default)$kernel) kernels ## gaussian kernel.fun(x = 0,kernel=kernels[1],deriv.order=0) kernel.fun(x = 0,kernel=kernels[1],deriv.order=1) ## silverman kernel.fun(x = 0,kernel=kernels[9],deriv.order=0) kernel.fun(x = 0,kernel=kernels[9],deriv.order=1) } \keyword{nonparametric} \keyword{kernel} kedd/man/claw.Rd0000644000176200001440000000256212614740640013162 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{Claw, Bimodal, Kurtotic, Outlier, Trimodal} \alias{claw} \alias{bimodal} \alias{kurtotic} \alias{outlier} \alias{trimodal} \docType{data} \title{ Datasets } \description{ A random sample of size 200 from the claw, bimodal, kurtotic, outlier and trimodal Gaussian density. } \usage{ data(claw) data(bimodal) data(kurtotic) data(outlier) data(trimodal) } \format{ Numeric vector with length 200. } \details{ Generate 200 random numbers, distributed according to a normal mixture, using \code{\link[nor1mix]{rnorMix}} in package \CRANpkg{nor1mix}. \preformatted{ ## Claw density claw <- rnorMix(n=200, MW.nm10) plot(MW.nm10) ## Bimodal density bimodal <- rnorMix(n=200, MW.nm7) plot( MW.nm7) ## Kurtotic density kurtotic <- rnorMix(n=200, MW.nm4) plot(MW.nm4) ## Outlier density outlier <- rnorMix(n=200, MW.nm5) plot( MW.nm5) ## Trimodal density trimodal <- rnorMix(n=200, MW.nm9) plot(MW.nm9) } } \source{ Randomly generated a normal mixture with the function \code{\link[nor1mix]{rnorMix}} in package \CRANpkg{nor1mix}. } \references{ Martin, M. (2013). \CRANpkg{nor1mix}: Normal (1-d) mixture models (S3 classes and methods). \emph{\R package version 1.1-4}. } \keyword{datasets} kedd/man/plot.h.bcv.Rd0000644000176200001440000000341512614740640014207 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.bcv} \alias{plot.h.bcv} \alias{lines.h.bcv} \title{ Plot for Biased Cross-Validation } \description{ The \code{\link{plot.h.bcv}} function loops through calls to the \code{\link{h.bcv}} function. Plot for biased cross-validation function for 1-dimensional data. } \usage{ \method{plot}{h.bcv}(x, seq.bws=NULL, \dots) \method{lines}{h.bcv}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.bcv} (output from \code{\link{h.bcv}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the biased cross-validation function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d biased cross-validation function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{bcv}{the values of the biased cross-validation function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.bcv}}. } \examples{ ## EXAMPLE 1: plot(h.bcv(trimodal, whichbcv = 1, deriv.order = 0),main="",sub="") lines(h.bcv(trimodal, whichbcv = 2, deriv.order = 0),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("black","red"),inset = .015) ## EXAMPLE 2: plot(h.bcv(trimodal, whichbcv = 1, deriv.order = 1),main="",sub="") lines(h.bcv(trimodal, whichbcv = 2, deriv.order = 1),col="red") legend("topright", c("BCV1","BCV2"),lty=1,col=c("black","red"),inset = .015) } \keyword{plot} kedd/man/h.ccv.Rd0000644000176200001440000001102012614740640013222 0ustar liggesusers\name{h.ccv} \alias{h.ccv} \alias{h.ccv.default} \alias{print.h.ccv} \title{ Complete Cross-Validation for Bandwidth Selection } \description{ The (S3) generic function \code{h.ccv} computes the complete cross-validation bandwidth selector of r'th derivative of kernel density estimator one-dimensional. } \usage{ h.ccv(x, \dots) \method{h.ccv}{default}(x, deriv.order = 0, lower = 0.1 * hos, upper = hos, tol = 0.1 * lower, kernel = c("gaussian", "triweight", "tricube", "biweight", "cosine"), \dots) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{x}{vector of data values.} \item{deriv.order}{derivative order (scalar).} \item{lower, upper}{range over which to minimize. The default is almost always satisfactory. \code{hos} (Over-smoothing) is calculated internally from an \code{kernel}, see details.