## 5 Experimental Results
### Performance Analysis  
In order to verify the effectiveness of the proposed models, this subsection compares four commonly used neural networks, i.e., BP neural networks, RBF neural networks, ELM neural networks, and LSTM neural networks. Two commonly used statistical indices, the root mean square error (RMSE) and the mean absolute error (MAE), are adopted for model validation to quantitatively evaluate the prediction performance. Their formulations are as follows:  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left\|\mathbf{y}_{i}-\mathbf{y}_{i}\right\|_{2}^ {2}} \tag{24}\]  
Figure 11: Cluster results of solar radiation in four seasons. (**a**) Spring: (**b**) Summer; (**c**) Autumn; (**d**) Winter. \[MAE=\frac{1}{N}{\sum\limits_{i=1}^{N}{\left\|{{{\hat{y}}_{i}}-{{\mathbf{y}}_{i}}}\right\| _{1}}} \tag{25}\]  
where \({{\mathbf{y}}_{i}}\) is the actual value of the output data, and \({{\hat{{\mathbf{y}}}_{i}}}\) is the corresponding predicted result. Tables 1-8 list the prediction results of BP, RBF, ELM and LSTM in the situations of with or without clustering. For the case of without clustering, the RMSE/MAE values of 15 days in the test set of each season are reported. In the case of clustering, the prediction errors are recorded for the three clusters, respectively. The last columns in these tables give the average of the forecast results of the three clusters. Compared with the case of without clustering, the RMSE value of the BP network with clustering in the four seasons goes down respectively by 3.38, 4.51, 3.15 and 4.87, while MAE rises by 0.72, 1.96, 0.69 and 0.91, respectively. As for RBF, RMSE is respectively decreased by 40.64, 62.07, 65.83 and 28.96 and MAE is decreased by 21.07, 42.90, 19.83 and 25.18, respectively. When using ELM, RMSE is respectively reduced by 3.89, 12.51, 13.75 and 10.68 and MAE is reduced by 1.31, 5.04, 1.31 and 5.09. At last, RMSE of LSTM is improved respectively by 133.56, 41.38, 104.40 and 115.75, and MAE is improved by 145.04, 111.64, 80.6 and 120.29. The experimental results demonstrate that the performance of the four neural network models are enhanced via spectral clustering, which indicates that machine learning models are significative to improving the prediction results of short-time solar radiation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**without Clustering**} & \multicolumn{4}{c}{**with Clustering**} \\ \hline  & & **Cluster 1** & **Cluster 2** & **Cluster 3** & **Average** \\ \hline BP & 28.33 & 27.95 & 43.09 & 16.02 & 29.02 \\ \hline RBF & 64.20 & 50.90 & 45.58 & 36.63 & 44.37 \\ \hline ELM & 42.01 & 39.80 & 46.53 & 35.77 & 40.70 \\ \hline LSTM & 158.67 & 39.38 & 77.85 & 116.81 & 78.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MAE of solar radiation forecast errors in autumn. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**without Clustering**} & \multicolumn{4}{c}{**with Clustering**} \\ \hline  & & **Cluster 1** & **Cluster 2** & **Cluster 3** & **Average** \\ \hline BP & 34.54 & 50.83 & 9.20 & 49.46 & 36.50 \\ \hline RBF & 88.71 & 61.76 & 9.81 & 65.85 & 45.81 \\ \hline ELM & 43.84 & 50.91 & 9.11 & 56.38 & 38.80 \\ \hline LSTM & 182.42 & 60.71 & 57.80 & 93.82 & 70.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: MAE of solar radiation forecast errors in summer. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**without Clustering**} & \multicolumn{4}{c}{**with Clustering**} \\ \hline  & & **Cluster 1** & **Cluster 2** & **Cluster 3** & **Average** \\ \hline BP & 34.66 & 23.95 & 54.90 & 10.51 & 29.79 \\ \hline RBF & 62.16 & 28.14 & 54.28 & 17.18 & 33.20 \\ \hline ELM & 46.89 & 31.66 & 62.30 & 14.68 & 36.21 \\ \hline LSTM & 155.57 & 40.44 & 43.50 & 35.50 & 39.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: RMSE of solar radiation forecast errors in winter.