\documentclass[nojss]{jss}
\usepackage{amsmath, amssymb, graphicx}

%\VignetteIndexEntry{mixtools for mixture models} 

%% macros (Didier)
\newcommand{\CF}{{\mathcal F}}
\newcommand{\CN}{{\mathcal N}}

\def\Bg{\mathbf{g}}
\def\Bh{\mathbf{h}}
\def\Bk{\mathbf{k}}
\def\Bx{\mathbf{x}}
\def\By{\mathbf{y}}
\def\Bc{\mathbf{c}}
\def\BC{\mathbf{C}}
\def\Bz{\mathbf{z}}

\newcommand{\argmax}{\mathop{\mbox{argmax}}}
\def\bP{\mathbb{P}} % Probability
\def\I{\mathbb I}   % indicator function
\def\bE{\mathbb{E}} % expectation
\def\bR{\mathbb{R}} % real line
\newcommand{\f}{{\vec\theta}}
\newcommand{\lb}{{\lambda}}

\def\post{{p}}
\def\defn{{\stackrel{\rm def}{=}}}
\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle\bf#1$}}
{\mbox{\boldmath$\textstyle\bf#1$}}
{\mbox{\boldmath$\scriptstyle\bf#1$}}
{\mbox{\boldmath$\scriptscriptstyle\bf#1$}}}

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%% almost as usual
\author{Tatiana Benaglia \\ Pennsylvania State University \And
  Didier Chauveau \\ Universit\'e d'Orl\'eans \AND David R.~Hunter \\
  Pennsylvania State University \And Derek S. Young \\ Pennsylvania
  State University} \title{\pkg{mixtools}: An \proglang{R} Package for
  Analyzing Finite Mixture Models}

%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Tatiana Benaglia, Didier Chauveau, David R.~Hunter, Derek
  Young} %% comma-separated
\Plaintitle{mixtools: An R Package for Analyzing Mixture
  Models} %% without formatting
\Shorttitle{mixtools for Mixture Models} %% a short title (if necessary)

%% an abstract and keywords
\Abstract{ The \pkg{mixtools} package for \proglang{R} provides a set of
  functions for analyzing a variety of finite mixture models.  These
  functions include both traditional methods, such as EM algorithms for
  univariate and multivariate normal mixtures, and newer methods that
  reflect some recent research in finite mixture models.  In the latter
  category, \pkg{mixtools} provides algorithms for estimating parameters
  in a wide range of different mixture-of-regression contexts, in
  multinomial mixtures such as those arising from discretizing
  continuous multivariate data, in nonparametric situations where the
  multivariate component densities are completely unspecified, and in
  semiparametric situations such as a univariate location mixture of
  symmetric but otherwise unspecified densities.  Many of the algorithms
  of the \pkg{mixtools} package are EM algorithms or are based on
  EM-like ideas, so this article includes an overview of EM algorithms
  for finite mixture models.  } \Keywords{cutpoint, EM algorithm,
  mixture of regressions, model-based clustering, nonparametric mixture,
  semiparametric mixture, unsupervised clustering}
%, keywords, comma-separated, not capitalized, \proglang{Java}}
\Plainkeywords{keywords, comma-separated, not capitalized,
  Java} %% without formatting
%% at least one keyword must be supplied

%% publication information
%% NOTE: Typically, this can be left commented and will be filled out by the technical editor
%% \Volume{13}
%% \Issue{9}
%% \Month{September}
%% \Year{2004}
%% \Submitdate{2004-09-29}
%% \Acceptdate{2004-09-29}

%% The address of (at least) one author should be given
%% in the following format:
\Address{
Didier Chauveau\\
Laboratoire MAPMO - UMR 7349 - F\'ed\'eration Denis Poisson\\
Universit\'e d'Orl\'eans\\
BP 6759, 45067 Orl\'eans cedex 2, FRANCE.\\
E-mail: \email{didier.chauveau@univ-orleans.fr} \\
URL: \url{http://www.univ-orleans.fr/mapmo/membres/chauveau/}\\
\\
David R.~Hunter\\
Department of Statistics\\
326 Thomas Building\\
Pennsylvania State University\\
University Park, PA 16802\\
Telephone:  +1/814-863-0979\\
Fax: +1/814-863-7114\\
E-mail: \email{dhunter@stat.psu.edu} \\
URL: \url{http://www.stat.psu.edu/~dhunter/}\\
\\
Tatiana Benaglia\\
Department of Statistics, Penn State (see above)\\
E-mail:  \email{tab321@stat.psu.edu} \\
\\
Derek Young\\
Department of Statistics, Penn State (see above)\\
E-mail:  \email{dsy109@psu.edu}
}
%% It is also possible to add a telephone and fax number
%% before the e-mail in the following format:
%% Telephone: +43/1/31336-5053
%% Fax: +43/1/31336-734

%% for those who use Sweave please include the following line (with % symbols):
%% need no \usepackage{Sweave.sty}

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\begin{document}

%% include your article here, just as usual
%% Note that you should use the \pkg{}, \proglang{} and \code{} commands.

\section[Introduction to finite mixtures and mixtools]{Introduction to
  finite mixtures and \pkg{mixtools}}
%% Note: If there is markup in \(sub)section, then it has to be escape
%% as above.
\label{s:intro}

Authors' note: The original version of this vignette was produced using
an article that appears in the {\it Journal of Statistical Software}
(URL:  \url{http://www.jstatsoft.org/}); see
\citet{Benaglia+Chauveau+Hunter+Young:2009}.

Populations of individuals may often be divided into subgroups.  Yet
even when we observe characteristics of these individuals that provide
information about their subgroup memberships, we may not actually
observe these memberships {\em per se}. The basic goal of the tools in
the \pkg{mixtools} package (version 0.4.3, as of this writing) for
\proglang{R} \citep{r2009} is to examine a sample of measurements to
discern and describe subgroups of individuals, even when there is no
observable variable that readily indexes into which subgroup an
individual properly belongs.  This task is sometimes referred to as
``unsupervised clustering'' in the literature, and in fact mixture
models may be generally thought of as comprising the subset of
clustering methods known as ``model-based clustering''.  The
\pkg{mixtools} package is available from the Comprehensive \proglang{R}
Archive Network at \url{http://CRAN.R-project.org/package=mixtools}.

Finite mixture models may also be used in situations beyond those for
which clustering of individuals is of interest.  For one thing, finite
mixture models give descriptions of entire subgroups, rather than
assignments of individuals to those subgroups (though the latter may be
accomplished using mixture models).  Indeed, even the subgroups may not
necessarily be of interest; sometimes finite mixture models merely
provide a means for adequately describing a particular distribution,
such as the distribution of residuals in a linear regression model where
outliers are present.

Whatever the goal of the modeler when employing mixture models, much of
the theory of these models involves the assumption that the subgroups
are distributed according to a particular parametric form --- and quite
often this form is univariate or multivariate normal.  While
\pkg{mixtools} does provide tools for traditional fitting of finite
mixtures of univariate and multivariate normal distributions, it goes
well beyond this well-studied realm.  Arising from recent research whose
goal is to relax or modify the assumption of multivariate normality,
\pkg{mixtools} provides computational techniques for finite mixture
model analysis in which components are regressions, multinomial vectors
arising from discretization of multivariate data, or even distributions
that are almost completely unspecified.  This is the main feature that
distinguishes \pkg{mixtools} from other mixture-related \proglang{R}
packages, also available from the Comprehensive \proglang{R} Archive
Network at \url{http://CRAN.R-project.org/}, such as \pkg{mclust}
\citep{Fraley+Raftery:2009} and \pkg{flexmix} \citep{jss:Leisch:2004,
  Grun+Leisch:2008}.  We briefly mention these two packages in
Sections~\ref{section:EMexample} and \ref{section:pdmp}, respectively.

To make the mixture model framework more concrete, suppose the possibly
vector-valued random variables $\vec X_1, \ldots, \vec X_n$ are a simple
random sample from a finite mixture of $m>1$ arbitrary distributions,
which we will call {\em components} throughout this article.  The
density of each $\vec X_i$ may be written
\begin{equation}
\label{mvmixture}
g_{\f}(\vec x_i) = \sum_{j=1}^m\lambda_j\phi_j(\vec x_i), \quad
\vec x_i\in\bR^r,
\end{equation}
where $\f=(\vec\lambda, \vec \phi) = (\lambda_1, \ldots, \lambda_m,
\phi_1, \ldots, \phi_m)$ denotes the parameter and the $\lambda_m$ are
positive and sum to unity.  We assume that the $\phi_j$ are drawn from
some family $\cal F$ of multivariate density functions absolutely
continuous with respect to, say, Lebesgue measure.  The representation
\eqref{mvmixture} is not identifiable if no restrictions are placed on
$\cal F$, where by ``identifiable'' we mean that $g_{\f}$ has a {\em
  unique} representation of the form \eqref{mvmixture} and we do not
consider that ``label-switching'' --- i.e., reordering the $m$ pairs
$(\lambda_1, \phi_1), \ldots, (\lambda_m, \phi_m)$ --- produces a
distinct representation.

In the next sections we will sometimes have to distinguish between {\em
  parametric} and more general {\em nonparametric} situations.  This
distinction is related to the structure of the family $\CF$ of
distributions to which the component densities $\phi_j$ in model
\eqref{mvmixture} belong.  We say that the mixture is {\em parametric}
if $\CF$ is a parametric family, $\CF = \{\phi(\cdot|\vec\xi),
\vec\xi\in\bR^d\}$, indexed by a ($d$-dimensional) Euclidean parameter
$\vec\xi$.  A parametric family often used is the univariate Gaussian
family $\CF = \{\phi(\cdot|\mu,\sigma^2)=\mbox{density of
}\CN(\mu,\sigma^2), (\mu,\sigma^2)\in\bR\times\bR^+_*\}$, in which case
the model parameter reduces to $\f = (\vec \lambda,
(\mu_1,\sigma^2_1),\ldots,(\mu_m,\sigma^2_m))$.  For the multivariate
case, a possible parametric model is the {\em conditionally i.i.d.\
  normal model}, for which $\CF=\{\phi(\vec x_i) = \prod_{k=1}^r
f(x_{ik}), \mbox{$f(t)$ density of $\CN(\mu,\sigma^2)$}\}$ (this model
is included in \pkg{mixtools}; see Section~\ref{ss:nbcomp}).  An example
of a (multivariate) nonparametric situation is $\CF=\{\phi(\vec x_i) =
\prod_{k=1}^r f(x_{ik}), \mbox{$f(t)$ a univariate density on $\bR$}\}$, in
which case $\vec\f$ consists in a Euclidean part ($\vec\lb$) and a
nonparametric part $(f_1,\ldots,f_m)$.

