## 5 Results and Discussion
### Reservoir Diversity and Error Performance  
Quantitative performance comparisons (ASE, training NRMSE, and testing NRMSE) of unpruned DeepESN and DeepESN, which were pruned by IPMA and different similarity estimation criterion-based NS-IPMA methods, are illustrated in Figure 8-10. The results on the two datasets were similar but different. On the whole, as reservoir size reduced, the ASE would decrease and training NRMSE would increase, confirming that the model with more neurons has a better information representation ability and a better training effect. Initially, testing NRMSE was greater than training NRMSE; this phenomenon is called overfitting. As the reservoir size reduced, the training error went down, and the overfitting was improved, confirming that reducing the network size is a directly effective approach to improving the generalization performance of the network. There is no doubt that pruning neurons will lead to a reduction in network size, which is the reason why the unpruned DeepESN is tested as a benchmark, and a successful pruning algorithm should outperform this benchmark. Furthermore, the minimum testing NRMSE condition is chosen to compare the extreme generalization performance of each experiment group, which was recorded in Table 1.  
Figure 7: The figure shows the number of neurons remaining in each layer of DeepESN, pruned by different similarity estimation criterion-based NS-IPMA methods. The vertical axis indicates the layer index, the horizontal axis indicates the percent of pruned neurons in the initial total number of neurons, the mesh color indicates the number of remaining neurons, and the darker color indicates more neurons remaining. (**a**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(MG\) dataset; (**b**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(MG\) dataset; (**c**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(LZ\) dataset; (**d**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(LZ\) dataset. Figure 8: ASE comparison of unpruned DeepESN and DeepESN pruned by IPMA, ED-IPMA, PC-IPMA, KC-IPMA, AC-IPMA. (**a**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(MG\) dataset; (**b**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(MG\) dataset; (**c**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(LZ\) dataset; (**d**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(LZ\) dataset. Figure 9: Training NRMSE comparison of unpruned DeepESN and DeepESN pruned by IPMA, ED-IPMA, PC-IPMA, SC-IPMA, KC-IPMA. (**a**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(MG\) dataset; (**b**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(MG\) dataset; (**c**): Initial 4 layer reservoirs with 100 neurons in each reservoir on \(LZ\) dataset; (**d**): Initial 8 layer reservoirs with 50 neurons in each reservoir on \(LZ\) dataset. Similarly, from Figure 8, the DeepESN pruned by ED-IPMA maintained a minimum ASE loss as the number of neurons decreases, and we suspect that there were some hidden relationships between the Euler distance and Renyi's quadratic entropy. From Table 1, the unpruned DeepESNs had a greater standard deviation of the minimum testing NRMSE compared to the pruned DeepESN, indicating that the NS-IPMA method has good robustness. On the MG dataset: From Figure 10a,b, in the early stage, the testing error of DeepESN pruned by SC-IPMA and KC-IPMA dropped rapidly; later, DeepESN pruned by ED-IPNA and PC-IPMA performed best when the majority of neurons had been pruned. From Table 1, although ED-IPMA achieved the best extreme generalization performance, the pruned DeepESNs had no obvious improvement in the mean value of minimum testing NRMSE compared to the unpruned DeepESN.