Systems, Man & Cybernetics - April 2017 - 24
Differences Between
point in a multidimensional space.
Clustering and Coclustering
That is, the variable for each ele-
Although a data set
Clustering is a commonly used meth-
ment in the vector represents an
od in unsupervised learning. Given
axis in the multidimensional space.
can be very large,
the input data as a matrix, the task of
The dimension of the vector is the
it is possible that
clustering is to partition the matrix
number of elements in the vector.
meaningful and
into clusters along either the row or
However, in image processing and
the column direction, but not both
computer programming, we use
coherent patterns
at the same time [Figure 2(a)-(b)].
the terms like 2-D and 3-D images,
embedded in the
The rows and columns correspond
and 2-D and 3-D data arrays. Here,
to samples and features, respec-
dimension means the number of
data array are much
tively, in pattern recognition ap -
directions or indices in an image or
smaller in size.
pl i cations. Similar to clustering,
data array. In tensor algebra, each
coclustering deals with unlabeled
direction is called a mode, and the
data and performs unsupervised pat-
term multiway data array is often
tern classification. However, there is
used. For simplicity, we will use
a fundamental difference between clustering and coclustering.
the word multidimensional for both vector spaces and
In coclustering, the data are partitioned in both row and col-
data arrays when the context is clear. That is, for real
umn directions. Therefore, for 2-D data, a cluster consists of a
numbers, a multidimensional space corresponds to the set
M
subset of rows (or columns) with all columns (rows) present in
R , where M is the number of elements or dimensionality
each row (column), while a cocluster consists of a subset of
of the vectors in the space. An N-dimensional data array
rows and a subset of columns [Figures 2(c)].
(N-way data array, or Nth order, or N-mode tensor)
There are several other differences between clustering and
belongs to the set R M # M # g # M , where M 1, M 1, ..., M N are
coclustering [3], [16]. In hard clustering, the clusters do not
the numbers of elements in corresponding directions.
have any overlap [Figure 2(a)-(b)]. That is, a sample is
In the previous discussion, we used the term coherent
assigned to one and only one cluster unless it is considered as
pattern, which means that elements in the rows or columns
an outlier and is excluded in classification. In soft clustering, a
of the pattern are correlated and display certain common
sample is assigned to each cluster with a membership value in
properties. We will explain later that a coherent pattern can
fuzzy logic modeling. In coclustering, several coclusters can
be represented as a low-rank matrix or tensor and that our
partially overlap. For example, in Figure 2(c), cocluster 3 over-
tasks of coclustering are to extract one or more coherent
laps with all the other three coclusters. In addition, an element
patterns embedded in the matrix or tensor.
in the data matrix may belong to none, one, or more coclus-
ters. As shown in Figure 1(b), a large matrix may contain
much smaller coclusters, and most elements in the matrix are
irrelevant. We need to identify the row and column indices of
relevant elements to detect the coclusters.
In hard clustering [Figure 2(a)-(b)], we can always
exchange the rows or columns so that each cluster occupies a
contiguous region. However, in coclustering, we may not be
able to display all coclusters as contiguous regions at the same
time. This is illustrated in Figure 2(c). If we exchange rows so
that three regions of cocluster 3 become contiguous, then
cocluster 2 or cocluster 4 would be split into more than one
region. Therefore, it is more difficult to visualize all coclusters
than clusters.
The concept of coclustering was introduced in the 1970s,
but it has gained much attention only recently due to its use-
ful applications to genomic data analysis [1]-[26]. In a
(a)
(b)
(c)
genome, genes usually work together to perform a biological
function. In gene expression data, the samples represent
Figure 1. an example of a cocluster. (a) a large
genes and features conditions. A condition can be a time
matrix that appears to contain just random data.
(b) a cocluster exists in the matrix. the elements of
point, an organ, a species, or a different experiment setup. A
the cocluster can be put together to show a coherent
cocluster represents a subset of genes that coexpress (or are
pattern by rearranging rows and columns in (a).
coregulated) under a subset of conditions. Coclustering anal-
(c) the locations of the elements in the cocluster in
ysis is useful for understanding diseases caused by aberrant
the original matrix in (a), with all elements outside
coexpressed genes.
the cocluster shown as a uniform gray background.
1
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IEEE SyStEmS, man, & CybErnEtICS magazInE A pri l 2017
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