# Fast Numerical Wind Turbine Candidate Site Evaluation
## 2 Methods
### Model Evaluation
#### 2.7.2 Quantitative  
Figures of merit enable us to quantitatively evaluate the results of the wind speed models for wind resource characterization and to benchmark the experimental models against NWP.  
Pearson correlation coefficient R, coefficient of determination R\({}^{2}\), mean square error (MSE), and mean standardised log loss (MSLL) values are used in evaluating the results of the models.  
Pearson correlation coefficient is defined as  
\[\text{R}=\frac{\text{cov}(\mathbf{y},\mu)}{\sigma_{\mathbf{y}}\sigma_{\mu}}, \tag{2}\]  
where \(\mathbf{y}\) is the vector of measured values, \(\mu\) is the vector of predicted values, cov is covariance, \(\sigma_{\mathbf{y}}\) is the standard deviation of the measured value, and \(\sigma_{\mu}\) is the standard deviation of the predicted (mean) value. The value is between \(-1\) and \(1\), and the more positive value is better.  
The coefficient of determination is defined as  
\[\text{R}^{2}=1-\frac{\left\|\mathbf{y}-\mu\right\|^{2}}{N\sigma_{\mathbf{y}}^ {2}}, \tag{3}\]  
where \(N\) is the number of the test samples and \(\sigma_{\mathbf{y}}^{2}\) is the variance of the measured value. The value is between \(0\) and \(1\), and a bigger value is better.  
MSE is defined as  
\[\text{MSE}=\frac{\left\|\mathbf{y}-\mu\right\|^{2}}{N}; \tag{4}\]  
It is always positive and smaller is better.  
MSLL is defined as [38]  
\[\text{MSLL}=\frac{1}{2N}\sum_{i=1}^{N}\Bigg{[}\ln\left(\sigma_{i}^{2}\right)-\ln \left(\sigma_{\mathbf{\gamma}}^{2}\right)+\frac{\left(E(\hat{y}_{i})-y_{i}\right)^{ 2}}{\sigma_{i}^{2}}-\frac{\left(y_{i}-E(\mathbf{\gamma})\right)^{2}}{\sigma_{\mathbf{ \gamma}}^{2}}\Bigg{]}, \tag{5}\]  
where \(y_{i}\) is the measured value, \(E(\hat{y}_{i})\) is the mean prediction, and \(\sigma_{i}^{2}\) is the predictive variance. The summation includes all of the test samples and the index \(i\) corresponds to the sample. MSLL takes the predictive variance into account. A lower MSLL value corresponds to a better model, and the values are typically negative.  
R, R\({}^{2}\), and MSE are popular standard figures of merit. They do not provide completely independent information, and the main reason for listing several of them is for easier comparison with other studies. MSLL is the figure of merit that is used to evaluate the predictive variance in addition to the predicted mean value.  
To evaluate the time-series results over the whole sample of the models, the figure of merit for each modelling method is averaged over the 12 models that were obtained with different training months.  
Estimating the annual energy production or related derivatives from the wind speed distribution is specific to the turbine [9] and is not attempted. The average wind speed \(\bar{v}\) and the average cube of the wind speed \(\bar{v}^{3}\), averaged over time throughout the test period, are used to quantify the distributions instead. The model-predicted values of \(\bar{v}\) and \(\bar{v^{3}}\) are computed from the predicted expected values of the wind speed. They are compared to the values of \(\bar{v}\) and \(\bar{v^{3}}\) that are calculated from the field measurements over the same period. For each modelling method, the MSE of either quantity over the 12 different choices of the training month is computed.