## 5 Time-Series Feature Prediction
### Time-Series Mode Prediction
PANFIS is also capable of crafting a classical-transparent fuzzy rule from a high-dimensional ellipsoidal rule through fuzzy rule transformation (14). An example of PANFIS's rule is exemplified in variance prediction:  
\[R_{1}:\textbf{If} Variance=\left[ \begin{array}{c}Variance_{n-1}\\ Variance_{n-2}\end{array} \right]\textit{ is close to }C_{1}=\left[ \begin{array}{c}0.290\\ 0.292\end{array} \right]\textit{,}\sum_{1}{}^{-1}=\left[ \begin{array}{c}7.4, 0.19\\ 0.19,7.4\end{array} \right] \tag{20}\] \[\textbf{Then}\ y=0.03+0.17\textit{variance}_{n-1}+0.04\textit{variance} _{n-2}\]  
Figure 9: A schematic diagram of time series mode prediction. This fuzzy rule is rather vague and is unable to be associated directly with linguistic labels because of the absence of atomic clauses. This issue can be addressed using the fuzzy set transformation strategy (14) and this leads to the traditional expression of the fuzzy rule as follows:  
\[ \begin{array}{l}R_{1}:\textbf{If}\ Variance_{n-1}\ is\ close\ to\ c_{11}=0.29,\sigma_{11}=0.11\\ \textbf{and}\ Skewness_{n-2}\ is\ close\ to\ c_{12}=0.292,\sigma_{12}=0.11\\ \textbf{Then}\ y=0.03+0.17\text{variance}_{n-1}+0.04\text{variance}_{n-2} \end{array}  \tag{21}\]  
This rule is more readable than (20) because each fuzzy set corresponds to a specific linguistic label. This fuzzy rule is generated under the time-series mode prediction. Another example of PANFIS prediction for histogram lower feature under the time-series mode is presented in Figure 12 and the fuzzy rule evolution is presented in Figure 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Input Feature** & **Description** & **PANFIS** & **eTS** & **Simp\_eTS** & **ANFIS** \\ \hline \multirow{4}{*}{RMS} & RMSE (\%) & 0.1 & 0.11 & 0.13 & 0.1 \\  & Rule & 2 & 5 & 2 & 2 \\  & Fuzzy set & 2-2 & 5-5 & 2-2 & 2-2 \\  & Time (s) & 0.73 & 0.54 & 1.54 & N/A \\ \hline \multirow{4}{*}{Variance} & RMSE (\%) & 0.09 & 0.11 & 0.14 & 0.09 \\  & Rule & 6 & 4 & 2 & 2 \\  & Fuzzy set & 6-5 & 4-4 & 2-2 & 2-2 \\  & Time (s) & 1.1 & 0.17 & 1.36 & N/A \\ \hline \multirow{4}{*}{Skewness} & RMSE (\%) & 0.08 & 0.09 & 0.09 & 0.12 \\  & Rule & 2 & 5 & 4 & 2 \\  & Fuzzy set & 2-2 & 5-5 & 4-4 & 2-2 \\  & Time (s) & 0.78 & 0.12 & 1.42 & N/A \\ \hline \multirow{4}{*}{Kurtosis} & RMSE (\%) & 0.13 & 0.13 & 0.18 & 0.11 \\  & Rule & 16 & 6 & 3 & 2 \\  & Fuzzy set & 16-16 & 6-6 & 3-3 & 2-2 \\  & Time (s) & 0.49 & 0.09 & 1.72 & N/A \\ \hline \multirow{4}{*}{Shape factor} & RMSE (\%) & 0.11 & 0.09 & 0.09 & 0.06 \\  & Rule & 4 & 5 & 2 & 2 \\  & Fuzzy set & 4-3 & 5-5 & 2-2 & 2-2 \\  & Time (s) & 0.55 & 0.17 & 1.4 & N/A \\ \hline \multirow{4}{*}{Crest factor} & RMSE (\%) & 0.09 & 0.14 & 0.14 & 0.12 \\  & Rule & 3 & 6 & 7 & 2 \\  & Fuzzy set & 3-3 & 6-6 & _7-7_ & 2-2 \\  & Time (s) & 0.49 & 0.15 & 2.1 & N/A \\ \hline \multirow{4}{*}{Entropy} & RMSE (\%) & 0.11 & 0.14 & 0.15 & 0.11 \\  & Rule & 6 & 6 & 5 & 2 \\  & Fuzzy set & 6-6 & 6-6 & 5-5 & 2-2 \\  & Time (s) & 0.51 & 0.12 & 1.81 & N/A \\ \hline \multirow{4}{*}{Histogram upper} & RMSE (\%) & 0.13 & 0.15 & 0.16 & 0.12 \\  & Rule & 7 & 8 & 6 & 2 \\  & Fuzzy set & 7-7 & 8-8 & 6-6 & 2-2 \\  & Time (s) & 0.69 & 0.19 & 1.7 & N/A \\ \hline \multirow{4}{*}{Histogram lower} & RMSE (\%) & 0.14 & 0.16 & 0.2 & 0.14 \\  & Rule & 2 & 8 & 5 & 2 \\ \cline{1-1}  & Fuzzy set & 2-2 & 8-8 & 5-5 & 2-2 \\ \cline{1-1}  & Time (s) & 0.52 & 0.15 & 1.5 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results using time-series scenario. Figure 11: Fuzzy rule evolution of variance feature. Figure 12: PANFIS prediction of histogram lower feature. Figure 10: PANFIS prediction of variance feature.