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\begin{document}

\title{The \Rpackage{clustComp} package}
\author{Alvis Brazma and Aurora Torrente}
\maketitle





\section{Introduction}

This document presents an overview to the \Rpackage{clustComp} package, 
which is a collection of tools developed for the comparison and the 
visualisation of relationships between two clusterings, either flat versus
flat or flat versus hierarchical. Both situations are addressed by 
representing clusters as nodes in a weighted bipartite graph, where 
each layer corresponds to one of the clusterings under comparison, 
and the edge weights are given by the number of elements in the 
intersection between any two (connected) clusters.\\[2mm]

\noindent To illustrate the use of the package, we use the publicly available 
dataset included in the experiment data package, \Rpackage{colonCA}, on 
Bioconductor, which contains an exprSet instance for the Alon et al. (1999) 
colon cancer data, including 40 tumour samples and 22 normal samples, 
measured at 2000 probeset. We first install and load the experiment data 
package, which requires the \Rpackage{Biobase} package:

<<loadData>>=
library(Biobase)
library(colonCA)
data(colonCA)
@

For visualisation purposes, we are going to rename some rownames of some 
of the dataset, which are too long:

<<renameLongNames>>=
featureNames(colonCA)[39:42] <- paste("HSAC07", 0:3)
featureNames(colonCA)[50:53] <- paste("UMGAP", 0:3)
featureNames(colonCA)[260:263] <- paste("i", 0:3)
@

\noindent Next, we load the \Rpackage{clustComp} package using:

<<loadPackage>>=
library(clustComp)
@

\noindent and build the clusterings to be compared. We arbitrarily choose 
flat clusterings of seven and six clusters, respectively, and two hierarchical 
trees built with the complete-linkage and the Ward methods, respectively:

<<>>=
set.seed(0)
# seven flat clusters:
flat1 <- paste("A", kmeans(exprs(colonCA), 7)$cluster, sep = "")   
# six flat clusters:
flat2 <- paste("B", kmeans(exprs(colonCA), 6)$cluster, sep = "")  
# dendrograms with 2000 leaves:
hierar <- hclust(dist(exprs(colonCA)))   
hierar2 <- hclust(dist(exprs(colonCA)), method = 'ward.D')
@

\section{Comparison of two flat clusterings}


The key algorithm for the comparison of clusterings is the \textbf{barycentre 
algorithm}, which tries to minimise heuristically the number of edge 
crossings in the bipartite graph, represented by default in a vertical 
layout. The size of the nodes and the width of the edges can be adjusted
by the user, with the arguments \verb|point.sz| and \verb|line.wd|; also a 
minimum distance \verb|h.min| between consecutive nodes can be set 
by the user to avoid overlapping.

\noindent To compare two flat clusterings, use the function \Rfunction
{flatVSflat}, which combines the barycentre algorithm with an additional 
heuristic consisting in swapping consecutive nodes if that leads to a reduced 
number of edge crossings. The following run of the function decreases the 
number of edge crossings from the initial 228422 to the final 9604:

<<comparisonFlatVsFlat1,fig=FALSE>>=
flatVSflat(weights = table(flat1,flat2), line.wd = 5, h.min = 0.3)
@
<<fig01, results = hide, echo = FALSE, fig = FALSE>>=
bitmap("fig01.png", height = 4, width = 4, pointsize = 10, res = 300)
flatVSflat(table(flat1, flat2), line.wd = 5, h.min = 0.3)
dev.off()
@

\vspace*{5mm}
\begin{center}
\fbox{\includegraphics[width=0.45\textwidth]{fig01}}
\end{center}


\noindent One can easily see that this layout is not the optimal one, but it 
can be used to find it, by setting an appropriate initial ordering of the 
nodes:



<<comparisonFlatVsFlat1a,fig=FALSE,results=hide>>=
flatVSflat(weights=table(flat1,flat2), coord1=c(6,3,4,7,5,1,2), 
    coord2=c(5,1,4,6,3,2))
@
<<fig01a, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig01a.png",height=4,width=4,pointsize=10,res=300)
flatVSflat(table(flat1,flat2),coord1=c(6,3,4,7,5,1,2), coord2=c(5,1,4,6,3,2))
dev.off()
@

\vspace*{5mm}
\begin{center}
\fbox{\includegraphics[width=0.45\textwidth]{fig01a}}
\end{center}


