## 4 Experimental Results
### The Effect of the Combination of Submodule Inputs on the DNFS  
To reveal the effect of the combination of submodule inputs on the DNFS, this paper introduced DNFS5-2-Random. The implementation steps of DNFS5-2-Random are as follows: each time DNFS5-2 is executed, the input of each layer is randomly scrambled. After DNFS5-2 is executed in this way 10 times, the optimal input order is taken as the final order of DNFS5-2, and the corresponding results are obtained. Meanwhile, in order to validate the effectiveness and superiority of DNFS, MBGD-RDA, which is the latest training algorithm for TSK FSs, was introduced to be compared with DNFS. MBGD-RDA was implemented by the MATLAB implementation given in [26], and its initial learning parameters were consistent with [26], which proved to be the optimal one. Besides, the radial basis function (RBF) [38], generalized regression neural network (GRNN) [39], and long-short term memory (LSTM) [40] were also introduced to further \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Methods** & **AIC** & **MAE** & **RMSE** & **SMAPE** & **Score** & **Layers** & **No. of Submodules** & **No. of Params** & **Time(s)** \\ \hline DNFS5-0 & \(-\)**409.42** & 0.035025 & 0.049632 & 5.4848\% & 9 & 3 & **9** & **480** & 2.1238 \\ DNFS5-1 & \(-\)84.53 & 0.036351 & 0.053814 & 5.5005\% & 5 & 3 & 11 & 624 & **1.7390** \\ DNFS5-2 & 97.92 & **0.034611** & **0.045798** & **5.4495\%** & **10** & 3 & 14 & 752 & 2.1080 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The performance of DNFSs with different shared inputs on Test Dataset No. 1. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Methods** & **AIC** & **MAE** & **RMSE** & **SMAPE** & **Score** & **Layers** & **No. of Submodules** & **No. of Params** & **Time(s)** \\ \hline DNFS5-0 & \(-\)**14,761.41** & 0.073618 & 0.098158 & 7.5000\% & 9 & **3** & **22** & **1904** & **78.6101** \\ DNFS5-1 & \(-\)14,065.16 & 0.074622 & 0.098456 & 7.6021\% & 5 & **3** & 26 & 2240 & 92.0680 \\ DNFS5-2 & \(-\)12,060.34 & **0.072505** & **0.096758** & **7.4269\%** & **10** & 4 & 40 & 3312 & 134.6429 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The performance of DNFSs with different shared inputs on Test Dataset No. 3. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Methods** & **AIC** & **MAE** & **RMSE** & **SMAPE** & **Score** & **Layers** & **No. of Submodules** & **No. of Params** & **Time(s)** \\ \hline DNFS5-0 & \(-\)**10,667.72** & 0.037904 & 0.051602 & 3.8449\% & 7 & **2** & **6** & 1632 & 99.3428 \\ DNFS5-1 & \(-\)**11,899.62** & 0.039198 & 0.051779 & 4.0155\% & 6 & **2** & **6** & **1008** & **56.0007** \\ DNFS5-2 & \(-\)10,960.29 & **0.036194** & **0.048819** & **3.6826\%** & **11** & 3 & 10 & 1616 & 75.7929 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The performance of DNFSs with different shared inputs on Test Dataset No. 1.  
reveal the superiority of the DNFS, which represent general machine learning models. Their implementation were mainly realized by calling the _newbr_, _newgrnn_, and _trainNetwork_ functions of the deep learning toolbox in MATLAB 2020a, whose learning parameters were mainly selected as the default values of the functions. Figure 4: Prediction chart of DNFSs with different shared inputs on the three test datasets. (**a**) Test Dataset No. 1. (**b**) Test Dataset No. 2. (**c**) Test Dataset No. 3. _The analysis of the effect of different combinations of submodule inputs_: By analyzing the structure of the DNFS, the combination of inputs for each submodule of the first layer played a key role in improving the performance of the DNFS. Therefore, we performed the Pearson correlation analysis on the confusion inputs of the first layer obtained in the experiments with the output variable. Figure 5 shows the correlation analysis diagram between the output and inputs sequentially and randomly on the five datasets.