IEEE Systems, Man and Cybernetics Magazine - April 2023 - 41

the direction and magnitude of the noise. The third variable
is random to output. The uncertainty is heteroscedastic,
and changes over X2. As the random number generator
provides uniform distribution over the range, the uncertainty
is symmetric. The noise magnitude
( )Me
for this
dataset is 0.5, representing medium uncertainty. There
exists a trigonometric relationship between inputs (X1,
and X2) and the output. Figure 2(a) represents a sine wave
with an uncertainty of magnitude (0.5). In Figure 2(b) and
for X2 close to zero, the target range is [−1, /],1122+
as
the uncertainty range is ([0, 0.5 × sin( /4r )] = [0, /])12 2
for X20= and the deterministic portion
has a range of [0, 1] based on the value of
X1. In Figure 2(c), the target is random
toward the value of X3. However, the target
density is higher in the range [−1, 1].
Dataset-3 has more samples near the output
range [−1, 1], which is the characteristic
of the dataset and that has no relation
to the value of X3. Even if we put another
related or unrelated input on the x-axis of
this plot, there will be no target out of the
[−1.5, 1.5] range.
Dataset-4 has three inputs and one out1
0.5
-0.5
put.
The first variable is responsible for
both the prediction and uncertainty. The
second variable is also partly responsible
for the uncertainty. Figure 3 presents relationships
between targets and input variables.
A heteroscedastic and asymmetric
uncertainty is observable in Figure 3(a).
As the second variable is partly responsible
for the uncertainty, some minor
-1
024
Input (X1, Arbitrary Unit)
(a)
6 024
Input (X1, Arbitrary Unit)
(b)
Figure 1. Relationship between target (T) and input (X1) on
(a) Dataset-1 and (b) Dataset-2.
6
0.5
-0.5
-1
changes are observable toward the x-axis of Figure 3(b).
As X3 is random to output, no observable difference is
found in the distribution of targets toward the x-axis
of Figure 3(c).
Dataset-5 has three inputs and one output. M 4=e
for
this dataset. The relationship between inputs and the output
is the combination of triangular and rectangular signal
equations. Figure 4 presents the distributions of
targets (T) for values of (X1, X2, and X3). In Dataset-5, the
magnitude of uncertainty becomes zero and four when
Xround X22 "
^h,
is negative and positive, respectively.
Data-1 Training Set
1.5
1
Data-2 Training Set
Training Set
Training Set
1.5
0.5
1
-0.5
-1
-1.5
024
Input (X1, Arbitrary Unit)
(a)
1.5
0.5
1
-0.5
-1
-1.5
60 24
Input (X2, Arbitrary Unit)
(b)
6
1.5
0.5
1
-0.5
-1
-1.5
02 4
Input (X3, Arbitrary Unit)
(c)
6
Training Set
Figure 2. Relationship between target (T) and inputs (X1, X2, and X3) on Dataset-3. (a) The first input (X1)
highly influences the point prediction. (b) The second input (X2) highly influences the level of uncertainty.
(c) The target is not related to the third input (X3).
April 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 41
Target (T, Arbitrary Unit)
Target (T, Arbitrary Unit)
Target (T, Arbitrary Unit)
Target (T, Arbitrary Unit)
Target (T, Arbitrary Unit)
IEEE Systems, Man and Cybernetics Magazine - April 2023

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