Systems, Man & Cybernetics - April 2017 - 29

Therefore, we find that a cocluster
◆◆ analysis of chemical compo-
exists in {1, 3, 4, 6, 8} # {1, 4, 5}.
nents in food to determine a
Coclustering provides
subset of components (nutri-
In this example, the column-pair-
ents or toxicants) in a subset of
based method works well. For a
classifications
food types
matrix with M 2 columns, there are
simultaneously
in
◆◆ analysis of web documents to
M 2 ( M 2 - 1) column pairs, and to
all directions of a
combine the results from all these
search for a subset of documents
column pairs can be complicated.
that contain a subset of keywords
multidimensional
One may think many column pairs
in a subset of locations
data array.
are redundant. In the example here,
◆◆ analysis of training data in
we can obtain the column indices {1,
machine learning to extract a
4, 5} from column pairs 1-4 and 1-5
subset of features that are rele-
without considering 4-5. However,
vant to a subset of features.
in more complex cases, due to noisy and multiple and over-
In [8], we analyzed foreign exchange data over 120
lapping coclusters, the column pairs may not produce the
months and found currencies with correlations in some
same results; therefore, it is difficult to determine which pairs
but not all time periods. Some of the coclusters reveal
are redundant.
that several currencies had similar rapid movement pat-
We can consider all columns at the same time using
terns in the early stage of the Asian financial crisis in late
HDSVS. First, we compute the SVD of the input matrix. The
1990s. In [26], we used coclustering to select Gabor wave-
singular values of the example in Figure 6(a) are 358.56, 89.39,
let features of images for facial expression classification.
54.81, 23.69, and 12.84. It is reasonable to retain the first three
There are thousands and even tens of thousands of fea-
large singular values and remove the two smallest ones to fil-
tures to compute for a full set of Gabor wavelets with dif-
ter out noise. Although the cocluster has a rank of two, we
ferent scales and orientations in different locations of a
usually obtain better results by retaining slightly more singu-
face image. With the type of facial expression in one
lar values. In Figure 6(d), we consider singular vectors u 1, u 2,
direction and the Gabor wavelet feature in another direc-
tion, we performed cocluster analysis of the feature data.
and u 3 corresponding to the three retained singular values. In
From the coclusters, we can retain relevant features for a
the u 1-u 2-u 3 space, we detect a hyperplane close to
facial expression and discard all other features. With
^u 11, u 12, u 13 h, ^u 31, u 32, u 33 h , ^u 41, u 42, u 43 h, ^u 61, u 62, u 63 h, and
only 20% of the features selected this way, we have
^u 81, u 82, u 83 h. Similarly, in the v 1-v 2-v 3 space, we detect a hyper-
obtained an even higher classification rate than that from
plane close to ^v 11, v 12, v 13 h, ^v 41, v 42, v 43 h, and ^v 51, v 52, v 53 h [Fig-
the entire set of features. The reason for the improved
ure 6(e)]. Therefore, a cocluster exists in {1, 3, 4, 6, 8} # {1, 4, 5}.
performance is that, through coclustering, we can
In HDSVS, the removal of small singular values can reduce
remove irrelevant and noisy features and enhance the
the number of variables for hyperplane detection. If the
feature quality. It is interesting that a reasonable accura-
matrix is large and the coclusters are small in size, the reduc-
cy can still be achieved even when we retain only 0.9% of
tion will be significant. Hyperplane detection is a key step in
the original features.
HDSVS. This can be done using the HT or the linear grouping
This example of facial expression feature selection
algorithm [25].
is especially encouraging. In traditional machine-learn-
Although the matrix in this example only has the size of
ing and pattern-recognition methods, all features are
eight by five for the purpose of illustration, the computation-
used to build a classifier. In practical applications, if
al procedure for a larger data size is the same. In our work on
our input data contain a large number of features, a sub-
gene expression data analysis, many experiments were car-
set of features may be useful for only a subset of class-
ried out with large data sets, and more examples can be
es, and another subset of features may be useful for
found in [6], [7], [9], [14], [15], [19], and [25]. To improve com-
just another subset of classes. In these cases, feature
putational speed, graphics processing units and specially
selection is a combinatorial problem. Coclustering pro-
designed hardware, such as field programmable gate arrays,
vides a systematic and robust method to deal with big
can be employed [22], [23].
data sets and select relevant and important features.
This procedure can be used to filter out noise, improve
Discussions and Conclusions
classification performance, and reduce the computing
Coclustering is useful for genomic data analysis. In add-
time significantly.
ition, it has many other applications [3], [16], some of
In summary, coclustering provides classifications
which are
simultaneously
in all directions of a multidimensional
◆◆ analysis of consumer spending patterns to discover a sub-
data array. It may be used to detect coherent patterns
set of consumers who like to purchase products from a
that are embedded in a large matrix or tensor and con-
subset of companies
tain a subset of elements in each direction. It can be
◆◆ analysis of financial data to find a subset of sectors that have
performed effectively in singular-vector spaces based
similar economic growth patterns in a subset of regions
Ap ri l 2017

IEEE SyStEmS, man, & CybErnEtICS magazInE

29



Table of Contents for the Digital Edition of Systems, Man & Cybernetics - April 2017

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