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  pdftitle = {A Handbook of Statistical Analyses Using R},
  pdfsubject = {Book},
  pdfauthor = {Brian S. Everitt and Torsten Hothorn},
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\renewcommand{\refname}{References \addcontentsline{toc}{chapter}{References}}
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\begin{document}

%% Title page

\title{A Handbook of Statistical Analyses Using \R}

\author{Brian S. Everitt and Torsten Hothorn}

\maketitle
%%\VignetteIndexEntry{Chapter Density Estimation}
%%\VignetteDepends{flexmix}
\setcounter{chapter}{6}


\SweaveOpts{prefix.string=figures/HSAUR,eps=FALSE,keep.source=TRUE} 

<<setup, echo = FALSE, results = hide>>=
rm(list = ls())
s <- search()[-1]
s <- s[-match(c("package:base", "package:stats", "package:graphics", "package:grDevices",
                "package:utils", "package:datasets", "package:methods", "Autoloads"), s)]
if (length(s) > 0) sapply(s, detach, character.only = TRUE)
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HSAURpkg <- require("HSAUR")
if (!HSAURpkg) stop("cannot load package ", sQuote("HSAUR"))
rm(HSAURpkg)
a <- Sys.setlocale("LC_ALL", "C")
book <- TRUE
refs <- cbind(c("AItR", "SI", "CI", "ANOVA", "MLR", "GLM", 
                "DE", "RP", "SA", "ALDI", "ALDII", "MA", "PCA", 
                "MDS", "CA"), 1:15)
ch <- function(x, book = TRUE) {
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    } else {
        return(paste("Chapter~\\\\ref{", ch[2], "}", sep = ""))
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}
@

\pagestyle{headings}

<<DE-setup, echo = FALSE, results = hide>>=
x <- library("KernSmooth")
x <- library("flexmix")
x <- library("boot")
@


\chapter[Density Estimation]{Density Estimation: Erupting Geysers and Star
Clusters \label{DE}}

\section{Introduction}


\section{Density Estimation}


The three kernel functions are implemented in \R{} as shown in lines 1--3 of 
Figure~\ref{DE-kernel-fig}. For some grid \Robject{x}, the kernel functions
are plotted using the \R{} statements in lines 5--11 (Figure~\ref{DE-kernel-fig}).

\numberSinput
\begin{figure}
\begin{center}
<<DE-kernel-figs, echo = TRUE, fig = TRUE, pdf = FALSE, png = TRUE>>=
rec <- function(x) (abs(x) < 1) * 0.5
tri <- function(x) (abs(x) < 1) * (1 - abs(x))
gauss <- function(x) 1/sqrt(2*pi) * exp(-(x^2)/2)
x <- seq(from = -3, to = 3, by = 0.001)
plot(x, rec(x), type = "l", ylim = c(0,1), lty = 1, 
     ylab = expression(K(x)))
lines(x, tri(x), lty = 2)
lines(x, gauss(x), lty = 3)
legend(-3, 0.8, legend = c("Rectangular", "Triangular", 
       "Gaussian"), lty = 1:3, title = "kernel functions", 
       bty = "n")
@
\caption{Three commonly used kernel functions. \label{DE-kernel-fig}}
\end{center}
\end{figure}
\rawSinput

<<DE-options, echo = FALSE, results = hide>>=
w <- options("width")$w
options(width = 66)
@

The kernel estimator $\hat{f}$ is a sum of `bumps' placed at the observations. %'
The kernel function determines the shape of the bumps while the 
window width $h$ determines their width.  
\index{Windows, in kernel density estimation}
Figure~\ref{DE-bumps} \citep[redrawn from a similar plot in][]{HSAUR:Silverman1986} 
shows the individual bumps $n^{-1}h^{-1} K((x - x_i) / h)$, as well as the estimate $\hat{f}$
obtained by adding them up for an artificial set of data points
<<DE-x-bumps-data, echo = TRUE>>=
x <- c(0, 1, 1.1, 1.5, 1.9, 2.8, 2.9, 3.5)
n <- length(x)
@
For a grid
<<DE-x-bumps-gaussian, echo = TRUE>>=
xgrid <- seq(from = min(x) - 1, to = max(x) + 1, by = 0.01) 
@
on the real line, we can compute the contribution of each measurement in
\Robject{x}, with $h = 0.4$, by the Gaussian kernel (defined in 
Figure~\ref{DE-kernel-fig}, line 3) as follows;
<<DE-x-bumps-bumps, echo = TRUE>>=
h <- 0.4
bumps <- sapply(x, function(a) gauss((xgrid - a)/h)/(n * h))
@
A plot of the individual bumps and their sum, the kernel density estimate
$\hat{f}$, is shown in Figure~\ref{DE-bumps}.

