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%\VignetteIndexEntry{A Trimming Approach to Cluster Analysis}
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\author{Heinrich Fritz\\Department of Statistics\\ and Probability Theory\\ Vienna University of Technology \And
        Luis A. Garc\'{\i}a-Escudero\\Departamento de Estad\'{\i}stica\\ e Investigaci\'{o}n Operativa\\ Universidad de Valladolid \And
        Agust\'{\i}n Mayo-Iscar\\Departamento de Estad\'{\i}stica\\ e Investigaci\'{o}n Operativa\\ Universidad de Valladolid }
\title{\pkg{tclust}: An \proglang{R} Package for a Trimming Approach to Cluster Analysis }

%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Heinrich Fritz, Luis A. Garc\'{\i}a-Escudero, Agust\'{\i}n Mayo-Iscar}
\Plaintitle{tclust: An R Package for a Trimming Approach to Cluster Analysis}
%\Shorttitle{\pkg{tclust}: An \proglang{R} package for a trimming approach to cluster analysis} %% a short title (if necessary) % this is optional. no use if not different from "real" title

%% an abstract and keywords
\Abstract{This introduction to the \proglang{R} package \pkg{tclust} is a (slightly)
modified version of \cite{FriG12}, published in the
\emph{Journal of Statistical Software}.

Outlying data can heavily influence standard clustering
methods. At the same time, clustering principles can be useful when
robustifying statistical procedures. These two reasons motivate the
development of feasible robust model-based clustering approaches.
With this in mind, an R package for performing non-hierarchical
robust clustering, called \pkg{tclust}, is presented here. Instead
of trying to ``fit'' noisy data, a proportion $\alpha$ of the most
outlying observations is trimmed. The \pkg{tclust} package
efficiently handles different cluster scatter constraints. Graphical
exploratory tools are also provided to help the user make sensible
choices for the trimming proportion as well as the number of
clusters to search for.} \Keywords{Model-based clustering, trimming,
heterogeneous clusters}
\Plainkeywords{Model-based clustering, trimming, heterogeneous clusters} %% 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}

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\Address{
  Luis A. Garc\'{\i}a-Escudero\\
  Departamento de Estad\'{\i}stica e Investigaci\'{o}n Operativa. Universidad de Valladolid\\
  Prado de la Magdalena s/n, 47005 Valladolid, SPAIN\\
  E-mail: \email{lagarcia@eio.uva.es}\\
  %URL: \url{http://www.eio.uva.es/infor/personas/langel.html}
}

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\begin{document}

\section{Introduction to robust clustering and tclust}\label{sec4:int}

Methods for cluster analysis attempt to detect homogeneous clusters
with large heterogeneity among them. As happens with other
(non-robust) statistical procedures, clustering methods may be
heavily influenced by even a small fraction of outlying data. For
instance, two or more clusters might be joined artificially, due to
outlying observations, or ``spurious'' non-informative clusters may
be composed of only a few outlying observations \citep[see,
e.g.,][]{GarG99,GarG10}. Therefore, the application of robust methods
in this context is very advisable, especially in fully automatic
clustering (unsupervised learning) problems. Certain relations
between cluster analysis and robust methods
\citep{RocW02,HarR04,GarG03,WooR04} also motivate interest in robust
clustering techniques. For example, robust clustering techniques can
be used to handle ``clusters'' of highly concentrated outliers which
are especially dangerous in (non-robust) estimation. \cite{GarG10}
provides a recent survey of robust clustering methods.

The \pkg{tclust} package for the \proglang{R} environment for
statistical computing \citep{R} implements different robust
non-hierarchical clustering algorithms where trimming plays a key
role. This package is available at
\texttt{http://CRAN.R-project.org/package=tclust}. When trimming
allows the removal of a fraction $\alpha$ of the ``most outlying''
data, the strong influence of outlying observations can be avoided.
This trimming approach to clustering has been introduced in
\cite{CueG97}, \cite{GalM02}, \cite{GalR05} and \cite{GarG08}.
Trimming also serves to identify potentially interesting anomalous
observations.

Trimming is not a new concept in statistics. For instance, the
trimmed mean for one-dimensional data removes a proportion
$\alpha/2$ each of the largest and smallest observations before
computing the mean. However, it is not straightforward to extend
this philosophy to cluster analysis, because most of these problems
are of multivariate nature. Moreover, it is often the case that
``bridge points'' lying between clusters ought to be trimmed. Instead
of forcing the statistician to define the regions to be trimmed in
advance, the procedures implemented in \pkg{tclust} take the whole
data structure into account in order to decide which parts of the
sample should be discarded. By considering this type of trimming,
these procedures are even able to trim outlying bridge points. The
``self-trimming'' philosophy behind these procedures is exactly the
same as adopted by some well-known high breakdown-point methods
\citep[see, e.g.,][]{RouL87}.

As a first example of this trimming approach, let us consider the
trimmed $k$-means method introduced in \cite{CueG97}. The function
\code{tkmeans} from the \pkg{tclust} package implements this method.
In the following example, this function is applied to a bivariate
data set based on the Old Faithful geyser called \code{geyser2} that
accompanies the \pkg{tclust} package. The code given below creates
Figure~\ref{fig4:f1}.
<<label=intro, echo=false, results=hide>>=

options(prompt = "R> ", continue = "+  ", width = 70, useFancyQuotes = FALSE)

## standard margins
mmar <- c(5.1, 4.1, 4.1, 1.1)
og <- grey(0)   ## color for outliers
@

<<label=fig-1, fig=TRUE, include=FALSE, width=7.14, height=7.8>>=
library(tclust)
set.seed(100)
data(geyser2)
clus <- tkmeans(geyser2, k = 3, alpha=0.03)
plot(clus, col = c(og, 2:4), tol.lwd = 1, tol.lty = 2)
@

In the data set \code{geyser2}, we are searching for $k=3$ clusters
and a proportion $\alpha=0.03$ of the data is trimmed. The
clustering results are shown in Figure~\ref{fig4:f1}. Among this $3\%$ of
trimmed data, we can see 6 anomalous ``short followed by short''
eruptions lengths. Notice that an observation situated between the
clusters is also trimmed.

\begin{figure}[t!]
\centering
\includegraphics{tclust-fig-1}
\caption{Trimmed $k$-means results with $k=3$ and $\alpha=0.03$ for
the bivariate Old Faithful Geyser data. Trimmed observations are
denoted by the symbol ``$\circ$'' (a convention followed in all the
figures in this work).} \label{fig4:f1}
\end{figure}

The package presented here adopts a ``crisp'' clustering approach,
meaning that each observation is either trimmed or fully assigned to
a cluster. In comparison, mixture approaches estimate a cluster
pertinence probability for each observation. Robust mixture
alternatives have also been proposed where noisy data is tried to be
fitted through additional mixture components. For instance, package
\pkg{mclust} \citep{FraR12, BanR93,FraR98} and the Fortran program
\pkg{emmix} \citep{EMMIX99, LacP00} implement such robust mixture fitting
approaches. Mixture fitting results can be easily converted into a
``crisp'' clustering result by converting the cluster pertinence
probabilities into 0-1 probabilities. Contrary to these mixture
fitting approaches, the procedures implemented in the \pkg{tclust}
package simply remove outlying observations and do not intend to fit
them at all. Package \pkg{tlemix} \citep[see][]{NeyF12, NeyF07} also
implements a closely related trimming approach. As described in
Section~\ref{sec4:cons}, the \pkg{tclust} package focuses on
offering adequate cluster scatter matrix constraints leading to a
wide range of clustering procedures depending on the chosen
constraint, and avoiding the occurrence of spurious non-interesting
clusters.

The outline of the paper is as follows: In Section~\ref{sec4:tri} we
briefly review the so-called ``spurious outliers'' model and show how
to derive two different clustering criteria from it. Different
constraints on the cluster scatter matrices and their implementation
in the \pkg{tclust} package are addressed in Section~\ref{sec4:cons}.
Section~\ref{sec4:num} presents the numerical
output returned by this package. Section~\ref{sec4:alg} provides
some brief comments concerning the algorithms implemented, and a
comparison of \pkg{tclust} and several other robust clustering
approaches are given in Section~\ref{sec4:comp}.
Section~\ref{sec4:kal} shows some graphical outputs that help advise the
choice of the number of clusters and the trimming proportion. Other
useful plots summarizing the robust clustering results are shown in
Section~\ref{sec4:gra}. Finally, Section~\ref{sec4:exa} applies the
\pkg{tclust} package to a well-know real data set.