As a simple example of a dataset to which mixture models may be applied,
consider the sample depicted in Figure \ref{geyser}. In the Old Faithful
dataset, measurements give time in minutes between eruptions of the Old
Faithful geyser in Yellowstone National Park, USA. These data are
included as part of the \pkg{datasets} package in \proglang{R}
\citep{r2009}; type \code{help("faithful")} in \proglang{R} for more
details.

<<faithful, echo=TRUE, results=hide>>=
library(mixtools)
data(faithful)
attach(faithful)
@

\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}[h]
  \centering
<<geyser, fig=TRUE, eps=FALSE>>=
hist(waiting, main="Time between Old Faithful eruptions",
     xlab="Minutes",  ylab="", cex.main=1.5, cex.lab=1.5, cex.axis=1.4)
@
\caption{The Old Faithful dataset is clearly suggestive of a
  two-component mixture of symmetric components.}
\label{geyser}
\end{figure}

For the Old Faithful eruption data, a two-component mixture
model is clearly a reasonable model based on the bimodality evident in
the histogram. This example is analyzed by \citet{hunter2007ims}, who
compare a standard normal-mixture method for fitting it with a novel
semiparametric approach.  Both approaches are included in
\pkg{mixtools}; see Sections \ref{section:EMexample} and
\ref{section:SPexample} of this article.

In Section~\ref{section:EM} of the current article we review the
well-known class of EM algorithms for finite mixture models, a common
thread that runs throughout much of the rest of the article.  The
remaining sections discuss various categories of functions found in the
\pkg{mixtools} package, from cutpoint methods that relax distributional
assumptions for multivariate data by discretizing the data
(Section~\ref{section:cut}), to semi- and non-parametric methods that
eliminate distributional assumptions almost entirely depending on what
the identifiability of the model allows (Section~\ref{section:np}), to
methods that handle various mixtures of regressions
(Section~\ref{section:reg}).  Finally, Section \ref{section:misc}
describes several miscellaneous features of the \pkg{mixtools} package.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{EM algorithms for finite mixtures}
\label{section:EM}


\subsection{Missing data setup}

Much of the general methodology used in \pkg{mixtools} involves the
representation of the mixture problem as a particular case of maximum
likelihood estimation (MLE) when the observations can be viewed as
incomplete data.  This setup implies consideration of two sample spaces,
the sample space of the (incomplete) observations, and a sample space of
some ``complete'' observations, the characterization of which being that
the estimation can be performed explicitly at this level.  For instance,
in parametric situations, the MLE based on the complete data may exist
in closed form.  Among the numerous reference papers and monographs on
this subject are, e.g., the original EM algorithm paper by
\citet{dempster1977mli} and the finite mixture model book by
\citet{mclachlan2000fmm} and references therein.  We now give a brief
description of this setup as it applies to finite mixture models in
general.

The (observed) data consist of $n$ i.i.d. observations $\vec x = (\vec
x_1,\ldots,\vec x_n)$ from a density $g_\f$ given by
\eqref{mvmixture}. It is common to denote the density of the sample by
$\Bg_\f$, the $n$-fold product of $g_\f$, so that we write simply
$\Bx\sim \Bg_\f$. In the missing data setup, $\Bg_\f$ is called the
incomplete-data density, and the associated log-likelihood is
$L_{\Bx}(\f) = \sum_{i=1}^n \log g_\f(\vec x_i)$. The (parametric) ML
estimation problem consists in finding $\hat\f_{\Bx} =
\argmax_{\f\in\Phi} L_{\Bx}(\f)$, or at least finding a local maximum
--- there are certain well-known cases in which a finite mixture model
likelihood is unbounded \citep{mclachlan2000fmm}, but we ignore these
technical details for now. Calculating $\hat\f_{\Bx}$ even for a
parametric finite mixture model is known to be a difficult problem, and
considering $\Bx$ as incomplete data resulting from non-observed
complete data helps.

The associated complete data is denoted by $\Bc = (\vec c_1,\ldots, \vec
c_n)$, with density $\Bh_\f(\Bc)=\prod_{i=1}^n h_\f(\vec c_i)$ (there
exists a many-to-one mapping from $\Bc$ to $\Bx$, representing the loss
of information).  In the model for complete data associated with
model~\eqref{mvmixture}, each random vector $\vec C_i = (\vec X_i,\vec
Z_i)$, where $\vec Z_i = (Z_{ij},j=1,\ldots m)$, and $Z_{ij}\in\{0,1\}$
is a Bernoulli random variable indicating that individual $i$ comes from
component $j$. Since each individual comes from exactly one component,
this implies $\sum_{j=1}^m Z_{ij}=1$, and
$$
\Prob(Z_{ij} = 1) = \lambda_{j},\quad (\vec X_i|Z_{ij}=1) \sim \phi_j,
\quad j=1,\ldots,m.
$$
The complete-data density for one observation is thus
$$
h_\f(\vec c_i) = h_\f(\vec x_i,\vec z_i) = \sum_{j=1}^m \I_{z_{ij}}\lb_j
\phi_j (\vec x_i),
$$
In the parametric situation, i.e.\ when $\CF$ is a parametric family, it
is easy to check that the complete-data MLE $\hat\f_{\Bc}$ based on
maximizing $\log \Bh_\f(\Bc)$ is easy to find, provided that this is the
case for the family $\CF$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{EM algorithms}
\label{sec:EM}

An EM algorithm iteratively maximizes, instead of the observed
log-likelihood $L_{\Bx}(\f)$, the operator
$$
Q(\f | \f^{(t)}) = \E \left[\log \Bh_\f(\BC)|\Bx,\f^{(t)} \right],
$$
where $\f^{(t)}$ is the current value at iteration~$t$, and the
expectation is with respect to the distribution $\Bk_\f(\Bc|\Bx)$ of
$\Bc$ given $\Bx$, for the value $\f^{(t)}$ of the parameter. The
iteration $\f^{(t)} \to \f^{(t+1)}$ is defined in the above general
setup by
\begin{enumerate}
\item E-step: compute $Q(\f | \f^{(t)})$

\item M-step: set $\f^{(t+1)} = \argmax_{\f\in\Phi}Q(\f | \f^{(t)})$
\end{enumerate}

For finite mixture models, the E-step does not depend on the structure
of $\CF$, since the missing data part is only related to the $\Bz$'s:
$$
\Bk_\f(\Bc|\Bx) = \prod_{i=1}^n k_\f(\vec z_i|\vec x_i).
$$
The $\Bz$ are discrete, and their distribution is given via Bayes'
theorem. The M-step itself can be split in two parts, the maximization
related to $\vec\lb$, which does not depend on $\CF$, and the
maximization related to $\vec \phi$, which has to be handled
specifically (say, parametrically, semi- or non-parametrically) for each
model.  Hence the EM algorithms for the models handled by the
\pkg{mixtools} package share the following common features:
\begin{enumerate}
\item{\bf E-step:\ } Calculate the ``posterior'' probabilities
  (conditional on the data and $\vec\theta^{(t)}$) of component
  inclusion,
\begin{equation}\label{posteriors}
  \post_{ij}^{(t)} \, \defn \,
  \Prob_{\vec\theta^{(t)}}(Z_{ij}=1| \vec x_i)
  = \frac{\lambda_j^{(t)} \phi_{j}^{(t)}(\vec x_{i})}
  {\sum_{j'=1}^m\lambda_{j'}^{(t)} \phi_{j'}^{(t)}(\vec x_{i})}
\end{equation}
for all $i=1,\ldots, n$ and $j=1, \ldots, m$.  Numerically, it can be
dangerous to implement equation (\ref{posteriors}) exactly as written
due to the possibility of the indeterminant form $0/0$ in cases where
$\vec x_i$ is so far from any of the components that all
$\phi_{j'}^{(t)}(\vec x_i)$ values result in a numerical underflow to
zero.  Thus, many of the routines in \pkg{mixtools} actually use the
equivalent expression
\begin{equation}\label{altposteriors}
  \post_{ij}^{(t)}
  = \left[ 1 +
    \sum_{j'\ne j} \frac{ \lambda_{j'}^{(t)} \phi_{j'}^{(t)}(\vec x_{i})}
    {\lambda_j^{(t)} \phi_{j}^{(t)}(\vec x_{i})} \right]^{-1}
\end{equation}
or some variant thereof.
\item{\bf M-step for $\vec\lb$:\ } Set
\begin{equation}\label{lambda}
  \lambda_j^{(t+1)} = \frac1n\sum_{i=1}^n \post_{ij}^{(t)} ,
  \quad\mbox{for $j=1, \ldots, m$.}
\end{equation}
\end{enumerate}


\subsection{An EM algorithm example}
\label{section:EMexample}

As an example, we consider the univariate normal mixture analysis of the
Old Faithful waiting data depicted in Figure \ref{geyser}. This fully
parametric situation corresponds to a mixture from the univariate
Gaussian family described in Section~\ref{s:intro}, where the $j$th
component density $\phi_j(x)$ in \eqref{mvmixture} is normal with mean
$\mu_j$ and variance $\sigma_j^2$.  This is a special case of the
general mixture-of-normal model that is well-studied in the literature
and for which other software, such as the \pkg{mclust}
\citep{Fraley+Raftery:2009} package for \proglang{R}, may also be used
for parameter estimation.  The M-step for the parameters
$(\mu_j,\sigma^2_j)$, $j=1,\ldots,m$ of this EM algorithm for such
mixtures of univariate normals is straightforward, and can be found,
e.g., in \citet{mclachlan2000fmm}.  The function \code{normalmixEM}
implements the algorithm in \pkg{mixtools}.  Code for the Old Faithful
example, using most of the default values (e.g., stopping criterion,
maximum number of iterations), is simply