\noindent The ordinates of each flat cluster are yielded by the barycentre 
algorithm; these can be overridden to have evenly distributed nodes 
(preserving the ordering); also a horizontal layout is allowed:


<<comparisonFlatVsFlat2,fig=FALSE, results=hide>>=
flatVSflat(weights=table(flat1,flat2), evenly=TRUE, horiz=TRUE)
@
<<fig02, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig02.png",height=4,width=4,pointsize=10,res=300)
flatVSflat(table(flat1,flat2),evenly=TRUE,horiz=TRUE)
dev.off()
@

\begin{center}
\fbox{\includegraphics[width=0.45\textwidth]{fig02}}
\end{center}
\vspace*{1cm}

\subsection{The greedy algorithm}


The \textbf{greedy algorithm} constructs a one-to-one mapping between 
groups of clusters from each partitioning. These new groups are named 
superclusters; they correspond to the p connected components in the 
subgraph found by considering only the thickest edge among those which are 
incident with each node. Ties are not broken at random; instead, all edges 
with maximum weight are taken into account for constructing the mapping.
\\[2mm]

\noindent The two sets of p superclusters are assigned new labels: 'S1',
'S2',...,'Sp' for the first clustering, and 'T1', 'T2',...,'Tp' for the second
one, as shown below:

<<greedyAlgorithm, fig=FALSE>>=
myMapping<-SCmapping(flat1, flat2)
@
<<fig03, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig03.png",height=4,width=4,pointsize=10,res=300)
SCmapping(flat1,flat2)
dev.off()
@
\begin{center}
\fbox{\includegraphics[width=0.45\textwidth]{fig03}}
\end{center}

\noindent  The initial clusters that form each supercluster are also 
indicated in the plot, in brackets. After merging them to form the 
superclusters, all the edges are drawn in the bi-graph (otherwise, it 
would contain only parallel edges).


\section{Comparison of flat and hierarchical clusterings}

To compare a hierarchical and a flat clusterings we compute cutoffs at 
different heights of the dendrogram and collapse the resulting branches to 
form flat clusters, that are compared to the non-hierarchical clustering with 
the barycentre algorithm, and optionally, with the greedy algorithm.\\[1mm] 

\noindent We explore the tree by depth-first search, starting at the root. Each 
comparison can be sequentially plotted (the default), and then the code 
needs the interaction of the user to continue, or alternatively, only the final 
comparison, after all split decisions have been made, is shown. The criterium 
used to decide whether we should divide a given branch is that splitting that 
branch yields a better value of a certain scoring function (namely, 
\verb|"it"|, a more stringent one, based on information theory, or 
\verb|"crossing"|, based on the layout aesthetics). In other words, we 
compute a certain score for the 'parent tree' (with the branch under 
study still collapsed) and for the 'children tree' (with the branch replaced 
by its descendants), and keep the best one. 
Note that the number of descendants might be different from 2 if we are 
using the look-ahead strategy.\\[1mm] 

\noindent The plots, either intermediate or final, display the pruned tree 
on the left, with branches evenly separated, and the flat clustering on the 
right, with coordinates given by the barycentre algorithm. Each collapsed 
branch is labelled with 'B' followed by a number in brackets. That 
number follows the notation of dendrograms in R: if it is negative, it is a 
leaf (therefore it cannot be split as it has no children); if it is positive, 
it indicates the stage at which the given branch was agglomerated when 
computing the tree (i.e. it has two children). The flat clusters on the right 
are labelled with 'F' followed by their original label, in brackets. 
Additionally, a box with as many divisions as branches/clusters indicates 
the sizes of the corresponding groups.\\[1mm]

\noindent The following pictures illustrate the use of the function \Rfunction
{flatVShier} for the comparison of the hierarchical clustering of the colon 
data, and the flat clustering with 7 clusters.

\begin{itemize}
\item \textit{Aesthetics-based scoring function, with no look ahead}: the 
optimal set of cutoffs yields 19 branches; the greedy algorithm computes 7 
superclusters, identified in the plot by different colours/symbols.
<<aestheticsNoLookAhead,  fig=FALSE, results=hide>>=
comparison.flatVShier.1<-flatVShier(hierar, flat1, verbose=FALSE, 
    pausing=FALSE, score.function="crossing", greedy.colours=1:4, 
    look.ahead=0)
@
<<fig04, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig04.png",height=6,width=8,pointsize=9,res=300)
flatVShier(hierar,flat1,pausing=FALSE,verbose=FALSE, 
    score.function="crossing",greedy.colours=1:4,look.ahead=0)
dev.off()
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig04}}
\end{center}

