# WRNFS: Width Residual Neuro Fuzzy System, a Fast-Learning Algorithm with High Interpretability
## 3 Results and Discussion  
### Dataset 1: Mackey-Glass (MG) Time-Delay Differential Equation  
[PERSON] and [PERSON] (1977) established first-order differential delay dynamics equations for physiological system models. The equation is a nonlinear delay differential equation, one special form is described as follows:  
\[\frac{ds(t)}{dt}=\frac{0.2s(t-\tau)}{1+s^{10}(t-\tau)}-0.1s(t)\]  
When \(\tau>17\), the whole sequence is chaotic, aperiodic, and does not converge or diverge. The Mackey-Glass time series is one of the benchmark problems of time series prediction in a neural network and fuzzy logic. In this section, 1000 data points are intercepted in the MG sequence. The first 500 data points are taken as the training set and the last 500 data points are taken as the test set. Using the common construction method of time series prediction problem, every five data points in the series are regarded as a group of four inputs and one output dataset. In addition, for each group, the first four data points are regarded as input and the fifth data point as output, and thus a learning task with four inputs and one output is constructed.  
The obtained data set was used to test the performance of WRNFS after dimensionality reduction. In other words, four input ANFIS system were divided into two two-input ANFIS systems, and the residual error is used to transmit between ANFIS systems (WRNFS). The performance of the ANFIS and WRNFS is list in Table 1.  
\[\text{Performance}=\text{RMSE}\times\text{Parameter Numbers}\times\text{ Time Efficiency}.\]  
We used the first data set to show the differences between the original ANFIS and our WRNFS. From Table 1, although the accuracy decreased by using WRNFS, a better comprehensive performance is obtained. As shown in Figures 4 and 5, the complexity of the entire system is notably simplified; we plotted a line-chart to compare the precision of ANFIS and WRNFS in Figure 6, in which the red line represents the ANFIS and the blue line represents the WRNFS. ANFIS fits the yellow base-line better, but WRNFS has enough precision, too.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithm** & \multicolumn{3}{c}{**ANFIS**} & \multicolumn{3}{c}{**WRNFS**} \\ \hline Total Input numbers & & & & & & & & \\ \hline Membership Functions & 3 & & & 4 & & 3 & & 4 \\ \hline Group & Train & Test & Train & Test & Train & Test & Train & Test \\ \hline RMSE 1  & 0.0014 & 0.0013 & 0.0006 & 0.0006 & 0.0268 & 0.0260 & 0.0245 & 0.0237 \\ \hline Parameter Numbers 2  & 441 & & 1328 & & 90 & 144 \\ \hline Time Efficiency 3  & 4.5469 & & \(>\)27 & & 0.6719 & 1.0469 \\ \hline Performance 4  & 2.81 & & 21.5 & & 1.62 & 3.69 \\ \hline \hline \end{tabular}
\end{table}
Table 1: ANFIS and WRNFS in the 4-input system(3 MF).  
To further certify the result, besides three membership systems, we also experimented with four membership systems. The results are also listed in Table 1.  
From data set 1, we can draw two conclusions:  
1. The training error and the test error of the same system are basically the same, which showed that WRNFS and ANFIS have similar robustness in the four-dimensional input system.
2. When other system parameters are set to be the same, dividing the input dimensions will reduce the accuracy of WRNFS compared with the original ANFIS; however, the interpretability and system complexity will be significantly reduced, and thus the comprehensive performance will be improved.