## 3 Methodology
### Classical Normal Linear Regression Model
Using the linearity assumption, we can predict a point estimate of the dependent variable as the conditional expected value for any vector of independent input \(\overrightarrow{x}=\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)^{T}\) where the predictors are given in (4):  
\[\hat{y}=E\!\left[Y\middle|\overrightarrow{x}\right]=\beta_{1}X_{1}+\beta_{2} X_{2}+\ldots+\beta_{p}X_{p} \tag{7}\]  
The part (7) from (5) is aka systematic component, whereas the error variable, \(\varepsilon\), in (5) is aka stochastic component [40]. Let as calculate the \(p\) regressors for the \(i\)th records in the dataset using (4):  
\[X_{ij}=g_{j}(x_{i1},x_{i2},x_{i3},x_{i4},x_{5})\,\ \text{for}\ i=1,2,\ldots n\ \text{and}\ j=1,2,\ldots,p \tag{8}\]  
Then, the information in the experimental dataset can be compactly described by the \(n\)-dimensional column vector of the observed values given in (9) and the \([n\times p]\)-dimensional design matrix \(\mathbf{X}\) shown in (10):  
\[\overrightarrow{y}=(y_{1},y_{2},\ldots,y_{n})^{T} \tag{9}\]  
\[\mathbf{X}=\left[ \begin{array}{cccc}X_{11}&X_{12}&\cdots&X_{1p}\\ X_{21}&X_{22}&\cdots&X_{2p}\\ \vdots&\vdots&\ldots&\vdots\\ X_{n1}&X_{n2}&\cdots&X_{np}\end{array} \right] \tag{10}\]  
In the MLR model, initially the training data set is the whole dataset which means that \(n=126\). To find an estimate of the unknown parameter vector we can write the MLR model (5) for all records in the data set:  
\[\overrightarrow{y}=\mathbf{X}\overrightarrow{\beta}+\overrightarrow{e}\ \text{where}\ \overrightarrow{e}=(e_{1},e_{2},\ldots,e_{n})^{T} \tag{11}\]  
In (11), the \(n\)-dimensional column vector of residuals \(\overrightarrow{e}\) is a function of the unknown parameter vector \(\overrightarrow{\beta}\). Since we want the residuals to be as small as possible, we can find the point estimate \(\overrightarrow{b}=\left(b_{1},b_{2},\ldots,b_{p}\right)^{T}\) of the parameter vector \(\overrightarrow{\beta}=\left(\beta_{1},\beta_{2},\ldots,\beta_{p}\right)^{T}\) as:  
\[\overrightarrow{b}=arg\underset{\overrightarrow{\beta}}{min}\left\{nRMSE^ {2}\!\left(\overrightarrow{\beta}\right)\right\}=arg\underset{ \overrightarrow{\beta}}{min}\left\{\sum\limits_{i=1}^{n}\left[y_{i}- \overrightarrow{y}_{i}\!\left(\overrightarrow{\beta}\right)\right]^{2}\right\} \tag{12}\]  
The identification of the coefficient estimates (12) is the essence of the OLS method. The optimization problem (12) has a closed solution [40]:  
\[\overrightarrow{b}=\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\overrightarrow {y} \tag{13}\]In (13), the matrix \((\mathbf{X}^{T}\mathbf{X})\) is a square \([p\times p]\) symmetric matrix (we will call it the information matrix for simplicity). As \(p<<n\), the information matrix is expected to have an inverse. According to the Gauss-Markov theorem, (13) is the best linear unbiased estimator (BLUE) for the parameter vector [41] (pp. 358-360). Plugging the point estimate of the parameter vector into (7) we can obtain the MLR model for hydrodynamic efficiency of the OWC WEC as required in Figure 2:  
\[\hat{y}=b_{1}X_{1}+b_{2}X_{2}+\ldots+b_{p}X_{p}=\sum\limits_{j=1}^{p}b_{j} \hat{y}_{j}(x_{1},x_{2},x_{3},x_{4},x_{5}) \tag{14}\]  
The resubstitution OLS residuals are calculated utilizing (9), (10), (11) and (14):  
\[\overset{\rightarrow}{e}=\left(e_{1},e_{2},\ldots,e_{n}\right)^{T}=\overset{ \rightarrow}{y}-\mathbf{X}\overset{\rightarrow}{b} \tag{15}\]  
The \(n\) components, \(e_{i}\), of the vector of residuals (15) form a sample containing \(n\) known variates of the error variable \(\varepsilon\) in (5).