<<normmixEM>>=
wait1 <- normalmixEM(waiting, lambda = .5, mu = c(55, 80), sigma = 5)
@

The code above will fit a 2-component mixture (because \code{mu} is a
vector of length two) in which the standard deviations are assumed equal
(because \code{sigma} is a scalar instead of a vector).  See
\code{help("normalmixEM")} for details about specifying starting values
for this EM algorithm.
<<geyserEM, eval=FALSE>>=
plot(wait1, density=TRUE, cex.axis=1.4, cex.lab=1.4, cex.main=1.8,
     main2="Time between Old Faithful eruptions", xlab2="Minutes")
@

\setkeys{Gin}{width=0.49\textwidth}
\begin{figure}[!h]
  \centering
<<geyserEM, echo=FALSE, results=tex>>=
for(i in 1:2){
  file=paste("geyserEM", i, ".pdf", sep="")
  pdf(file=file, paper="special", width=6, height=6)
  plot(wait1, whichplots=i, cex.axis = 1.4, cex.lab = 1.4, cex.main =
       1.8,   main2 = "Time between Old Faithful eruptions", xlab2 =
       "Minutes")
  dev.off()
  cat("\\includegraphics{", file, "}\n", sep="")
}
@
\caption{The Old Faithful waiting data fitted with a parametric EM
  algorithm in \pkg{mixtools}. Left: the sequence of log-likelihood
  values; Right: the fitted Gaussian components.}
\label{geyserEM}
\end{figure}

The \code{normalmixEM} function returns an object of class
\code{"mixEM"}, and the \code{plot} method for these objects delivers
the two plots given in Figure \ref{geyserEM}: the sequence $t\mapsto
L_{\Bx}(\f^{(t)})$ of observed log-likelihood values and the histogram
of the data with the $m$ ($m=2$ here) fitted Gaussian component
densities of $\CN(\hat\mu_j,\hat\sigma^2_j)$, $j=1,\ldots,m$, each
scaled by the corresponding $\hat\lambda_j$, superimposed.  The
estimator $\hat{\vec\theta}$ can be displayed by typing, e.g.,
<<geyserestimates>>=
wait1[c("lambda", "mu", "sigma")]
@

Alternatively, the same output may be obtained using the \code{summary}
method:
<<geysersummary>>=
summary(wait1)
@

\section{Cutpoint methods}
\label{section:cut}

Traditionally, most literature on finite mixture models has assumed that
the density functions $\phi_j(\vec x)$ of equation (\ref{mvmixture})
come from a known parametric family.  However, some authors have
recently considered the problem in which $\phi_j(\vec x)$ is unspecified
except for some conditions necessary to ensure the identifiability of
the parameters in the model.  One such set of conditions is as follows:

\citet{hettmansperger2000ani}; \citet{cruzmedina2004smm}; and
\citet{elmore2004ecc} treat the case in which $\phi_j(\vec x)$ equals
the product $f_j(x_i)\cdots f_j(x_r)$ for some univariate density
function $f_j$.  Thus, conditional on knowing that $\vec X$ comes from
the $j$th mixture component, the coordinates of $\vec X$ are independent
and identically distributed.  For this reason, this case is called the
conditionally i.i.d.\ model.

The authors named above have developed an estimation method for the
conditionally i.i.d.\ model.  This method, the {\em cutpoint approach},
discretizes the continuous measurements by replacing each
$r$-dimensional observation, say $\vec X_i= (x_{i1}, \ldots, x_{ir})$,
by the $p$-dimensional multinomial vector $(n_1, \ldots, n_p)$, where
$p\ge2$ is chosen by the experimenter along with a set of cutpoints
$-\infty = c_0 < c_1 < \cdots < c_p=\infty$, so that for $a=1, \ldots,
p$,
\[
n_a = \sum_{k=1}^r I\{c_{a-1} < x_{ik} \le c_a\}.
\]
Note that the multinomial distribution is guaranteed by the conditional
i.i.d.\ assumption, and the multinomial probability of the $a$th
category is equal to $\theta_a \equiv P_{}(c_{a-1}<X_{ik}\le c_a)$.

The cutpoint approach is completely general in the sense that it can be
applied to any number of components $m$ and any number of repeated
measures $r$, just as long as $r \ge 2m-1$, a condition that guarantees
identifiability \citep{elmore2003}.  However, some information is lost
in the discretization step, and for this reason it becomes difficult to
obtain density estimates of the component densities.  Furthermore, even
if the assumption of conditional independence is warranted, the extra
assumption of identically distributed coordinates may not be; and the
cutpoint method collapses when the coordinates are not identically
distributed.

As an illustration of the cutpoint approach applied to a dataset, we
show here how to use \pkg{mixtools} to reconstruct---almost---an example
from \citet{elmore2004ecc}.  The dataset is \code{Waterdata}, a
description of which is available by typing \code{help("Waterdata")}.
This dataset contains 8 observations on each of 405 subjects, where the
observations are angle degree measurements ranging from $-90$ to $90$
that describe the subjects' answers to a series of 8 questions related
to a conceptual task about how the surface of a liquid would be oriented
if the vessel containing it were tipped to a particular angle.  The
correct answer is 0 degree in all cases, yet the subjects showed a
remarkable variety of patterns of answers.  \citet{elmore2004ecc}
assumed the conditionally i.i.d.\ model (see \citet{benaglia2009} for an
in-depth discussion of this assumption and this dataset) with both $m=3$
and $m=4$ mixture components.  \citet{elmore2004ecc} summarized their
results by providing plots of estimated empirical distribution functions
for the component distributions, where these functions are given by
\begin{equation}\label{ecdf}
  \tilde F_j(x) = \frac{1}{mn\lambda_j}\sum_{i=1}^n \sum_{\ell=1}^r
  p_{ij}I\{x_{i\ell}\le x\}.
\end{equation}
In equation (\ref{ecdf}), the values of $\lambda_j$ and $p_{ij}$ are the
final maximum likelihood estimates of the mixing proportions and
posterior component membership probabilities that result from fitting a
mixture of $m$ multinomials (note in particular that the estimates of
the multinomial parameters $\theta_a$ for each component are not used in
this equation).

We cannot obtain the exact results of \citet{elmore2004ecc} because
those authors do not state specifically which cutpoints $c_a$ they use;
they merely state that they use thirteen cutpoints.  It appears from
their Figures 1 and 2 that these cutpoints occur approximately at
intervals of 10.5 degrees, starting at $-63$ and going through $63$;
these are the cutpoints that we adopt here.  The function
\code{makemultdata} will create a multinomial dataset from the original
data, as follows:
<<cutpoint>>=
data("Waterdata")
cutpts <- 10.5*(-6:6)
watermult <- makemultdata(Waterdata, cuts = cutpts)
@

Once the multinomial data have been created, we may apply the
\code{multmixEM} function to estimate the multinomial parameters via an
EM algorithm.
<<multmixEM>>=
set.seed(15)
theta4 <- matrix(runif(56), ncol = 14)
theta3 <- theta4[1:3,]
mult3 <- multmixEM(watermult, lambda = rep(1, 3)/3, theta = theta3)
mult4 <- multmixEM (watermult, lambda = rep (1, 4) / 4, theta = theta4)
@

Finally, \code{compCDF} calculates and plots the estimated distribution
functions of equation (\ref{ecdf}).  Figure \ref{WDcutpoint} gives plots
for both a 3-component and a 4-component solution; these plots are very
similar to the corresponding plots in Figures 1 and 2 of
\citet{elmore2004ecc}.
<<echo=TRUE, eval=FALSE>>=
cdf3 <- compCDF(Waterdata, mult3$posterior, lwd=2, lab=c(7, 5, 7),
                xlab="Angle in degrees", ylab="Component CDFs",
                main="Three-Component Solution")
cdf4 <- compCDF(Waterdata, mult4$posterior, lwd=2, lab=c(7, 5, 7),
                xlab="Angle in degrees", ylab="Component CDFs",
                main="Four-Component Solution")
@

<<cutpointplots, results=hide, echo=FALSE>>=
pdf(file="WDcutpoint3comp.pdf", paper="special", width=8, height=8)
cdf3 <- compCDF(Waterdata, mult3$posterior, lwd=3,
                xlab="Angle in degrees", lab=c(7, 5, 7), ylab="Component CDFs",
                main="Three-Component Solution", cex.axis=1.4, cex.lab=1.5,
                cex.main=1.5)
ltext <- paste(round(mult3$lam*100, 1), "%", sep="")
legend("bottomright", legend=ltext, pch=15:17, cex=1.5, pt.cex=1.35)
y <- compCDF(Waterdata, mult3$posterior, x=cutpts, makeplot=F)
for(i in 1:3) points(cutpts, y[i,], pch=14+i, cex=1.35)
dev.off()
pdf(file="WDcutpoint4comp.pdf", paper="special", width=8, height=8)
cdf4 <- compCDF(Waterdata, mult4$posterior, lwd=3,
                xlab="Angle in degrees", lab=c(7, 5, 7),
                ylab="Component CDFs", main="Four-Component Solution",
                cex.axis=1.4, cex.lab=1.5, cex.main=1.5)
ltext <- paste(round(mult4$lam*100,1), "%", sep="")
legend("bottomright", legend=ltext, pch=15:18, cex=1.5, pt.cex=1.35)
y <- compCDF(Waterdata, mult4$posterior, x=cutpts, makeplot=F)
for(i in 1:4) points(cutpts, y[i,], pch=14+i, cex=1.35)
dev.off()
@

\begin{figure}[!h]
  \centering
  \includegraphics[width=0.49\textwidth]{WDcutpoint3comp}
  \includegraphics[width=0.49\textwidth]{WDcutpoint4comp}
  \caption{Empirical cumulative distribution function (CDF) estimates
    for the three- and four-component multinomial cutpoint models for
    the water-level data; compare Figures 1 and 2 of
    \citet{elmore2004ecc}.  The 13 cutpoints used are indicated by the
    points in the plots, and the estimated mixing proportions for the
    various components are given by the legend.  }
  \label{WDcutpoint}
\end{figure}

As with the output of \code{normalmixEM} in Section~\ref{section:EM},
it is possible to summarize the output of the \code{multmixEM} function using
the \code{summary} method for \code{mixEM} objects:
<<summarymult4>>=
summary(mult4)
@


\section{Nonparametric and semiparametric methods}
\label{section:np}

In this section, we consider nonparametric multivariate finite mixture
models.  The first algorithm presented here was introduced by
\citet{benaglia2009} as a generalization of the stochastic
semiparametric EM algorithm of \citet{bordes2007sas}. Both algorithms
are implemented in \pkg{mixtools}.