\section{Trimming and the spurious outliers model}\label{sec4:tri}
\cite{GalM02} and \cite{GalR05} propose the ``spurious outliers
model'' as a probabilistic framework for robust crisp clustering. Let
$f(\cdot;\mu,\Sigma)$ denote the probability density function of the
$p$-variate normal distribution with mean $\mu$ and covariance
matrix $\Sigma$. The ``spurious-outlier model'' is defined through
``likelihoods'' like
\begin{equation}\label{eqn4:spu}
  \bigg[\prod_{j=1}^{k}\prod_{i\in R_j}f(x_i;\mu_j,\Sigma_j)\bigg]\bigg[\prod_{i \in R_0}g_i(x_i)\bigg]
\end{equation}
with $\{R_0,...,R_k\}$ being a partition of the set of indices
$\{1,2,...,n\}$ such that $\#R_0=\rounding{n\alpha}$. $R_0$ are the
indices of the ``non-regular'' observations  generated by other (not
necessarily normal) probability density functions $g_i$.
``Non-regular'' observations can be clearly considered as ``outliers''
if we assume certain sensible assumptions for the $g_i$ \citep[see
details in][]{GalM02,GalR05}. Under these assumptions, the search of
a partition $\{R_0,...,R_k\}$ with $\#R_0=\rounding{n\alpha}$,
vectors $\mu_j$ and positive definite matrices $\Sigma_j$ maximizing
(\ref{eqn4:spu}) can be simplified to the same search (of a
partition, vectors and positive definite matrices) by just
maximizing
\begin{equation}\label{eqn4:e3}
 \sum_{j=1}^{k}\sum_{i\in R_j}\log f(x_i;\mu_j,\Sigma_j).
\end{equation}
Notice that observations $x_i$ with $i\in R_0$ are not taken into
account in (\ref{eqn4:e3}). Maximizing (\ref{eqn4:e3}) with $k=1$ yields the
Minimum Covariance Determinant (MCD) estimator \citep{RouP85}.

Unfortunately, the direct maximization of (\ref{eqn4:e3}) is not a
well-defined problem when $k>1$. It is easy to see that (\ref{eqn4:e3})
is unbounded without any constraint on the cluster scatter matrices
$\Sigma_j$. The \code{tclust} function from the \pkg{tclust} package
approximately maximizes (\ref{eqn4:e3}) under different cluster scatter
matrix constraints which will be shown in Section~\ref{sec4:cons}.

The maximization of (\ref{eqn4:e3}) implicitly assumes equal cluster
weights. In other words, we are ideally searching for clusters with
equal sizes. The function \code{tclust} provides this option by
setting the argument \code{equal.weights = TRUE}. The use of this
option does not guarantee that all resulting clusters exactly
contain the same number of observations, but the method hence
prefers this type of solutions.

Alternatively, different cluster sizes or cluster weights can be
considered by searching for a partition $\{R_0,...,R_k\}$ (with
$\#R_0=\rounding{n\alpha}$), vectors $\mu_j$, positive definite
matrices $\Sigma_j$ and weights $\pi_j\in[0,1]$ maximizing
\begin{equation}\label{eqn4:e4}
 \sum_{j=1}^{k}\sum_{i\in R_j}(\log \pi_j + \log f(x_i;\mu_j,\Sigma_j)).
\end{equation}
The (default) option \code{equal.weights = FALSE} is used in this
case. Again, the scatter matrices also have to be constrained such
that the maximization of (\ref{eqn4:e4}) becomes a well-defined
problem. Note that (\ref{eqn4:e4}) simplifies to (\ref{eqn4:e3})
when assuming \code{equal.weights = TRUE} and all weights are
equally set to $\pi_j=1/k$.

\newpage
\section{Constraints on the cluster scatter matrices}\label{sec4:cons}

As already mentioned, the function \code{tclust} implements
different algorithms aimed at approximately maximizing
(\ref{eqn4:e3}) and (\ref{eqn4:e4}) under different types of
constraints which can be applied on the scatter matrices $\Sigma_j$.
The type of constraint is specified by the argument \code{restr} of
the \code{tclust} function. Table~\ref{tab4:app.ovv} gives an
overview of the different clustering approaches implemented by the
\code{tclust} function depending on the chosen type of constraint.

\begin{table}[t!]
  \centering
\begin{tabular}{||l|l|l||}
  \hline\hline
  &\code{equal.weights = TRUE}&\code{equal.weights = FALSE}\\ \hline
 \code{restr = "eigen"} & \hspace{-1.8mm}$\begin{array}{l} k\textit{-means} \\ \text{\cite{CueG97}} \end{array}$ & \cite{GarG08} \\\hline
 \code{restr = "deter"} & \text{\cite{GalM02}} & \text{This work}\\\hline
 \hline\hline
\end{tabular}
  \caption{\label{tab4:app.ovv}Clustering methods handled by \pkg{tclust}. Names in cursive letters are untrimmed ($\alpha=0$) methods.}
\end{table}

Imposing constraints is compulsory because maximizing
(\ref{eqn4:e3}) or (\ref{eqn4:e4}) without any restriction is not a
well-defined problem. Notice that an almost degenerated scatter
matrix $\Sigma_j$ would cause trimmed log-likelihoods
(\ref{eqn4:e3}) and (\ref{eqn4:e4}) to tend to infinity. This issue
can cause a (robust) clustering algorithm of this type to end up
finding ``spurious'' clusters almost lying in lower-dimensional
subspaces. Moreover, the resulting clustering solutions might
heavily depend on the chosen constraint. The strength of the
constraint is controlled by the argument \code{restr.fact} $\geq 1$
in the \code{tclust} function. The larger \code{restr.fact} is
chosen, the looser is the restriction on the scatter matrices,
allowing for more heterogeneity among the clusters. On the contrary,
small values of \code{restr.fact} close to 1 imply very ``equally
scattered'' clusters. This idea of constraining cluster scatters to
avoid spurious solutions goes back to \cite{HatR85}, who proposed it
in mixture fitting problems.

Also arising from the spurious outlier model, other types of
constraints have recently been introduced by \cite{GalR09,GalR10}.
These (closely related) constraints also serve to avoid degeneracy
of trimmed likelihoods but they are not implemented in the current
version of the \pkg{tclust} package.

\subsection{Constraints on the eigenvalues}\label{sec4:consteig}
Based on the eigenvalues of the cluster scatter matrices, a scatter
similarity constraint may be defined. With $\lambda_l(\Sigma_j)$ as
the eigenvalues of the cluster scatter matrices $\Sigma_j$ and
\begin{equation}\label{eqn4:e5}
    M_n=\max_{j=1,...,k}\,\max_{l=1,...,p}\lambda_l(\Sigma_j)\text{
and } m_n=\min_{j=1,...,k}\,\min_{l=1,...,p}\lambda_l(\Sigma_j)
\end{equation}
as the maximum and minimum eigenvalues, the restriction
\texttt{restr = "eigen"} constrains the ratio $M_n/m_n$  to be
smaller or equal than a fixed value \code{restr.fact} $\geq 1$. A
theoretical study of the properties of this approach with
\code{equal.weights = FALSE} can be found in \cite{GarG08}.

This type of constraint limits the relative size of the axes of the
equidensity ellipsoids defined through the obtained $\Sigma_j$ when
assuming normality. This way we are simultaneously controlling the
relative group sizes and also the deviation from sphericity in each
cluster.

Setting \code{equal.weights = TRUE}, \code{restr = "eigen"} and
\code{restr.fact = 1} implies the most constrained case. In this
case, the \code{tclust} function tries to solve the trimmed
$k$-means problem as introduced by \cite {CueG97}. This problem
simplifies to the well-known $k$-means clustering criterion when no
trimming is done (i.e., \code{alpha = 0}). The \code{tkmeans}
function directly implements this most constrained application of
the \code{tclust} function.

\subsection{Constraints on the determinants}\label{sec4:constdet}
Another way of restricting cluster scatter matrices is constraining
their determinants. Thus, if
$$
M_n=\max_{j=1,...,k}|\Sigma_j|\text{ and
}m_n=\min_{j=1,...,k}|\Sigma_j|
$$
are the maximum and minimum
determinants, we attempt to maximize (\ref{eqn4:e3}) or (\ref{eqn4:e4}) by
constraining the ratio $M_n/m_n$ to be smaller or equal than a fixed
value \code{restr.fact}. This is done in the function \code{tclust}
by using the option \code{restr = "deter"}.

Now, this type of constraint limits the relative volumes of the
mentioned equidensity ellipsoids, but not the cluster shapes. The
use of this type of constraint is particularly advisable when affine
equivariance is required because this property is satisfied when
\code{restr = "deter"}.

The untrimmed case \code{alpha = 0}, \code{restr = "deter"} and
\code{restr.fact = 1} was already outlined in \cite{MarJ74}, as the
only sensible way to avoid (Mahalanobis distance modified) $k$-means
type algorithms to return clusters of a few almost collinear
observations. The possibility of trimming data was also considered
in \cite{GalM02} who implicitly assumed $|\Sigma_1|=...=|\Sigma_k|$
(and so \code{restr.fact = 1}). The package presented here extends
her approach to more general cases (\code{restr.fact} $>1$).

