# A Compound Approach for Monitoring the Variation in Wind Turbine Power Performance with SCADA Data
## 2 Data Pre-Processing Methodology
### Feature Extraction and Dimensionality Reduction  
#### 2.2.1 KPCA Methodology  
During the data pre-processing stage, feature extraction and data dimensionality reduction play a crucial role in improving the model's representation capability. Feature extraction involves generating or identifying distinctive features from the set of input variables, while data dimensionality reduction aims to streamline the number of features without compromising accuracy. Addressing the limitation of principal component analysis  
Figure 1: The data cleaning process of a 2.0 MW wind turbine.  
algorithms in handling nonlinear datasets, [PERSON] et al. [33] proposed a kernel principal component analysis (KPCA) method based on the original algorithm. A kernel function was employed to map nonlinear data from the original space to a high-dimensional space; then, linear PCA was utilized to extract the underlying nonlinear correlations between variables for achieving data dimensionality reduction [34].  
Assume that there is a training data sample \(X=\begin{bmatrix}x_{1},&x_{2},&\cdots,&x_{N}\end{bmatrix}^{T}\in R^{N\times M}\), where \(M\) is the number of dimensions and \(N\) is the number of data points. Define the nonlinear mapping function: \(\Phi(x)\in F\), \(\Phi:R^{M}\to F\), where \(F\) is the eigenspace. The covariance matrix of \(F\) space is calculated as follows:  
\[C^{F}=\frac{1}{N}{\sum_{j=1}^{N}\Phi\big{(}X_{j}\big{)}\Phi\big{(}X_{j}\big{)}^ {T}} \tag{1}\]  
In this study, the Gaussian radial basis function \(k(x,y)=e^{\frac{\|x-y\|^{2}}{2\sigma^{2}}}\) is selected as the kernel function for eigenvalue decomposition of Equation (1):  
\[\lambda V=C^{F}V=\frac{1}{N}{\sum_{j=1}^{N}\langle\Phi(X_{j})^{T},V\rangle\Phi (X_{j})\lambda V} \tag{2}\]  
where \(<\cdot>\) denotes the dot product operator; \(\lambda\), \(V\) are the eigenvalues and eigenvectors of the covariance matrix \(C^{F}\), \(V=\sum_{j=1}^{N}\[PERSON]\big{(}X_{j}\big{)}\). Additionally, the kernel function matrix \(K\in R^{N\times M}\), which satisfies the condition \(K_{ij}=\langle\[PERSON](X_{i}),\[PERSON]\big{(}X_{j}\big{)}\rangle\), is introduced; it is incorporated into the following expression:  
\[\frac{1}{N}{\sum_{i=1}^{N}\alpha_{i}\langle\[PERSON](X_{k}),\sum_{j=1}^{N}\[PERSON] \big{(}X_{j}\big{)}\rangle\langle\[PERSON]\big{(}X_{j}\big{)},\[PERSON](X_{i})\rangle}= \frac{1}{N}{\sum_{i=1}^{N}\alpha_{i}}{\sum_{i=1}^{N}K_{kj}K_{ji}} \tag{3}\]  
The data \(\Phi(x)\) are mapped to the feature vector \(V_{k}\), and the principal element \(t\) is given by:  
\[t_{k}=\langle V_{k},\Phi(x)\rangle={\sum_{i=1}^{N}\alpha_{i}^{k}\langle\[PERSON](x _{i}),\[PERSON](x)\rangle}\;\;k=1,\cdots,p \tag{4}\]  
where \(p\) is the number of nuclear primitives.  
The formula to calculate the Cumulative Percent Variance (CPV) based on the cumulative contribution of variance is as follows:  
\[CPV(p)={\sum_{i=1}^{p}\lambda_{i}}/{\sum_{i=1}^{M}\lambda_{i}}\times 100\% \tag{5}\]  
where \(\lambda_{i}\) is the eigenvalue of the covariance matrix, and \(M\) is the number of all variables.