\subsection{EM-like algorithms for mixtures of unspecified densities}
\label{section:EMlike}

Consider the mixture model described by equation \eqref{mvmixture}. If
we assume that the coordinates of the $\vec X_i$ vector are {\em
  conditionally independent}, i.e. they are independent conditional on
the subpopulation or component ($\phi_1$ through $\phi_m$) from which
$\vec X_i$ is drawn, the density in \eqref{mvmixture} can be rewritten
as:
\begin{equation}
  \label{mvmixture2}
  g_{\vec\theta}(\vec x_i) =
  \sum_{j=1}^m\lambda_j\prod_{k=1}^rf_{jk}(x_{ik}),
\end{equation}
where the function $f(\cdot)$, with or without subscripts, will always
denote a univariate density function. Here we do not assume that
$f_{jk}(\cdot)$ comes from a family of densities that may be indexed by
a finite-dimensional parameter vector, and we estimate these densities
using nonparametric density techniques. That is why we say that this
algorithm is a fully nonparametric approach.

The density in equation \eqref{mvmixture2} allows for a different
distribution for each component and each coordinate of $\vec
X_i$. Notice that if the density $f_{jk}(\cdot)$ does not depend on $k$,
we have the case in which the $\vec X_i$ are not only conditionally
independent but identically distributed as well. These are the two
extreme cases. In order to encompass both the conditionally i.i.d. case
and the more general case \eqref{mvmixture2} simultaneously in one
model, we allow that the coordinates of $\vec X_i$ are conditionally
independent and there exist {\em blocks} of coordinates that are also
identically distributed. If we let $b_k$ denote the block to which the
$k$th coordinate belongs, where $1\le b_k\le B$ and $B$ is the total
number of such blocks, then equation \eqref{mvmixture2} is replaced by
\begin{equation}\label{rmgeneral}
  g_{\vec\theta} (\vec x_i) = \sum_{j=1}^m \lambda_j \prod_{k=1}^r
  f_{j{b_k}} (x_{ik}).
\end{equation}

The indices $i$, $j$, $k$, and $\ell$ will always denote a generic
individual, component (subpopulation), coordinate (repeated
measurement), and block, respectively. Therefore, we will always have
$1\le i\le n$, $1\le j\le m$, $1\le k\le r$, and $1\le\ell\le B$.

The EM algorithm to estimate model \eqref{rmgeneral} has the E-step and
M-step described in Section~\ref{sec:EM}.  In equation
(\ref{posteriors}), we have $\phi_j^{(t)}(\vec x_i) = \prod_{k=1}^r
f_{jb_k}^{(t)}(x_{ik})$, where $f_{j\ell}^{(t)}(\cdot)$ is obtained by a
weighted nonparametric (kernel) density estimate, given by:
\begin{enumerate}
  \addtocounter{enumi}{2}
\item{\bf Nonparametric (Kernel) density estimation step:\ } For any
  real $u$, define for each component $j\in\{1, \ldots, m\}$ and each
  block $\ell\in\{1, \ldots, B\}$
  \begin{equation}
    \label{densest}
    f_{j\ell}^{t+1}(u) = \frac
    {1}{nh_{j\ell} C_\ell\lambda_{j}^{t+1}} \sum_{k=1}^r \sum_{i=1}^n
    \post_{ij}^{(t)} I\{b_k=\ell\}
    K\left(\frac{u-x_{ik}}{h_{j\ell}}\right),
  \end{equation}
  where $K(\cdot)$ is a kernel density function, $h_{j\ell}$ is the
  bandwidth for the $j$th component and $\ell$th block density estimate,
  and $C_\ell$ is the number of coordinates in the $\ell$th block.
\end{enumerate}

The function \code{npEM} implements this algorithm in
\pkg{mixtools}. This function has an argument \code{samebw} which, when
set to \code{TRUE} (the default), takes $h_{j\ell} = h$, for all $1 \le
j \le m$ and $1\le\ell\le B$, that is, the same bandwidth for all
components and blocks, while \code{samebw = FALSE} allows a different
bandwidth for each component and each block, as detailed in
\citet{bch:festchrift2009}.  This function will, if called using
\code{stochastic = TRUE}, replace the deterministic density estimation
step (\ref{densest}) by a {\em stochastic} density estimation step of
the type proposed by \citet{bordes2007sas}: First, generate $\vec
Z^{(t)}_{i} = (Z^{(t)}_{i1}, \ldots, Z^{(t)}_{im})$ as a multivariate
random vector with a single trial and success probability vector $\vec
p_i^{(t)} = (p_{i1}^{(t)}, \ldots, p_{1m}^{(t)})$, then in the M-step
for $\lambda_{j}^{t+1}$ in equation~(\ref{lambda}), replace
$p^{(t)}_{ij}$ by $Z^{(t)}_{ij}$ and let
\[
f_{j\ell}^{t+1}(u) = \frac {1}{nh_{j\ell} C_\ell\lambda_{j}^{t+1}}
\sum_{k=1}^r \sum_{i=1}^n Z_{ij}^{(t)} I\{b_k=\ell\}
K\left(\frac{u-x_{ik}}{h_{j\ell}}\right).
\]
In other words, the stochastic versions of these algorithms re-assign
each observation randomly at each iteration, according to the
$p_{ij}^{(t)}$ values at that iteration, to one of the $m$ components,
then the density estimate for each component is based only on those
observations that have been assigned to it.  Because the stochastic
algorithms do not converge the way a deterministic algorithm often does,
the output of \code{npEM} is slightly different when \code{stochastic =
  TRUE} than when \code{stochastic = FALSE}, the default.  See the
corresponding help file for details.

\citet{benaglia2009} also discuss specific cases of model
(\ref{rmgeneral}) in which some of the $f_{jb_k}(\cdot)$ densities are
assumed to be the same except for a location and scale change.  They
refer to such cases as semiparametric since estimating each
$f_{jb_k}(\cdot)$ involves estimating an unknown density as well as
multiple location and scale parameters.  For instance, equation (17) of
\citet{benaglia2009} sets
\begin{equation}
  \label{spEM}
  f_{j\ell}(x) = \frac{1}{\sigma_{j\ell}}f
  \left( \frac{x-\mu_{j\ell}}{\sigma_{j\ell}} \right),
\end{equation}
where $\ell=b_k$ for a generic $k$.

The \pkg{mixtools} package implements an algorithm for fitting model
(\ref{spEM}) in a function called \code{spEM}.  Details on the use of
this function may be obtained by typing \code{help("spEM")}.
Implementation of this algorithm and of that of the \code{npEM} function
requires updating the values of $f_{jb_k}(x_{ik})$ for all $i$, $j$, and
$k$ for use in the E-step (\ref{posteriors}).  To do this, the
\code{spEM} algorithm keeps track of an $n\times m$ matrix, called
$\Phi$ here, where
\[
\Phi_{ij} \equiv \phi_j(\vec x_i) = \prod_{k=1}^r f_{jb_k}(x_{ik}).
\]
The density estimation step of equation (\ref{densest}) updates the
$\Phi$ matrix for the $(t+1)$th iteration based on the most recent
values of all of the parameters.  For instance, in the case of model
(\ref{spEM}), we obtain
\begin{eqnarray*}
  \Phi_{ij}^{t+1}
  &=& \prod_{\ell=1}^B\prod_{k:b_k=\ell} \frac{1}{\sigma_{j\ell}^{t+1}}
  f^{t+1}
  \left( \frac{x-\mu_{j\ell}^{t+1}}{\sigma_{j\ell}^{t+1}} \right) \\
  &=& \prod_{\ell=1}^B
  \prod_{k:b_k=\ell} \frac{1}{\sigma_{j\ell}^{t+1}} \sum_{i'=1}^n
  \frac{p_{ij}^{t+1}}{hrn\lambda_j^{t+1}}
  \sum_{k'=1}^r
  K\left[
    \frac{\left(\frac{x_{ik}-\mu_{j\ell}^{t+1}}{\sigma_{j\ell}^{t+1}} \right) - (x_{i'k'} -
      \mu_{j\ell}^{t+1})}
    {h\sigma_{j\ell}^{t+1}}
  \right].
\end{eqnarray*}

\subsection{A univariate symmetric, location-shifted semiparametric
  example}
\label{section:SPexample}

Both \citet{hunter2007ims} and \citet{bordes2006set} study a particular
case of model ({\ref{mvmixture}) in which $x$ is univariate and
\begin{equation}
  \label{spmodel}
  g_{\vec \theta}(x) = \sum_{j=1}^m\lambda_j \phi(x-\mu_j),
\end{equation}
where $\phi(\cdot)$ is a density that is assumed to be completely
unspecified except that it is symmetric about zero.  Because each
component distribution has both a nonparametric part $\phi(\cdot)$ and a
parametric part $\mu_j$, we refer to this model as semiparametric.

Under the additional assumption that $\phi(\cdot)$ is absolutely
continuous with respect to Lebesgue measure, \citet{bordes2007sas}
propose a stochastic algorithm for estimating the model parameters,
namely, $(\vec\lambda, \vec\mu, \phi)$.  This algorithm is implemented
by the \pkg{mixtools} function \code{spEMsymloc}.  This function also
implements a nonstochastic version of the algorithm, which is the
default and which is a special case of the general algorithm described
in Section~\ref{section:EMlike}.