<<DE-reoptions, echo = FALSE, results = hide>>=
options(width = w)
@

\numberSinput
\begin{figure}
\begin{center}
<<DE-x-bumps, echo = TRUE, fig = TRUE, leftpar = TRUE>>=
plot(xgrid, rowSums(bumps), ylab = expression(hat(f)(x)),
     type = "l", xlab = "x", lwd = 2)
rug(x, lwd = 2)
out <- apply(bumps, 2, function(b) lines(xgrid, b))
@
\caption{Kernel estimate showing the contributions of Gaussian 
         kernels evaluated for the individual observations with bandwidth $h =
         0.4$. \label{DE-bumps}}
\end{center}
\end{figure}
\rawSinput


\begin{figure}
\begin{center}
<<DE-epakernel-fig, echo = TRUE, fig = TRUE>>=
epa <- function(x, y) 
    ((x^2 + y^2) < 1) * 2/pi * (1 - x^2 - y^2)
x <- seq(from = -1.1, to = 1.1, by = 0.05)
epavals <- sapply(x, function(a) epa(a, x))
persp(x = x, y = x, z = epavals, xlab = "x", ylab = "y", 
      zlab = expression(K(x, y)), theta = -35, axes = TRUE, 
      box = TRUE)
@
\caption{Epanechnikov kernel for a grid between $(-1.1, -1.1)$ and $(1.1, 1.1)$. 
         \label{DE-epakernel-fig}}
\end{center}
\end{figure}


\section{Analysis Using \R{}}


\numberSinput
\begin{figure}
\begin{center}
<<DE-faithful-density, echo = TRUE, fig = TRUE, height = 4>>=
data("faithful", package = "datasets")
x <- faithful$waiting
layout(matrix(1:3, ncol = 3))
hist(x, xlab = "Waiting times (in min.)", ylab = "Frequency",
     probability = TRUE, main = "Gaussian kernel", 
     border = "gray")
lines(density(x, width = 12), lwd = 2)
rug(x)
hist(x, xlab = "Waiting times (in min.)", ylab = "Frequency",
     probability = TRUE, main = "Rectangular kernel", 
     border = "gray")
lines(density(x, width = 12, window = "rectangular"), lwd = 2)
rug(x)
hist(x, xlab = "Waiting times (in min.)", ylab = "Frequency",
     probability = TRUE, main = "Triangular kernel", 
     border = "gray")
lines(density(x, width = 12, window = "triangular"), lwd = 2)
rug(x)
@
\caption{Density estimates of the geyser eruption data imposed on a histogram
of the data. \label{DE:faithfuldens}}
\end{center}
\end{figure}
\rawSinput


\begin{figure}
\begin{center}
<<DE-CYGOB1-contour, echo = TRUE, fig = TRUE>>=
library("KernSmooth")
data("CYGOB1", package = "HSAUR")
CYGOB1d <- bkde2D(CYGOB1, bandwidth = sapply(CYGOB1, dpik))
contour(x = CYGOB1d$x1, y = CYGOB1d$x2, z = CYGOB1d$fhat,
        xlab = "log surface temperature", 
        ylab = "log light intensity")
@
\caption{A contour plot of the bivariate density estimate of the \Robject{CYGOB1} data,
         i.e., a two-dimensional graphical display for a three-dimensional
         \newline problem.
         \label{DE:CYGOB12Dcontour}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
<<DE-CYGOB1-persp, echo = TRUE, fig = TRUE>>=
persp(x = CYGOB1d$x1, y = CYGOB1d$x2, z = CYGOB1d$fhat,
      xlab = "log surface temperature", 
      ylab = "log light intensity",
      zlab = "estimated density", 
      theta = -35, axes = TRUE, box = TRUE)
@
\caption{The bivariate density estimate of the \Robject{CYGOB1} data, here shown in a
         three-dimensional fashion using the \Rcmd{persp} function.
         \label{DE:CYGOB12Dpersp}}
\end{center}
\end{figure}

