## 4 Case Study and Discussion
### Case Study
In BEMD, the signal in relatively high frequency is first extracted using the sifting process, and the residue can be considered as the trend from the point of view of timeseries analysis. The above fact illustrates that first decomposed subprocesses have more complicated dynamic behavior, which naturally calls for identifiers \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Season**} & \multicolumn{4}{c}{**Model Structure**} \\ \cline{2-5}  & **BP Network** & **RNN** & **TCN (_k-i-n_)** & **Hammerstein Model** \\ \hline Spring & 8-10-10-12 & 6-11-15 & 8-8-4 & \(\frac{1.38z^{-1}+0.91z^{-2}-0.65z^{-3}}{1-1.83z^{-1}+1.19z^{-2}-0.67z^{-3}+0.46z ^{-4}}\) \\ \hline Summer & 12-15-8 & 8-16 & 4-6-3 & \(\frac{-0.49z^{-1}-0.41z^{-2}}{1-2.21z^{-1}+2.2z^{-2}+0.65z^{-3}-0.43z^{-4}}\) \\ \hline Autumn & 5-10-17 & 9-14 & 3-7-3 & \(\frac{-1.09z^{-3}}{1-0.98z^{-1}-0.06z^{-1}+0.04z^{-3}}\) \\ \hline Winter & 10-16-9 & 6-12-10 & 8-16-4 & \(\frac{-0.72z^{-1}-0.28z^{-2}}{1-1.78z^{-1}+0.86z^{-2}-0.28z^{-3}+0.19z^{-4}}\) \\ \hline \end{tabular} Note: (1) Sigmoid network is chosen as the nonlinearity in the above Hammerstein model, (2) For BP network and RNN network, the number of input and output units are omitted, e.g., 12-15-8 denotes a network that contains three hidden layers with 12, 15 and 8 units, respectively.  
\end{table} 
Table 2: Optimal structure of candidate models. Figure 6: BEMD results in spring. (**Upper**) IMFs and residue for limestone feed flow. (**Bottom**) IMFs and residue for slurry pH. with greater structural complexity and hyperparameters are generally chosen by trial and error. With each decomposed subprocess at hand, corresponding TCN model is set up to identify it, and Table 3 summarizes the training performance for each subprocess with the use of suitably designed TCN models. At this stage, for each season, a model set that consists of TCN models is established. Using (3) in conjunction with the model set, the final identification result on the test set can be easily obtained; corresponding results for four seasons are presented below. It can be clearly observed in Figure 7 that the proposed identification approach can fully capture complex non-linear effects in IFGD dynamics. The reason for this is twofold: on one hand, the originally highly complex process is decomposed into several subprocesses through BEMD, which have far fewer complicated frequency components and non-stationary properties, and are thus more simpler to identify; on the other hand, TCN has superior capability of representing nonlinearities and long-term dependencies while keeping the number of parameters in a low level. For a proper comparison, identification results of all approaches are listed in Table 4 below. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Season**} & \multirow{2}{*}{**Metrics**} & \multicolumn{6}{c}{**Identification Performance for BEMD Results**} \\ \cline{3-10}  & & **IMF1 1** & **IMF2** & **IMF3** & **IMF4** & **IMF5** & **IMF6** & **Residue** \\ \hline \multirow{3}{*}{Spring} & MAPE & 0.9970 & 0.1130 & 0.0362 & 0.0067 & 0.0072 & 0.0268 & 0.00015 \\  & RMSE & 0.0023 & 0.00078 & 0.00024 & 0.00013 & 0.00086 & 0.00063 & 0.0011 \\  & MAPE & 1.8731 & 0.4375 & 0.0201 & 0.0080 & 0.0152 & 0.1886 & 0.0016 \\ \multirow{3}{*}{Summer} & RMSE & 0.0041 & 0.00066 & 0.00021 & 0.000061 & 0.00019 & 0.0023 & 0.00013 \\  & MAPE & 2.0012 & 0.1180 & 0.0112 & 0.0220 & 0.0155 & 0.1302 & 0.0237 \\ \multirow{3}{*}{Autumn} & RMSE & 0.0047 & 0.0012 & 0.00074 & 0.00027 & 0.00023 & 0.00074 & 0.0052 \\  & MAPE & 1.3712 & 0.4398 & 0.0211 & 0.0135 & 0.0148 & 0.2443 & 0.0076 \\ \multirow{3}{*}{Winter} & RMSE & 0.0050 & 0.00076 & 0.00011 & 0.00032 & 0.00019 & 0.0030 & 0.00073 \\ \hline \hline \end{tabular}  
* Subprocess is conventionally represented as IMF in the above table.  
\end{table} 
Table 3: Identification results for decomposed subprocesses in four different seasons. Figure 7: Identification results for IFGD process using BEMD-TCN approach.