IEEE Systems, Man and Cybernetics Magazine - April 2023 - 45

total number of samples and the range of targets, respectively,
and the upper bound is always greater than the
lower bound ().yyj
number ()10c
10
-
e =
2 Marin et al. [29] consider PICP,
PINAW, and the deviation from the mid interval as important
quality criteria of PI. They denote the deviation from
the mid interval with e notation. The definition e
can be written as follows:
e =- 2
N t
1
N
|
j=1
where we divide the sum square by N to avoid a large
e for large N. Also, in their article, they take deviation
from the point prediction, and they consider Gaussian and
symmetric uncertainty. Midinterval and the point predictions
are the same for Gaussian and symmetric uncertainty.
All notations in (11) have the same meaning as (9).
According to (11), the value of e is equal to the root
mean square deviation between target and the mid interval.
Kabir et al. [30] consider PICP, PINAW, and the PI normalized
average failure distance (PINAFD) as important quality
criteria of PI. The definition of PINAFD is as follows:
N
PINAFD=
|^1 jj j
j=1
-- -
=
Rc
ct t
h# min yyj
# (| 2jh
_
N
^1-+ec
j
1
where the upper bound is always greater than the lower
bound (),yyj
(9). When ^h the PI does not cover the target. At
that situation, min
2 and cj
11cj
-= ,
tty yjjjj
_ -- i results in the mini,
mum
distance of bounds to the target, that is, the gap
between the nearest bound to the target. Total failure distances
are normalized by the range R and the total number
of PI noncoverage (( )).c1j
Rn
=1
- j
We add a small constant
1.5
1
0.5
-0.5
-1
-1.5
0123
Input (X1, Arbitrary Unit)
Point Prediction
45 6
Prediction Interval Target
Figure 8. Targets, point predictions, and prediction
intervals on the test subset of Dataset-2 and for the
shallow NN.
contains the same meaning as
j
,
i
(12)
Technical Validation: Model Training
for Initial Performance
An initial result on a new dataset encourages many algorithm
developers to develop a better algorithm and report
the result [33]. Moreover, the publicly available code of
the initial algorithm helps many researchers in understanding
the dataset. Therefore, we train shallow NNs
and RVFL networks [34] on the proposed datasets and
provide training details with publicly available codes
(Links https://www.kaggle.com/dipuk0506/shallow
-nn-on-toy-datasets, https://www.kaggle.com/dipuk0506/
rvfl-on-synthetic-dataset).
1.5
1
0.5
-0.5
-1
-1.5
0123
Input (X1, Arbitrary Unit)
Point Prediction
45 6
Prediction Interval Target
Figure 9. Targets, point predictions, and prediction
intervals on the test subset of Dataset-2 and for the
RVFL network.
April 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 45
Data Records
We generate ten synthetic datasets with the help of the
following python script in the Kaggle: https://www.kaggle.com/dipuk0506/toy-dataset-for-regression-and-uq.
This
notebook also presents the characteristics of each
dataset. The " Datasets and Their Characteristics " section
of this article explains all ten generated datasets and
their characteristics. We also upload datasets to GitHub at
the following link: https://github.com/dipuk0506/UQ-Data.
We also upload datasets to Figshare at the following
link: https://figshare.com/articles/dataset/Synthetic_Datasets_for_Numeric_Uncertainty_Quantification/16528650.
j
yyj
+ j
2
(11)
with the denominator. It helps the
computation machine to avoid divide-by-zero error when
all targets are covered.
Researchers have developed many cost functions based
on different combinations of these criteria [28], [29], [30],
[31], [32]. Different cost function works well in different
datasets. Therefore, we compute and present these four
basic quality criteria. Future researchers can easily compute
cost functions from basic criteria.
Target and Predictions (Arbitrary Unit)
Target and Predictions (Arbitrary Unit)

https://www.kaggle.com/dipuk0506/toy-dataset-for-regression-and-uq https://www.kaggle.com/dipuk0506/toy-dataset-for-regression-and-uq https://github.com/dipuk0506/UQ-Data ttps://figshare.com/articles/dataset/Synthetic_Datasets_for_Numeric_Uncertainty_Quantification/16528650 ttps://figshare.com/articles/dataset/Synthetic_Datasets_for_Numeric_Uncertainty_Quantification/16528650 https://www.kaggle.com/dipuk0506/shallow-nn-on-toy-datasets https://www.kaggle.com/dipuk0506/rvfl-on-synthetic-dataset https://www.kaggle.com/dipuk0506/shallow-nn-on-toy-datasets https://www.kaggle.com/dipuk0506/rvfl-on-synthetic-dataset

IEEE Systems, Man and Cybernetics Magazine - April 2023

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