\subsection{A Parametric Density Estimate for the Old Faithful Data
\label{DE-waiting}}


<<DE-faithful-optim, echo = TRUE, results = hide>>=
logL <- function(param, x) {
    d1 <- dnorm(x, mean = param[2], sd = param[3])
    d2 <- dnorm(x, mean = param[4], sd = param[5])
    -sum(log(param[1] * d1 + (1 - param[1]) * d2))
}

startparam <- c(p = 0.5, mu1 = 50, sd1 = 3, mu2 = 80, sd2 = 3)
opp <- optim(startparam, logL, x = faithful$waiting, 
             method = "L-BFGS-B",
             lower = c(0.01, rep(1, 4)),
             upper = c(0.99, rep(200, 4)))
opp
@
<<DE-faithful-optim-print, echo = FALSE>>=
print(opp[names(opp) != "message"])
@

Of course, optimising the appropriate likelihood `by hand' %'
is not very convenient. In fact, (at least) two packages offer high-level
functionality for estimating mixture models. The first one is package
\Rpackage{mclust} \citep{PKG:mclust} implementing the methodology described
in \cite{HSAUR:FraleyRaftery2002}. Here, a Bayesian information criterion
(BIC) is applied to choose the form of the mixture model:
\index{Bayesian Information Criterion (BIC)}
<<DE-attach-mclust, echo = FALSE, results = hide>>=
library("mclust")
@
<<DE-faithful-mclust, echo = TRUE>>=
library("mclust")
mc <- Mclust(faithful$waiting)
mc
@
and the estimated means are
<<DE-faithful-mclust-mu, echo = TRUE>>=
mc$parameters$mean
@
with estimated standard deviation (found to be equal within both groups)
<<DE-faithful-mclust-para, echo = TRUE>>=
sqrt(mc$parameters$variance$sigmasq)
@
The proportion is $\hat{p} = \Sexpr{round(mc$parameters$pro[1], 2)}$. The second package is called
\Rpackage{flexmix} whose functionality is described by
\cite{HSAUR:Leisch2004}. 
A mixture of two normals can be fitted using
<<DE-faithful-flexmix, echo = TRUE>>=
library("flexmix")
fl <- flexmix(waiting ~ 1, data = faithful, k = 2)
@
with $\hat{p} = \Sexpr{round(fl@prior, 2)}$ and estimated parameters
<<DE-faithful-flexmix-parameters, echo = TRUE>>=
parameters(fl, component = 1)
parameters(fl, component = 2)
@

\begin{figure}
\begin{center}
<<DE-faithful-2Dplot, echo = TRUE, fig = TRUE>>=
opar <- as.list(opp$par)
rx <- seq(from = 40, to = 110, by = 0.1)
d1 <- dnorm(rx, mean = opar$mu1, sd = opar$sd1)
d2 <- dnorm(rx, mean = opar$mu2, sd = opar$sd2)
f <- opar$p * d1 + (1 - opar$p) * d2
hist(x, probability = TRUE, xlab = "Waiting times (in min.)",
     border = "gray", xlim = range(rx), ylim = c(0, 0.06), 
     main = "")
lines(rx, f, lwd = 2)
lines(rx, dnorm(rx, mean = mean(x), sd = sd(x)), lty = 2, 
      lwd = 2)
legend(50, 0.06, lty = 1:2, bty = "n",
       legend = c("Fitted two-component mixture density",
                  "Fitted single normal density"))
@
\caption{Fitted normal density and two-component normal mixture for geyser
eruption data. \label{DE:2Dplot}}
\end{center}
\end{figure}