\subsection{Example}
\begin{figure}[t!]
\centering
%\includegraphics[width= 0.65 \textwidth]{Plots_R-chunk2}
%\includegraphics[width=10cm]{Plots_R-chunk2a}
<<fig=TRUE,echo=false,results=hide,label=chunk2,width=7.14,height=7.8>>=

################
##  Figure 2  ##
################

data(M5data)
cl <- M5data[, "cluster"]
plot(M5data[, 1:2], col = cl + 1, pch = cl + 1, main = "The M5data data set")

@
\caption{ A scatter plot of the \code{M5data} data set. Different
symbols are used for the data points generated by each of the three
bivariate normal components and ``$\circ$'' for the added
outliers.}\label{f2a}
\end{figure}
In this example, we examine the influence of different constraints
by applying the function \code{tclust} to the so-called
\code{M5data} data set. This data set, which accompanies the
\pkg{tclust} package, has been generated following the simulation
scheme M5 introduced in \cite{GarG08}. Thus it is a bivariate
mixture of three simulated gaussian components with very different
scatters and a clear overlap between two of these components.
\begin{figure}[t!] \centering
%\includegraphics[width= 0.77 \textwidth]{Plots_R-chunk3}
<<fig=TRUE,echo=false,results=hide,label=chunk3,width=8,height=9>>=

################
##  Figure 3  ##
################

data(M5data)
x <- M5data[, 1:2]
set.seed(100)

## applying tclust with restr.fact = 1, restr= "eigen"
res.a <- tclust(x, k = 3, alpha=0.1, restr.fact = 1, restr= "eigen",
                 equal.weights = TRUE)

## applying tclust with restr.fact = 50, restr= "eigen"
res.b <- tclust(x, k = 3, alpha=0.1, restr.fact = 50, restr= "eigen",
                 equal.weights = FALSE)

## applying tclust with restr.fact = 1, restr= "deter"
res.c <- tclust(x, k = 3, alpha=0.1, restr.fact = 1, restr= "deter",
                 equal.weights = TRUE)

## applying tclust with restr.fact = 50, restr= "deter"
res.d <- tclust(x, k = 3, alpha=0.1, restr.fact = 50, restr= "deter",
                 equal.weights = TRUE)

op <- par(mfrow = c (2, 2), mar = c (2.1, 2.1, 3.6, 1.1))
col <- c(og, 4, 2, 3)
pch <- c(1, 4, 2, 3)

  ## plotting results..
plot(res.a, col = col, pch = pch, tol.lty = 2, xlab = "",
     ylab = "", main.pre ="(a)", main = "/r")

plot(res.b, col = col, pch = pch, tol.lty = 2, xlab = "",
     ylab = "", main.pre ="(b)", main = "/r")

plot(res.c, col = col, pch = pch, tol.lty = 2, xlab = "",
     ylab = "", main.pre ="(c)", main = "/r")
plot(res.d, col = col, pch = pch, tol.lty = 2, xlab = "",
     ylab = "", main.pre ="(d)", main = "/r")
par(op)


@

\caption{Results of the clustering processes for the \code{M5data}
data set for different constraints on the cluster scatter matrices
and the parameters $\alpha = 0.1$ and $k= 3$. Different colors and
symbols represent each observation's individual cluster assignment.}
\label{fig4:f2}
\end{figure}
A $10\%$ proportion of outliers is also added in the outer region of
the bounding rectangle enclosing the three gaussian components. See
Figure~\ref{f2a} for a graphical representation and \cite{GarG08}
for more details on the structure of this \code{M5data} data set.
Executing the following code yields Figure~\ref{fig4:f2}.
\begin{Code}
R > data ("M5data")
R > x <- M5data[, 1:2]
R > res.a <- tclust(x, k = 3, alpha = 0.1, restr.fact = 1,  restr = "eigen",
    + equal.weights = TRUE)
R > res.b <- tclust(x, k = 3, alpha = 0.1, restr.fact = 50, restr = "eigen",
    + equal.weights = FALSE)
R > res.c <- tclust(x, k = 3, alpha = 0.1, restr.fact = 1,  restr = "deter",
    + equal.weights = TRUE)
R > res.d <- tclust(x, k = 3, alpha = 0.1, restr.fact = 50, restr = "deter",
    + equal.weights = FALSE)
R > plot(res.a, main = "/r")
R > plot(res.b, main = "/r")
R > plot(res.c, main = "/r")
R > plot(res.d, main = "/r")
\end{Code}
Although different constraints are imposed, we are searching for
$k=3$ clusters and the trimming proportion is set to $\alpha=0.1$ in
all the cases. Note that only the clustering procedure introduced in
\cite{GarG08}, shown in Figure~\ref{fig4:f2},(d), with a sufficiently
large value of \code{restr.fact} approximately returns the three
original clusters in spite of the very different cluster scatters
and different cluster sizes. Moreover, this clustering procedure
adequately handles the severe overlap of two clusters. The value
\code{restr.fact = 50} has been chosen in this case because the
eigenvectors of the covariance matrices of the three gaussian
components satisfy restriction (\ref{eqn4:e5}) for this value. Due to
their underlying assumptions, the other three clustering methods
(trimmed $k$-means in Figure~\ref{fig4:f2},(a), \cite{GalR05} in (b),
\cite{GalM02} in (c)) return rather similarly structured clusters.
In fact, we found spherical clusters in (a), clusters with the same
scatter matrix in (b) and clusters with the same cluster scatter
matrix determinant in (c). The \code{M5data} is perhaps a very
``extreme'' situation and restriction settings in (a), (b) and (c)
can be useful (and easier to be interpreted) with not so extreme
data sets and where the assumptions implied by these restriction
settings hold.

\section{Numerical output}\label{sec4:num}

The function \code{tclust} returns an S3 object containing the
cluster centers $\mu_j$ by columns (\code{\$centers}), scatter
matrices $\Sigma_j$ as an array (\code{\$cov}), the weights
(\code{\$weights}), the number of observations in each group
(\code{\$size}) and the maximum value found for the trimmed
log-likelihood objective function (\ref{eqn4:e3}) or (\ref{eqn4:e4})
(\code{\$obj}). The vector \code{\$cluster} provides the cluster
assignment of each observation, whereas an artificial cluster
``\code{0}'' (without location and scatter information) is introduced
which holds all trimmed data points.


\begin{figure}[t!]
\centering
%\includegraphics[width= 0.65 \textwidth]{Plots_R-chunk4}
<<fig=TRUE,echo=false,results=hide,label=chunk4,width=7.14,height=7.8>>=

################
##  Figure 4  ##
################

  ## data generation
library(MASS)
set.seed (10)
x <- rbind(MASS::mvrnorm(200, c (0, 0), diag (2)),
           MASS::mvrnorm(200, c (5, 0), diag (2)))

  ## applying tclust
clus.4 <- tclust(x, k = 3, alpha = 0.0, restr.fact = 1)

  ## plotting results..
plot(clus.4, xlim = c (-3, 9), ylim = c (-6, 6))


@
\caption{Applying \code{tclust} with $k = 3$ and $\alpha = 0$ on a
simulated data set which originally consists of 2 clusters when
\code{equal.weights = FALSE}.} \label{fig4:f3a}
\end{figure}

Sometimes, (\ref{eqn4:e3}) and (\ref{eqn4:e4}) maximize with some
clusters remaining empty (see Figure~\ref{fig4:f3a}). In this case,
only information on the non-empty groups is returned. Notice that,
if we are searching for $k$ clusters, empty clusters can be found
when a clustering solution for a number of clusters strictly smaller
than $k$ attains a higher value for (\ref{eqn4:e3}) or
(\ref{eqn4:e4}) than the best solution found with $k$ clusters. In
this case, artificial empty clusters may be defined by considering
sufficiently remote centers $\mu_j$ and scatter matrices $\Sigma_j$
satisfying the specific constraints that are assigned to these empty
clusters. They are chosen such that $f(\cdot;\mu_j,\Sigma_j)$ gives
almost null density to all the observations in the sample. These
artificially added centers and scatter matrices are not returned as
output by the \code{tclust} function and a warning is issued. For
instance, let us consider the following code
\begin{Code}
R > set.seed(10)
R > x <- rbind(MASS::mvrnorm(200, c (0, 0), diag (2)),
    +          MASS::mvrnorm(200, c (5, 0), diag (2)))
R > clus <- tclust(x, k = 3, alpha = 0, restr.fact = 1)
R > plot(clus)
\end{Code}
Although we are searching for $k=3$ clusters, Figure~\ref{fig4:f3a} and
the issued warning show that only 2 clusters are found. Notice that
$k=2$ is surely a more sensible choice for the number of clusters
than $k=3$ for this generated data set. Therefore, the detection of
empty clusters, or clusters with few data points, can be helpful,
providing valuable tools for making sensible choices for $k$ as we
will see in Section~\ref{sec4:kal}. On the other hand, the detection
of empty clusters is very unlikely to happen when the argument
\code{equal.weights = TRUE} is provided in the call to
\code{tclust}.

\section{Algorithms}\label{sec4:alg}
The maximization of (\ref{eqn4:e3}) or (\ref{eqn4:e4}) considering different
cluster scatter matrix constraints is not straightforward because of
the combinatorial nature of the associated maximization problems.