<<spsymmplots, results=hide, echo=FALSE>>=
pdf(file="spsymmfig1.pdf", paper="special", width=8, height=8)
par(mar=0.1+c(5,4.2,4,1.8))
plot(wait1, which = 2, cex.axis = 1.4, cex.lab = 1.5, cex.main = 1.5,
     main2 = "Time between Old Faithful eruptions", xlab2 = "Minutes")
wait2 <- spEMsymloc(waiting, mu0 = c(55, 80))
plot(wait2, lty = 2, newplot = FALSE, addlegend = FALSE)
dev.off()
pdf(file="spsymmfig2.pdf", paper="special", width=8, height=8)
par(mar=0.1+c(5,4.2,4,1.8))
wait2a <- spEMsymloc(waiting, mu0 = c(55, 80), bw = 1)
wait2b <- spEMsymloc(waiting, mu0 = c(55, 80), bw = 6)
plot(wait2a, lty = 1, addlegend = FALSE, cex.axis = 1.4, cex.lab = 1.5,
     cex.main = 1.5, title = "Time between Old Faithful eruptions",
     xlab = "Minutes")
plot(wait2b, lty = 2, newplot = FALSE, addlegend = FALSE)
dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[height=3in,width=3in]{spsymmfig1}
  \includegraphics[height=3in,width=3in]{spsymmfig2}
  \caption{The Old Faithful dataset, fit using different algorithms in
    \pkg{mixtools}. Left: the fitted Gaussian components (solid) and a
    semiparametric fit assuming model (\ref{spmodel}) with the default
    bandwidth of $4.0$ (dashed); Right: the same model (\ref{spmodel})
    using bandwidths of $1.0$ (solid) and $6.0$ (dashed).}
  \label{spsymmfig}
\end{figure}

As noted in Figure \ref{geyser}, model (\ref{spmodel}) appears to be an
appropriate model for the Old Faithful waiting times dataset.  Here, we
provide code that applies the \code{spEMsymloc} function to these data.
First, we display the normal mixture solution of Figure \ref{geyserEM}
with a semiparametric solution superimposed, in Figure
\ref{spsymmfig}(a):
<<plotspsymm, eval=FALSE>>=
plot(wait1, which = 2, cex.axis = 1.4, cex.lab = 1.4, cex.main = 1.8,
     main2 = "Time between Old Faithful eruptions", xlab2 = "Minutes")
wait2 <- spEMsymloc(waiting, mu0 = c(55, 80))
plot(wait2, lty = 2, newplot = FALSE, addlegend = FALSE)
@

Because the semiparametric version relies on a kernel density estimation
step (\ref{densest}), it is necessary to select a bandwidth for this
step.  By default, \code{spEMsymloc} uses a fairly simplistic approach:
It applies ``Silverman's rule of thumb'' \citep{silverman1986des} to the
entire dataset using the \code{bw.nrd0} function in \proglang{R}.  For
the Old Faithful waiting time dataset, this bandwidth is about~$4$:
<<bandwidth>>=
bw.nrd0(waiting)
@

But the choice of bandwidth can make a big difference, as seen in Figure
\ref{spsymmfig}(b).
<<plotbweffect, eval=FALSE>>=
wait2a <- spEMsymloc(waiting, mu0 = c(55, 80), bw = 1)
wait2b <- spEMsymloc(waiting, mu0 = c(55, 80), bw = 6)
plot(wait2a, lty = 1, addlegend = FALSE, cex.axis = 1.4,
     cex.lab = 1.4, cex.main = 1.8, xlab = "Minutes",
     title = "Time between Old Faithful eruptions")
plot(wait2b, lty = 2, newplot = FALSE, addlegend = FALSE)
@

We find that with a bandwidth near $2$, the semiparametric solution
looks quite close to the normal mixture solution of Figure
\ref{geyserEM}.  Reducing the bandwidth further results in the
``bumpiness'' exhibited by the solid line in Figure \ref{spsymmfig}(b).
On the other hand, with a bandwidth of 8, the semiparametric solution
completely breaks down in the sense that algorithm tries to make each
component look similar to the whole mixture distribution.  We encourage
the reader to experiment by changing the bandwidth in the above code.

\subsection{A trivariate Gaussian example}
\label{ss:trigauss}

As a first simple, nonparametric example, we simulate a Gaussian
trivariate mixture with independent repeated measures and a shift of
location between the two components in each coordinate, i.e., $m=2$,
$r=3$, and $b_k=k$, $k=1,2,3$.  The individual densities $f_{jk}$ are
the densities of $\CN(\mu_{jk},1)$, with component means $\vec\mu_1 =
(0,0,0)$ and $\vec\mu_2=(3,4,5)$. This example was introduced by
\citet{hall2005nim} then later reused by \citet{benaglia2009} for
comparison purposes.  Note that the parameters in this model are
identifiable, since \citet{hall2003nec} showed that for two components
($m=2$), identifiability holds in model~\eqref{mvmixture} is under mild
assumptions as long as $r\ge3$, even in the most general case in which
$b_k=k$ for all $k$.

A function \code{ise.npEM} has been included in \pkg{mixtools} for
numerically computing the integrated squared error (ISE) relative to a
user-specified true density for a selected estimated density $\hat
f_{jk}$ from \code{npEM} output.  Each density $\hat f_{jk}$ is computed
using equation~(\ref{densest}) together with the posterior probabilities
after convergence of the algorithm, i.e., the final values of the
$\post_{ij}^t$ (when \code{stochastic = FALSE}).  We illustrate the
usage of \code{ise.npEM} in this example by running a Monte Carlo
simulation for $S$ replications, then computing the square root of the
mean integrated squared error (MISE) for each density, where
\[ {\rm MISE} = \frac{1}{S}\sum_{s=1}^S \int \left(\hat
  f_{jk}^{(s)}(u)-f_{jk}(u)\right)^2\,du,\quad j=1,2 \mbox{ and }
k=1,2,3.
\]

For this example, we first set up the model true parameters with $S=100$
replications of $n=300$ observations each:
<<gaussexample>>=
m <- 2; r <- 3; n <- 300; S <- 100
lambda <- c(0.4, 0.6)
mu <- matrix(c(0, 0, 0, 3, 4, 5), m, r, byrow = TRUE)
sigma <- matrix(rep(1, 6), m, r, byrow = TRUE)
@

Next, we set up ``arbitrary'' initial centers, a matrix for storing sums
of integrated squared errors, and an integer storing the number of
suspected instances of label switching that may occur during the
replications:
<<gaussinitial>>=
centers <- matrix(c(0, 0, 0, 4, 4, 4), 2, 3, byrow = TRUE)
ISE <- matrix(0, m, r, dimnames = list(Components = 1:m, Blocks = 1:r))
nblabsw <- 0
@
Finally, we run the Monte Carlo simulation, using the \code{samebw =
  FALSE} option since it is more appropriate for this location-shift
model:
<<sqMISE>>=
set.seed(1000)
for (mc in 1:S) {
  x <- rmvnormmix(n, lambda, mu, sigma)
  a <- npEM(x, centers, verb = FALSE, samebw = FALSE)
  if (a$lambda[1] > a$lambda[2]) nblabsw <- nblabsw + 1
  for (j in 1:m) {
     for (k in 1:r) {
     ISE[j, k] <- ISE[j, k] + ise.npEM(a, j, k, dnorm,
         lower = mu[j, k] - 5, upper = mu[j, k] + 5, plots = FALSE,
         mean = mu[j, k], sd = sigma[j, k])$value
    }
  }
}
MISE <- ISE/S
print(sqMISE <- sqrt(MISE))
@

We can examine the \code{npEM} output from the last replication above
using
<<summarygauss>>=
summary(a)
@

We can also get plots of the estimated component densities for each
block (recall that in this example, block $\ell$ consists only of
coordinate $\ell$) using the \code{plot} function. The resulting plots
are given in Figure~\ref{fig:gausstrivariate}.
<<plotgauss3rm, eval=FALSE>>=
plot(a)
@

<<gauss3rm, echo=FALSE, results=hide>>=
pdf("gauss3rm.pdf", paper="special", width=10, height=5)
par(mfrow=c(1,3))
plot(a)
dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[width=.99\textwidth]{gauss3rm}
  \caption{Output of the \code{npEM} algorithm for the trivariate
    Gaussian model with independent repeated measures.}
  \label{fig:gausstrivariate}
\end{figure}


\subsection{A more general multivariate nonparametric example}
\label{sec:generalmv}

In this section, we fit a more difficult example, with non-multimodal
mixture densities (in block \#2), heavy-tailed distributions, and
different scales among the coordinates. The model is multivariate with
$r=5$ repeated measures and $m=2$ components (hence identifiability
holds; cf.\ \citet{hall2003nec} as cited in
Section~\ref{ss:trigauss}). The $5$ repeated measures are grouped into
$B=2$ blocks, with $b_1=b_2=b_3=1$ and $b_4=b_5=2$.  Block $1$
corresponds to a mixture of two noncentral Student $t$ distributions,
$t'(2,0)$ and $t'(10,8)$, where the first parameter is the number of
degrees of freedom, and the second is the non-centrality. Block~2
corresponds to a mixture of Beta distributions, ${\cal B}(1,1)$ (which
is actually the uniform distribution over $[0,1]$) and ${\cal
  B}(1,5)$. The first component weight is $\lambda_1 = 0.4$. The true
mixtures are depicted in Figure~\ref{fig:true5rm}.

