# WRNFS: Width Residual Neuro Fuzzy System, a Fast-Learning Algorithm with High Interpretability
## 4 Conclusions  
The WRNFS we proposed in this paper is a new system structure based on ANFIS, which is established by using several low dimensional ANFISs and connecting these systems through residuals. Through the experiments, we concluded that WRNFS has better performance than ANFIS and the application fields of WRNFS are more variable than ANFIS. The proposed WRNFS has obtained good accuracy and performance only based on ANFIS. The program in this article is easy for researchers to write even if they are not algorithm practitioners. Unlike the parameter number of ANFIS grows exponentially, the parameter number of WRNFS almost grows on a linear scale. When the total dimension of the input is divisible by the input dimension of the ANFIS model at each layer, there is no need to reuse the input data. In this case, the WRNFS system complexity increases discretely and linearly. When the total dimension of the input is not divisible by the input dimension of the ANFIS model at each layer, the data of some dimensions need to be reused. In this case, the complexity of the WRNFS system should increase as a step function along the upper edge of the linear growth curve described above. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**WRNFS**} \\ \hline System Configuration & \multicolumn{6}{c}{Total Input Number: 12} & \multicolumn{6}{c}{Membership Functions: 3} \\ \hline Inputs of Low-D-ANFIS & 2 & \multicolumn{6}{c}{3} & \multicolumn{6}{c}{4} & \multicolumn{6}{c}{6} \\ \hline Number of Low-D-ANFIS & 6 & \multicolumn{6}{c}{4} & \multicolumn{6}{c}{3} & \multicolumn{6}{c}{2} \\ \hline Group & Train & Test & Train & Test & Train & Test & Train & Test \\ \hline RMSE & 0.0167 & 0.0175 & 0.0164 & 0.0312 & 0.0152 & 0.0201 & 0.0035 & 0.3863 \\ \hline MRPE & 6.4036 & 2.7321 & 7.5228 & 2.8870 & 10.4399 & 3.5540 & 7.8706 & 31.2332 \\ \hline Parameter Numbers & 270 & \multicolumn{6}{c}{540} & \multicolumn{6}{c}{1323} & \multicolumn{6}{c}{10,314} \\ \hline Rule Numbers & 54 & \multicolumn{6}{c}{108} & \multicolumn{6}{c}{243} & \multicolumn{6}{c}{1458} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: WRNFS in the 12-input system (3 MF). Figure 7: The Error Decrease Curve of WRNFS. The WRNFS we proposed has a great potential of further promotion because of its simple structure. There are several places can be optimized. First, the connection between subsystems is established by addition; it can be replaced by methods such as linear least square regression. Second, we did not discuss how to arrange the distribution of each input dimension; we made every subsystem have the same input dimension for the convenience of discussion and assigned the inputs to subsystems only by one random plan. In fact, if the arrangement of dimensions is based on some interpretable algorithms, WRNFS may have better precision. Third, the WRNFS is now short of pre-processing methods, or in other words, short of forestage systems. With the help of other interpretable algorithms, WRNFS will have more possibilities in application. Conceptualization, L.K. and D.C.; methodology, D.C.; software, L.K. ; validation, L.K. ; formal analysis, L.K., D.C. and R.C. ; investigation, L.K. ; resources, D.C.; data curation, L.K. ; writing--original draft preparation, L.K. ; writing--review and editing, D.C. and R.C. ; visualization, L.K. and R.C. ; supervision, D.C. and R.C. ; project administration, D.C.; funding acquisition, D.C. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China under Grant 61976055, the special fund for education and scientific research of Fujian Provincial Department of Finance under Grant GY-Z21001 and the open project of State Key Laboratory of Management and Control for Complex Systems under Grant 20210116. Not applicable. The authors declare no conflict of interest.