IEEE Systems, Man and Cybernetics Magazine - April 2023 - 46

Table 1. Performance of trained shallow NN models. Models are trained on
the training subset, validated in the validation dataset in each 200 epochs,
and then tested on the test dataset.
Size of NN
Data
Data-1
Data-2
Data-3
Data-4
Data-5
Data-6
Data-7
Data-8
Data-9
Data-10
Number
of layers
2
2
2
2
2
2
2
2
2
2
Neurons
per layer
300
500
500
500
500
500
700
1200
1000
400
MSE
1.039e-3
1.072e-2
1.785e-2
9.368e0
1.030e0
2.164e-2
2.816e-2
4.656e-1
2.803e-2
1.912e-2
We train shallow NNs and RVFL networks on the train
Dataset and validate them in a 200-epoch interval. Before
training these models, we normalize the dataset. According
to both theory and our observation, the dataset normalization
statistically brings uniform performances over
the input range and improves the overall performance
[35], [36]. Figure 8 visualizes a representative performance
of trained NN models that are investigated in this article.
Figure 9 visualizes a representative performance of trained
PICP (%)
96.85
94.65
95.05
93.95
92.75
99.95
94.95
100
96
86.5
PINAW
5.984e-2
1.433e-1
1.766e-1
2.752e-1
8.972e-1
2.307e-1
2.436e-1
1.219e0
3.037e-1
1.608e-1
|| e ||
3.223e-2
1.035e-1
1.336e-1
3.060e0
1.015e0
1.471e-1
1.678e-1
6.823e-1
1.674e-1
1.383e-1
PINAFD
2.964e-3
3.018e-3
4.851e-3
1.384e-2
1.181e-1
7.563e-3
1.045e-3
2.950e-2
RVFL models that are investigated in this article. We have
drawn similar plots for all datasets in Kaggle notebooks.
Since all plots are similar, the article does not present
other plots. We present details of NNs and performances in
each dataset in Table 1. We present RVFL networks and
their performance in Table 2. We train networks for point
prediction. Then, we compute the upper and lower bounds
with a Gaussian and homoscedastic uncertainty assumption
[9], [37]. In the Gaussian and homoscedastic uncertainty
assumption, the PI is presented as
zz ,
6 -+ @ . In this representation, n
nvnv
Table 2. Performance of trained shallow RVFL models.
Models are trained on the training subset, validated
in the validation dataset in each 200 epochs, and then
tested on the test dataset.
Data
Data-1
Data-2
Data-3
Data-4
Data-5
Data-6
Data-7
Data-8
Data-9
Data-10
Neurons
1000
1000
1000
1000
2000
1000
1000
1000
1000
1000
MSE
1.255e-3
2.094e-2
2.19e-1
1.418e1
1.114e0
2.159e-2
3.416e-2
4.478e-1
1.943e-2
4.485e-3
PICP (%)
95.85
93.5
97.1
88.95
92.65
100
95.45
100
92
96.5
PINAW
6.633e-2
2.009e-1
6.187e-1
2.731e-1
9.316e-1
2.305e-1
2.930-1
1.196e0
2.529e-1
1.211e-1
46 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE April 2023
|| e ||
3.543e-2
1.447e-1
4.680e-1
3.766e0
1.055e0
1.469e-1
1.848e-1
6.692e-1
1.393e-1
6.197e-2
PINAFD
9.827e-4
2.009e-3
2.462e-2
2.006e-2
1.235e-1
2.475e-2
3.832e-2
3.537e-3
is the mean or the mean of expected outcomes,
v is the variance, and z = 1.96 for
95% PICP.
Point prediction and overall standard
deviation values compute uncertainty
bounds for 95% coverage probability. Some
of these proposed datasets have heteroscedasticity
and asymmetry. We consider
homoscedastic and symmetric uncertainty.
Moreover, we apply simple and lightweight
networks. There exist ample opportunities
to improve results with the consideration
of heteroscedasticity, asymmetry, deeper
NN, and so on. Moreover, the developer of
novel models may consider these initial
results for comparison.
Usage Notes
In this study, we propose ten different synthetic
datasets for numeric UQ. One can
investigate the strength of the model in
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