IEEE Systems, Man and Cybernetics Magazine - April 2020 - 24

closely, they are there. The SL hierarchy is easily extracted
by applying a back pass that cuts edges in the iVAT MST
that reordered X 10 . Figure 6(c) includes a dendrogram of
the clusters produced by extracting the SL hierarchy of
clusters this way. We see that the anomalous waveform
(x 4) is reluctant to join the SL hierarchy, coming in as the
last merger on the dendrogram. The evolution of clusters
in Figure 6(c) is clearly visible in the iVAT image. Figure 6(b) and (c) makes the relationship between iVAT and
SL quite transparent: an iVAT image can be interpreted as
a visual front end to SL clustering.
Although SL is a natural clustering algorithm to use
after VAT reordering because it does not require any additional computation, other choices have also been applied
in the literature. Fuzzy clustering algorithms, such as
fuzzy c-means, were used in [36]-[39] after clustering tendency was visually assessed with the VAT algorithm. Hathaway et  al. [36] extend the popular kernelized clustering
algorithms to relational data by proposing a kernelized
form of the non-Euclidean relational fuzzy c-means algorithm using a VAT image as a preliminary step for clustering. Sledge et  al. [37] used VAT images to determine the
number of clusters in the data set before applying their
reformulated fuzzy possibilistic c-means (PCM) algorithm,
which can be applied to A-norm relational data.
Some VAT-based clustering algorithms have also been
proposed in the literature. Prasad and Reddy [40] extended
VAT as a complete clustering method called the visualized
clustering approach (VCA) by using a Khun-Munkres function (a combinatorial optimization algorithm that solves the
assignment problem in polynomial time). VCA can effectively access the number of clusters and discover the clustering
results. An extension of the VCA approach, called the context-aware graph-based VAC, was also proposed by the
same authors. This scheme computes the context-aware dissimilarity matrix (CAD) of the data set using pairwise and
k-nearest neighbor hypergraphs for a set of objects. The
CAD is then used as an input for VAT and the VCA clustering approach. The VAT reordering of the points in a data set
was used as a preprocessing step in [41] to mitigate the
ordering effects for the clustering structures formed by the
fuzzy adaptive resonance theory. This approach is especially
useful when performing offline incremental learning to
improve clustering performance, reduce the number of categories, and decrease variability in the clustering outcome.
Application to Cluster Validity
The third and final step in the cluster-analysis task (after
assessing the clustering tendency before clustering and
applying the clustering algorithm to the data to generate clusters) is to verify the "correctness" of a particular set of
clusters in a given data set, commonly known as the clustervalidity problem. Cluster validity is a widely studied problem. The vast majority of validation methods attempt to
assess the quality of generated clusters by a scalar measure
of partition quality. One problem inherent in this approach is
24	

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

that representing the correctness of particular cluster analysis by a single real number invariably loses much information.
Bezdek and Hathaway [42], [43] took the opposite
approach to the cluster-validity problem and proposed a
VAT-based visual display of fit approach, which they called
visual cluster validity (VCV), using all of the information
produced by the clustering method. Their method was
inspired by the SHADE approach introduced in [44], and it
applies to all prototype generator clustering methods. The
VCV approach retains and organizes the information that
is lost through the massive aggregation of information by
scalar validity measures. VCV, although it provides good
visual tools for validating clustering results for "object
data," that is, data points being represented by feature vectors, cannot be applied to relational data because it needs
object prototype parameters-specifically, the mean vectors-from prototype generator clustering methods, which
are unavailable from relational algorithms.
To solve this problem, Ding and Harrison [45] presented
a relational VCV (RVCV) method based on VCV. RVCV uses
relational prototype parameters, distances, and membership values and follows the steps of VCV; however, it permits the reordering of clusters at the crucial stage
(corresponding to the first stage in VCV), thus permitting
generalization to relational data. RVCV presents relational
cluster validity results in a natural, visual form and fills a
gap in the body of visual cluster-validity theory initiated by
Bezdek and Hathaway.
Gunnersen et al. [46] noticed that both VAT and VCV result
in a visualization of Euclidean distance, either between the
data points themselves or between the data point and the
cluster prototype. They proposed a new visual cluster membership validity (VCMV) algorithm, which extends the VCV
algorithm by visualizing class memberships produced by an
external fuzzy clustering algorithm rather than Euclidean distance. They also combined the VCMV algorithm with self-tuning spectral clustering [47] to create SpecVCMV, which
simultaneously utilizes the advantages of spectral clustering,
addresses the chaining phenomenon [20] found in VAT and
the underlying assumptions found in VCV to create a robust
algorithm that behaves consistently across data sets and
yields more useful results in complex data sets.
Huband and Bezdek proposed another VCV algorithm
they called VCV2 in [48], which compares the partitions
found using any clustering algorithm with the VAT image
of the unlabeled input data. The VCV2 method matches the
VAT RDI image with the transformed VAT-like image of the
(reordered) partition matrix generated by the clustering
algorithm. The stronger the visual match, the more confident we are that the candidate partition is a useful representation of substructure in the data.
Figure 7, taken from [48], shows the VAT image I (R ))
and VCV2 images I (U )) for different values of c for a data
set consisting of five distinct Gaussian clusters. For the
VCV2 images for c = 2 and c = 4 in Figure 7(b) and (c), the
dark blocks differ visibly from those in the VAT image, so



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