IEEE Systems, Man and Cybernetics Magazine - April 2020 - 33

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 13. The siVAT and siVAT+ RDIs (D l*n) and runtimes for each of the data sets (parameter values: k l = 50

and n = 500): (a) GM1: siVAT, 725 s; (b) GM2: siVAT, 738 s; (c) KDD: siVAT, 76.5 s; (d) forest: siVAT, 10.8 s; (e) census:
siVAT, 65.5 s; (f) BigCross: siVAT, 2,504 s; (g) GM1: siVAT+, 49 s; (h) GM2: siVAT+, 48 s; (i) KDD: siVAT+, 6 s; (j) forest:
siVAT+, 0.8 s; (k) census: siVAT+, 8.1 s; and (l) BigCross: siVAT+, 44.7 s.

processed. The order of the dark blocks along the diagonal may be different, but this is unimportant; their
number and size are almost the same.
◆◆ siVAT+ produces this information 10-50 times faster
than siVAT.
The pseudocode for siVAT+ is given in algorithm S21 in
"Pseudocode for Various Algorithms Belonging to the Visual Assessment of Tendency Family."
However, like sVAT, siVAT+ only suggests the number of
clusters in the data set; it cannot find the actual partition.
To find clusters in large volumes of high-dimensional data
while simultaneously overcoming both the "curse of dimensionality" problem due to high dimensions and scalability
problems due to large sample size, Rathore et al. [91] proposed a new fast clustering algorithm called fast ensemble
siVAT (FensiVAT), which is a hybrid, ensemble-based clustering algorithm that uses fast data-space reduction and an
intelligent sampling strategy.
FensiVAT aggregates multiple distance matrices, computed in a lower-dimensional space, to obtain an approximate iVAT image in a fast and efficient manner, which
provides visual evidence about the number of clusters to
seek in the original data set. MMRS sampling picks distinguished objects from the data set; therefore, it requires relatively very few samples, compared with random sampling,
to yield a diverse subset of the big data that represents the
cluster structure in the original (big) data set. To be computationally efficient, FensiVAT performs a near-MMRS
sampling, which is done in a randomly projected down
space. The samples are then lifted by transferring the sample indices to the (input) up space. An ensemble of Q n # n
distance matrices computed from Q sets of near-MMRS
samples in the up space is then used to obtain a reliable
output iVAT image, which visually suggests the number of
clusters, k, in the data set.
	

(a)

(b)

(c)

(d)

Figure 14. The ClusiVAT and FensiVAT images and
runtimes for GM1 and GM2 data sets (parameter
values: k l = 9, n = 205 for GM1 and k l = 12, n = 206 for
GM2; down-space dimensions for FensiVAT: q = 20 for
GM1 and q = 50 for GM2): (a) GM1: clusiVAT, 20.1 s;
(b) GM1: FensiVAT, 0.35 s; (c) GM2: clusiVAT, 21.3 s; and
(d) GM2: FensiVAT, 0.84 s.

The iVAT images obtained using clusiVAT and FensiVAT
for the GM1 and GM2 data sets described previously are
included in Figure 14 (see Figure 4 in [91]). Figure 14(a)
and  (b) shows that clusiVAT and FensiVAT exhibit three
(well-separated) dark blocks along the diagonal, suggesting k = 3 for GM1. The ground truth partition for GM2 is
not compact and well separated because of overlapping
clusters; for this data set, FensiVAT produces three dark
Ap ri l 2020

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE	

33



IEEE Systems, Man and Cybernetics Magazine - April 2020

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