## 4 Experimental Results
### The Effect of the Combination of Submodule Inputs on the DNFS
It can be seen obviously that the input variables with high correlation values were relatively concentrated both sequentially and in the random cases on Dataset No. 1 and No. 3. On the contrary, the inputs with high correlation values were more scattered in the random case than that of the sequential case on Dataset No.2, No. 4, and No. 5. Meanwhile, the results of Tables 8-12 indicate that DNFS5-2-Random achieved better performance than DNFS5-2 on Dataset No. 2, No. 4, and No. 5. Compared with DNFS5-2, DNFS5-2-Random decreased by 38.8%, 27.7%, and 38.5% the MAE, RMSE, and SMAPE for Test Dataset No. 2, while it decreased them by 19.9%, 16.7%, and 18.9% for Test Dataset No. 4, and by 72.4%, 71.8%, and 57.7% for Test Dataset No. 5, respectively. However, their performance was not significantly differences on the other two datasets. Through the analysis of the results obtained, it can be concluded that the features that had a higher correlation with the output should be dispersed into each submodule, so that the performance of each submodule can be balanced and the overall performance improved. Figure 5: Correlation analysis comparison of sequential inputs with random inputs. (**a**) The random inputs were obtained from Dataset No. 1. (**b**) The random inputs were obtained from Dataset No. 2. (**c**) The random inputs were obtained from Dataset No. 3. (**d**) The random inputs were obtained from Dataset No. 4. (**e**) The random inputs were obtained from Dataset No. 5. LSTM, the average MAE decreased by 40.1%, 36.7%, 43.3%, and 35.8%, respectively, the average RMSE decreased by 34.1%, 36.4%, 36.8%, and 27.7%, respectively, and the average SMAPE decreased by 40.4%, 39.2%, 43.9%, and 36.8% respectively. The performance of DNFS5-2 was second only to DNFS5-2-Random. Compared to MBGD-RDA, RBF, GRNN, and LSTM, the average MAE of DNFS5-2 was reduced by 21.4%, 16.8%, 25.6%, and 15.6%, respectively, the average RMSE decreased by 14.1%, 17.2%, 17.6%, and 5.8%, respectively, and the average SMAPE decreased by 18.9%, 17.3%, 23.7%, and 14.1%, respectively. Hence, there is no doubt that the DNFS outperformed the other algorithms in prediction accuracy. Meanwhile, Tables 8-12 show that DNFS5-2 had fewer parameters and a simpler structure than RBF, GRNN, and LSTM, which validated that DNFS5-2 has less complexity and higher interpretability. On the contrary, the parameters of the three machine learning models increased exponentially with the dimensionality of the input, which led to their interpretability being very poor. In addition, MBGD-RDA constrained the maximum input dimensionality five, so that its parameters were equal to \(212(5*2*2+25*(5+1)=212)\), while the MF was Gaussian and the number for each input domain was two. On all five datasets, the parameters of MBGD-RDA were far fewer than the other algorithms, resulting in the AIC being minimum. However, the prediction accuracy of MBGD-RDA was not optimal, which was probably because of the loss of important features due to PCA, so the performance was greatly affected. The average computational time that DNFS5-2, MBGD-RDA, RBF, GRNN, and LSTM spent on the five datasets was 52.99 s, 50.39 s, 13.31 s, 1.23 s, and 18.22 s respectively. Obviously, GRNN was the most efficient method among all the algorithms. Although the efficiency of DNFS5-2 was relatively poor, it could achieve the best performance within a valid period of time, which could not be achieved by previous FSs. Additionally, the computational cost for each algorithm increased with the features and samples based on Tables 8-12. Figure 6: Prediction charts of different algorithms on Test Dataset No. 1. Figure 8: Prediction charts of different algorithms on Test Dataset No. 3. Figure 7: Prediction charts of different algorithms on Test Dataset No. 2. Figure 10: Prediction charts of different algorithms on Test Dataset No. 5. Figure 9: Prediction charts of different algorithms on Test Dataset No. 4.