## 5 Time-Series Feature Prediction
### Time-Series Mode Prediction  
The time-series mode aims to deliver one-step-ahead prediction based on the previous two consecutive data samples \(y_{n}=f(y_{n-1},y_{n-2})\). The progression of the slew bearing wear is traced by the predictive model. All samples are fed to the training process and the predictive quality is evaluated by the training error, which pinpoints to what extent the predictive model learns given training representation. A schematic illustration of variance feature prediction based on PANFIS time-series mode is presented in Figure 9. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Feature** & **Description** & **PANFIS** & **eTS** & **Simp\_eTS** & **ANFIS** \\ \hline \multirow{4}{*}{RMS} & RMSE (\%) & 0.03 & 0.04 & 0.25 & 0.32 \\  & Rule & 1 & 5 & 2 & 2 \\  & Fuzzy set & 8 & 40 & 16 & 16 \\  & Time (s) & 0.41 & 0.12 & 0.15 & N/A \\ \hline \multirow{4}{*}{Variance} & RMSE (\%) & 0.04 & 0.04 & 0.21 & 0.11 \\  & Rule & 1 & 5 & 2 & 2 \\  & Fuzzy set & 8 & 40 & 16 & 16 \\  & Time (s) & 0.43 & 0.13 & 0.15 & N/A \\ \hline \multirow{4}{*}{Skewness} & RMSE (\%) & 0.11 & 0.21 & 0.25 & 0.33 \\  & Rule & 8 & 5 & 2 & 2 \\  & Fuzzy set & 64 & 40 & 16 & 16 \\  & Time (s) & 0.61 & 0.09 & 0.15 & N/A \\ \hline \multirow{4}{*}{Kurtosis} & RMSE (\%) & 0.05 & 0.27 & 0.43 & 0.09 \\  & Rule & 1 & 5 & 2 & 2 \\  & Fuzzy set & 8 & 40 & 16 & 16 \\  & Time (s) & 0.49 & 0.13 & 1.1 & N/A \\ \hline \multirow{4}{*}{Shape factor} & RMSE (\%) & 0.006 & 0.05 & 0.11 & 0.33 \\  & Rule & 1 & 5 & 2 & 2 \\  & Fuzzy set & 8 & 40 & 16 & 16 \\  & Time (s) & 0.52 & 0.19 & 1.36 & N/A \\ \hline \multirow{4}{*}{Crest factor} & RMSE (\%) & 0.14 & 0.23 & 0.79 & 8.69 \\  & Rule & 5 & 5 & 2 & 2 \\  & Fuzzy set & 40 & 40 & 16 & 16 \\  & Time (s) & 0.35 & 0.1 & 1.29 & N/A \\ \hline \multirow{4}{*}{Entropy} & RMSE (\%) & 0.14 & 0.22 & 0.79 & 0.15 \\  & Rule & 5 & 5 & 2 & 2 \\  & Fuzzy set & 39 & 40 & 16 & 16 \\  & Time (s) & 0.64 & 0.13 & 1.4 & N/A \\ \hline \multirow{4}{*}{Histogram upper} & RMSE (\%) & 0.08 & 0.3 & 0.28 & 0.1 \\  & Rule & 1 & 5 & 2 & 2 \\  & Fuzzy set & 8 & 40 & 16 & 16 \\  & Time (s) & 0.57 & 0.15 & 1.01 & N/A \\ \hline \multirow{4}{*}{Histogram lower} & RMSE (\%) & 0.05 & 0.37 & 0.35 & 0.54 \\  & Rule & 1 & 5 & 2 & 2 \\ \cline{1-1}  & Fuzzy set & 8 & 40 & 16 & 16 \\ \cline{1-1}  & Time (s) & 0.4 & 0.15 & 1.2 & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results using direct partition scenario. PANFIS is compared against three prominent algorithms, known as eTS [2], simp_eTS [12], and ANFIS [1]. eTS and simp_eTS are counterparts of PANFIS, which features both structural and parameter learning scenarios in the online manner. This comparison is needed to confirm the learning performance of PANFIS with respect to similar algorithms. ANFIS is a pioneer of the fuzzy neural network (FNN) which is more traditional than PANFIS, eTS, and simp_eTS. It adopts an offline learning scenario where the training process is repeated over multiple epochs. Ten epochs are applied in our study. This comparison aims to demonstrate that although PANFIS works fully in the one-pass learning scenario, it produces a comparable predictive quality. Table 3 presents the numerical results of the consolidated algorithm in time-series mode. The advantage of PANFIS is more obvious in the direct mode prediction (as shown in Table 2) than that in the time-series mode prediction, as displayed in Table 3. However, it still outperforms the other three algorithms in terms of RMSE, rule, and fuzzy set in almost all features prediction. Figure 10 visualizes PANFIS's prediction of the variance feature under the time-series mode. It is seen that PANFIS perfectly models the variance attribute using only its previous two time steps. Figures 8 and 11 demonstrate the adaptive trait of PANFIS. It keeps changing on demand, which follows the nonlinear and time-varying properties of the given problem.