IEEE Systems, Man and Cybernetics Magazine - April 2020 - 26

of clustering (not exactly the number of clusters) automatically. They generated a series of threshold plots by summing all of the black pixels of the VAT RDI for different
threshold values (representing the probability of occurrence of black pixels in the VAT image). These threshold
plots were then combined to generate a clustering score for
the particular data set. Later works by Sledge et al. [52], [53]
automatically (algorithmically) determined k, the number
of clusters from the VAT image. They experimented on a
variety of ineffective techniques before proposing a cluster
count extraction (CCE) algorithm, which implemented frequency domain correlation and feature recognition to
count the number of clusters automatically.
Inspired by the CCE technique, Wang and Leckie [54] proposed a new method called dark block extraction (DBE) for
automatically estimating the number of clusters from the
VAT RDI using several image- and signal-processing techniques. The main steps of the DBE algorithm are 1) dissimilarity transformation and image segmentation, 2) directional
morphological filtering of the binary image, 3)  distance
transform and diagonal projection of the filtered image, and
4) detection of major peaks and valleys in the projection signal. The pseudocode for the DBE algorithm is given in algorithm S5 of "Pseudocode for Various Algorithms Belonging
to the Visual Assessment of Tendency Family."
Figure 8 shows the output of different steps of DBE when
applied to a data set consisting of five Gaussian clusters in a
2D space. The views of Figure 8 are self-descriptive; more
information about them can be found in [54]. Prabhu and
Duraiswamy [55] extended DBE to a new method called
enhanced DBE, which relies on E-VAT [56], DBE, distance
measures for diverse type of attributes, and basic image- and
signal-processing techniques.
The iVAT algorithm proposed in [14] and [15] generates
much sharper RDIs than VAT; therefore, it is easier to infer
the number of clusters from the iVAT image manually or
automatically. To this end, Wang et  al. [14] proposed an
automated VAT (aVAT) algorithm to automatically determine the number of clusters suggested by the iVAT RDI
using common image-processing techniques (binarization,
edge map, dissimilarity transform, and so on). A comparison of aVAT with CCE and DBE showed that aVAT was
somewhat better then CCE and DBE since it explicitly
showed the number of clusters, positions, and ranges of
each block (or clusters) within the image itself.
Automatic Assessment Based on
Spectral Graph Theory
Another category of techniques for automatically extracting the number of clusters in a data set using VAT without
any human intervention are based on spectral graph theory. The SpecVAT algorithm discussed in the "Improvements in the VAT RDI for Complex Cluster Structure and
Noisy Data" section belong to this category. The detailed
implementation of SpecVAT, including an example showing
how to obtain the best value of k without human
26	

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

intervention, is given in the "Improvements in the VAT RDI
for Complex Cluster Structure and Noisy Data" section
(see Figure 4 and the corresponding discussion).
Automatic Assessment Based on
Off-Diagonal Dissimilarity
The off-diagonal values of the reordered dissimilarity
matrix have been used by some researchers to automatically determine the cluster structure of a data set. Hu and
Hathaway [57], [58] introduced the concept of tendency
curves to identify possible diagonal blocks in the RDI by
using various averages of values of the w subdiagonal
band (excluding the diagonal) of the RDI, which are stored
as vectors and displayed as curves. The possible cluster
borders are then seen as the high-low patterns on the tendency curves, which can be caught not only by human eyes
but also by the computer using a suitable threshold. An
enhanced technique called visual assessment of cluster
tendency using diagonal tracing (VATdt) was proposed in
[59], which extensively experimented on a particular type
of tendency curve called a d-curve and concluded that a
d-curve is effective in determining the number of clusters
in a data set, even in those cases where the human eyes see
no structure from the visual outputs of VAT.
Automatic Assessment Based on
Optimization Techniques
Some optimization-based techniques have also been proposed to automatically determine k from the VAT RDI.
Havens et al. [60] defined an objective function by combining
the measures of contrast and edginess of the VAT RDI to recognize the blocky structure in reordered data. The objective
function is optimized when the boundaries in the VAT RDI
are matched by those in an aligned partition of the objects.
The authors proved that the set of aligned partitions is exponentially smaller than the set of all partitions that must be
searched if clusters are sought in the raw data, thus making
the optimization problem tractable. They propose an extraction of clusters from the ordered dissimilarity data (CLODD)
algorithm, which uses particle-swarm optimization to find
optimal clusters from the aligned partitions.
Another technique, suggested by Pakhira and Dutta [61],
proposes using genetic algorithms to automatically determine the number of clusters from VAT images. Similar to
Havens et  al. [60], Pakhira and Dutta generated a set of
aligned partitions as candidate cluster memberships and
used a variable-string-length genetic algorithm, where chromosomes are real-coded with cluster centers, and Dunn's
cluster validity index is used as the objective function. A
related article by the same authors [62] proposed optimizing
another cluster validity measure, the Pakhira, Bandyopadhyay, and Maulik (PBM) index [63], instead of Dunn's index
as proposed in [61], using the same variable-string-length
genetic algorithm optimization technique. A better and less
computationally expensive approach to computing the PBM
index value directly from the VAT image based on a robust



IEEE Systems, Man and Cybernetics Magazine - April 2020

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