The algorithm presented in \cite{GarG08} can be adapted to
approximately solve all these problems. The methods implemented in
\pkg{tclust} could be seen as Classification EM algorithms
\citep{Sch76,CelG92}, whereas a certain type of ``concentration''
steps \citep[see the fast-MCD algorithm in][]{RouD98} is also
applied. In fact, the concentration steps applied by the package
\pkg{tclust} can be considered as an extension of those applied by
the batch-mode $k$-means algorithm \citep{Ste56,For65}. It can be
seen that the target function always increases throughout the
application of concentration steps, whereas several random start
configurations are needed in order to avoid ending trapped in local
maxima. Therefore, \code{nstart} random initializations and
\code{iter.max} concentration steps are considered. The probability
that the algorithm converges close to the global optimum maximizing
(\ref{eqn4:e3}) or (\ref{eqn4:e4}) increases with larger values of
\code{nstart} and \code{iter.max}. The drawback of high values of
\code{nstart} and \code{iter.max} obviously is the increasing
computational effort.

In the concentration step, the centers and scatter matrices are
updated by considering the cluster sample means and cluster sample
covariance matrices. New cluster assignments are obtained by
gathering the ``closest'' observations to the new centers.
Mahalanobis distances, based on the computed cluster sample
covariance matrices, are used in order to decide which are the
closest observations to each center. If needed, in the updating
step, the cluster sample covariance matrices are modified as little
as possible but in such a way that they satisfy the desired
constraints \citep{GarG08}. The main idea behind these
``constrained'' concentration steps is to replace the eigenvalues of
the sample covariance matrices by optimally truncated eigenvalues,
which satisfy the desired constraint. A more detailed description of
the algorithm applied by \code{tclust} and the way the restrictions
are forced onto the cluster scatter matrices can be found in
\cite{FriG11}.

\section{Comparison with other robust clustering
proposals}\label{sec4:comp}

In this section, we briefly compare the performance of the
clustering procedures implemented in the \pkg{tclust} package with
respect to other robust clustering proposals in the literature.

The Partitioning Around Medoids (PAM) clustering method
\citep{KauR90} has been proposed as a robust alternative to
$k$-means clustering. It can be seen that the effect of the 6
anomalous ``short followed by short'' eruptions lengths in the lower
left corner of Figure~\ref{fig4:f1} do not affect the position of the
$k$-medoid centers (see Figure~\ref{fig4:pam},(a)) too much, nor the
resulting clusters. However, in Figure~\ref{fig4:pam},(b), we see
that the clustering results with $k=3$ are strongly affected when
moving these 6 anomalous points toward a more distant position. On
the other hand, in Figure~\ref{fig4:pam},(c), we can see that these
outlying data points do not affect the trimmed $k$-means based
clustering at all once that they are trimmed.

\begin{figure}[t!]
\centering
%\includegraphics[width= 1 \textwidth]{plots_R-chunk5}
<<fig=TRUE,echo=false,results=hide,label=chunk5,width=9,height=5>>=
################
##  Figure 5  ##
################

library(cluster)
data(geyser2)

set.seed(0)

op <- par(mfrow = c (1, 3), mar = c (3.1, 3.1, 3.1, 2.1))

xlab <- "" #names (geyser2)[1]
ylab <- "" #names (geyser2)[2]

  ## applying PAM on the geyser2 data set
clus <- pam(geyser2, 3)
plot(geyser2, xlab = xlab, ylab = ylab, col = clus$clustering + 1,
      main = "(a) PAM with Geyser")
points(clus$medoids, pch = "X", cex = 1.4)

  ## modifying the geyser2 data set
geyser2.mod <- geyser2
idx <- c(16,  21, 36, 171, 236, 265)
geyser2.mod[idx,] <- geyser2[idx,] - 12

  ## applying PAM on the modified geyser2 data set
clus <- pam(geyser2.mod, 3)
plot(geyser2.mod, xlab = xlab, ylab = ylab, col = clus$clustering + 1,
      main = "(b) PAM with modified Geyser")

  ## applying tkmeans on the modified geyser2 data set
clus <- tkmeans(geyser2.mod, k = 3, alpha = 0.03)
plot(clus, xlab = xlab, ylab = ylab,
      main = "(c) TCLUST with modified Geyser", sub = "")
par(op)

@
\caption{PAM's clustering results for the \code{geyser2} data with
$3$-medoids denoted by the symbol ``$\times$'' in (a). PAM's
clustering results for a modified \code{geyser2} data set in (b) and
when applying \code{tkmeans} with $k=3$ and $\alpha=0.03$ in (c).}
\label{fig4:pam}
\end{figure}

In fact, only one single outlier placed in a very remote position is
able to completely break down the PAM method \citep{GarG99}. This
also happens when applying \pkg{emmix}, which has a breakdown point
of zero \citep{Hen04}.  The \pkg{emmix} approach is able to obtain
appropriate clustering results for the two data sets made of
mixtures of symmetric and asymmetric $t$ components as those shown
in Figure~\ref{fig4:elt}. These two data sets contain three main
clusters with some distant observations in the heavy tails of these
$t$ components, which would be considered as outliers when assuming
normality. When applying the classification EM algorithm without
trimming to these data sets, we are not able to find the three
cluster structures and two main clusters are artificially joined
together. However, when considering $\alpha =0.05$, \pkg{tclust}
perfectly avoids the harmful effect of the observations in the tails
and still discovers the three clusters. In fact, almost all
non-trimmed observations are correctly clustered in both, symmetric
and asymmetric, cases. Any  $\alpha >0$ discarding the most outlying
observations would give similar results. Moreover, it may be seen
that the shape of the elliptical clusters are essentially discovered
in the case of the symmetric-elliptical $t$ components. In this
example, we see that applying \pkg{tclust} to data sets including
non-normally distributed components as those in
Figure~\ref{fig4:elt} may result in proper clustering solutions,
this however cannot be guaranteed if the underlying distributions
differ too much from normality. The closely related \pkg{tlemix}
package allows to consider other non-normal models by taking
advantage of the flexibility provided by the \pkg{FlexMix} package
\citep{LeiF04}. On the other hand, \pkg{tclust} focuses on normally
distributed components and on the implementation of appropriate
cluster scatter matrix constraints while \pkg{tlemix} does not. The
\pkg{tlemix} mainly controls the minimum number of observations in a
cluster.

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.9 \textwidth]{plots_R-chunk6}
<<fig=TRUE,echo=false,results=hide,label=chunk6,width=9,height=6.5>>=
################
##  Figure 6  ##
################

fig.elt <- list()

library(sn)

op <- par(mfrow = c (2, 3), mar = c (3.1, 2, 3.1, 2))

set.seed (3)

n <- 200
xlab <- "" #"x1"
ylab <- "" #"x2"

## data generation: Mixture of elliptical t's
xi <- c (0.5, -1)
Omega <- diag (2) / 2 + 0.5
alpha <- c (0, 0)
rnd1 <- rmst (n, xi, Omega, alpha, nu = 2)
xi <- c (6, -3)
Omega <- matrix (c (4, 1, 1, 2), nrow = 2, ncol = 2)
alpha <- c (0, 0)
rnd2 <- rmst (n, xi, Omega, alpha, nu = 3)
xi <- c (-3, 3)
Omega <- matrix (c (2, 1, 1, 4), nrow = 2, ncol = 2)
alpha <- c (0, 0)
rnd3 <- rmst (n, xi, Omega, alpha, nu = 4)
rnd.1 <- rbind (rnd1, rnd2, rnd3)
real.clus1 <- c (rep (1, n), rep (2, n), rep (3, n))

ylim <- c (-25, 15)

 ## Plotting the real clusters
plot(rnd.1, xlim = c (-15, 80), ylim = ylim, xlab = xlab, ylab = ylab,
      col = real.clus1 + 1, pch = real.clus1 + 1)
title("(a) Elliptical t components", line = 1.6)

  ## applying tclust without trimming
clus <- tclust(rnd.1, k = 3, alpha = 0.0)
plot(clus, xlim = c (-15, 80), ylim = ylim, xlab = xlab, ylab = ylab,
      main.pre = "(b)")

 ## applying tclust with trimming (5%)
clus <- tclust(rnd.1, k = 3, alpha = 0.05)
plot(clus, xlim = c (-15, 80), ylim = ylim, xlab = xlab, ylab = ylab,
      main.pre = "(c)")

## data generation: Mixture of non-elliptical t's
xi <- c (0.5, -1)
Omega <- diag (2) / 2 + 0.5
alpha <- c (2, 80)
rnd1 <- rmst (n, xi, Omega, alpha, nu = 2)
xi <- c (6, -3)
Omega <- matrix (c (4, 1, 1, 2), nrow = 2, ncol = 2)
alpha <- c (2, 2)
rnd2 <- rmst (n, xi, Omega, alpha, nu = 3)
xi <- c (-3, 3)
Omega <- matrix (c (2, 1, 1, 4), nrow = 2, ncol = 2)
alpha <- c (4, 4)
rnd3 <- rmst (n, xi, Omega, alpha, nu = 4)

rnd.2 <- rbind (rnd1, rnd2, rnd3)
real.clus2 <- c (rep (1, n), rep (2, n), rep(3, n))

ylim <- c (-10, 20)
  ## Plotting the real clusters
plot (rnd.2, xlim = c (-10, 20), ylim = ylim, xlab = xlab, ylab = ylab,
      col = real.clus2 + 1, pch=real.clus2 + 1)
title ("(d) Non-elliptical t components", line = 1.6)

