## 3 Methodology
### Procedures for Model Construction
#### 3.3.3 Model Development  
For the development of the GPR model, five different models based on the squared exponential (Equation (24)), exponential (Equation (25)), rational quadratic (Equation (26)),matern 3/2 (Equation (27)), and matem 5/2 (Equation (28)), covariance function as well as the functions were developed. Each model had a constant MF.  
\[g\big{(}r,r^{\prime}\big{)}=v_{s}^{2}exp\bigg{[}\frac{-\|r-r^{\prime}\|}{2\chi^{ 2}}\bigg{]} \tag{24}\]  
\[g\big{(}r,r^{\prime}\big{)}=v_{s}^{2}exp\bigg{[}\frac{-\|r-r^{\prime}\|}{\chi} \bigg{]} \tag{25}\]  
\[g\big{(}r,r^{\prime}\big{)}=v_{s}^{2}exp\bigg{[}\frac{-\|r-r^{\prime}\|}{2 \beta\chi^{2}}\bigg{]}^{-\beta} \tag{26}\]  
\[g\big{(}r,r^{\prime}\big{)}=v_{s}^{2}exp\bigg{[}1+\frac{\sqrt{3}\|r-r^{\prime} \|}{\chi}\bigg{]}exp\bigg{[}-\frac{\sqrt{3}\|r-r^{\prime}\|}{\chi}\bigg{]} \tag{27}\]  
\[g\big{(}r,r^{\prime}\big{)}=v_{s}^{2}exp\bigg{[}1+\frac{\sqrt{5}\|r-r^{\prime }\|}{\chi}+\frac{5\|r-r^{\prime}\|^{2}}{3\chi^{2}}\bigg{]}exp\bigg{[}-\frac{ \sqrt{5}\|r-r^{\prime}\|}{\chi}\bigg{]} \tag{28}\]  
where \(\beta\) is the rational quadratic covariance's shape parameter, \(\chi\) is the length scale, and \(v_{s}^{2}\) is the covariance function's signal variance.  
The model with the lowest mean squared error and highest correlation coefficient on the test dataset was chosen as the optimum GPR model. For the BPNN model, a three-layered architecture was chosen--the first with the input layer, the second with a hidden layer and the thirdly with an output layer. A single hidden layer was used because it has been established to be a reliable predictor for any prediction problem [90]. Furthermore, in the case of hidden and output layers, hyperbolic and linear transfer functions were selected and used. The Levenberg-Marquardt algorithm was used to train this BPNN model. According to the suggested values by the previous researchers, a range of 1 to 40 for neurons was tried and the optimum number was the one that gives the lowest MSE on the test dataset [91; 92]. The optimum number of neurons in the hidden layer that resulted in the lowest MSE on the test dataset was determined using a sequential experimental procedure in the construction of the ELM model. In that regard, 1 to 20 neurons were tried. It is worth stating that, the building of the MARS model, entails the choice of the highest number of basis functions to be used in the forward selection stage as well as the maximum degree of interaction. These serve as constraints in the development process. Based on their levels of interaction, three independent MARS models were built in this study-zero-degree, first degree and second-degree. Furthermore, a maximum of 20 basis functions were selected for the forward selection stage. The model with the highest correlation coefficient and lowest mean squared error (MSE) was chosen as the optimum MARS model. The MVRA model was developed using the same dataset for the development and testing of the GPR, BPNN, ELM and MARS models. The MVRA solves the multilinear regression equations established for the various input parameters and PPV using the least square technique in order to find the regression coefficient (Equation (18)) for each input parameter as well as the intercept.