IEEE Systems, Man and Cybernetics Magazine - April 2020 - 32

120

1,000

100
Time (s)

1,500

500
0
-500 0 500
(a)

(b)

inc−VAT + inc−iVAT
VAT + iVAT

80
60
40
20

(c)

1,500

0
50
0
1,
00
0
1,
50
0
2,
00
0
2,
50
0
3,
00
0
3,
50
0
4,
00
0
4,
50
0
5,
00
0

0

1,000

Number of Data Points (n)
(a)

500
0
-500 0 500
(d)

120
(e)

Time (s)

1,500
1,000
500

80
60
40
20

0

1,500

0

0
50
0

0

00

1,

0

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1,

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2,

0

00
3,

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3,

0

00

4,

50

4,

00

(i)

0

0
(h)

5,

-500 0 500
(g)

Number of Data Points (n)
(b)

1,000

Figure 12. Time comparisons of the VAT, iVAT, inc-VAT,

500
0
-500 0 500
(j)

(k)

(l)

1,500
1,000
500
0
-500 0 500
(m)

(n)

(o)

Figure 11. The 2D data scatterplots and

(incrementally built) inc-VAT and inc-iVAT images of X
at n = 1,300; 2,300; 2,700; 4,300; and 5,000: (a) X1,300,
(b) inc-VAT (X1,300), (c) inc-iVAT (X1,300), (d) X2,300,
(e) inc-VAT (X2,300), (f) inc-iVAT (X2,300), (g) X2,700, (h)
inc-VAT (X2,700), (i) inc-iVAT (X2,700), (j) X4,300, (k) incVAT (X4,300), (l) inc-iVAT (X4,300), (m) X5,000, (n) inc-VAT
(X5,000), and (o) inc-iVAT (X5,000).

Clustering High-Dimensional Data
The clusiVAT algorithm was proven to be useful in determining the cluster structure of big data with a large
number of data points (high volume). However, these
approaches are time consuming when the data are large in
both the number of samples (N ) and the number of
dimensions ( p). To tackle this issue, Rathore et  al. [78]
introduced a fast, approximate, scalable iVAT algorithm
called siVAT+, which combines random sampling with the
original sampling scheme (MMRS) of siVAT to reduce the
computational cost of siVAT for large volumes of highdimensional data.
32	

dec−VAT + dec−iVAT
VAT + iVAT

100

(f)

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2020

inc-iVAT, dec-VAT, and dec-iVAT algorithms for the 5,000point 2D data set: (a) VAT + iVAT versus inc-VAT + inc-iVAT
and (b) VAT + iVAT versus dec-VAT + dec-iVAT.

The modification of the original sampling scheme is
called MMRS+, and it randomly selects an object from the
entire data set and designates it as the first distinguished
object. A second distinguished object is a maximin object
from a randomly generated sample of the big data set. A new
random subset of the data set is chosen, and a maximin
sample is chosen from it. Sampling this way continues until
the desired number of distinguished objects are obtained.
The MMRS+ samples are then used to build an approximate
siVAT image for the very large, high-dimensional data,
which provides visual evidence about the potential number
of clusters to seek in the big input data set.
To demonstrate siVAT+, Figure 13 (see Figure 2 in [78])
shows the comparison of RDI and computation times
between siVAT and siVAT+ on two synthetic data sets, Gaussian mixture (GM) 1 and GM2, each with N = 1,000,000 data
points in p = 1,000 dimensions, constructed by drawing
labeled samples from a mixture of k = 3 Gaussian distributions. Data set GM1 is a well-separated Gaussian mixture,
whereas GM2 has overlapping Gaussian clusters. Experiments were also performed on four publicly available real,
high-dimensional (large-volume) data sets: knowledge discovery from data (KDD)-99 cup data [87], forest-cover type data
[88], U.S. Census 1990 data [89], and BigCross data [90]. The
conclusions that can be made from Figure 13 are as follows.
◆◆ siVAT and siVAT+ contain essentially the same visual
information about cluster structure in the samples



IEEE Systems, Man and Cybernetics Magazine - April 2020

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