 ## applying tclust without trimming
clus <- tclust(rnd.2, k = 3, alpha = 0.0)
plot(clus, xlim = c (-10, 20), ylim = ylim, xlab = xlab, ylab = ylab,
      main.pre = "(e)")

 ## applying tclust with trimming (5%)
clus <- tclust(rnd.2, k = 3, alpha = 0.05)
plot(clus, xlim = c (-10, 20), ylim = ylim, xlab = xlab, ylab = ylab,
      main.pre = "(f)")

par(op)


@

\caption{A data set made up of three elliptical $t$ components is
shown in (a) and three asymmetrical $t$ components in (d). The
associated clustering results when applying the \code{tclust}
function with $k=3$ and $\alpha=0$ are shown in (b) and (e) and with
$k=3$ and $\alpha=0.05$ in (c) and (f).} \label{fig4:elt}
\end{figure}

The widely used \pkg{mclust} package considers a uniformly
distributed component for explaining outlying data points. As we can
see, this uniform component successfully accommodates the $10\%$
``background noise'' as seen in Figure~\ref{fig4:mclust},(a).
However, it is not able to cope with a more structured noise pattern
like the ``helix'' in Figure~\ref{fig4:mclust},(c) which also
accounts for $10\%$ of the data, although the information of a
$10\%$ contamination level was passed to \pkg{mclust}.
Alternatively, the \pkg{tclust} package with $k = 2$ and $\alpha =
0.1$ properly discovers the outlying data points without trying to
fit them.

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.7 \textwidth]{plots_R-chunk7}
<<fig=TRUE,echo=false,results=hide,label=chunk7,width=7.14,height=7.8>>=

################
##  Figure 7  ##
################

library(mclust)

  ## function for plotting mclust-objects in tclust-style.
mclustplottcluststyle2d <- function (x, clus, clus.perm, tol.lty = 3,
                                     size = sqrt (qchisq (0.95, 2)), ...)
{
  if (missing (clus.perm))
    cuse <- clus$classification
  else
    cuse <- clus.perm [clus$classification + 1]

  plot (x, col = cuse + 1, pch = cuse + 1, ...)

  k <- ncol (clus$parameters$mean)

  for (i in 1:k)
    tclust:::.doEllipses(cov = clus$parameters$variance$sigma[,, i],
                          center = clus$parameters$mean[, i], size = size,
                          lty = tol.lty)
}

op <- par (mfrow = c (2, 2), mar = c (3.1, 3.1, 3.1, 2.1))

  ## example 1 - uniformly distributed noise

  ## data generation

set.seed (0)
x <- rbind (
            MASS::mvrnorm(360, c (0.0,  0), matrix (c ( 1,  0,  0, 1), ncol = 2)),
            MASS::mvrnorm(540, c (5.0, 10), matrix (c ( 6, -2, -2, 6), ncol = 2)),
            cbind (runif (100, -5, 15), runif (100, -10, 20)))

  ## applying mclust
noiseInit <- sample (c (TRUE, FALSE), size = nrow (x),
                     replace = TRUE, prob = c (0.1, 0.9))
clus <- Mclust(x, initialization = list (noise = noiseInit), G = 2)
mclustplottcluststyle2d (x, clus, c (0, 2, 1), xlab = "", ylab = "",
                         main = "(a) mclust", tol.lty = 2)

  ##  applying tclust
clus <- tclust(x, k = 2, alpha = 0.1)
plot (clus, xlab = "", ylab = "", main = "(b) tclust", sub = "", tol.lty = 2)


  ## example 2 - structured noise

  ## data generation
set.seed (0)
v <- runif (100, -2 * pi, 2 * pi)
noise <- cbind (100 + 25 * sin (v), 10 + 5 * v)

x <- rbind (
            MASS::mvrnorm(360, c (0.0,  0), matrix (c (1,  0,  0, 1), ncol = 2)),
            MASS::mvrnorm(540, c (5.0, 10), matrix (c (6, -2, -2, 6), ncol = 2)),
            noise)

  ##  applying mclust
noiseInit <- sample (c (TRUE, FALSE), size = nrow (x),
                     replace = TRUE, prob = c (0.1, 0.9))
clus <- Mclust (x, initialization = list (noise = noiseInit), G = 2)
mclustplottcluststyle2d (x, clus, c (0, 2, 1), xlab = "", ylab = "",
                         main = "(c) mclust", tol.lty = 2)

  ##  applying tclust
clus <- tclust(x, k = 2, alpha = 0.1)
plot (clus, xlab = "", ylab = "", main = "(d) tclust", sub = "", tol.lty = 2)

par (op)


@

\caption{Clustering results for two simulated data sets when
applying \pkg{mclust} with $k=2$ in (a) and (c) and \pkg{tclust}
with $k=2$ and $\alpha=0.1$ in (b) and (d).} \label{fig4:mclust}
\end{figure}

Since groups of outliers may be considered as further clusters, it
could be argued that robust clustering problems can always be solved
by increasing the number of groups we are searching for. However, as
explained in \cite{GarG10}, this is not necessarily the best
strategy. Firstly, sometimes the researcher fixes the number of
clusters in advance, not being aware of the presence of a small
amount of outlying observations. Secondly, it could lead to a quite
large number of clusters when very scattered outliers are present in
the data set.

A clear limitation of \pkg{tclust} is that it is not applicable on
high-dimensional data sets, as the method in its current definition
definitely needs a data set containing more observations than
dimensions.

\section{Selecting the number of groups and the trimming size}\label{sec4:kal}

Perhaps one of the most complex problems when applying cluster
analysis is the choice of the number of clusters, $k$. In some cases
one might have an idea of the number of clusters in advance, but
usually $k$ is completely unknown. Moreover, in the approach
proposed here, the trimming proportion $\alpha$ has also to be
chosen without knowing the true contamination level.

As we will see through the following example, the choices for $k$
and $\alpha$ are related problems that should be addressed
simultaneously. It is important to see that a particular trimming
level implies a specific number of clusters and vice versa. This
dependency can be explained as entire clusters tend to be trimmed
completely when increasing $\alpha$. On the other hand, when
choosing $\alpha$ too low, groups of outliers might form new
spurious clusters and thus it appears that the number of clusters
found in the data set is higher. Moreover, the simultaneous choice
of $k$ and $\alpha$ depends on the  type of clusters we are
searching for and on the allowed differences between cluster sizes.
These two aspects can be controlled by the choice of arguments
\code{restr} and \code{restr.fact}.

To demonstrate the relation between $\alpha$, $k$ and
\code{restr.fact}, let us consider \code{restr = "eigen"} and the
data set in Figure~\ref{fig4:f3} which could either be interpreted as a
mixture of three components (a) or a mixture of two components (b)
with a $10\%$ outlier proportion. Both clustering solutions shown in
Figure~\ref{fig4:f3} are perfectly sensible and the final choice of
$\alpha$ and $k$ only depends on the value given to
\code{restr.fact}. The code used to obtain Figure~\ref{fig4:f3} is the
following:
\begin{Code}
R > sigma1 <- diag(2)          ## EigenValues: 1, 1
R > sigma2 <- diag(2) * 8 - 2  ## EigenValues: 8, 4
R > sigma3 <- diag(2) * 50     ## EigenValues: 50, 50
R > mixt <- rbind(
    + MASS::mvrnorm(360, mean = c (0.0,  0), sigma = sigma1),
    + MASS::mvrnorm(540, mean = c (5.0, 10), sigma = sigma2),
    + MASS::mvrnorm(100, mean = c (2.5,  5), sigma = sigma3))
R > plot(tclust(mixt, k = 3, alpha = 0.00, restr.fact = 50))
R > plot(tclust(mixt, k = 2, alpha = 0.05, restr.fact = 12))
\end{Code}

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.9 \textwidth]{Plots_R-chunk8}
<<fig=TRUE,echo=false,results=hide,label=chunk8,width=10,height=5.45>>=

################
##  Figure 8  ##
################

  ## data generation
set.seed (100)
mixt <- rbind(MASS::mvrnorm(360, c (  0,  0), matrix (c (1,  0,  0,  1), ncol = 2)),
              MASS::mvrnorm(540, c (  5, 10), matrix (c (6, -2, -2,  6), ncol = 2)),
              MASS::mvrnorm(100, c (2.5,  5), matrix (c (50, 0,  0, 50), ncol = 2)))

  ## applying tclust
set.seed (100)
clus.1 <- tclust(mixt, k = 3, alpha =  0.0, restr.fact = 50)
clus.2 <- tclust(mixt, k = 2, alpha = 0.05, restr.fact =  8)

  ## plotting results
op <- par (mfrow = c (1, 2))
plot(clus.1, by.clust = TRUE, col = c (og, 2, 3, 4), pch = c (1, 2, 3, 4),
     tol.lty = 2, main.pre = "(a)")
plot(clus.2, by.clust = TRUE, col = c (og, 2, 3), pch = c (1, 2, 3),
     tol.lty = 2, main.pre = "(b)")
par (op)


@
\caption{Clustering results for the simulated data set \code{mixt}
with $k=3$, $\alpha=0$ and \code{restr.fact} $= 50$ (a) and $k=2$,
$\alpha=0.1$ and \code{restr.fact} $=8$ (b).} \label{fig4:f3}
\end{figure}

Considering \code{sigma1} and \code{sigma2}, the quotient of the
largest and smallest eigenvalue is 8, whereas the maximal quotient
is 50 if we consider \code{sigma1}, \code{sigma2} and \code{sigma3}.
Thus \code{restr.fact} $=8$ would allow to consider two clusters
while \code{restr.fact} $=50$ would also allow to assume three
groups there. Although the proportion of ``contaminated'' data is
equal to $10\%$, the trimming level must be reduced to $5\%$, when
considering $k=2$, because the third (more scattered) gaussian
component partially overlaps with the other two components.