A more descriptive version of previous figure can be achieved by setting the 
parameter \verb|expanded| to TRUE; in that case, the full dendrogram is 
represented:
<<aestheticsNoLookAheadExpanded,  fig=FALSE, results=hide>>=
comparison.flatVShier.1<-flatVShier(hierar, flat1, verbose=FALSE, 
    pausing=FALSE, score.function="crossing", greedy.colours=1:4, 
    look.ahead=0, expanded=TRUE)
@
<<fig05, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig05.png",height=14,width=8,pointsize=9,res=300)
flatVShier(hierar,flat1,pausing=FALSE,verbose=FALSE, 
    score.function="crossing",greedy.colours=1:4,look.ahead=0, 
    expanded=TRUE)
dev.off()
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig05}}
\end{center}



\item \textit{Information theoretical-based scoring function, looking one step 
ahead}: the optimal set of cutoffs yields 7 branches; the greedy algorithm 
computes 4 superclusters.
<<itLookAhead1,  fig=FALSE, results=hide>>=
comparison.flatVShier.2<-flatVShier(hierar, flat1, verbose=FALSE, 
    pausing=FALSE, h.min=0.2, score.function="it", 
    greedy.colours=1:4, look.ahead=1)
@
<<fig06, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig06.png",height=6,width=8,pointsize=9,res=300)
flatVShier(hierar,flat1,pausing=FALSE,h.min=0.2,verbose=FALSE, 
    score.function="it",greedy.colours=1:4,look.ahead=1)
dev.off()
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig06}}
\end{center}

\item \textit{Information theoretical-based scoring function, looking two steps 
ahead}: the optimal set of cutoffs yields 12 branches; the greedy algorithm 
computes 5 superclusters.
<<itLookAhead2,  fig=FALSE, results=hide>>=
comparison.flatVShier.3<-flatVShier(hierar, flat1, verbose=FALSE, 
    pausing=FALSE, h.min=0.2, score.function="it", 
    greedy.colours=1:4, look.ahead=2)
@
<<fig07, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig07.png", height=6, width=8, pointsize=9, res=300)
flatVShier(hierar, flat1, pausing=FALSE, verbose=FALSE, score.function="it",
    greedy.colours=1:4, look.ahead=2,h.min=0.2)
dev.off()
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig07}}
\end{center}
\end{itemize}


\noindent The intermediate steps can be visualised by setting the argument 
\verb|pausing| equal to TRUE (it requires the interaction of the user to show 
all the iterations).\\[1mm]

\noindent Also, an expanded version of the comparison can be achieved by 
displaying the whole tree, which helps visualising the size of the clusters.

<<itLookAhead2Expanded,  fig=FALSE, results=hide>>=
comparison.flatVShier.4<-flatVShier(hierar2, flat1, verbose=FALSE, 
    pausing=FALSE, h.min=0.2, score.function="it", 
    greedy.colours=1:4, look.ahead=2, expanded=TRUE)
@
<<fig08, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig08.png", height=14, width=8, pointsize=9, res=300)
flatVShier(hierar2, flat1, pausing=FALSE, verbose=FALSE, 
    score.function="it", greedy.colours=1:4, look.ahead=2, 
    h.min=0.2, expanded=TRUE)
dev.off()
Sys.sleep(10)
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig08}}
\end{center}

\noindent In the expanded version, it is also possible to draw the heatmap, 
by providing the expression data as an argument.

<<itLookAhead2Heatmap,  fig=FALSE, results=hide>>=
comparison.flatVShier.5<-flatVShier(hierar2, flat1, verbose=FALSE, 
    pausing=FALSE, h.min=0.2, score.function="it", 
    greedy.colours=1:4, look.ahead=2, expanded=TRUE,
    expression=exprs(colonCA))
@
<<fig09, results=hide, echo=FALSE, fig=FALSE>>=
bitmap("fig09.png", height=14, width=8, pointsize=9, res=300)
flatVShier(hierar2, flat1, pausing=FALSE, verbose=FALSE, 
    score.function="it", greedy.colours=1:4, look.ahead=2, 
    h.min=0.2, expanded=TRUE, expression=exprs(colonCA))
dev.off()
Sys.sleep(10)
@
\begin{center}
\fbox{\includegraphics[width=0.65\textwidth]{fig09}}
\end{center}





\end{document}