<<true5rm, echo=FALSE, results=hide>>=
pdf("truepdf5rm_block1.pdf")
par(mar=0.1+c(5,4.2,4,1.5))
x <- seq(-10, 25, len=250)
plot(x, .4* dt(x, 2, 0) + .6 * dt(x, 10, 8), type="l", lwd=3, col=2,
     cex.axis=1.4, cex.lab=1.5, cex.main=1.5, main="Block 1", xlab="",
     ylab="Density")
lines (x, .4*dt(x, 2, 0), lwd=4, lty=2)
lines (x, .6*dt(x, 10, 8), lwd=4, lty=2)
dev.off()
pdf("truepdf5rm_block2.pdf")
par(mar=0.1+c(5,4.2,4,1.5))
x <- seq(0, 1, len=250)
plot(x, .4 + .6 * dbeta(x, 1, 5), type="l", lwd=3, col=2, cex.axis=1.4,
   cex.lab=1.5, cex.main=1.5, main="Block 2", xlab="", ylab="Density",
   ylim= c(0, 3.4))
lines (x, rep(.4, 250), lwd=4, lty=2)
lines (x, .6*dbeta(x, 1, 5), lwd=4, lty=2)
dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[height=2.5in,width=2.5in]{truepdf5rm_block1}
  \includegraphics[height=2.5in,width=2.5in]{truepdf5rm_block2}
  \caption{True densities for the mixture of
    Section~\ref{sec:generalmv}, with individual component densities
    (scaled by $\lambda_j$) in dotted lines and mixture densities in
    solid lines.  The noncentral $t$ mixture of coordinates 1 through 3
    is on the left, the beta mixture of coordinates 4 and 5 on the
    right.}
  \label{fig:true5rm}
\end{figure}


To fit this model in \pkg{mixtools}, we first set up the model
parameters:
<<parameters5rm>>=
m <- 2; r <- 5
lambda <- c(0.4, 0.6)
df <- c(2, 10); ncp <- c(0, 8)
sh1 <- c(1, 1) ; sh2 <- c(1, 5)
@

Then we generate a pseudo-random sample of size $n=300$ from this model:
<<generate5rm>>=
n <- 300; z <- sample(m, n, rep = TRUE, prob = lambda)
r1 <- 3; z2 <- rep(z, r1)
x1 <- matrix(rt(n * r1, df[z2], ncp[z2]), n, r1)
r2 <- 2; z2 <- rep(z, r2)
x2 <- matrix(rbeta(n * r2, sh1[z2], sh2[z2]), n, r2)
x <- cbind(x1, x2)
@

For this example in which the coordinate densities are on different
scales, it is obvious that the bandwidth in \code{npEM} should depend on
the blocks and components.  We set up the block structure and some
initial centers, then run the algorithm with the option \code{samebw =
  FALSE}:
<<npEM5rm>>=
id <- c(rep(1, r1), rep(2, r2))
centers <- matrix(c(0, 0, 0, 1/2, 1/2, 4, 4, 4, 1/2, 1/2), m, r,
                  byrow = TRUE)
b <- npEM(x, centers, id, eps = 1e-8, verb = FALSE, samebw = FALSE)
@
Figure~\ref{fig:npEM5rm} shows the resulting density estimates, which
may be obtained using the plotting function included in \pkg{mixtools}:

<<plot5rm, eval=FALSE>>=
plot(b, breaks = 15)
@

% plot(b, breaks = 15, cex.main = 1.5, cex.lab = 1.5, cex.axis = 1.4,
%     cex.legend = 1.5)

<<plot5rmcommands, echo=FALSE, results=hide>>=
pdf("npEM5rm.pdf", width=8, height=5)
par(mfrow=c(1,2))
plot(b, breaks = 15)

dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[width=.95\textwidth]{npEM5rm}
  \caption{Result of plotting \code{npEM} output for the example of
    Section~\ref{sec:generalmv}.  Since $n=300$, the histogram on the
    left includes 900 observations and the one on the right includes
    600.}
  \label{fig:npEM5rm}
\end{figure}

Finally, we can compute the ISE of the estimated density relative to the
truth for each block and component. The corresponding output is depicted
in Figure \ref{fig:ISEnpEM5rm}.
<<ISEnpEM5rm, eval=FALSE>>=
par(mfrow=c(2,2))
for (j in 1:2){
  ise.npEM(b, j, 1, truepdf = dt, lower = ncp[j] - 10,
           upper = ncp[j] + 10, df = df[j], ncp = ncp[j])
  ise.npEM(b, j, 2, truepdf = dbeta, lower = -0.5,
           upper = 1.5,  shape1 = sh1[j], shape2 = sh2[j])
}
@ 

<<plotISEnpEM5rm, echo=FALSE, results=hide>>=
options(warn=-1)
pdf("ISEnpEM5rm.pdf", width=8, height=8)
par(mfrow = c(2, 2))
for (j in 1:2){
  ise.npEM(b, j, 1, truepdf = dt, lower = ncp[j] - 10,
           upper = ncp[j] + 10, df = df[j], ncp = ncp[j])
  ise.npEM(b, j, 2, truepdf = dbeta, lower = -0.5,
           upper = 1.5,  shape1 = sh1[j], shape2 = sh2[j])
}
dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[height=5in,width=6in]{ISEnpEM5rm}
  \caption{\code{ise.npEM} output for the 5-repeated measures example;
    the true densities are $f_{11}\equiv t'(2,0)$, $f_{21}\equiv
    t'(10,8)$, $f_{12}\equiv {\cal U}_{(0,1)}$, $f_{22}\equiv {\cal
      B}(1,5)$.}
  \label{fig:ISEnpEM5rm}
\end{figure}

\section{Mixtures of regressions}
\label{section:reg}

\subsection{Mixtures of linear regressions}

Consider a mixture setting where we now assume $\textbf{X}_{i}$ is a
vector of covariates observed with a response $Y_{i}$.  The goal of
mixtures of regressions is to describe the conditional distribution of
$Y_{i}|\textbf{X}_{i}$. Mixtures of regressions have been extensively
studied in the econometrics literature and were first introduced by
\citet{quandt1972sr} as the \textit{switching regimes} (or
\textit{switching regressions}) problem. A switching regimes system is
often compared to \textit{structural change} in a system
\citep{quandtram1978sr}. A structural change assumes the system depends
deterministically on some observable variables, but switching regimes
implies one is unaware of what causes the switch between regimes. In the
case where it is assumed there are two heterogeneous classes,
\citet{quandt1972sr} characterized the switching regimes problem ``by
assuming that nature chooses between regimes with probabilities
$\lambda$ and $1-\lambda$''.

Suppose we have $n$ independent univariate observations,
$y_{1},\ldots,y_{n}$, each with a corresponding vector of predictors,
$\textbf{x}_{1},\ldots,\textbf{x}_{n}$, with
$\textbf{x}_{i}=(x_{i,1},\ldots,x_{i,p})^\top$ for $i=1,\ldots,n$. We
often set $x_{i,1}=1$ to allow for an intercept term. Let
$\textbf{y}=(y_{1},\ldots,y_{n})^\top$ and let $\underline{\textbf{X}}$
be the $n\times p$ matrix consisting of the predictor vectors.

Suppose further that each observation $(y_{i}, \vec x_i)$ belongs to one
of $m$ classes. Conditional on membership in the $j$th component, the
relationship between $y_{i}$ and $\textbf{x}_{i}$ is the normal
regression model
\begin{equation}\label{regmodel}
  y_{i}=\textbf{x}_{i}^\top\vec{\beta}_{j}+\epsilon_{i},
\end{equation}
where $\epsilon_{i}\thicksim \CN(0,\sigma^{2}_{j})$ and
$\vec{\beta}_{j}$ and $\sigma_{j}^{2}$ are the $p$-dimensional vector of
regression coefficients and the error variance for component $j$,
respectively.

Accounting for the mixture structure, the conditional density of
$y_{i}|\vec x_i$ is
\begin{equation}\label{mor}
  g_{\vec\theta}(y_{i}|\textbf{x}_{i})=\sum_{j=1}^{m}\lambda_{j}
  \phi(y_{i} | \textbf{x}_{i}^\top\vec{\beta}_{j},\sigma^{2}_{j}),
\end{equation}
where $\phi(\cdot|\textbf{x}^\top\vec{\beta}_{j},\sigma^{2}_{j})$ is the
normal density with mean $\textbf{x}^\top\vec\beta$ and variance
$\sigma^2$.  Notice that the model parameter for this setting is
$\vec\theta=(\vec\lambda,(\vec{\beta}_{1},\sigma^2_{1}),\ldots,(\vec{\beta}_{m},\sigma^2_{m}))$.
The mixture of regressions model (\ref{mor}) differs from the well-known
mixture of multivariate normals model
$(Y_{i},\textbf{X}_{i}^\top)^\top\thicksim
\sum_{j=1}^{m}\lambda_{j}\CN_{p+1}(\vec{\mu}_{j},\Sigma_{j})$ because
model (\ref{mor}) makes no assertion about the marginal distribution of
$\textbf{X}_{i}$, whereas the mixture of multivariate normals specifies
that $\textbf{X}_{i}$ itself has a mixture of multivariate normals
distribution.

<<gnpdata, echo=FALSE, results=hide>>=
data("CO2data")
attach(CO2data)
pdf("gnpdata.pdf")
par(mar=0.1+c(5,4.2,4,1.5))
plot(GNP, CO2, xlab="Gross National Product",
     ylab=expression(paste(CO[2]," per Capita")),
     cex.lab=1.5, cex.main=1.5, cex.axis=1.4,
     main="1996 GNP and Emissions Data")
text(GNP, CO2, country, adj=c(.5,-.5))
dev.off()
@

\begin{figure}[h]
  \centering
  \includegraphics[height=3in,width=3in]{gnpdata.pdf}
  \caption{1996 data on gross national product (GNP) per capita and
    estimated carbon dioxide ($\textrm{CO}_{2}$) emissions per capita.
    Note that ``CH'' stands for Switzerland, not China.}
  \label{gnpdata}
\end{figure}

As a simple example of a dataset to which a mixture of regressions
models may be applied, consider the sample depicted in Figure
\ref{gnpdata}. In this dataset, the measurements of carbon dioxide
($\textrm{CO}_{2}$) emissions are plotted versus the gross national
product (GNP) for $n=28$ countries. These data are included
\pkg{mixtools}; type \code{help("CO2data")} in \proglang{R} for more
details.  \citet{hurn} analyzed these data using a mixture of
regressions from the Bayesian perspective, pointing out that ``there do
seem to be several groups for which a linear model would be a reasonable
approximation.''  They further point out that identification of such
groups could clarify potential development paths of lower GNP countries.