Let us assume first that \code{restr} and \code{restr.fact} have
been fixed in advance by the researcher who applies the robust
clustering method. Even with this information and assuming
$\alpha=0$, choosing the appropriate number of clusters is not an
easy task. The careful monitoring of the maximum value attained by
log-likelihoods like those in (\ref{eqn4:e3}) and (\ref{eqn4:e4}) while
changing $k$ has traditionally been applied as a method for choosing
the number of clusters when $\alpha=0$. Moreover \cite{BryP91}
stated that the use of ``weighted'' log-likelihoods (\ref{eqn4:e4}) is
preferred to the use of log-likelihoods assuming equal weights
(\ref{eqn4:e3}). Notice that increasing $k$ always causes the maximized
log-likelihood (\ref{eqn4:e3}) to increase too, and this could lead to
``overestimate'' the appropriate number of clusters
\citep[see][]{GarG11}.

In this trimming framework, let us consider
$\mathcal{L}_{\texttt{restr.fact}}^{\Pi}(\alpha,k)$ as the maximum
value reached by (\ref{eqn4:e4}) for each combination of a given set of
values for $k$ and $\alpha$. \cite{GarG11} propose to monitor the
``classification trimmed likelihoods'' functionals
$$(\alpha,k)\mapsto \mathcal{L}_{\texttt{restr.fact}}^{\Pi}(\alpha,k)$$
while altering $\alpha$ and $k$, which yields an exploratory
graphical tool for making sensible choices for parameters $\alpha$
and $k$. In fact, it is proposed to choose the number of clusters as
the smallest value of $k$ such that
\begin{equation}\label{eqn4:diff}
    \mathcal{L}_{\texttt{restr.fact}}^{\Pi}(\alpha,k+1)-\mathcal{L}_{\texttt{restr.fact}}^{\Pi}(\alpha,k)
\end{equation}
is (close to) 0 except for small values of $\alpha$. Once the number
of clusters is fixed, a good choice for the trimming level is the
first $\alpha_0$ such that (\ref{eqn4:diff}) is (close to) 0 for every
$\alpha\geq \alpha_0$. Although we are convinced that monitoring the
classification trimmed likelihoods functionals is very informative,
no theoretical statistical procedures are available yet for
determining when (\ref{eqn4:diff}) can be formally considered as ``close
to 0''.

The function \code{ctlcurves} in package \pkg{tclust} approximates
the classification trimmed likelihoods by successively applying the
\code{tclust} function for a sequence of values of $k$ and $\alpha$.
A default value  \code{restr.fact = 50} is considered but, if
desired, other values of \code{restr.fact} can be passed to
\code{tclust} via \code{ctlcurves} too.

For instance, the following code applied to the previously simulated
\code{mixt} data set
\begin{Code}
R > plot (ctlcurves (mixt, k = 1:4, alpha = seq (0, 0.2, by = 0.05)))
\end{Code}
results in Figure~\ref{fig4:f4}. This figure shows that increasing
$k$ from 2 to 3 is needed when $\alpha=0$, as the objective
functions value differs noticeably between $k = 2$ and $k = 3$. On
the other hand, increasing $k$ from 2 to 3 is not needed anymore as
the third (more scattered) ``cluster'' vanishes when trimming 5\% of
the most outlying observations. Thus, there is no discernable
difference of the objective functions value with $\alpha \geq
\alpha_0=0.05$ and $k \geq 2$. Increasing $k$ from 3 to 4 is not
needed in any case.

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.65 \textwidth]{Plots_R-chunk9}
<<fig=TRUE,echo=false,results=hide,label=chunk9,width=7.14,height=6.25>>=

################
##  Figure 9  ##
################

  ## computing ctlcurves
ctl <- ctlcurves(mixt, restr.fact = 50, alpha = seq (0, 0.2, by = 0.05))

op <- par(mfrow = c (1, 1))
plot (ctl)
par (op)

@

\caption{Classification trimmed likelihoods with $k=1,...,4$,
$\alpha=0,$ $0.05$, ..., $0.2$ and \code{restr.fact} $=50$ for the
\code{mixt} data set in Figure~\ref{fig4:f3}.} \label{fig4:f4}
\end{figure}

The previously described procedure for making sensible choices for
parameters $k$ and $\alpha$ requires an active role from the
researcher. The type of restriction and the allowed restriction
factor, which do not necessarily depend on the given data set, must
be specified in advance. For instance, some specific clustering
applications like ``location-facilities'' problems require almost
spherical clusters that can be obtained by setting \code{restr =
"eigen"} and a \code{restr.fact} close to 1. The researcher's
decision on the restriction consequently modifies the proper
determination of parameters $k$ and $\alpha$.


Due to the important role of the statement of \code{restr} and
\code{restr.fact}, some general guideline for fixing them will be
given here. For instance, as already commented, fixing \code{restr =
"deter"} is recommended when only the relative cluster sizes shall
be constrained, or when affine equivariance is clearly needed. On
the other hand, using \code{restr = "eigen"} is advised when we want
to simultaneously constrain relative cluster sizes and shapes.

With respect to the choice of \code{restr.fact}, we recommend to
initially use large values when applying \code{ctlcurves}, thus,
providing high flexibility to the clustering method. The default
value \code{restr.fact = 50} is suggested for \code{ctlcurves}, as
it worked well with a lot of data sets (especially if the variables
have been properly standardized through the \code{scale} argument in
the \code{tclust} function). The so obtained ``sensible'' values for
$k$ and $\alpha$ and their associated clustering solutions must be
explored carefully. For instance, \code{tclust} issues a warning
when the returned clustering solution has been ``artificially
restricted'' by the algorithm, as shown in Section~\ref{sec4:exa}.
This means, that the values $M_n$ and $m_n$ (see
Section~\ref{sec4:consteig} and Section~\ref{sec4:constdet}) derived
from the returned scatter matrices satisfy $M_n/m_n =$
\code{restr.fact}, because the algorithm has forced the chosen
constraint, since the (unconstrained) group sample covariance
matrices do not satisfy $M_n/m_n \leq$ \code{restr.fact}. In this
situation, if no specific constraints are required,
\code{restr.fact} may be increased stepwise until this warning
disappears. Moreover, printing the object returned by the
\code{ctlcurves} function points out all ``artificially restricted''
solutions for each computed combination of parameters $k$ and
$\alpha$. In this way, if desired, we can easily search for
clustering solutions which are not artificially restricted and do
not contain spurious clusters. Finally, the exploratory tools in
Section~\ref{sec4:gra} also help to evaluate whether all these
parameters are reasonably chosen.

Note that arguments \code{nstart} and \code{iter.max} may be
provided in the call to \code{ctlcurves} and they are internally
passed to function \code{tclust}.

The curves presented in \cite{GarG03} can be considered as
precedents of those we obtain by using the \code{ctlcurves}
function. Trimmed likelihoods have also been taken into account in
\cite{NeyF07} for choosing $k$ and $\alpha$ by using a BIC
criterion.

\section{Graphical displays}\label{sec4:gra}
As seen in previous examples, the package \pkg{tclust} provides
functions for visualizing the computed cluster assignments.
One-dimensional, two-dimensional and higher-dimensional cases are
visualized differently:
\begin{description}
\item[$p = 1$:] The one-dimensional data set with the corresponding cluster assignments is
 displayed along the $x$-axis. Setting the argument \code{jitter = TRUE} jitters
  the data along the $y$-axis in order to increase the visibility of the actual
 data structure. Additionally, a (robust) scatter
 estimation of each cluster is also displayed.
\item[$p = 2$:] Tolerance ellipsoids are plotted additionally in order to
visualize the estimated cluster scatter matrices.
\item[$p > 2$:] The first two Fisher's canonical coordinates are displayed in
this case, which are computed based on the estimated cluster scatter
matrices. Notice that trimmed observations are not taken into
account when computing these coordinates, since they have been
completely discarded. The implementation of these canonical
coordinates is derived from the function \code{discrcoord} as
implemented in the package \pkg{fpc} \citep{HenC10}.
\end{description}

A simple example demonstrates how the \code{plot} function works in
different dimensions. The code:
\begin{Code}
R > geyser1 <- geyser2[, 1, drop = FALSE]
R > geyser3 <- cbind (geyser2, rnorm (nrow (geyser2)))
R > plot (tkmeans (geyser1, k = 2, alpha = 0.03), jitter = TRUE)
R > plot (tkmeans (geyser3, k = 3, alpha = 0.03))
\end{Code}
yields Figure~\ref{fig4:f5a}. For demonstrating the different plotting
modes, we have selected one single variable from \code{geyser2} to
obtain a one-dimensional data set (\code{geyser1}), and, added an
additional normally distributed variable to \code{geyser2}, yielding
a three-dimensional data set (\code{geyser3}). Figure~\ref{fig4:f5a}
plots the results of the trimmed $k$-means robust clustering method
for these two generated data sets.

