## 3 Methodology  
### Performance Indicators for Black Box Models  
Given the dataset described in Section 2, we can develop different BB models for OWC WEC, which will accept as input the values of the five independent variables from Table 1, organized in a vector of independent input \(\overrightarrow{x}=(K,B,d,D,\theta)^{T}\), and as an output will predict the value of the dependent variable \(\widehat{y}\) (the predicted hydrodynamic efficiency of the OWC WEC) as shown in Figure 2.  
Each model can be created using a training set \(\{\left\langle\stackrel{{\rightarrow}}{{x}}_{i},y_{i}\right\rangle\), for \(i\) = 1, 2,..., \(n\)} consisting of \(n\) records from the dataset, where \(n\) is selected suitable for the particular model. If the model predicts \(\widehat{y}_{i}\) for the input of \(\overrightarrow{x}_{i}\), then the quantities \(e_{i}=y_{i}-\widehat{y}_{i}\) are called residuals. During the training of the model, three measures of the residual magnitudes are used traditionally: the root mean square error (\(RMSE\)), the mean absolute error (\(MAE\)) and the coefficient of determination (\(R^{2}\)):  
\[RMSE=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}(e_{i})^{2}}}=\sqrt{\frac{1}{n}{ \sum\limits_{i=1}^{n}\left(y_{i}-\widehat{y}_{i}\right)^{2}}} \tag{1}\]  
\[MAE=\frac{1}{n}{\sum\limits_{i=1}^{n}|e_{i}|}=\frac{1}{n}{\sum\limits_{i=1}^{n }\left|y_{i}-\widehat{y}_{i}\right|} \tag{2}\]  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}(e_{i})^{2}}{\sum\limits_{i=1}^{n}\left(y_ {i}-\overline{y}_{i}\right)^{2}}=1-\frac{\sum\limits_{i=1}^{n}\left(y_{i}- \widehat{y}_{i}\right)^{2}}{\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y} \right)^{2}}\text{, where }\overline{y}=\frac{1}{n}{\sum\limits_{i=1}^{n}y_{i}} \tag{3}\]  
In (3), \(R^{2}\) is the ratio of the explained by the model \(y\)-variance and initial variance of the dependent variable. It can take any value between negative infinity and 1. Negative \(R^{2}\) shows that the model produces residuals which are generally greater that the \"constant response model\" which predicts the mean value \(\overline{y}\) for any input \(\overrightarrow{x}=(x_{1},x_{2},x_{3},x_{4},x_{5})^{T}=(K,B,d,D,\theta)^{T}\). The performance indicators (1), (2), and (3) are measures of the resubstitution error of the BB model because the same training set is used to estimate the performance of the model [35]. That is why the former are optimistic in a sense that they indicate better model performance than the actual one. To remedy this situation, we can use a test set \(\{\left\langle\stackrel{{\rightarrow}}{{x}}_{i,HO},y_{i,HO}\right\rangle\), for \(i\) = 1, 2,..., \(\mu_{HO}\}\) consisting of \(n_{HO}\) records from the dataset which (as a rule) do not belong to the training set for the particular BB model. If the latter predicts \(\widehat{y}_{i,HO}\) for the input of \(\overrightarrow{x}_{i,HO}\), then the quantities \(e_{i,HO}=y_{i,HO}-\widehat{y}_{i,HO}\) are called predicted residuals [36] (pp. 411-412). If the performance indicators (1), (2), and (3) are estimated for the test set we will get \(RMSE_{HO}\), \(MAE_{HO}\), and \(R_{HO}^{2}\). Those three indicators measure  
Figure 2: The input-output diagram for a black box (BB) model for hydrodynamic efficiency predictions. the holdout (HO) error of the BB model because the test set to measure the performance of the model has never been used in its training [35]. Hence, \(RMSE_{HO}\), \(MAE_{HO}\), and \(R_{HO}^{2}\) are unbiased estimates of the actual BB model performance, but their variance increases with the decrease of \(n_{HO}\). As a result, the partition of the dataset into two sets containing \(n\) and \(n_{HO}\) observations is a question of compromise--we want \(n_{HO}\) to be small (to increase maximally its complement \(n\) for better training of the model), but simultaneously we want \(n_{HO}\) to be large (to decrease the variance of the performance indicators). Usually, this problem is solved by selecting widespread \(n\):\(n_{HO}\) partitions such as 70:30, or 80:20.