\subsection{EM algorithms for mixtures of regressions}

A standard EM algorithm, as described in Section~\ref{section:EM}, may
be used to find a local maximum of the likelihood surface.  The E-step
is the same as for any finite mixture model EM algorithm; i.e., the
$p_{ij}^{(t)}$ values are updated according to equation
(\ref{posteriors})---or, in reality, equation
(\ref{altposteriors})---where each $\phi_j^{(t)}(\vec x_i)$ is replaced
in the regression context by $\phi(y_i | \vec x_i^\top\vec\beta_j,
\sigma_j^2)$:
\begin{equation}\label{regposteriors}
  \post_{ij}^{(t)} = \left[ 1 + \sum_{j'\ne j} \frac{ \lambda_{j'}^{(t)} \phi(y_i | \vec x_i^\top\vec\beta_{j'}, \sigma_{j'}^2)}{\lambda_j^{(t)} \phi(y_i | \vec x_i^\top\vec\beta_j, \sigma_j^2)} \right]^{-1}
\end{equation}
The update to the $\lambda$ parameters in the M-step, equation
(\ref{lambda}), is also the same.  Letting
$\textbf{W}_{j}^{(t)}=\textrm{diag}(\post_{1j}^{(t)},\ldots,\post_{nj}^{(t)})$,
the additional M-step updates to the $\vec\beta$ and $\sigma$ parameters
are given by
\begin{eqnarray}\label{betaEM}
  \vec{\beta}_{j}^{(t+1)} &=& (\underline{\textbf{X}}^\top\textbf{W}_{j}^{(t)}\underline{\textbf{X}})^{-1}\underline{\textbf{X}}^\top
  \textbf{W}_{j}^{(t)}\textbf{y}
  \quad \mbox{and} \\
\label{sigma}
\sigma_{j}^{2(t+1)} &=& \frac{\biggl\|\textbf{W}_{j}^{1/2(t)}(\textbf{y}-\underline{\textbf{X}}^\top\vec{\beta}_{j}^{(t+1)})\biggr\|^{2}}{\mbox{tr}(\textbf{W}_{j}^{(t)})},
\end{eqnarray}
where $\|\textbf{A}\|^{2}=\textbf{A}^\top\textbf{A}$ and
$\mbox{tr}(\textbf{A})$ means the trace of the matrix $\textbf{A}$.
Notice that equation (\ref{betaEM}) is a weighted least squares (WLS)
estimate of $\vec{\beta}_{j}$ and equation (\ref{sigma}) resembles the
variance estimate used in WLS.

Allowing each component to have its own error variance $\sigma_j^2$
results in the likelihood surface having no maximizer, since the
likelihood may be driven to infinity if one component gives a regression
surface passing through one or more points exactly and the variance for
that component is allowed to go to zero.  A similar phenomenon is
well-known in the finite mixture-of-normals model where the component
variances are allowed to be distinct \citep{mclachlan2000fmm}.  However,
in practice we observe this behavior infrequently, and the
\pkg{mixtools} functions automatically force their EM algorithms to
restart at randomly chosen parameter values when it occurs.  A local
maximum of the likelihood function, a consistent version of which is
guaranteed to exist by the asymptotic theory as long as the model is
correct and all $\lambda_j$ are positive, usually results without any
restarts.

The function \code{regmixEM} implements the EM algorithm for mixtures of
regressions in \pkg{mixtools}.  This function has arguments that control
options such as adding an intercept term, \code{addintercept = TRUE};
forcing all $\vec\beta_j$ estimates to be the same, \code{arbmean =
  FALSE} (for instance, to model outlying observations as having a
separate error variance from the non-outliers); and forcing all
$\sigma_j^2$ estimates to be the same, \code{arbvar = FALSE}.  For
additional details, type \code{help("regmixEM")}.

As an example, we fit a 2-component model to the GNP data shown in
Figure \ref{gnpdata}.  \citet{hurn} and \citet{youngphd} selected 2
components for this dataset using model selection criteria, Bayesian
approaches to selecting the number of components, and a bootstrapping
approach.  The function \code{regmixEM} will be used for fitting a
2-component mixture of regressions by an EM algorithm:
<<results=hide>>=
data("CO2data")
attach(CO2data)
@ 
<<CO2reg>>=
CO2reg <- regmixEM(CO2, GNP, lambda = c(1, 3) / 4,
                   beta = matrix(c(8, -1, 1, 1), 2, 2), sigma = c(2, 1))
@

We can then pull out the final observed log-likelihood as well as
estimates for the 2-component fit, which include $\hat{\lambda}$,
$\hat{\vec{\beta}}_{1}$, $\hat{\vec{\beta}}_{2}$, $\hat{\sigma}_{1}$,
and $\hat{\sigma}_{2}$:
<<summaryCO2reg>>=
summary(CO2reg)
@

The reader is encouraged to alter the starting values or let the
internal algorithm generate random starting values.  However, this fit
seems appropriate and the solution is displayed in Figure \ref{co2EM}
along with 99\% Working-Hotelling Confidence Bands, which are
constructed automatically by the \code{plot} method in this case by
assigning each point to its most probable component and then fitting two
separate linear regressions:

<<plotCO2reg, eval=FALSE>>=
plot(CO2reg, density = TRUE, alpha = 0.01, cex.main = 1.5, cex.lab = 1.5,
     cex.axis = 1.4)
@

\setkeys{Gin}{width=0.49\textwidth}
\begin{figure}[!h]
  \centering
<<trueplotCO2reg, echo=FALSE, results=tex>>=
for(i in 1:2){
    file=paste("CO2reg", i, ".pdf", sep="")
    pdf(file=file, paper="special", width=6, height=6)
    plot(CO2reg, whichplots=i, alpha = 0.01, cex.main = 1.5, cex.lab = 1.5,
         cex.axis = 1.4)
    dev.off()
    cat("\\includegraphics{", file, "}\n", sep="")
}
@
\caption{The GNP data fitted with a 2-component parametric EM
  algorithm in \pkg{mixtools}. Left: the sequence of log-likelihood
  values, $L_{\Bx}(\f^{(t)})$; Right: the fitted regression lines with
  99\% Working-Hotelling Confidence Bands.}
\label{co2EM}
\end{figure}


\subsection{Predictor-dependent mixing proportions}
\label{section:pdmp}

Suppose that in model (\ref{mor}), we replace $\lambda_j$ by
$\lambda_{j}(\textbf{x}_{i})$ and assume that the mixing proportions
vary as a function of the predictors $\textbf{x}_{i}$.  Allowing this
type of flexibility in the model might be useful for a number of
reasons.  For instance, sometimes it is the proportions $\lambda_j$ that
are of primary scientific interest, and in a regression setting it may
be helpful to know whether these proportions appear to vary with the
predictors.  As another example, consider a \code{regmixEM} model using
\code{arbmean = FALSE} in which the mixture structure only concerns the
error variance: In this case, $\lambda_j(\vec x)$ would give some sense
of the proportion of outliers in various regions of the predictor space.

One may assume that $\lambda_{j}(\textbf{x})$ has a particular
parametric form, such as a logistic function, which introduces new
parameters requiring estimation. This is the idea of the
\textit{hierarchical mixtures of experts} (HME) procedure
\citep{jacobsall}, which is commonly used in neural networks and which
is implemented, for example, in the \pkg{flexmix} package for
\proglang{R} \citep{jss:Leisch:2004, Grun+Leisch:2008}.  However, a
parametric form of $\lambda_{j}(\textbf{x})$ may be too restrictive; in
particular, the logistic function is monotone, which may not
realistically capture the pattern of change of $\lambda_j$ as a function
of $\vec x$.  As an alternative, \citet{young2009mor} propose a
nonparametric estimate of $\lambda_{j}(\textbf{x}_{i})$ that uses ideas
from kernel density estimation.

The intuition behind the approach of \citet{young2009mor} is as follows:
The M-step estimate (\ref{lambda}) of $\lambda_j$ at each iteration of a
finite mixture model EM algorithm is simply an average of the
``posterior'' probabilities $p_{ij}=\E(Z_{ij}|\mbox{data})$.  As a
substitute, the nonparametric approach uses local linear regression to
approximate the $\lambda_j(\textbf{x})$ function.  Considering the case
of univariate $x$ for simplicity, we set $\lambda_j(x) =
\hat\alpha_{0j}(x)$, where
\begin{equation}\label{lambdax}
(\hat\alpha_{0j}(x), \hat\alpha_{1j}(x))=
\arg\min_{(\alpha_0, \alpha_1)} \sum_{i=1}^n
K_h(x-x_i) \left[ p_{ij} - \alpha_0 - \alpha_1(x-x_i) \right]^2
\end{equation}
and $K_h(\cdot)$ is a kernel density function with scale parameter
(i.e., bandwidth) $h$.  It is straightforward to generalize equation
(\ref{lambdax}) to the case of vector-valued $\vec x$ by using a
multivariate kernel function.

\citet{young2009mor} give an iterative algorithm for estimating mixture
of regression parameters that replaces the standard $\lambda_j$ updates
(\ref{lambda}) by the kernel-weighted version (\ref{lambdax}).  The
algorithm is otherwise similar to a standard EM; thus, like the
algorithm in Section~\ref{section:EMlike} of this article, the resulting
algorithm is an EM-like algorithm.  Because only the $\lambda_j$
parameters depend on $\vec x$ (and are thus ``locally estimated''),
whereas the other parameters (the $\vec\beta_j$ and $\sigma_j$) can be
considered to be globally estimated, \citet{young2009mor} call this
algorithm an iterative global/local estimation (IGLE) algorithm.
Naturally, it replaces the usual E-step (\ref{regposteriors}) by a
modified version in which each $\lambda_j$ is replaced by
$\lambda_j(x_i)$.

