## 3 Novel Broad Echo State Network
### Experimental Environment and Settings  
The experiments are conducted on a platform with a 64-bit Windows system. Its memory is 16 GB, and the processor is AMD R7 4800H (2.9 GHz). The deep learning framework is based on Tensorflow 2.0 and Keras 2.4.3. The code is in Python 3.7 programming language.  
To demonstrate the performance of the proposed CMBESN model, some typical methods in time series prediction are set as the contrast. The contrast models include the GRU, BLS, ESN, and CMBLS. The GRU is a classical model of RNN. The BLS, ESN, and CMBLS are the basic models for the proposed CMBESN. Then the four models are selected as the contrast.  
The models above are trained for the two datasets to obtain relatively good results. The models are determined after the training. Tables 2 and 3 show the parameters of models for the two datasets. The parameters include the number of nodes in the mapping layer, the number of nodes in the enhancement layer, the size of the reservoir in the ESN cell, the spectral radius of the reservoir, the leakage rate, and the sparsity of the reservoir.  
For the prediction of time series, the evaluation metrics of model regression are usually chosen to verify the performance of the models. In this paper, the following metrics are taken, mean absolute deviation (MAE) [47], mean square root error (RMSE) [48], symmetric mean absolute percentage error (SMAPE) [49], and coefficient of determination \(R^{2}\)[50]. MAE, RMSE, and SMAPE reflect the deviation between the predicted and actual values. The smaller these three values are, the better the model performance is. \(R^{2}\) demonstrates the reasonableness of the final prediction model, and the closer \(R^{2}\) is to 1, the better the fit of the prediction model is. The formulas of the evaluation indexes are as follows:  
\[\text{MAE}=\frac{1}{n}\sum\limits_{k=1}^{n}\left|y_{r}(k)-y_{p}(k)\right| \tag{25}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum\limits_{k=1}^{n}\left(y_{  
To show the prediction results more obviously, the errors are drawn in Figure 10. In the figure, each bar indicates the absolute error value between the predicted and real values. It can be seen that the overall error value of CMBESN is small.  
To quantitatively analyze the performance of CMBESN, the evaluation metrics of ESN, BLS, CMBLS, GRU, and CMBESN are presented in the form of numerical values, as shown in Table 4.  
Figure 11 shows the decreased percentage of SMAPE, MAE, and RMSE and the increased percentage of \(R^{2}\) when the CMBESN is compared with other models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Training Time(s)** & **SMAPE** & **MAE** & **RMSE** & \(R^{2}\) \\ \hline ESN & 3.7242 & 0.0525 & 0.0512 & 0.0634 & 0.9999 \\ BLS & 0.5952 & 0.0369 & 0.0326 & 0.0412 & 0.9995 \\ GRU & 71.6812 & 0.0360 & 0.0360 & 0.0409 & 0.9996 \\ CMBLS & 0.4175 & 0.0381 & 0.0335 & 0.0426 & 0.9995 \\ CMBESN & 109.6323 & 0.0349 & 0.0325 & 0.0405 & 0.9996 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evaluation metrics for each model in the MSO dataset.  
Figure 10: Errors of each model on the MSO dataset.  
Figure 9: Prediction results of each model on the MSO dataset.  
From the error plot in Figure 10 and the evaluation metrics in Table 4, it can be found that CMBESN improves the prediction accuracy. It can also be seen from Figure 11 that the CMBESN model has the smallest prediction error.