IEEE Computational Intelligence Magazine - August 2021 - 64

Experimental results demonstrate that the DNR
can provide satisfactory and stable forecasting
performances for wind speed time series.
the regression ability. Thus, DNR is more suitable for
solving prediction problems than the DNM. In addition, a
comparison of DNR-SMS and DNR-BP is presented in
Tables IV and V. DNR-SMS is significantly superior to
DNR-BP in most cases, suggesting that the SMS algorithm,
as a global optimization approach that does not easily
become trapped in local minima, can offer enhanced
optimization performance compared to that achieved using
the BP algorithm.
Furthermore, the convergence curves of DNR-SMS for
the wind speed time series are provided in Fig. 4.
Although the maximum number of iterations of the SMS
algorithm was set to 1000, convergence was attained in less
than 200 iterations. Thus, the DNR model trained using
the SMS algorithm could achieve prompt convergence,
indicating that the SMS algorithm is an efficient optimization
tool for real-world prediction problems. In addition,
Figures 5 and 6 show the fit graphs and corresponding
correlation coefficient graphs of DNR-SMS for long-term
and short-term wind speed prediction,
respectively. Figures 5 and 6 indicate that
long-term wind speed time series prediction
is more complex than short-term prediction.
Although the forecast curves of DNR-SMS
closely fit the actual curves for both longterm
and short-term wind speed prediction with different
partition ratios, the long-term prediction performance of
DNR-SMS is higher than the short-term prediction performance,
as indicated by the correlation coefficient graphs. In
addition, for both long-term and short-term wind speed prediction,
the performance of the DNR model trained using
the SMS algorithm was compared with that of DNR models
trained using other state-of-the-art evolutionary algorithms,
namely, a genetic algorithm (GA), particle swarm optimization
(PSO), adaptive differential evolution with an optional
external archive (JADE) [61] and differential evolution with
success-history-based parameter adaptation and linear population
size reduction (L-SHADE) [62]. The experimental
results are presented in Tables VI and VII. The SMS algorithm
notably outperforms the other algorithms. Consequently,
the SMS algorithm is recommended as the main
optimization approach for DNR. These experimental results
demonstrate that the proposed approach can enable satisfactory
and stable forecasting for wind speed time series.
TABLE VI Long-term prediction performance of DNR models trained using different evolution algorithms
for the experimental wind speed time series.
LONG-TERM WIND SPEED TIME SERIES WITH A PARTITION RATIO OF 1:1
ALGORITHM
DNR-GA
DNR-PSO
DNR-JADE
MSE (MEAN ± STD
(OR SD))
1.91E-02±1.55E-03
1.73E-02±6.64E-04
1.74E-02±6.89E-04
DNR-L-SHADE 1.81E-02±2.15E-03
DNR-SMS
1.69E-02±2.04E-04
ALGORITHM
DNR-GA
DNR-PSO
DNR-JADE
MSE (MEAN ± STD
(OR SD))
1.91E-02±1.43E-03
1.74E-02±4.84E-04
1.78E-02±1.01E-03
DNR-L-SHADE 1.85E-02±1.66E-03
DNR-SMS
1.71E-02±1.84E-04
ALGORITHM
DNR-GA
DNR-PSO
DNR-JADE
MSE (MEAN ± STD
(OR SD))
2.46E-02±2.20E-03
2.35E-02±2.80E-03
2.30E-02±1.19E-03
DNR-L-SHADE 2.35E-02±1.49E-03
DNR-SMS
2.21E-02±3.55E-04
P-VALUE
1.01E-06
1.01E-02
9.13E-07
1.46E-02
-
RMSE (MEAN ± STD
(OR SD))
1.38E-01±5.50E-03
1.32E-01±2.50E-03
1.32E-01±2.59E-03
1.34E-01±7.48E-03
1.30E-01±7.85E-04
P-VALUE
9.13E-07
1.50E-02
2.65E-05
1.03E-05
-
RMSE (MEAN ± STD
(OR SD))
1.38E-01±5.03E-03
1.32E-01±1.82E-03
1.34E-01±3.66E-03
1.36E-01±5.93E-03
1.31E-01±7.05E-04
P-VALUE
2.25E-06
2.27E-03
8.25E-04
2.13E-05
-
RMSE (MEAN ± STD
(OR SD))
1.57E-01±6.81E-03
1.53E-01±8.46E-03
1.52E-01±3.91E-03
1.53E-01±4.80E-03
1.49E-01±1.20E-03
P-VALUE
1.01E-06
1.06E-02
9.12E-07
1.54E-02
-
MAE (MEAN ± STD
(OR SD))
1.84E+00±1.16E-01
1.72E+00±1.73E-02
1.73E+00±4.62E-02
1.76E+00±1.08E-01
1.72E+00±1.07E-02
LONG-TERM WIND SPEED TIME SERIES WITH A PARTITION RATIO OF 4:1
P-VALUE
9.13E-07
1.54E-02
2.54E-05
1.13E-05
-
MAE (MEAN ± STD
(OR SD))
1.79E+00±9.18E-02
1.69E+00±1.97E-02
1.72E+00±3.98E-02
1.73E+00±6.58E-02
1.67E+00±9.46E-03
LONG-TERM WIND SPEED TIME SERIES WITH A PARTITION RATIO OF 9:1
P-VALUE
2.25E-06
2.27E-03
9.49E-04
2.13E-05
-
MAE (MEAN ± STD
(OR SD))
1.97E+00±1.05E-01
1.91E+00±6.54E-02
1.90E+00±4.89E-02
1.90E+00±4.06E-02
1.86E+00±1.35E-02
P-VALUE
3.66E-06
5.51E-05
1.86E-03
6.52E-05
-
R (MEAN ± STD
(OR SD))
4.34E-01±6.40E-02
4.66E-01±5.01E-02
4.82E-01±2.57E-02
4.67E-01±3.16E-02
4.94E-01±9.41E-03
P-VALUE
1.01E-06
6.67E-04
7.13E-06
7.13E-06
-
R (MEAN ± STD
(OR SD))
4.66E-01±5.43E-02
5.06E-01±1.40E-02
5.00E-01±3.17E-02
4.81E-01±4.08E-02
5.14E-01±6.22E-03
P-VALUE
4.89E-06
3.48E-01
9.12E-07
8.41E-02
-
R (MEAN ± STD
(OR SD))
5.02E-01±5.98E-02
5.46E-01±1.68E-02
5.46E-01±2.10E-02
5.27E-01±6.00E-02
5.53E-01±5.73E-03
64 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2021
IEEE Computational Intelligence Magazine - August 2021

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