IEEE Systems, Man and Cybernetics Magazine - April 2020 - 34

blocks along the diagonal for GM2, whereas clusiVAT
shows three light blocks, including many tiny blocks (data
points) along the diagonal.
Although clusiVAT and FensiVAT both show three
blocks for GM1 and GM2, FensiVAT provides a more convincing assessment because of the sharper contrast
between diagonal blocks and the background. Moreover,
FensiVAT takes only a small fraction of the time needed
by clusiVAT for both data sets. The sizes of the diagonal
blocks in all four images show the relative size of each
cluster accurately, which supports the claim that nearMMRS sampling replicates (approximately) the same
cluster distribution in the sample as the MMRS sampling
used by clusiVAT. Finally, SL clustering of the samples
and NPR extension to the rest of the data set is performed in the down space for different random projections, and majority-voting-based schemes are used to
assign the final cluster labels. The pseudocode for FensiVAT is given in algorithm S22 in "Pseudocode for Various Algorithms Belonging to the Visual Assessment of
Tendency Family."
Coclustering
The VAT family of algorithms tackles the problem of clustering tendency assessment and subsequent clustering
for an n # n (square) dissimilarity (or, in more general
terms, relational) matrix. An even more general form of
relational data is rectangular. These data are represented
by an m # n dissimilarity matrix D, where the entries
are the pairwise dissimilarity values between m-row
objects O r and n-column objects O c . An example comes
from Web-document analysis, where the row objects are
m webpages, the columns are n words, and the (dis)
similarity entries are occurrence measures of words in
the webpages [92].
Another important problem involving rectangular
relational data is the analysis of gene-expression data,
where the m rows correspond to genes and the n columns correspond to tissue samples or conditions [93]. In
each case, the row and column objects may be nonintersecting sets, so structural relations between the row (or
column) objects are unknown. Conventional relational
clustering algorithms are ill equipped to deal with rectangular data. Additionally, the definition of a cluster as a
group of similar objects takes on a new meaning. There
can be groups of similar objects that are composed of
only row objects, only column objects, only mixed
objects (often called coclusters), and, finally, clusters in
the union of all of the row and column objects. In other
words, a rectangular dissimilarity matrix comprises four
different clustering problems.
Bezdek et al. [94] developed an approach for visually
assessing cluster tendency for the objects represented by
a rectangular relational data matrix D by assuming that
D is an m # n (sub)matrix containing only m # n of
the (m + n) # (m + n) possible pairwise dissimilarities
34	

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

between objects in O = O r , O c . The full distance matrix
D r , c of O was assumed to be of the form
Dr,c = ;

Dr D
E,
DT Dc

where D T is the transpose of D.
The coVAT coclustering VAT (coVAT) algorithm proposed
in [94] first generates an estimate of D r and D c by interpreting the m rows and n columns of D as feature vectors
representing m row objects and n column objects, respectively, and imputing the missing values using the (Euclidean)
distance between them. The VAT algorithm is then applied
to D r , c to generate the reordering indices of the objects in
O = O r , O c . The coVAT algorithm then unshuffles the row
objects from the column objects based on their indices to
generate individual row and column reordering arrays: RP
(for row permutation) and CP (for column permutation).
The coVAT image is then produced by displaying a (scaled)
u = [du i j] = [d RP(i),CP( j )], for
version of the rectangular matrix D
1 # i # m and 1 # j # n , obtained by reordering the rows
and columns of the original matrix D using the indices
stored in RP and CP, respectively. Just as with VAT, dark
u ) (not along any diagonal, and not necessarily
blocks in I (D
square) suggest the existence of coclusters. The pseudocode for coVAT is given in algorithm S23 in "Pseudocode for
Various Algorithms Belonging to the Visual Assessment of
Tendency Family."
The basic idea of coVAT is embodied in Figure 15,
which is excerpted from [94] (Figures 1 and 3). Figure 15(a)
depicts a set of n = 20 points labeled as row objects (the
circles) and m = 40 points labeled as column objects (the
squares). It may be helpful to imagine the circles as
women (5) and the squares as men (4) who have congregated at the locations shown in five small groups.
Three of the groups are "pure," or unmixed: the two sets
of squares at the top and the centrally located set of circles at the bottom. The lower left and lower right clusters
are mixed groups of circles and squares, that is, coclusters. The spatial coordinates of these points are used only
to compute Euclidean distances between the circles and
squares, yielding a rectangular dissimilarity matrix for
input to coVAT. Evidently, there are three clusters (O r) in
the row objects (ignoring the column objects), four clusters (O c) in the column objects (ignoring the row objects),
five clusters (O r , c) in the union of the row and column
objects, and two mixed coclusters in O r , c .
Figure 15(b)-(e) shows the coVAT images built from the
rectangular input data for each of these four cases. The
numbers of clusters for each of the four clustering problems are seen in the images as dark blocks: diagonal for
the three square subproblems and nondiagonal for the
coclustering problem. In this simple example, coVAT images provide a good visual estimate for possible cluster
structure in all four problems. The coVAT algorithm was
extended to the coiVAT algorithm in [95] by applying a
path-based distance transform used by the iVAT algorithm.



IEEE Systems, Man and Cybernetics Magazine - April 2020

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