} \item{tol}{the convergence tolerance for \code{\link{optimize}}.} \item{kernel}{a character string giving the smoothing kernel to be used, with default \code{"gaussian"}.} \item{\dots}{further arguments for (non-default) methods.} } \details{ \code{h.ccv} complete cross-validation implements for choosing the bandwidth \eqn{h} of a r'th derivative kernel density estimator.\cr Jones and Kappenman (1991) proposed a so-called complete cross-validation (CCV) in kernel density estimator. This method can be extended to the estimation of derivative of the density, basing our estimate of integrated squared density derivative (Peter and Marron 1987) on the \eqn{\bar{\theta}_{r}(h)}{bar(theta)(h;r)}'s, we get the following, start from \eqn{R\left(\hat{f}_{h}^{(r)}\right) - \bar{\theta}_{r}(h)}{R(hat(f)(h;r)) - bar(theta)(h;r)} as an estimate of MISE. Thus, \eqn{\hat{h}^{(r)}_{CCV}}{h(r)_(CCV)}, say, is the \eqn{h} that minimises: \deqn{CCV(h;r)=R\left(\hat{f}_{h}^{(r)}\right)-\bar{\theta}_{r}(h)+\frac{1}{2}\mu_{2}(K) h^{2} \bar{\theta}_{r+1}(h)+\frac{1}{24}\left(6\mu_{2}^{2}(K) -\delta(K)\right)h^{4}\bar{\theta}_{r+2}(h)}{CCV(h;r)= R(K(x;r))/ n h^(2r+1) + R(hat(f)(h;r))- bar(theta)(h;r) + 0.5 mu(K(x)) h^2 bar(theta)(h;r+1) + 1/24 (6 mu(K(x))^2 - delta(K(x))) h^4 bar(theta)(h;r+2)} with \deqn{R\left(\hat{f}_{h}^{(r)}\right) = \int \left(\hat{f}_{h}^{(r)}(x)\right)^{2} dx = \frac{R\left(K^{(r)}\right)}{nh^{2r+1}} + \frac{(-1)^{r}}{n (n-1) h^{2r+1}} \sum_{i=1}^{n}\sum_{j=1;j \neq i}^{n} K^{(r)} \ast K^{(r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{R(hat(f)(h;r)) = int (hat(f)(x;r))^2 dx =R(k(x;r))/n h^(2r+1) + (-1)^r / n (n-1) h^(2r+1) sum(sum(K(.;r)*K(.;r)(x(j)-x(i)/h)), i=1...n, j=1...n, j != i)} and \deqn{\bar{\theta}_{r}(h)= \frac{(-1)^r}{n(n-1) h^{2r+1}} \sum_{i=1}^{n} \sum_{j=1;j \neq i}^{n} K^{(2r)} \left(\frac{X_{j}-X_{i}}{h}\right)}{bar(theta)(h;r) = (-1)^r / n(n-1) h^(2r+1) sum(sum(K((x(j)-x(i)/h);2r)), i=1...n, j=1...n, j != i)} and \eqn{K^{(r)} \ast K^{(r)} (x)}{K(x;r)*K(x;r)} is the convolution of the r'th derivative kernel function \eqn{K^{(r)}(x)}{K(x;r)} (see \code{\link{kernel.conv}} and \code{\link{kernel.fun}}); \eqn{R\left(K^{(r)}\right) = \int_{R} K^{(r)}(x)^{2} dx}{R(K(x;r)) = int K(x;r)^2 dx} and \eqn{\mu_{2}(K) = \int_{R}x^{2} K(x) dx}{mu(K(x)) = int x^2 K(x) dx}, \eqn{\delta(K) = \int_{R}x^{4} K(x) dx}{delta(K(x)) = int x^4 K(x) dx}.\cr The range over which to minimize is \code{hos} Oversmoothing bandwidth, the default is almost always satisfactory. See George and Scott (1985), George (1990), Scott (1992, pp 165), Wand and Jones (1995, pp 61). } \value{ \item{x}{data points - same as input.} \item{data.name}{the deparsed name of the \code{x} argument.} \item{n}{the sample size after elimination of missing values.} \item{kernel}{name of kernel to use} \item{deriv.order}{the derivative order to use.} \item{h}{value of bandwidth parameter.} \item{min.ccv}{the minimal CCV value.} } \references{ Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. \emph{Scandinavian Journal of Statistics}, \bold{19}, 337--349. Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. \emph{Statistics and Probability Letters}, \bold{6}, 109--115. } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{plot.h.ccv}}. } \examples{ ## Derivative order = 0 h.ccv(kurtotic,deriv.order = 0) ## Derivative order = 1 h.ccv(kurtotic,deriv.order = 1) } \keyword{smooth} \keyword{bandwidth selection} \keyword{nonparametric} kedd/man/kedd-package.Rd0000644000176200001440000003624112614740640014535 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \newcommand{\HR}{\href{http://www.usthb.dz/en/}} \name{kedd-package} \alias{kedd-package} \alias{kedd} \docType{package} \title{ Kernel Estimator and Bandwidth Selection for Density and Its Derivatives } \description{ Smoothing techniques and computing bandwidth selectors of the r'th derivative of a probability density for one-dimensional data. } \details{ \tabular{ll}{ Package: \tab kedd\cr Type: \tab Package\cr Version: \tab 1.0.3\cr Date: \tab 2015-10-30\cr License: \tab GPL (>= 2) \cr } There are four main types of functions in this package: \enumerate{ \item Compute the derivatives and convolutions of a kernel function (1-d). \item Compute the kernel estimators for density and its derivatives (1-d). \item Computing the bandwidth selectors (1-d). \item Displaying kernel estimators. } } \section{Main Features}{ \bold{Convolutions and derivatives in kernel function:}\cr\cr In non-parametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. The kernels functions \eqn{K(x)}{K(x)} are used in derivatives of kernel density estimator to estimate \eqn{\hat{f}^{(r)}_{h}(x)}{hat(f)(x;r)}, satisfying the following three requirements: \enumerate{ \item \eqn{\int_{R} K(x) dx = 1}{ int K(x) dx = 1} \item \eqn{\int_{R} xK(x) dx = 0}{ int x K(x) dx = 0} \item \eqn{\mu_{2}(K) = \int_{R}x^{2} K(x) dx < \infty}{mu(K(x)) = int x^2 K(x) dx < inf} } Several types of kernel functions \eqn{K(x)} are commonly used in this package: Gaussian, Epanechnikov, Uniform (rectangular), Triangular, Triweight, Tricube, Biweight (quartic), Cosine.\cr The function \code{\link{kernel.fun}} for kernel derivative \eqn{K^{(r)}(x)}{K(x;r)} and \code{\link{kernel.conv}} for kernel convolution \eqn{K^{(r)}\ast K^{(r)} (x)}{K(x;r) * K(x;r)}, where the write formally: \deqn{K^{(r)}(x) = \frac{d^{r}}{d x^{r}} K(x)}{K(x;r) = d^r/d x^r (K(x))} \deqn{K^{(r)} \ast K^{(r)} (x) = \int_{-\infty}^{+\infty} K^{(r)}(y)K^{(r)}(x-y)dy}{K(x;r) * K(x;r) = int K(y;r) K(x-y;r) dy} for \eqn{r = 0, 1, 2, \dots}\cr \bold{Estimators of r'th derivative of a density function:}\cr\cr \dfn{A natural estimator of the r'th derivative of a density function \eqn{f(x)} is: \deqn{\hat{f}^{(r)}_{h}(x)= \frac{d^{r}}{d x^{r}} \frac{1}{nh} \sum_{i=1}^{n} K\left(\frac{x-X_{i}}{h}\right) = \frac{1}{nh^{r+1}}\sum_{i=1}^{n} K^{(r)}\left(\frac{x-X_{i}}{h}\right)}{hat(f)(x;r) = n^-1 h^-(r+1) Sum ( K(x-X(i)/h ;r) ,i = 1...n)} Here, \eqn{X_{1}, X_{2}, \dots,X_{n}}{X(1), X(2),...,X(n)} is an i.i.d, sample of size \eqn{n} from the distribution with density \eqn{f(x)}, \eqn{K(x)} is the kernel function which we take to be a symmetric probability density with at least \eqn{r} non zero derivatives when estimating \eqn{f^{(r)}(x)}{f(x;r)}, and \eqn{h} is the bandwidth, this parameter is very important that controls the degree of smoothing applied to the data.}\cr The case \eqn{(r=0)} is the standard kernel density estimator (e.g. Silverman 1986, Wolfgang 1991, Scott 1992, Wand and Jones 1995, Jeffrey 1996, Bowman and Azzalini 1997, Alexandre 2009), properties of such derivative estimators are well known e.g. Sheather and Jones (1991), Jones and Kappenman (1991), Wolfgang (1991). For the case \eqn{(r > 0)}, is derivative of kernel density estimator (e.g. Bhattacharya 1967, Schuster 1969, Alekseev 1972, Wolfgang et all 1990, Jones 1992, Stoker 1993) and for applications which require the estimation of density derivatives can be found in Singh (1977).