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.9 \textwidth]{Plots_R-chunk10}
<<label=fig-10, fig=TRUE, echo=false, results=hide, width=10, height=5.45>>=
#################
##  Figure 10  ##
#################

data (geyser2)
op <- par (mfrow = c (1, 2))
set.seed (10)

  ## creating a one dimensional data "matrix"
geyser1 <- geyser2[, 1, drop = FALSE]
  ## applying tkmeans
plot(tkmeans(geyser1, k = 2, alpha = 0.03), jitter = TRUE, tol.lwd = 2,
      main.pre = "(a)")

  ## adding a random dimension to geyser2
geyser3 <- cbind(geyser2, rnorm (nrow (geyser2)))
  ## applying tkmeans
plot(tkmeans(geyser3, k = 3, alpha = 0.03), main.pre = "(b)")

par(op)

@
\caption{Trimmed $k$-means clustering results for \code{geyser1}
(one-dimensional) in (a) and for \code{geyser3} (three-dimensional)
in (b). These two data sets are based on \code{geyser2}. $k=2$ is
fixed in (a) and $k = 3$ in (b) while $\alpha=0.03$ is fixed in both
cases.} \label{fig4:f5a}
\end{figure}

Given a \code{tclust} object, some additional exploratory graphical
tools can be applied in order to evaluate the quality of the cluster
assignments and the trimming decisions. This is done by applying the
function \code{DiscrFact}.

Let $\widehat{R}=\{\widehat{R_0},\widehat{R_1},...,\widehat{R_k}\}$,
$\widehat{\theta}=(\widehat{\theta_1},...,\widehat{\theta_k})$ and
$\widehat{\pi}=(\widehat{\pi_1},...,\widehat{\pi_k})$ be the values
obtained by maximizing  (\ref{eqn4:e3}) or (\ref{eqn4:e4}) (we set
$\widehat{\pi_j}=1/k$ when maximizing (\ref{eqn4:e3})).
$D_j(x_i;\widehat{\theta},\widehat{\pi})=\widehat{\pi_j}\phi(x_i,\widehat{\theta_j})$
is a measure of the degree of affiliation of observation $x_i$ with
cluster $j$. These values can be ordered as
$D_{(1)}(x_i;\widehat{\theta},\widehat{\pi}) \leq ... \leq
D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})$. Thus the quality of
the assignment decision of a non trimmed observation $x_i$ to the
cluster $j$ with
$D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})=D_{j}(x_i;\widehat{\theta},\widehat{\pi})$
can be evaluated by comparing its degree of affiliation with cluster
$j$ to the best second possible assignment. That is
$$\text{DF}(i)=\log\big(D_{(k-1)}(x_i;\widehat{\theta},\widehat{\pi})/D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})\big)
\text{ for }x_i\text{ not trimmed}.$$

Let $x_{(1)},... ,x_{(n)}$ be the observations in the sample after
being sorted according to their
$D_{(k)}(\cdot;\widehat{\theta},\widehat{\pi})$ values, i.e., $
D_{(k)}(x_{(1)};\widehat{\theta},\widehat{\pi}) \leq ... \leq
D_{(k)}(x_{(n)};\widehat{\theta},\widehat{\pi}).$ It is not
difficult to see that $x_{(1)},... , x_{(\rounding{n\alpha})}$ are
the trimmed observations which are not assigned to any cluster.
Nevertheless, it is possible to compute the degree of affiliation
$D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})$ of a trimmed
observation $x_i$ to its nearest cluster. Thus, the quality of the
trimming decision on this observation can be evaluated by comparing
$D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})$ to
$D_{(k)}(x_{(\rounding{n\alpha}+1)};\widehat{\theta},\widehat{\pi})$,
with $x_{(\rounding{n\alpha}+1)}$ being the non-trimmed observation
with smallest value of
$D_{(k)}(\cdot;\widehat{\theta},\widehat{\pi})$. That is
$$\text{DF}(i)=\log\big(
D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})/D_{(k)}(x_{(\rounding{n\alpha}+1)};\widehat{\theta},\widehat{\pi})\big)\text{
for }x_i\text{ trimmed}.
$$
Hence, discriminant factors $\text{DF}(i)$ $\leq 0$ are obtained for
every observation in the data set, whether trimmed or not.

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.9 \textwidth]{Plots_R-chunk11}
<<label=fig-11, fig=TRUE, echo=false, results=hide, width=9, height=6.5>>=

#################
##  Figure 11  ##
#################

## data generation
set.seed (100)
mixt2 <- rbind (
  MASS::mvrnorm(360, c (0,   0), matrix (c (1,  0,  0,  1), ncol = 2)),
  MASS::mvrnorm(540, c (5,  10), matrix (c (6, -2, -2,  6), ncol = 2)),
  MASS::mvrnorm(100, c (2.5, 5), matrix (c (50, 0,  0, 50), ncol = 2)))

  ## applying tclust
set.seed (100)
clus.w <- tclust(mixt2, k = 3, alpha = 0.1, restr.fact = 1,
                  equal.weights = TRUE)

  ## applying DiscrFact
discr.clus.w <- DiscrFact(clus.w)

  ## plotting results
op <- par(mfrow = c(1, 3), mar = mmar)

plot(clus.w, col = c(og, 2, 3, 4), tol.lty = 2, main.pre = "(a)")
tclust:::plot_DiscrFact_p2(discr.clus.w, xlim = c (-70, 0), main.pre = "(b)")
tclust:::plot_DiscrFact_p3(discr.clus.w, tol.lty = 2, main.pre = "(c)")

par(op)
@
\caption{Graphical displays based on the $\text{DF}(i)$ values for a
\code{tclust} cluster solution obtained with $k=3$, $\alpha=0.1$,
\code{restr.fact} $= 1$ and \code{equal.weights = TRUE} for the
\code{mixt} data set.} \label{fig4:f5}
\end{figure}

Observations with large $\text{DF}(i)$ values (i.e., values close to
zero) indicate doubtful assignments or trimming decisions. The use
of this type of discriminant factors was already suggested in
\cite{AelW06} in a clustering problem without trimming.
``Silhouette'' plots \citep{RouP87} can be used for summarizing the
obtained ordered discriminant factors. Clusters in the silhouette
plot with many large $\text{DF}(i)$ values indicate the existence of
not very ``well-determined'' clusters. The most ``doubtful''
assignments with $\text{DF}(i)$ larger than a
$\log(\texttt{threshold})$ value are highlighted by the function
\code{DiscrFact}.

Figure~\ref{fig4:f5} shows the result of applying the \code{DiscrFact}
function to a clustering solution found for the \code{mixt} data set
appearing in Figure~\ref{fig4:f3}. The following code is used to obtain
this figure:

\begin{Code}
R > clus.w <- tclust (mixt, k = 3, alpha = 0.1, restr.fact = 1,
    + equal.weights = TRUE)
R > discr.clus.w <- DiscrFact (clus.w, threshold = 0.1)
R > plot (discr.clus.w)
\end{Code}

The choice \code{threshold} $= 0.1$ means that a decision on a
particular observation $x_i$ is considered as doubtful, if the
quality of the second best possible decision
($D_{(k-1)}(x_i;\widehat{\theta},\widehat{\pi})$ or
$D_{(k)}(x_{(\rounding{n\alpha}+1)};\widehat{\theta},\widehat{\pi})$
for trimmed observations) is larger than one tenth of the quality
of the actually made decision
($D_{(k)}(x_i;\widehat{\theta},\widehat{\pi})$).

Although Figure~\ref{fig4:f4} suggests to choose $k = 2$, $k$ has been
increased to $3$ in order to show how such a change leads to
doubtful cluster assignment decisions which can be visualized by
\code{DiscrFact}. Figure~\ref{fig4:f5},(a) simply illustrates the cluster
assignments and trimming decisions. The mentioned silhouette plot is
presented in (b), whereas the doubtful decisions are marked in (c).
All observations with $\text{DF}(i) \geq \log(0.1)$ are highlighted
as they are plotted darker/in color. Most of the doubtful decisions
are located in the overlapping area of the two artificially found
clusters (highlighted symbols ``$\times$'' and ``$+$''). Some
doubtfully trimmed observations (highlighted symbol ``$\circ$'') are
located in the boundaries of these two clusters.

