IEEE Systems, Man and Cybernetics Magazine - April 2020 - 25

these two candidates are rejected. The VCV2 images in Figure 7(e) and (f) have roughly the same visual structure as
the VAT image I (R )), the principal difference being the
second block in the image for c = 6, which is somewhat
lighter in color, indicating that a very small subset of
points (possibly a singleton) was chosen as the sixth cluster. Based on visual image similarity, the VCV2 algorithm
clearly suggests that c = 5 is the preferred solution.
A few papers have tried to understand the relationship
between the VAT-based VCV indices described previously
and the traditional numeric cluster validity indices. Havens
et al. [49] addressed the relationship between the VAT algorithm and Dunn's cluster validity index [50]. Their experiments on a variety of data sets demonstrate that the
effectiveness of VAT in showing cluster tendency is strongly
related to Dunn's index, which provides a measure of contrast between the dark diagonal blocks in the VAT RDI and
the bright background regions. Similarly, a framework that
couples the possibilistic Rand index and the VAT algorithm
to estimate the number of clusters and identify coincident
clusters found by the PCM algorithm was discussed in [38].
Automating VAT/iVAT for Clustering
Tendency Assessment
The RDI generated by the VAT algorithm provides an excellent tool to (visually) interpret possible cluster structure in
a data set. A human observer can (sometimes) count the

number of dark blocks along the diagonal of a VAT image
to get an estimate for k, the number of clusters for which
to look. For data sets with compact, well-separated clusters, the dark blocks along the diagonal of the VAT image
are clear and easily countable, but as the data become
more and more mixed, the VAT image will degrade considerably. Although humans can usually deduce the suggested
number of clusters from a VAT image in all but the most
incorrigible data sets, different humans may see different
values, especially when the clusters have significant overlap or strange geometries, both of which lead to VAT images with nondistinct diagonal block boundaries.
Due to these shortcomings, visual methods, such as VAT,
have been criticized for being subjective and requiring human
input, which becomes impractical and seems somewhat
archaic in the current climate of automation created by
advances in artificial intelligence. Several groups have tackled this problem and designed methods to automatically
detect the number of clusters without requiring human input
to interpret a VAT/iVAT RDI. These techniques, classified by
the technique used in them to automatically determine k
from the VAT RDI, are discussed next.
Automatic Assessment Based on Image
Processing of the VAT RDI
Keller and Sledge [51] were the first to develop an algorithm
that takes the VAT RDI as an input to determine the degree

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Figure 7. The VAT I (R *) and VCV2 images I (U *) for different values of c for a five-cluster data set: (a) input
data with five clusters; (b) a VAT-ordered image of I (R *); and VCV2 images of (c)  I (U *) for c = 2, (d)  I (U *) for
c = 4, (e)  I (U *) for c = 5, and (f)  I (U *) for c = 6.

	

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IEEE Systems, Man and Cybernetics Magazine - April 2020

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