The function \code{regmixEM.loc} implements the IGLE algorithm in
\pkg{mixtools}. Like the \code{regmixEM} function, \code{regmixEM.loc}
has the flexibility to include an intercept term by using
\code{addintercept = TRUE}.  Moreover, this function has the argument
\code{kern.l} to specify the kernel used in the local estimation of the
$\lambda_{j}(\textbf{x}_{i})$.  Kernels the user may specify include
\code{"Gaussian"}, \code{"Beta"}, \code{"Triangle"}, \code{"Cosinus"},
and \code{"Optcosinus"}.  Further numeric arguments relating to the
chosen kernel include \code{kernl.g} to specify the shape parameter for
when \code{kern.l = "Beta"} and \code{kernl.h} to specify the bandwidth
which controls the size of the window used in the local estimation of
the mixing proportions. See the corresponding help file for additional
details.

For the GNP and emissions dataset, Figure \ref{co2EM} indicates that the
assumption of constant weights for the component regressions across all
values of the covariate space may not be appropriate.  The countries
with higher GNP values appear to have a greater probability of belonging
to the first component (i.e., the red line in Figure \ref{co2EM}).  We
will therefore apply the IGLE algorithm to this dataset.

We will use the triweight kernel in equation (\ref{lambdax}), which is
given by setting $\gamma=3$ in
\begin{equation}\label{beta}
  K_{h}(x)=\frac{1}{hB(1/2,\gamma+1)}\left(1-\frac{x^2}{h^2}\right)^{\gamma}_{+},
\end{equation}
where $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$ is the beta function.  For
the triweight, $B(1/2, 4)$ is exactly $32/35$.  This kernel may be
specified in \code{regmixEM.loc} with \code{kern.l = "Beta"} and
\code{kernl.g = 3}. The bandwidth we selected was $h=20$, which we
specify with \code{kernl.h = 20}.

For this implementation of the IGLE algorithm, we set the parameter
estimates obtained from the mixture of regressions EM algorithm as
starting values for $\hat{\vec{\beta}}_{1}$, $\hat{\vec{\beta}}_{2}$,
$\hat{\sigma}_{1}$, and $\hat{\sigma}_{2}$, and set the starting values
for $\lambda(x_{i})$ to be 0.5 for all $x_{i}$.
<<CO2igle>>=
CO2igle <- regmixEM.loc(CO2, GNP, beta = CO2reg$beta, sigma = CO2reg$sigma,
                        lambda = matrix(.5, 28, 2), kern.l = "Beta",
                        kernl.h = 20, kernl.g = 3)
@

We can view the estimates for $\hat{\vec{\beta}}_{1}$,
$\hat{\vec{\beta}}_{2}$, $\hat{\sigma}_{1}$, and $\hat{\sigma}_{2}$.
Notice that the estimates are comparable to those obtained for the
mixture of regressions EM output and the log-likelihood value is
slightly higher.
<<CO2iglesummary>>=
summary(CO2igle)
@

Next, we can plot the estimates of $\lambda(x_{i})$ from the IGLE
algorithm.

<<lamplot, eval=FALSE>>=
plot(GNP, CO2igle$post[,1], xlab = "GNP", cex.axis = 1.4, cex.lab = 1.5,
  ylab = "Final posterior probabilities")
lines(sort(GNP), CO2igle$lambda[order(GNP), 1], col=2)
abline(h = CO2igle$lambda[1], lty = 2)
@

<<truelamplot, echo=FALSE, results=hide>>=
pdf("lamplot.pdf")
plot(GNP, CO2igle$post[,1], xlab = "GNP", cex.axis = 1.4, cex.lab = 1.5,
  ylab = "Final posterior probabilities")
lines(sort(GNP), CO2igle$lambda[order(GNP), 1], col=2, lwd=2)
abline(h = CO2igle$lambda[1], lty = 2, lwd=2)
dev.off()
@

This plot is given in Figure \ref{lamplot}.  Notice the curvature
provided by the estimates from the IGLE fit.  These fits indicate an
upward trend in the posteriors.  The predictor-dependent mixing
proportions model provides a viable way to reveal this trend since
the regular mixture of regressions fit simply provides the same
estimate of $\lambda$ for all $x_{i}$.

\begin{figure}[h]
  \centering
  \includegraphics[height=3in,width=3in]{lamplot.pdf}
  \caption{Posterior membership probabilities $p_{i1}$ for component
    one versus the predictor GNP along with estimates of
    $\lambda_1(x)$ from the IGLE algorithm (the solid red curve) and
    $\lambda_1$ from the mixture of linear regressions EM algorithm
    (the dashed black line).}
  \label{lamplot}
\end{figure}


\subsection{Parametric bootstrapping for standard errors}

With likelihood methods for estimation in mixture models, it is possible
to obtain standard error estimates by using the inverse of the observed
information matrix when implementing a Newton-type method. However, this
may be computationally burdensome. An alternative way to report standard
errors in the likelihood setting is by implementing a parametric
bootstrap.  \citet{eftib} claim that the parametric bootstrap should
provide similar standard error estimates to the traditional method
involving the information matrix.  In a mixture-of-regressions context,
a parametric bootstrap scheme may be outlined as follows:

\begin{enumerate}
\item Use \code{regmixEM} to find a local maximizer $\hat{\vec\theta}$
  of the likelihood.
\item For each $\textbf{x}_{i}$, simulate a response value $y_{i}^{*}$
  from the mixture density $g_{\hat{\vec\theta}}(\cdot|\textbf{x}_{i})$.
\item Find a parameter estimate $\tilde{\vec{\theta}}$ for the bootstrap
  sample using \code{regmixEM}.
\item Use some type of check to determine whether label-switching
  appears to have occurred, and if so, correct it.
\item Repeat steps 2 through 4 $B$ times to simulate the bootstrap
  sampling distribution of $\hat{\vec\theta}$.
\item Use the sample covariance matrix of the bootstrap sample as an
  approximation to the covariance matrix of $\hat{\vec\theta}$.
\end{enumerate}
Note that step 3, which is not part of a standard parametric bootstrap,
can be especially important in a mixture setting.

The \pkg{mixtools} package implements a parametric bootstrap algorithm
in the \code{boot.se} function. We may apply it to the regression
example of this section, which assumes the same estimate of $\lambda$
for all $x_{i}$, as follows:
<<CO2boot, results=hide>>=
set.seed(123)
CO2boot <- boot.se(CO2reg, B = 100)
@

This output consists of both the standard error estimates and the
parameter estimates obtained at each bootstrap replicate.  An
examination of the slope and intercept parameter estimates of the 500
bootstrap replicates reveals that no label-switching is likely to have
occurred.  For instance, the intercept terms of component one range from
4 to 11, whereas the intercept terms of component two are all tightly
clumped around 0:
<<bootresults>>=
rbind(range(CO2boot$beta[1,]), range(CO2boot$beta[2,]))
@

We may examine the bootstrap standard error estimates by themselves as follows:
<<CO2bootse>>=
CO2boot[c("lambda.se", "beta.se", "sigma.se")]
@

\section[Additional capabilities of mixtools]{Additional capabilities of
  \pkg{mixtools}}
\label{section:misc}

\subsection{Selecting the number of components}
\label{ss:nbcomp}

Determining the number of components $k$ is still a major contemporary
issue in mixture modeling.  Two commonly employed techniques are
information criterion and parametric bootstrapping of the likelihood
ratio test statistic values for testing
\begin{eqnarray}\label{mixturetest}
\nonumber
H_{0}&:& k = k_{0} \\
H_{1}&:& k = k_{0}+1
\end{eqnarray}
for some positive integer $k_{0}$ \citep{mclach}.

The \pkg{mixtools} package has functions to employ each of these methods
using EM output from various mixture models. The information criterion
functions calculate An Information Criterion (AIC) of \citet{aic}, the
Bayesian Information Criterion (BIC) of \citet{schw}, the Integrated
Completed Likelihood (ICL) of \citet{biern}, and the consistent AIC
(CAIC) of \citet{boz}. The functions for performing parametric
bootstrapping of the likelihood ratio test statistics sequentially test
$k=k_{0}$ versus $k=k_{0}+1$ for $k_0=1, 2, \ldots$, terminating after
the bootstrapped $p$-value for one of these tests exceeds a specified
significance level.

Currently, \pkg{mixtools} has functions for calculating information
criteria for mixtures of multinomials (\code{multmixmodel.sel}),
mixtures of multivariate normals under the conditionally i.i.d.\
assumption (\code{repnormmixmodel.sel}), and mixtures of regressions
(\code{regmixmodel.sel}). Output from various mixture model fits
available in \pkg{mixtools} can also be passed to the function
\code{boot.comp} for the parametric bootstrapping approach.  The
parameter estimates from these EM fits are used to simulate data from
the null distribution for the test given in (\ref{mixturetest}).  For
example, the following application of the \code{multmixmodel.sel}
function to the water-level multinomial data from
Section~\ref{section:cut} indicates that either 3 or 4 components seems
like the best option (no more than 4 are allowed here since there are
only 8 multinomial trials per observation and the mixture of
multinomials requires $2m \le r+1$ for identifiability):
<<modelsel>>=
<<cutpoint>>
set.seed(10)
multmixmodel.sel(watermult, comps = 1:4, epsilon = 0.001)
@

\citet{youngphd} gives more
applications of these functions to real datasets.



\subsection{Bayesian methods}

Currently, there are only two \pkg{mixtools} functions relating to
Bayesian methodology and they both pertain to analyzing mixtures of
regressions as described in \citet{hurn}.  The \code{regmixMH} function
performs a Metropolis-Hastings algorithm for fitting a mixture of
regressions model where a proper prior has been assumed.  The sampler
output from \code{regmixMH} can then be passed to \code{regcr} in order
to construct credible regions of the regression lines.  Type
\code{help("regmixMH")} and \code{help("regcr")} for details and an
illustrative example.



\section*{Acknowledgments}
This research is partially supported by NSF Award SES-0518772.  DRH
received additional funding from Le Studium, an agency of the Centre
National de la Recherche Scientifique of France.


\bibliography{mixtools}


\end{document}
