IEEE Systems, Man and Cybernetics Magazine - April 2020 - 22

image corresponding to k = 3 is the clearest, with maximum
contrast between the diagonal dark blocks and the nondiagonal white region. The visual suggestion is that the most likely
number of clusters in the data set is k = 3. The pseudocode
for SpecVAT is given in algorithm S3 of "Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family." We will return to SpecVAT in "Applications of
the VAT Family to Different Domains," where we discuss the
extended SpecVAT (E-SpecVAT) algorithm [19].
Another factor that significantly deteriorates the quality of the VAT image is the presence of noise (especially inliers: bridge points between clusters). This shortcoming of
VAT is inherited from the SL algorithm, which is the backbone of VAT reordering. SL has a well-known defect called
the chaining effect [20], which happens when a few points
form a bridge between two clusters, causing an SL to (mistakenly) join the two clusters into one. This chaining effect
makes SL and, therefore, VAT sensitive to noise and bridge
points (inliers).
Excerpted from [21], Figure 5(a) shows a data set W consisting of two well-separated clusters (X shown by green
and brown points) and three inliers (the bridging points
between the clusters) " w 1, w 2, w 3 , added to it (shown in
black). The iVAT image of W is displayed in Figure  5(b),
which weakly suggests the presence of three clusters in the
data set. A solution to this problem was proposed by Kumar
et al. in [21], who used three new approaches to detect and
remove the inliers. The first method, distance modification
with local outlier factor (LOF) (DM LOF ) adjusts the distance
values using LOF scores so that the influence of inliers on
subsequent processing is reduced or eliminated. The other
two methods are data-removal approaches based on LOF and
maximin sampling anomaly scores, DR LOF and DR MM . These
two schemes identify and remove the inliers (data cleansing)
before subsequent processing begins. Figure 5(c)-(e) shows
the modified iVAT images after applying these three methods;
all three images now correctly indicate the presence of two
clusters in the data set.
Extension of VAT to Asymmetric/Incomplete
-Dissimilarity Input Data
VAT and the subsequent RDI enhancement techniques discussed in the previous section require complete information

w2

about the distance matrix between the data points and
require it to be symmetric for their respective algorithms to
work. The symmetric requirement for the distance matrix
assumes that a distance measure between two data points
(say, oi and o j) is commutative, that is, d (o i, o j) = d (o j, o i).
However, for many applications, there are dissimilarity measures, which are not symmetric, such as d (oi, o j) ! d (o j, oi).
This asymmetry of D is common in social network
data, where relationships are not reciprocal, and in other
domains, such as bioinformatics, where the popular basiclocal-alignment search tool similarity [22] between biological sequences, such as the amino acid sequences of
proteins or the nucleotides of DNA or RNA sequences, is
not symmetric. Sampson's monastery data [23] is an example of this type. For these data, Breiger et al. [24] give the
relationship from Bonhaven to Ambrose the value 2, but
the value from Ambrose to Bonhaven in the opposite
direction is 1. An extension of the VAT/iVAT algorithms to
asymmetric matrices [called asymmetric iVAT (asiVAT)]
was proposed by Havens et al. [25]. The extension is based
on replacing the asymmetric input data with its unique
least-squared error approximation by a symmetric matrix.
Given an asymmetric distance matrix D, the asiVAT algorithm first generates a symmetric approximation of D as
D ! (D + D T )/2 before applying VAT/iVAT to the (symmetric) transformed matrix.
Another challenge when determining the clustering tendency of many social network data is that, usually, the
relational matrix is incomplete; that is, the dissimilarity
between many pairs of objects cannot be determined, and
these missing values prohibit the direct application of
VAT/iVAT to such data. For example, for Sampson's monastery data, some relationships are "missing," giving rise to
an incomplete input-distance matrix. According to Wasserman and Faust [26], this is the most common form of
social network data.
The karate-club scenario is another popular real-life
example of social network data with missing values [27].
This famous data set is an asymmetric social network that
links 34 members of a university karate club. There are 156
links and 1,000 missing values. VAT/iVAT RDIs formed from
incomplete data do not offer a very rich interpretation of
cluster structure. To address this problem, Park et  al. [28]

w3
w1
(a)

(b)

(c)

(d)

(e)

Figure 5. The application of iVAT, DM LOF, DR LOF, and DR MM to a data set with inliers: (a) W = X , {w 1, w 2, w 3},

(b) the iVAT image I (D l*) of W, (c) DM LOF of W, (d) DR LOF of W, and (e) DR MM of W. (Source: Kumar et al. [21].)

22	

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IEEE Systems, Man and Cybernetics Magazine - April 2020

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