## 3 Weighted Least Square Support Vector Regression Model
### Weighted LSSVM  
Apply the improvements of Section 3.2 into LSSVM to form a weighted LSSVM, thus the objective function can be described as follows:  
\[min\frac{1}{2}\mathbf{w^{T}}\mathbf{w}+\frac{1}{2}\gamma\sum_{i=1}^{N}\mu_{i}\xi_{i}^{ 2} \tag{18}\]  
\[s\cdot t\ y_{i}=\mathbf{w}^{T}\varphi(x_{i})+b+\xi_{i},i=1,2,3,\cdots N; \tag{19}\]  
In order to solve the above problem, the Lagrange function is established:  
\[L(\mathbf{w},b,\xi_{i},\alpha_{i})=\frac{1}{2}\mathbf{w^{T}}\mathbf{w}+\frac{1}{2}\gamma \sum_{i=1}^{N}\mu_{i}\xi_{i}^{2}-\sum_{i=1}^{N}\alpha_{i}\Big{[}\mathbf{w}^{T} \varphi(\mathrm{x_{i}})+b+\xi_{i}-y_{i}\Big{]} \tag{20}\]  
where \(\alpha_{i}\) is Lagrange multiplier. Each variable of the function is derivated, and is made to zero:  
\[\left\{\begin{array}{c}\frac{\partial L}{\partial\mathbf{w}}=0\rightarrow\mathbf{w} =\sum\limits_{i=1}^{N}\alpha_{i}\varphi(x_{i})\\ \frac{\partial L}{\partial b}=0\rightarrow\sum\limits_{i=1}^{N}\alpha_{i}=0\\ \frac{\partial L}{\partial\xi_{i}}=0\rightarrow\alpha_{i}=\gamma\mu_{i}\xi_{i} \\ \frac{\partial L}{\partial\alpha}=0\rightarrow\mathbf{w}^{T}+b+\xi_{i}-y_{i}=0 \end{array}\right. \tag{21}\]  
Eliminate \(\mathbf{w}\) and \(\xi_{i}\), and transform it into the following problem:  
\[\left[\begin{array}{cc}0&\mathbf{e}_{n}^{T}\\ \mathbf{e}_{n}&\Omega+\gamma^{-1}\mu^{-1}\cdot I\end{array}\right]\cdot\left[ \begin{array}{c}b\\ a\end{array}\right]=\left[\begin{array}{c}0\\ y\end{array}\right] \tag{22}\]  
number of sparks \(M\), the constant determining the explosion radius \(\hat{R}\), the upper and lower limits of the individual searching range of fireworks \(V_{up}\) and \(V_{down}\). After the above preparation, apply the candidate features data into the feature selection process of FA, and obtain the optimal feature subset.  
Step 2: Apply the feature subset of each iteration to W-LSSVM, and the predicted accuracy of the feature subset \(r(i)\) will be calculated by learning the training samples. Then, the fitness function value of each iteration process \(fitness(i)\) can be calculated. By comparing the value of each particle's fitness function, the optimal subset \(subset(i)\) is obtained. If the stop criteria is not satisfied in this iteration, the new initial feature subset \(Newsubset(i)\), obtained by the subset selection rules, enters a new round of iteration until the final optimal subset is obtained. It is should be noted that parameters in the W-LSSVM model should be initialized. \(\gamma\), \(\sigma\) are assigned random values.  
Step 3: Use the final optimal feature subset to retrain the W-LSSVM based on the icing thickness data. Parameters determination of W-LSSVM has impact on the training and learning ability and has a strong relationship with the forecasting accuracy, so the parameters \(\gamma,\sigma\) of W-LSSVM should be optimized by the fireworks algorithm to find the optimal parameters to improve the forecasting accuracy, rather than be directly subjectively defined.  
\begin{table}
\begin{tabular}{l l} \hline \(C_{1},\cdots,C_{4}\) & \(T_{t-i,t}=0,1,2,4\), expresses temperature when time is \(t-i\). \\ \(C_{5},\cdots,C_{8}\) & \(H_{t-i,t}=0,1,2,4\), expresses humidity when time is \(t-i\). \\ \(C_{9},\cdots,C_{12}\) & \(WS_{t-i,t}=0,1,2,4\), expresses wind speed when time is \(t-i\). \\ \(C_{13},C_{14},C_{15}\) & \(WD\) expresses wind direction, \(S\) expresses sunlight intensity, \(AP\) expresses air pressure. \\ \(C_{16},C_{17},C_{18}\) & \(AL\) expresses altitude, \(C\) expresses condensation level, \(LD\) expresses line direction. \\ \(C_{19},C_{20}\) & \(LSH\) expresses line suspension height, \(LC\) represents load current. \\ \(C_{21},C_{22}\) & \(R\) expresses rainfall, \(CT\) expresses conductor temperature. \\ \hline \end{tabular}
\end{table}
Table 1: Candidate features table.  
Figure 2: A flowchart of the Weighted least square support vector machine with fireworks algorithm for feature selection (W-LSSVM-FA) framework for icing forecasting.