## 5 Methodologies  
### SVM Models--Theoretical Descriptions  
The SVM is a non-linear, data-driven technique, gaining popularity due to superior performance as compared to traditional ERM as used by conventional neural networks [49]. With the use of SRM, the upper bound on the anticipated risk minimises the effect of a reduction in training data error and gives SVM greater capacity to generalise function as compared to neural networks. Initially, SVMs were constructed to provide objective/optimal classification called SVC but more recently have been applied to regression and termed as SVR. In this section, the theoretical description of SVM regression models [36; 50] for WT power curves is described as follows. Let us consider \(N\) training vectors \(x_{i}\in\Re\) defined by a set of definite variables \(x_{i}=\{x_{i1},x_{i2},x_{i3},\ldots,x_{ip}\}\) and by the class response \(y_{i}\in\Re\). To use non-linear functions for regression of the data \(x\), a non-linear map \(\emptyset\): \(x\rightarrow\emptyset(x)\) into a high dimensional space is suggested to allow linear regression in that space \((f(x)=\langle w,\emptyset(x)\rangle+b)\). Dot products of \(\langle x_{i},x\rangle\) found in calculating linear SVR are in the non-linear case replaced by the dot products \(\langle\emptyset(x_{i}),\emptyset(x)\rangle\). This is the symmetric function in \(x_{i}\) and \(x\) that should satisfy [PERSON]'s condition and is called a kernel: \((x_{i},x)=\langle\emptyset(x_{i}),\emptyset(x)\rangle\). The SVR technique is formulated as a minimisation of the following function:  
\[\min_{w,b}\frac{1}{2}\|\ w\ \|^{2} \tag{5}\]  
subject to:  
\[\|\ f(x_{i})-y_{i}\ \|\leq\varepsilon,\ i\ =\ 1,2\ldots,N \tag{6}\]  
where  
\[f(x_{i})\ =\ \langle w,\varnothing(x_{i})\rangle+b \tag{7}\]  
Here, \(w\) is a space weight coefficient vector, and \(b\) is a bias. The SVM is a supervised machine learning technique whose sole goal is to design a hyperplane that classifies all training vectors into two classes. The most appropriate hyperplane is one that leaves the maximum margin between both classes. Therefore, minimising the term \(w\) maximises the separability. Minimisation of \(w\) is a non-linear, optimisation task, which can be solved by the  
Figure 3: Air density corrected filtered power curve. Karush-Kuhn-Tucker (KKT) approach [50], using Lagrange function/multipliers to find a \(\xi\)-insensitive loss function, by the following function:  
\[f(x)\ =\ \sum\
olimits_{i\ =\ 1}^{N}\big{(}\alpha_{i}-\alpha_{i}^{*}\big{)}K(x_{ i},x)+b \tag{8}\]  
with the property:  
\[w\ =\ \sum\
olimits_{i\ =\ 1}^{N}\big{(}\alpha_{i}-\alpha_{i}^{*}\big{)} \varnothing(x_{i}) \tag{9}\]  
where \(b\) is the bias term (a scalar), and \(\alpha_{i}\) and \(\alpha^{*}\geq 0\) are the Lagrange multipliers. The sample point that appears with non-zero coefficients \(\alpha_{i}\) is called the support vector. A slack variable \(\mathbb{C}\) is introduced into (5) to generate an optimal SVR model by relaxing the margin constraints and then neglecting a controlled part of the data. This allows the optimisation problem to be reformulated as Equation (10) below,  
\[\min_{w,\ b,\xi^{-},\xi^{+}}\frac{1}{2}\parallel w\parallel^{2}+C\sum \
olimits_{k\ =\ 1}^{N}\big{(}\xi_{k}^{-}+\xi_{k}^{+}\big{)} \tag{10}\]  
subject to:  
\[y_{i}-w^{T}\varnothing(x_{i})-b\leq\varepsilon+\xi_{k}^{-},\ i\ =\ 1,2\ldots,N\]  
and  
\[-y_{i}+w^{T}\varnothing(x_{i})+b\leq\varepsilon+\xi_{k}^{+},\ i\ =\ 1,2\ldots,N\]  
where \(\xi_{k}^{-},\ \xi_{k}^{+}\geq 0\) are slack variables that cause a penalty term, which is weighted by \(C\) and are used to measure the deviations of the samples outside the \(\xi\)-insensitive zone. The addition of a slack variable lies in the following range: \(0\leq\alpha_{i},\ \alpha_{i}^{*}\leq C\). Weight of misclassifications increases with the increase in \(C\) values, which leads to a higher cost of the misclassified data points that cause strict separation of data.