\cr For r'th derivatives of kernel density estimator one-dimensional, the main function is \code{\link{dkde}}. For display, its plot method calls \code{\link{plot.dkde}}, and if to add a plot using \code{\link{lines.dkde}}. \preformatted{ R> data(trimodal) R> dkde(x = trimodal, deriv.order = 0, kernel = "gaussian") Data: trimodal (200 obs.); Kernel: gaussian Derivative order: 0; Bandwidth 'h' = 0.1007 eval.points est.fx Min. :-2.91274 Min. :0.0000066 1st Qu.:-1.46519 1st Qu.:0.0669750 Median :-0.01765 Median :0.1682045 Mean :-0.01765 Mean :0.1723692 3rd Qu.: 1.42989 3rd Qu.:0.2484626 Max. : 2.87743 Max. :0.4157340 R> dkde(x = trimodal, deriv.order = 1, kernel = "gaussian") Data: trimodal (200 obs.); Kernel: gaussian Derivative order: 1; Bandwidth 'h' = 0.09094 eval.points est.fx Min. :-2.87358 Min. :-1.740447 1st Qu.:-1.44562 1st Qu.:-0.343952 Median :-0.01765 Median : 0.009057 Mean :-0.01765 Mean : 0.000000 3rd Qu.: 1.41031 3rd Qu.: 0.415343 Max. : 2.83828 Max. : 1.256891 } \bold{Bandwidth selectors:}\cr\cr The most important factor in the r'th derivative kernel density estimate is a choice of the bandwidth \eqn{h}{h} for one-dimensional observations. Because of its role in controlling both the amount and the direction of smoothing, this choice is particularly important. We present the popular bandwidth selection (for more details see references) methods in this package: \itemize{ \item Optimal Bandwidth (AMISE); with \code{deriv.order >= 0}, name of this function is \code{\link{h.amise}}.\cr For display, its plot method calls \code{\link{plot.h.amise}}, and to add a plot used \code{\link{lines.h.amise}}. \item Maximum-likelihood cross-validation (MLCV); with \code{deriv.order = 0}, name of this function is \code{\link{h.mlcv}}.\cr For display, its plot method calls \code{\link{plot.h.mlcv}}, and to add a plot used \code{\link{lines.h.mlcv}}. \item Unbiased cross validation (UCV); with \code{deriv.order >= 0}, name of this function is \code{\link{h.ucv}}.\cr For display, its plot method calls \code{\link{plot.h.ucv}}, and to add a plot used \code{\link{lines.h.ucv}}. \item Biased cross validation (BCV); with \code{deriv.order >= 0}, name of this function is \code{\link{h.bcv}}.\cr For display, its plot method calls \code{\link{plot.h.bcv}}, and to add a plot used \code{\link{lines.h.bcv}}. \item Complete cross-validation (CCV); with \code{deriv.order >= 0}, name of this function is \code{\link{h.ccv}}.\cr For display, its plot method calls \code{\link{plot.h.ccv}}, and to add a plot used \code{\link{lines.h.ccv}}. \item Modified cross-validation (MCV); with \code{deriv.order >= 0}, name of this function is \code{\link{h.mcv}}.\cr For display, its plot method calls \code{\link{plot.h.mcv}}, and to add a plot used \code{\link{lines.h.mcv}}. \item Trimmed cross-validation (TCV); with \code{deriv.order >= 0}, name of this function is \code{\link{h.tcv}}.\cr For display, its plot method calls \code{\link{plot.h.tcv}}, and to add a plot used \code{\link{lines.h.tcv}}. } \preformatted{ R> data(trimodal) R> h.bcv(x = trimodal, whichbcv = 1, deriv.order = 0, kernel = "gaussian") Call: Biased Cross-Validation 1 Derivative order = 0 Data: trimodal (200 obs.); Kernel: gaussian Min BCV = 0.004511636; Bandwidth 'h' = 0.4357812 R> h.ccv(x = trimodal, deriv.order = 1, kernel = "gaussian") Call: Complete Cross-Validation Derivative order = 1 Data: trimodal (200 obs.); Kernel: gaussian Min CCV = 0.01985078; Bandwidth 'h' = 0.5828336 R> h.tcv(x = trimodal, deriv.order = 2, kernel = "gaussian") Call: Trimmed Cross-Validation Derivative order = 2 Data: trimodal (200 obs.); Kernel: gaussian Min TCV = -295.563; Bandwidth 'h' = 0.08908582 R> h.ucv(x = trimodal, deriv.order = 3, kernel = "gaussian") Call: Unbiased Cross-Validation Derivative order = 3 Data: trimodal (200 obs.); Kernel: gaussian Min UCV = -63165.18; Bandwidth 'h' = 0.1067236 } For an overview of this package, see \code{vignette("kedd")}. } \section{Requirements}{ \R version >= 2.15.0 } \author{ Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} (Dept. Probability and Statistics, \HR{USTHB}, Algeria).