## 4 Discussion
### Significance Test between Forecasting Values and Actual Data  
To evaluate the significance level of forecasting errors between the proposed combined model and the other models, a classical hypothesis test method, the Diebold-Mariano Test [55], was used to measure the significance of prediction errors for different models. Theoretical support for this model is as follows. Given a certain probability of coming to a wrong conclusion \(\alpha\), the original hypothesis \(H_{0}\) will not be rejected as long as the forecasting capacity of the proposed model possesses no evident distinction when contrasted with the comparison model; if the inverse is true, \(H_{0}\) will be rejected and \(H_{1}\) accepted. The hypothetical form is:  
\[ \begin{array}{l}H_{0}:E[L(error^{1})]=E[L(error^{2})]\\ H_{1}:E[L(error^{1})]\
eq E[L(error^{2})]\end{array}  \tag{1}\]  
where \(L\) represents the loss function for forecasting errors, and \(error^{1}\) and \(error^{2}\) are the forecasting errors of the combined model with five sub-models and those of other combined models and single models, respectively. Further, the DM statistical magnitude can be expressed by:  
\[\text{DM}=\frac{\sum_{i=1}^{n}\left(L(error^{1})-L(error^{2})\right)/n}{\sqrt {S^{2}/n}}\mathbf{s}^{2} \tag{2}\]  
In the above equation, \(S^{2}\) is the estimated variance of \(\mathbf{d}=L(error^{1})-L(error^{2})\). After obtaining the calculated DM value, a comparison between the DM statistics and a critical value \(Z_{\alpha/2}\) is conducted. Once the DM statistic is greater than \(Z_{\alpha/2}\) or less than \(-Z_{\alpha/2}\), the original hypothesis will be rejected. Moreover, it may be concluded that there is an evident distinction between our developed model and the compared model. The Wilcoxon rank-sum test [56; 57] is a nonparametric test that can be used to determine whether two independent samples have been selected from populations with the same distribution. If \(h=1\), the null hypothesis that there is no difference between two samples at the 5% significance level is rejected. If \(h=0\), there is a failure to reject the null hypothesis at the 5% significance level. The results of the Diebold-Mariano test and Wilcoxon rank-sum test for wind speed and wind power forecasting are shown in Table 8. The following conclusions were made: First, the test results of DM test for wind speed showed little variation in the forecasting error among the different combined models, and the Wilcoxon rank-sum test results showedno difference between the forecasting results of each model and the actual wind speed at the 5% significance level. When the combined models were compared, the combined model with five sub-models was found to be evidently different from the other models at the 1% significance level for wind power forecasting. However, the DM test between the combined model and the sub-model revealed that some sub-models obtained DM values that were higher than the threshold at a 5% significance level. For example, in the first season, the SSAWD-Elman sub-model had a DM value of 1.6448 for wind power forecasting. The test results for the other sub-models are shown in Table A5. The Wilcoxon rank-sum test result of each combined model showed no differences between the actual wind power values and the forecasting results of each combined model. Meanwhile, the Wilcoxon rank-sum test results presented in Table A5 show that some sub-models rejected the null hypothesis at the 5% significance level, which indicates that the forecasting results of some sub-models differed from the actual data. For the wind speed and wind energy forecasting error test results, there was little variation among the combined models at the 1% significance level, while there were greater differences between the combined models and the sub-models. Comparing the actual data with those forecast by each model, it was found that the wind speed data forecast by each sub-model and the real data represented the same sample at the 5% significance level.