## 3 Operational Data Analysis: Methods and Results
The procedure goes as follows: The data set D\({}_{\text{bef}}\) is divided \(J\) times randomly in two subsets. \((K-1)/K\) of the data are used for training and the remaining \(1/K\) are used for validation. \(K\) is selected to be 10 for this study, because short validation folds stretch the capabilities of the model to the limit. J has been set to 300 to increase the statistical significance of the study. The performances of the linear model are quantified for each of the \(J\) runs of the cross-validation for a given \(p\)-value in terms of the mean absolute error between model estimates and measurements. In this case, the main feature of the results of the cross-validation is the following: The order of magnitude of the mean absolute error does not vary considerably when varying the \(p\)-value, while the input selection sensibly depends on the \(p\)-value. In particular, for a large \(p\)-value, the model is not as stable as for a small \(p\)-value: When the \(p\)-value is large, several different final input selections are obtained along the \(J\) runs of the cross-validation. This behavior (in particular the fact that the mean absolute error saturates quickly to a lower bound) can probably be interpreted as the effect of the remarkable complexity of the terrain and therefore of the wind flow (discussed by other points of view, for example, in References [32; 33]). Therefore, the final decision is for a stable model that is obtained when \(p\)-value \(\leq 10^{-8}\). Notice that \(p\)-values up \(10^{-15}\) have been tested: Therefore, the choice should be considered solid. The selected input variables for modeling the power of T7 are:  
* The power of T6;
* the power of T9;
* the rotor rpm of T8. This input variable selection is reasonable: In the wind energy literature, it is widely discussed that the low precision of nacelle cup anemometers leads to the idea of using wind turbine rotors (and therefore, power and/or rpm) as probes of the wind conditions [36]. Furthermore, T6, T8, and T9 are the nearest wind turbines with respect to T7, although T8 and T9 are sited at a quite different altitude. The resulting model can easily be interpreted as a generalization of the power-power method, because a model has been obtained for the power of the target wind turbine (T7) as a function of meaningful variables of three references nearby wind turbines (T6, T8, T9), instead of only one. The procedure therefore goes as follows: Input measurements are normalized and organized in a matrix \(\underline{x}_{train}\). The pseudo-inverse of \(\underline{x}_{train}\) is used to compute the weights of the linear regression model as:  
\[\mathbf{W}=\underline{x}_{train}^{\intercal}\cdot\mathbf{y}_{train}, \tag{1}\]  
where \(\mathbf{W}\) is a column vector and \(\mathbf{y}_{train}\) is the column vector of the T7 power measurements in the training data set. Finally, the estimated power output vector \(\mathbf{\hat{y}}_{valid}\) on a given validation data set is computed as a linear combination of the input variables, weighted using \(\mathbf{W}\):  
\[\mathbf{\hat{y}}_{valid}=\underline{x}_{valid}\cdot\mathbf{W}. \tag{2}\]* \(D_{bef}\) is randomly divided in two subsets: D0 (\(\frac{2}{3}\) of the data) and D1 (\(\frac{1}{3}\) of the data). D0 is used for training the model and constructing the weight matrix \(\mathbf{W}\), and D1 is used for the pre-upgrade data set employed for validating the model. * \(D_{aft}\) is the post-upgrade data set employed for validating the model. For notation consistency, it is also referred to equivalently as D2. The residuals between the measurement \(y\) and the simulation \(\hat{y}\), for the data sets D1 and D2, are studied. The focus is on how the residuals vary after the upgrade. Therefore, consider Equation (3) with \(i=1,2\).