\cr Please send comments, error reports, etc. to the author via the addresses email mentioned above. } \section{Licence}{ This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package. } \references{ Alekseev, V. G. (1972). Estimation of a probability density function and its derivatives. \emph{Mathematical notes of the Academy of Sciences of the USSR}. \bold{12}(5), 808--811. Alexandre, B. T. (2009). \emph{Introduction to Nonparametric Estimation}. Springer-Verlag, New York. Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of kernel density estimates. \emph{Biometrika}, \bold{71}, 353--360. Bowman, A. W. and Azzalini, A. (1997). \emph{Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations}. Oxford University Press, Oxford. Bowman, A.W. and Azzalini, A. (2003). Computational aspects of nonparametric smoothing with illustrations from the \CRANpkg{sm} library. \emph{Computational Statistics and Data Analysis}, \bold{42}, 545--560. Bowman, A.W. and Azzalini, A. (2013). \CRANpkg{sm}: Smoothing methods for nonparametric regression and density estimation. \emph{\R package version 2.2-5.3}. Ported to \R by B. D. Ripley. Bhattacharya, P. K. (1967). Estimation of a probability density function and Its derivatives. \emph{Sankhya: The Indian Journal of Statistics, Series A}, \bold{29}, 373--382. Duin, R. P. W. (1976). On the choice of smoothing parameters of Parzen estimators of probability density functions. \emph{IEEE Transactions on Computers}, \bold{C-25}, 1175--1179. Feluch, W. and Koronacki, J. (1992). A note on modified cross-validation in density estimation. \emph{Computational Statistics and Data Analysis}, \bold{13}, 143--151. George, R. T. (1990). The maximal smoothing principle in density estimation. \emph{Journal of the American Statistical Association}, \bold{85}, 470--477. George, R. T. and Scott, D. W. (1985). Oversmoothed nonparametric density estimates. \emph{Journal of the American Statistical Association}, \bold{80}, 209--214. Habbema, J. D. F., Hermans, J., and Van den Broek, K. (1974) A stepwise discrimination analysis program using density estimation. \emph{Compstat 1974: Proceedings in Computational Statistics}. Physica Verlag, Vienna. Heidenreich, N. B., Schindler, A. and Sperlich, S. (2013). Bandwidth selection for kernel density estimation: a review of fully automatic selectors. \emph{Advances in Statistical Analysis}. Jeffrey, S. S. (1996). \emph{Smoothing Methods in Statistics}. Springer-Verlag, New York. Jones, M. C. (1992). Differences and derivatives in kernel estimation. \emph{Metrika}, \bold{39}, 335--340. Jones, M. C., Marron, J. S. and Sheather,S. J. (1996). A brief survey of bandwidth selection for density estimation. \emph{Journal of the American Statistical Association}, \bold{91}, 401--407. Jones, M. C. and Kappenman, R. F. (1991). On a class of kernel density estimate bandwidth selectors. \emph{Scandinavian Journal of Statistics}, \bold{19}, 337--349. Loader, C. (1999). \emph{Local Regression and Likelihood}. Springer, New York. Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). \emph{NIST Handbook of Mathematical Functions}. Cambridge University Press, New York, USA. Peter, H. and Marron, J.S. (1987). Estimation of integrated squared density derivatives. \emph{Statistics and Probability Letters}, \bold{6}, 109--115. Peter, H. and Marron, J.S. (1991). Local minima in cross-validation functions. \emph{Journal of the Royal Statistical Society, Series B}, \bold{53}, 245--252. Radhey, S. S. (1987). MISE of kernel estimates of a density and its derivatives. \emph{Statistics and Probability Letters}, \bold{5}, 153--159. Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. \emph{Scandinavian Journal of Statistics}, \bold{9}, 65--78. Scott, D. W. (1992). \emph{Multivariate Density Estimation. Theory, Practice and Visualization}. New York: Wiley. Scott, D.W. and George, R. T. (1987). Biased and unbiased cross-validation in density estimation. \emph{Journal of the American Statistical Association}, \bold{82}, 1131--1146. Schuster, E. F. (1969) Estimation of a probability density function and its derivatives. \emph{The Annals of Mathematical Statistics}, \bold{40} (4), 1187--1195. Sheather, S. J. (2004). Density estimation. \emph{Statistical Science}, \bold{19}, 588--597. Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. \emph{Journal of the Royal Statistical Society, Series B}, \bold{53}, 683--690. Silverman, B. W. (1986). \emph{Density Estimation for Statistics and Data Analysis}. Chapman & Hall/CRC. London. Singh, R. S. (1977). Applications of estimators of a density and its derivatives to certain statistical problems. \emph{Journal of the Royal Statistical Society, Series B}, \bold{39}(3), 357--363. Stoker, T. M. (1993). Smoothing bias in density derivative estimation. \emph{Journal of the American Statistical Association}, \bold{88}, 855--863. Stute, W. (1992). Modified cross validation in density estimation. \emph{Journal of Statistical Planning and Inference}, \bold{30}, 293--305. Tarn, D. (2007). \CRANpkg{ks}: Kernel density estimation and kernel discriminant analysis for multivariate data in \R. \emph{Journal of Statistical Software}, \bold{21}(7), 1--16. Tristen, H. and Jeffrey, S. R. (2008). Nonparametric Econometrics: The \CRANpkg{np} Package. \emph{Journal of Statistical Software},\bold{27}(5). Venables, W. N. and Ripley, B. D. (2002). \emph{Modern Applied Statistics with S}. New York: Springer. Wand, M. P. and Jones, M. C. (1995). \emph{Kernel Smoothing}. Chapman and Hall, London. Wand, M.P. and Ripley, B. D. (2013). \CRANpkg{KernSmooth}: Functions for Kernel Smoothing for Wand and Jones (1995). \emph{\R package version 2.23-10}. Wolfgang, H. (1991). \emph{Smoothing Techniques}, \emph{With Implementation in S}. Springer-Verlag, New York. Wolfgang, H., Marlene, M., Stefan, S. and Axel, W. (2004). \emph{Nonparametric and Semiparametric Models}. Springer-Verlag, Berlin Heidelberg. Wolfgang, H., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. \emph{Journal of the Royal Statistical Society, Series B}, 223--232. } \keyword{package} \seealso{ \CRANpkg{ks}, \CRANpkg{KernSmooth}, \CRANpkg{sm}, \CRANpkg{np}, \CRANpkg{locfit}, \CRANpkg{feature}, \CRANpkg{GenKern}. } kedd/man/plot.h.amise.Rd0000644000176200001440000000255112614740640014533 0ustar liggesusers\newcommand{\CRANpkg}{\href{http://CRAN.R-project.org/package=#1}{\pkg{#1}}} \name{plot.h.amise} \alias{plot.h.amise} \alias{lines.h.amise} \title{ Plot for Asymptotic Mean Integrated Squared Error } \description{ The \code{\link{plot.h.amise}} function loops through calls to the \code{\link{h.amise}} function. Plot for asymptotic mean integrated squared error function for 1-dimensional data. } \usage{ \method{plot}{h.amise}(x, seq.bws=NULL, \dots) \method{lines}{h.amise}(x,seq.bws=NULL, \dots) } \arguments{ \item{x}{object of class \code{h.amise} (output from \code{\link{h.amise}}).} \item{seq.bws}{the sequence of bandwidths in which to compute the AMISE function. By default, the procedure defines a sequence of 50 points, from \code{0.15*hos} to \code{2*hos} (Over-smoothing).} \item{\dots}{other graphics parameters, see \code{\link[graphics]{par}} in package "graphics".} } \value{ Plot of 1-d AMISE function are sent to graphics window.\cr \item{kernel}{name of kernel to use.} \item{deriv.order}{the derivative order to use.} \item{seq.bws}{the sequence of bandwidths.} \item{amise}{the values of the AMISE function in the bandwidths grid.} } \author{Arsalane Chouaib Guidoum \email{acguidoum@usthb.dz} } \seealso{ \code{\link{h.amise}}. } \examples{ plot(h.amise(bimodal,deriv.order=0)) } \keyword{plot}