\index{Bootstrap approach|(}
We can get standard errors for the five parameter estimates 
by using a bootstrap approach \citep[see][]{HSAUR:EfronTibshirani1993}.
The original data are slightly perturbed by drawing $n$ out of $n$
observations \stress{with replacement} and those artificial replications of
the original data are called \stress{bootstrap samples}. Now, we can fit the
mixture for each bootstrap sample and assess the variability of the
estimates, for example using confidence intervals. 
\index{Confidence interval!derived from bootstrap samples}
Some suitable \R{} code based on the \Rcmd{Mclust} 
function follows. First, we define a function that,
for a bootstrap sample \Robject{indx}, fits a two-component mixture model
and returns $\hat{p}$ and the estimated means (note that we need to make
sure that we always get an estimate of $p$, not $1 - p$):
<<DE-faithful-boot, echo = TRUE>>=
library("boot")
fit <- function(x, indx) {
    a <- Mclust(x[indx], minG = 2, maxG = 2)$parameters
    if (a$pro[1] < 0.5)
        return(c(p = a$pro[1], mu1 = a$mean[1], 
                               mu2 = a$mean[2]))
    return(c(p = 1 - a$pro[1], mu1 = a$mean[2], 
                               mu2 = a$mean[1]))
}
@
The function \Rcmd{fit} can now be fed into the \Rcmd{boot} function \citep{PKG:boot} 
for bootstrapping (here $1000$ bootstrap samples are drawn)
\begin{Schunk}
\begin{Sinput}
R> bootpara <- boot(faithful$waiting, fit, R = 1000)
\end{Sinput}
\end{Schunk}

<<DE-faithful-bootrun, echo = FALSE>>=
bootparafile <- system.file("cache", "DE-bootpara.rda", package = "HSAUR")
if (file.exists(bootparafile)) {
    load(bootparafile)
} else {
    bootpara <- boot(faithful$waiting, fit, R = 1000)
}
@

We assess the variability of our estimates $\hat{p}$ by
means of adjusted bootstrap percentile (BCa) confidence intervals, which for
$\hat{p}$ can be obtained from
<<DE-faithful-p-ci, echo = TRUE>>=
boot.ci(bootpara, type = "bca", index = 1)
@
We see that there is a reasonable variability in the mixture model, however,
the means in the two components are rather stable, as can be seen from
<<DE-faithful-mu1-ci, echo = TRUE>>=
boot.ci(bootpara, type = "bca", index = 2)
@
for $\hat{\mu}_1$ and for $\hat{\mu}_2$ from
<<DE-faithful-mu2-ci, echo = TRUE>>=
boot.ci(bootpara, type = "bca", index = 3)
@
Finally, we show a graphical representation of both the bootstrap
distribution of the mean estimates \stress{and} the corresponding confidence
intervals. For convenience, we define a function for plotting, namely
<<DE-bootplot, echo = TRUE>>=
bootplot <- function(b, index, main = "") {
    dens <- density(b$t[,index])
    ci <- boot.ci(b, type = "bca", index = index)$bca[4:5]
    est <- b$t0[index]
    plot(dens, main = main)
    y <- max(dens$y) / 10
    segments(ci[1], y, ci[2], y, lty = 2)
    points(ci[1], y, pch = "(")
    points(ci[2], y, pch = ")")
    points(est, y, pch = 19)
}
@
The element \Robject{t} of an object created by \Rcmd{boot} contains the
bootstrap replications of our estimates, i.e., the values computed by
\Rcmd{fit} for each of the $1000$ bootstrap samples of the geyser data.
First, we plot a simple density estimate and then construct a line
representing the confidence interval.
We apply this function to the bootstrap distributions of our estimates
$\hat{\mu}_1$ and $\hat{\mu}_2$ in Figure~\ref{DE-bootplot}.

\begin{figure}
\begin{center}
<<DE-faithful-boot-plot, echo = TRUE, fig = TRUE, height = 4>>=
layout(matrix(1:2, ncol = 2))
bootplot(bootpara, 2, main = expression(mu[1]))
bootplot(bootpara, 3, main = expression(mu[2]))
@
\caption{Bootstrap distribution and confidence intervals for the mean estimates
         of a two-component mixture for the geyser data. \label{DE-bootplot}}
\end{center}
\end{figure}
\index{Bootstrap approach|)}


\bibliographystyle{LaTeXBibTeX/refstyle}
\bibliography{LaTeXBibTeX/HSAUR}   
\end{document}
