# Prediction of Air Pollution Concentration Based on mRMR and Echo State Network
## 4 Results and Discussion
### Experimental Results and Analysis  
Phase space reconstruction is performed on the optimal subset to obtain a 14-dimensional reconstructed time series, which are used for inputs of prediction model. For the LI-ESN model, the reasonable range of the leaking rate \(a\) is (0, 1). For the SLI-ESN model, we use the cross-validation \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Variables** & **PM\({}_{2.5}\)** & **PM\({}_{10}\)** & **NO\({}_{2}\)** & **CO** & **O\({}_{3}\)** & **SO\({}_{2}\)** & **T** & **P** & **H** & **WS** & **WD** \\ \hline \(\tau\) & **8** & **8** & 4 & **6** & 4 & 6 & 4 & 12 & **4** & **4** & 6 \\ \(m\) & **2** & **2** & 3 & **2** & 4 & 2 & 6 & 2 & **4** & **4** & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Phase space reconstruction parameters of variables. Abbreviations: hourly temperature (T), pressure (P), humidity (H), wind speed (WS), and wind direction (WD). Figure 4: Results of optimal subset selection. Figure 3: Correlation between different variables and PM\({}_{2.5}\). to select the two parameters \(a\) and \(b\). According to experience, the feasible and effective range of the supplementary factor \(b\) is mainly within (0, 0.1). In this paper, ESN, LI-ESN, ELM, H-ELM and SAE are selected as comparison methods. The specific one-step (1 hour) prediction results are shown in Table 2.  
It can be seen from Table 2 that the proposed method achieves better prediction results in one-step (1 hour) prediction. The one-step (1 hour) prediction result of PM\({}_{2.5}\) concentration is shown in Figure 5. And Figure 6 plots the fit of predicted value to actual data. It can be seen from the figures that the prediction has a good linear relationship with actual value. SLI-ESN performs satisfactorily at peaks and undulating moments, which mainly depends on the full utilization of the historical state of the reservoir and the effective information obtained by phase space reconstruction. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **RMSE** & **NRMSE** & **MAE** & **SMAPE** & **R** \\ \hline
**ESN** & 10.2020 & 0.0160 & 6.6948 & 0.1053 & 0.9936 \\
**LI-ESN** & 9.7063 & 0.0152 & 6.2322 & 0.1001 & 0.9943 \\
**ELM** & 11.7330 & 0.0184 & 7.1902 & 0.1082 & 0.9914 \\
**H-ELM** & 14.1520 & 0.0222 & 8.0575 & 0.1102 & 0.9876 \\
**SAE** & 32.1700 & 0.0505 & 20.1840 & 0.2764 & 0.9448 \\
**SLI-ESN** & 9.3953 & 0.0147 & 5.8447 & 0.0894 & 0.9945 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of one-step (1 hour) prediction results. Abbreviations: root mean square error (RMSE), normalized root mean square error (NRMSE), mean absolute error (MAE), symmetric mean absolute percentage error (SMAPE), Pearson correlation coefficient (R), echo state network (ESN), leaky integrator ESN (LI-ESN), extreme learning machine (ELM), hierarchical ELM (H-ELM), stacked auto-encoder (SAE), supplementary leaky integrator echo state network (SLI-ESN). Figure 5: PM\({}_{2.5}\) concentration one-step (1 hour) prediction curve. At the same time, the simulation results of the five-step (5 hours) prediction are given in Figure 7. The prediction curve can better track original input, and the medium-term prediction effect is also good. Table 3 gives the five-step (5 hours) prediction results. The ten-step (10 hours) prediction result of SLI-ESN is shown in Figure 8. As seen in Figure 8, in some of the peaks, the prediction curve is still able to roughly fit the fluctuation trend of the original data, which is precisely because SLI-ESN makes full use of the role of historical information. Its comparison with other algorithms is shown in Table 4. The proposed algorithm achieves the optimal value on all four parameter indicators, except for SMAPE, which fully demonstrates the effectiveness of SLI-ESN in long-term prediction. Figure 6: PM\({}_{2.5}\) concentration one-step (1 hour) predicted output fit curve. Figure 7: PM\({}_{2.5}\) concentration five-step (5 hours) prediction curve.