\section{Swiss Bank notes data}\label{sec4:exa}
\begin{figure}[t!]
\centering
%\includegraphics[width= 0.65 \textwidth]{plots_R-chunk12}
<<fig=TRUE,echo=false,results=hide,label=chunk12,width=7.14,height=6.25>>=
#################
##  Figure 12  ##
#################

data (swissbank)
set.seed (0)
fig6.ctl <- ctlcurves(swissbank, k = 1:4, alpha = seq (0, 0.3, by = 0.025),
                       restr.fact = 50, iter.max = 100, nstart = 100)

op <- par (mfrow = c (1, 1), mar = mmar)
plot (fig6.ctl)

par (op)
@
\caption{Classification trimmed likelihoods for $k=1,...,4$ and
$\alpha=0,$ $.025$, ..., $.3$ when \code{restr.fact} $=50$ for the
``Swiss Bank notes'' data set.} \label{fig4:f6}
\end{figure}
The well-known ``Swiss Bank notes'' data set includes 6 numerical
measurements (six-dimensi\-onal data set) made on 100 genuine and
100 counterfeit old Swiss 1000-franc bank notes \citep{FluR88}. The
following code can be used to obtain the classification trimmed
likelihoods shown in Figure~\ref{fig4:f6}.
\begin{Code}
R > data ("swissbank")
R > plot (ctlcurves (swissbank, k = 1:4, alpha = seq (0, 0.3, by = 0.025)))
\end{Code}
This figure indicates the clear existence of $k=2$ main clusters
(``genuine'' and ``forged'' bills). Moreover, considering the clear
difference between $\mathcal{L}_{\texttt{50}}^{\Pi}(0,3)$ and
$\mathcal{L}_{\texttt{50}}^{\Pi}(0,2)$, we can see that a further
cluster, i.e., $k=3$, is needed when no trimming is allowed. This
extra cluster can be justified by the heterogeneity of the group of
forgeries (perhaps due to the presence of different sources of
forged bills).

\begin{figure}[t!]
\centering
%\includegraphics[width= 0.9 \textwidth]{plots_R-chunk13}

<<fig=TRUE,echo=false,results=hide,label=chunk13,width=9,height=6.5>>=
#################
##  Figure 13  ##
#################

fig7.clus <- tclust(swissbank, k = 2, alpha = .1, restr.fact = 50)
fig7.discrfact <- DiscrFact(fig7.clus, threshold = .000125)

op <- par (mfrow = c (1, 3), mar = mmar)
plot (fig7.discrfact, enum = TRUE)
par (op)
@
\caption{Clustering results with $k=2$, $\alpha = 0.1$ and
\code{restr.fact} $=50$ summarized by the use of \code{DiscrFact}
function for the ``Swiss Bank notes'' data set. The threshold value
is chosen in order to highlight the 7 most doubtful cluster
assignments.} \label{fig4:f7}
\end{figure}

Considering Figure~\ref{fig4:f6}, the choice $k=2$ and a value of
$\alpha$ close to 0.1 also seem sensible. Notice that
$\mathcal{L}_{50}^{\Pi}(\alpha,3)$ is clearly larger than
$\mathcal{L}_{50}^{\Pi}(\alpha,2)$ for $\alpha < 0.1$ while these
differences are not so big when $\alpha \geq 0.1$. We can even see
smaller differences in the classification trimmed likelihood curves
when increasing $k$ from 3 to 4. However, these differences are less
significant than those previously commented. More spurious clusters
can be surely found but they have less entity and importance.

Figure~\ref{fig4:f7} shows the clustering results with $k=2$, $\alpha =
0.1$ and \code{restr.fact = 50} obtained by executing the code:
\begin{Code}
R > clus <- tclust (swissbank, k = 2, alpha = 0.1, restr.fact = 50)
R > plot (DiscrFact (clus, threshold = 0.0001))
\end{Code}

Notice that, in this example, we did not want to impose a specific
constraint on the solution. Thus, the default parameter
\code{restr.fact = 50} has initially been used in \code{ctlcurves}.
After choosing the combination $\alpha = 0.10$ and $k = 2$, we could
try to reduce the restriction factor which resulted in a warning:

\begin{Code}
R > tclust(swissbank, k = 2, alpha = 0.1, restr.fact = 40)
In .tclust.warn(warnings, ret):
  The result is artificially constrained due to restr.fact = 40.
\end{Code}

Thus the choice \code{restr.fact = 50} seems appropriate as it does
not artificially restrict the result, whereas a slightly smaller
restriction factor (40) does. By examining the sizes of the obtained
groups, we see that no spurious groups are found with
\code{restr.fact = 50}:
\begin{Code}
R > clus$size
  [1] 95 85
\end{Code}

We have used \code{restr = "eigen"} in this example, but \code{restr
= "deter"} can be also successfully applied  with smaller values of
\code{restr.fact}.

We also use the function \code{DiscrFact} to summarize the obtained
clustering results. The two first Fisher's canonical coordinates
derived from the final cluster assignments are plotted. The
threshold value 0.0001 is chosen in order to highlight the 7 most
doubtful decisions.

Finally, Figure~\ref{fig4:f8} shows a scatterplot of the fourth
(``Distance of the inner frame to lower border'') against the sixth
variable (``Length of the diagonal'') with the corresponding cluster
assignments. We use the symbols ``\code{G}'' for the genuine bills
and ``\code{F}'' for the forged ones. The 7 most doubtful decisions
(i.e., the observations with largest $\text{DF}(i)$ values that were
highlighted in Figure~\ref{fig4:f7},(c)) are surrounded by circles in
this figure.
\begin{figure}[t!]
\centering
%\includegraphics[clip,width= 0.9 \textwidth]{plots_R-chunk14}
<<label=fig-14, fig=TRUE, echo=false, results=hide, width=9, height=6.5>>=

#################
##  Figure 14  ##
#################


clus <- tclust(swissbank, k = 2, alpha=.1, restr.fact=50)
discrfact <- DiscrFact(clus)
pch <- c (rep ("G", 100), rep ("F", 100))
condition <- discrfact$assignfact > log (.000125)
cl <- clus$cluster
data (swissbank)
op <- par (mfrow = c (1, 3), mar = mmar)

xlab <- "Distance of the inner frame to lower border"
ylab <- "Length of the diagonal"

plot(swissbank[, 4], swissbank[, 6], col = "darkgrey", pch = pch,
      main = "(a) Cluster1", xlab = xlab, ylab = ylab)

cl1 <- cl == 1
points(swissbank[cl1, 4], swissbank[cl1, 6], pch = pch[cl1], col = 2)
idx <- (cl1) & condition
points(swissbank[idx, 4], swissbank[idx, 6], pch = 1, cex = 4, col = "blue")

plot(swissbank[, 4], swissbank[, 6], col = "darkgrey", pch = pch,
      main = "(b) Cluster2", xlab = xlab, ylab = ylab)
cl2 <- cl == 2
points(swissbank[cl2, 4], swissbank[cl2, 6], pch = pch[cl2], col = 3)
idx <- (cl2) & condition
points(swissbank[idx, 4], swissbank[idx, 6], pch = 1, cex = 4, col = "blue")

cl0 <- cl == 0
plot(swissbank[, 4], swissbank[, 6], col = "darkgrey", pch = pch,
      main = "(c) Trimmed", xlab = xlab, ylab = ylab)

points(swissbank[cl0, 4], swissbank[cl0, 6], pch = pch[cl0])
idx <- (cl0) & condition
points(swissbank[idx, 4], swissbank[idx, 6], pch = 1, cex = 4, col = "blue")
par(op)


@
\caption{Clustering results with $k=2$, $\alpha=.1$ and 
    \code{restr.fact} $=50$ for the ``Swiss Bank notes'' data set. 
    Only the fourth and sixth variables are plotted. The 7 most doubtful 
    decisions are rounded by a circle symbol.} \label{fig4:f8}
\end{figure}
We can see that ``Cluster 1'' essentially includes most of the
``forged'' bills while ``Cluster 2'' includes most of the ``genuine''
ones. Among the trimmed observations, we can find a subset of 15
forged bills following a clearly different forgery pattern that has
been previously commented by other authors \citep[see,
e.g.,][]{FluR88,CooP99}. These most doubtful assignments include 5
``genuine'' bills that have perhaps been wrongly trimmed.

\section{Conclusion}\label{sec4:con}
This paper presents a package called \pkg{tclust} for robust
(non-hierarchical) clustering. The implementation is flexible, so
only the restrictions on the cluster scatters have to be changed in
order to carry out different robust clustering algorithms.
Robustness is achieved by trimming a specific proportion of
observations which are identified as the ``most outlying'' ones.

Although this \proglang{R}-package implements robust clustering
approaches which have already been described in the literature, some
of these approaches have been extended to provide increased
flexibility. The package also provides some graphical tools which on
the one hand help to chose appropriate parameters (\code{ctlcurves})
and on the other hand help to estimate the adequacy of a particular
clustering solution (\code{DiscrFact}).

Future work on this package focuses on implementing additional types
of scatter restrictions, making the algorithm even more flexible,
and on providing numerical tools for automatically choosing the
number of clusters and the trimming proportion.

\section*{Acknowledgements:}This research is partially supported by
the Spanish Ministerio de Ciencia e Innovaci\'{o}n, grant
MTM2011-28657-C02-01. We would like to thank the reviewers and the
editor whose comments and suggestions greatly helped in improving
the content and the presentation of this paper.

\bibliography{tclust}

\end{document}
