## 4 Results and Discussions  
### Comparative Model Outputs  
As mentioned earlier, models using SVM, ANN, and MLR were tested in the current study to predict pressure gradients (Tables 2-4). The parameters for these models were fixed as per the judgement of the authors, except for weights and coefficients of SVM and ANN and the hidden neurons for ANN. Weights and coefficients were calculated as part of the learning process of the models. Hidden neurons for ANNs were determined on the basis of trial and error by comparison accuracy attained with a different number of neurons. Other parameters were fixed because of the fact that optimizing all parameters was not practically feasible for a single study. For each model, the mean square error (MSE), Mean Absolute Percent Error (MAPE) and coefficient of determination (R-square) between the predicted and experimental values were calculated to assess the accuracy of the model. MSE is an indicator of the magnitude of error, MAPE is a measure relative to the scale  
Figure 4: Performance of ANN with different number of hidden neurons. Figure 3: Structure of ANN Process. of model output, and CC denotes the ability of a model to capture the variation in the trend of data. All parameters were calculated separately for training and test datasets to evaluate the robustness of the model when used for a new dataset. The comparison of these parameters is given in Table 5. Due to the volatile nature of MLR models, three-fold cross-validation was applied and the results shown in Table 5 are the average of the three trials with different datasets. Table 6 shows the results of each individual trial in terms of MSE and T-square. Weights for support vectors of SVM and neurons of ANN are provided in the Appendix A. Table A1 in Appendix A provides weights (constants) and coefficients for the explanatory parameters in each support vector. Table A2 in Appendix A provides the threshold (constant) and coefficients for explanatory variables in hidden neurons and the same values for output neurons. It should be noted that the general functions for these models are given in Equations (2) and (3), and Figure 3. These parameters were obtained by developing the model using the training dataset while minimizing the error functions. When the MLR model was applied to the current dataset, the fraction of water had a statistically insignificant coefficient hence it was not part of that model. The variables for the MLR model were filtered based on the hypothesis that their coefficients would be statistically different from 'zero' at a probability of 5% (margin of error). The model presented in Equation (4) includes only the variables that have less than a 5% chance (\(p\)-value) of the coefficient being close to 'zero.' According to this MLR model, oil velocity, oil viscosity, and water density have negative effects on the pressure gradient, while other statistically significant parameters have positive impacts. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Model Parameters**} & **Estimate** & \multicolumn{1}{c}{\(p\)**-Value**} \\ \hline Intercept & \(-97.03\) & \(0.01\) \\ \hline Dia & \(-32.06\) & \(0.00\) \\ \hline Vel & \(0.77\) & \(0.00\) \\ \hline Oden & \(-0.01\) & \(0.00\) \\ \hline OVisc & \(0.07\) & \(0.00\) \\ \hline WDen & \(0.11\) & \(0.00\) \\ \hline WVisc & \(-1995.36\) & \(0.00\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: MLR Model. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Value/Description** \\ \hline Kernel type & Radial basis \\ \hline Number of support vectors & \(83\) \\ \hline \(\sigma\) & \(0.1\) \\ \hline B & \(0.14\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SVM Model.