## 3 Weighted Least Square Support Vector Regression Model
### W-LSSVM for Icing Forecasting  
Before all calculations are carried out, it is necessary to state that the methods presented in this paper do not exhibit any uncertainty in the prediction process. This is also the next important work in our future research. After obtaining the final features of the data, the inputs into the W-LSSVM to retrain and test the data are applied. In this paper, Matlab is used to run the self-written program and calculate the results; it is worth noting that the wavelet kernel function is chosen as the kernel function of W-LSSVM, and the parameters of W-LSSVM are obtained by using FA to optimize them. Parameters of fireworks  
Figure 4: The curve of convergence and global optimum. algorithm are set as shown in Section 4.3; through the calculation, the parameters of W-LSSVM are chosen as follows: \(\gamma=42.4756\), \(\sigma=18.8812\). In order to prove the performance of the proposed forecasting model for icing thickness, three well-known icing forecasting models including SVM (support vector machine), BPNN (BP neural networks), and MLRM (multi-variable linear regression model) are also applied to the data-set described in Section 4.2. In the single SVM forecasting model, three input vectors, including temperature, humidity, and wind speed, can be regarded as support vectors of the SVM model. Additionally, the parameters of \(C,\sigma,\varepsilon\) are obtained by cross validation on the former 576 training data. Through the training calculation, the parameters are chosen as follows:  
\[C=10.913,\quad\varepsilon=0.0012,\quad\sigma=2.4532.\]  
In the BPNN icing forecasting model, the number of input layers, hidden layers, and output layers are 5, 7, and 1 respectively. The Sigmoid function is selected as the transfer function. The maximum permissible error of model training is 0.001, and the maximum training time is 5000. Furthermore, this paper uses the relative error (RE), mean absolute percentage error (MAPE), and root mean square relative error (RMSE) as the final evaluation indicators:  
\[RE=\frac{y(i)-\hat{y}(i)}{y(i)}\times 100\% \tag{28}\]  
\[MAPE=\frac{1}{N}\underset{i=1}{\overset{N}{\underset{i=1}{\sum}}}\left|\frac {y(i)-\hat{y}(i)}{y(i)}\right|\times 100\% \tag{29}\]  
\[RMSE=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\underset{i=1}{\sum}}}}\left( \frac{y(i)-\hat{y}(i)}{y(i)}\right)^{2} \tag{30}\]  
where \(y(i)\) is the original icing thickness value, and \(\hat{y}(i)\) is the predictive value. All the icing thickness forecasting value with the BP, SVM, W-LSSVM-FA, and MLRM models are shown in Figure 5, part of which are given in Table 2. In addition, Figure 6 gives the forecasting errors with these models. Based on the above results, several patterns are observed. Firstly, the maximum and minimum gap between original icing value and forecasting value are captured from Figure 5. In the MLRM model, the maximum gap is 0.77 mm, and the minimum gap is 0.25 mm. In the BPNN model, the maximum and minimum gaps are 0.56 mm and 0.16 mm, respectively. Both values of BPNN are smaller than that of MLRM, which indicates that the forecasting accuracy of BPNN is much more precise. In the single SVM model, the maximum and minimum gaps are 0.47 mm and 0.11 mm, respectively. The deviation between the maximum and minimum gaps of the single SVM model is 0.36 mm, which is smaller than that of BPNN and MLRM; this result demonstrates that the SVM icing forecasting model is much more stable than that of BPNN and MLRM. In the proposed W-LSSVM-FA model, the maximum gap is only 0.2 mm and the minimum gap is only 0.064 mm. Compared with the other three models, both values of W-LSSVM-FA are smaller and the deviation between the maximum and minimum gap is 0.136 mm, which is also smaller than that of SVM, BPNN, and MLRM. This illustrates that the proposed W-LSSVM-FA model has a higher prediction accuracy and stability.