## 3 Methods
### Polynomial LASSO Regression  
Given \(M\) covariates \(x_{i}\), with \(i=1,\ldots,M\), the LASSO regression is a particular way of establishing a multivariate linear model (Equation (5)) between the set of \(x_{i}\) and the target \(y\). \[y=\mathbf{x}\beta+\epsilon, \tag{5}\]  
In this study, a model for each of the \(k\) clusters was set using the reference \(D_{1}\) data set. The target of the models is the power \(P\) of the test case wind turbine, and the covariates \(\mathbf{x}\) are listed in Table 2: the set of covariates includes the wind speed \(v_{c}\), the blade pitch \(\beta\), and the rotor speed \(\omega\) up to third power. This was done because the relation between the wind speed and the power is non-linear, and it can be approximated as cubic at least in a part of the power curve; furthermore, it is known that the power factor of a wind turbine depends non-linearly on the blade pitch \(\beta\)[38]. Therefore, the objective of the present regression structure is inquiring if it is possible to approximate with a polynomial the non-linear dependence of the power \(P\) on the wind speed and on the most influential working parameters (blade pitch \(\beta\) and rotor speed \(\omega\)). The LASSO regression algorithm selects the \(\beta\) coefficients by adding one covariate at a time, so that the added covariate has the highest correlation with the target \(y\). The coefficients are set by minimizing the cost function in Equation (6):  
\[J=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}(y_{i}-\hat{y}_{i})^{2}+\lambda\sum_{i=1}^ {M}|\beta_{i}|, \tag{6}\]  
where \(N_{1}\) is the number of observations in the training data set \(D_{1}\) and the estimation \(\hat{y}_{i}\) is given in Equation (7):  
\[\hat{y}_{i}=\sum_{j=1}^{M}x_{i}^{j}\beta_{j}. \tag{7}\]  
The LASSO regression, expressed in terms of the covariates, is basically a multivariate linear regression where the coefficients are set by minimizing the squared sum of errors with a bound on the norm of the coefficients of the regression. The rationale for selecting this method in the present work is that the penalty on the norm of the coefficients (Equation (6)) forces the coefficients of the covariates having a minor contribution to the model to be exactly zero: this substantially acts as a simple feature selection algorithm, which can easily be implemented in applications, and this is the reason why this was considered appropriate for the purposes of the present work. To set the hyperparameter \(\lambda\), \(k\)-fold (with \(k=10\)) cross-validation was performed [50]. Equation (7) was employed for simulating the output on the target data set \(D_{2}\), and \begin{table}
\begin{tabular}{c c} \hline \hline
**Covariate** & **Variable** \\ \hline \(x_{1}\) & \(v_{c}\) \\ \hline \(x_{2}\) & \(\beta\) \\ \hline \(x_{3}\) & \(\omega\) \\ \hline \(x_{4}\) & \(v_{c}^{2}\) \\ \hline \(x_{5}\) & \(\beta^{2}\) \\ \hline \(x_{6}\) & \(\omega^{2}\) \\ \hline \(x_{7}\) & \(v_{c}^{3}\) \\ \hline \(x_{8}\) & \(\beta^{3}\) \\ \hline \(x_{9}\) & \(\omega^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Covariates of the LASSO regression. the performance of the model, given the residuals between the model estimated and measurements (Equation (8)), was quantified through the most common metrics: \(MAE\) (Equation (9)), \(MAPE\) (Equation (10)), and \(RMSE\) (Equation (11)).