## 3 Methodology
### Performance Metrics  
The data collected from the flow loop experiment has a variety of different dimensional units and ranges. It is evident from Tables 2 and 3 that parameters such as e is unitless and ranging from 0 to 1, whereas, pressure loss is in the range of 0.196 to 381.23 kPa. In order to bring out a common scale to analyze the features in the data without distorting or losing the values, we have transformed the data in the range of \([0,1]\) using a transformation function, given in Equations (3) and (4).  
\[x_{std}=\frac{(x-x_{min})}{x_{max}-x_{min}} \tag{3}\]  
\[x_{scale}=x_{std}\times(max-min)+min \tag{4}\]  
where \(x\) is the dimensional data while \(x_{scale}\) is the scaled transformation of \(x\) using Equation (3) and Equation (4), \(x_{scale}\) was used as input for subsequent algorithms. Three performance metrics, namely Mean Absolute Error (MAE), Coefficient of Determination (R\({}^{2}\)), and Root Mean Squared Error (RMSE) are used to measure the effectiveness of the applied predictive algorithm. RMSE quantifies the disagreement between the predicted and observed data points that can be determined from Equation (5). MAE is indication of the relative divergence of the measured and the predicted values are computed from Equation (6).  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}{(y_{i}-\hat{y_{i}})}^{2}}{N}} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Annulus \#1** & **Annulus \#2** & **Annulus \#3** & **Annulus \#4** \\ \hline Pipe Diameter [m] & 0.009652 & 0.0127 & 0.017272 & 0.016002 \\ Hole Diameter [m] & 0.035052 & 0.035052 & 0.035052 & 0.020828 \\ Di/Do & 0.27 & 0.36 & 0.49 & 0.76 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimension of the annular sections used in the study.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Maximum** & **Minimum** \\ \hline (Di/Do) (\(\longrightarrow\)) & 0.76 & 0.27 \\ e (\(\longrightarrow\)) & 1 & 0 \\ n (\(\longrightarrow\)) & 0.49 & 0.35 \\ K (\(Pa.s^{\prime\prime}\)) & 4.49 & 0.64 \\ \(\tau_{y}\) (Pa) & 14.4 & 0 \\ Q (\(m^{3}/s\)) & \(1380.33\times 10^{-6}\) & \(1.55\times 10^{-6}\) \\ \(\Delta\)P (\(k\)\(Pa\)) & 381.23 & 0.196 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Range of experimental data.  
\[MAE=\frac{1}{N}\sum_{1}^{n}\bigl{[}y_{i}-\hat{y}_{i}\bigr{]} \tag{6}\]  
The coefficient of determination can be calculated using Equation (7). It is a statistical measure of the regression prediction and approximation with the real data point, where 1 indicates the regression prediction perfectly fits the data.  
\[R^{2}=1-\frac{\sum_{1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{1}^{n}\bigl{(}y_{i}- \overline{y}_{i}\bigr{)}^{2}} \tag{7}\]  
where \(\hat{y}_{1}\) represents the predicted values, number of observations is represented by n, y\({}_{i}\) is the given data point and \(\bar{Y}_{i}\) is the mean of the given values. The optimal condition for a predictive algorithm is when the RMSE = 0 and R\({}^{2}\) = 1, the lower the RMSE and MAE score the better the quality of the result.  
Four algorithms are used in this study to predict pressure drop in concentric and eccentric annulus, i.e., SVM, ANN, BNN, and RF.