## 3 Novel Broad Echo State Network  
### Network Based on the Cascade of Mapping Nodes  
As mentioned in Section 2.2, CMBLS develops from BLS. In CMBLS, the original data are mapped to the first mapping nodes, the first layer's output is set as the input of the second layer, and so on. The improved structure of CMBLS can strengthen the feature extraction ability. It is theoretically proved that CMBLS retains a good fitting function [19]. The mapping nodes and enhancement layers in both BLS and CMBLS are essential for a reliable regression of the time series data. Then we embed the ESN into the enhancement layer of CMBLS to form a novel broad echo state network. The new network is abbreviated as CMBESN (Broad Echo State Network with Cascade of Mapping nodes), and its structure is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Categories** & **Models** & **Features** & **Existing Problems** \\ \hline \multirow{6}{*}{Statistical approaches} & \multirow{3}{*}{ARMA [40]} & It treats the subsequent data as a linear combination of the data. & \\  & & It combines the advantages of the AR model and MA model. & It is difficult to reduce the value of the loss function, which is not applicable in real-world data. \\ \cline{2-4}  & & ARIMA [22] & A differential process is added to ARMA. \\ \hline \multirow{6}{*}{Typical deep learning networks} & \multirow{3}{*}{GRU [23]} & It improves the problems of gradient disappearance and explosion in RNN. & \\  & & It reduces the number of gate units in the LSTM. & Time cost and computing resources increase. \\ \cline{2-4}  & & It connects multiple echo state networks vertically with the error backpropagation training mechanism. & \\ \hline \multirow{6}{*}{Broad learning system} & \multirow{3}{*}{BLS [19]} & It is with horizontal scaling of the neural units. The incremental learning algorithms are designed for faster training. & The prediction accuracy is relatively low on complex time series data. \\ \cline{1-1} \cline{2-4}  & & It is with horizontal scaling of the neural units. The incremental learning algorithms are designed for faster training. & The prediction accuracy is relatively low on complex time series data. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of each model. The CMBESN model consists of the mapping, enhancement, and output layers. The ESN units are added to the enhancement layer with the incremental algorithm. The output of the mapping layer is shown in Formula (3). The output of the enhancement layer of the CMBESN is defined as:  
\[\mathbf{H}_{j}(t+1)=(1-\delta)\mathbf{H}_{j}(t)+\delta\cdot f\left(\mathbf{W}_{\eta}\mathbf{Z}+ \mathbf{W}_{j}\mathbf{H}_{j}(t)\right) \tag{10}\]  
\(\mathbf{H}_{j}\), \(\mathbf{W}_{\eta j}\), and \(\mathbf{W}_{j}\) denote the output of the first ESN unit in the enhancement layer, the connection weight of the \(j\)th ESN unit to the mapping layer output, and the inner weights of the reserve pool in the \(j\)th ESN unit, respectively. Meanwhile, \(\mathbf{Z}\) denotes the output of the mapping layer. The output and weight matrix of the CMBESN are calculated as follows. \[\begin{array}{ll}\mathbf{Y}&=[\mathbf{Z}_{1},\mathbf{Z}_{2},\cdots,\mathbf{Z}_{n}|\mathbf{H}_{1},\mathbf{H}_{2},\cdots,\mathbf{H}_{m}]\mathbf{W}_  
The formulas above show that it does not need to restart the training data or recalculate \(\left(\mathbf{A}_{n}{}^{m+1}\right)^{+}\) when the incremental algorithm updates the network structure. \(\left(\mathbf{A}_{n}{}^{m+1}\right)^{+}\) and \(\mathbf{W}_{n}{}^{m+1}\) are updated with a simple matrix operation. The incremental algorithm can rapidly update the network model [30]. The training algorithm of the CMBESN is shown in Algorithm 1. ```
Input: Input data (after abnormal data processing and normalization, custom data step length), number of nodes in the mapping layer, RMSE threshold, spectral radius of ESN, leakage factor, and reserve pool size. Output: Number of nodes in the enhancement layer, RMSE, training time.