## 3 Results
### The Symbolic Expression for Detection of Hepatitis C Patients  
The Table 3 the list of randomly chosen hyperparameters with which highest values of classification metrics were achieved on each dataset variation are shown. As seen from Table 3 the population size in all four cases is near the upper bound i.e., near 500 while the number_of_generations and the tournament_size are near the lower bound. In all four cases, the hoist and point mutations (values in the range of 0.31-0.69) were dominating over crossover and subtree mutations (values in the range of 0.057-0.26). The stopping criteria values were very small in all cases (range \(10^{-5}\)-\(10^{-6}\)) however, the majority of GPSC executions were stopped due to the maximum number of generations value was reached not because fitness value dropped below the stopping criteria value. The max_samples was near the lower bound i.e., 0.6 (Table 2). Although the parsimony coefficient value was very low in all four cases (range \(10^{-5}\)-\(10^{-6}\)) the size of the population members did grow however this growth contributed to lowering the fitness measure value so no bloating phenomenon occurred. The graphical representation of the mean values of accuracy, precision, recall, AUC, and F1-score with standard deviation is shown in Figure 10.  
Due to a small difference between mean and standard deviation values of evaluation metrics shown in Figure 10 all the results including the required average CPU time required to obtain these symbolic expressions are listed in Table 4.  
From the obtained results shown in Figure 10, it can be seen that all the symbolic expressions have very high classification accuracy since mean _ACC_, _AUC_, Precision, Recall and F1-score values are all above 0.98. The highest evaluation metric values were achieved in the case of SMOTE, followed by ADASYN, BorderlineSMOTE, and Random Oversampling. To select the best symbolic expression for the binary classification problem the symbolic expression length was also measured. This length is measured by counting the number of elements (mathematical functions, and variables) inside the symbolic expressions. Based on that measure the longest symbolic expression was obtained in the ADASYN case followed by the Random Oversampling, SMOTE, and BorderlineSMOTE case. Since the symbolic expression obtained in the case of the BorderlineSMOTE dataset has slightly lower evaluation metric values than in the SMOTE case the best symbolic expression in the binary classification problem was chosen to be the symbolic expression obtained in the case of BorderlineSMOTE due to the small size. \begin{table}
\begin{tabular}{l c} \hline \hline \multirow{3}{*}{**Dataset Type**} & **GPSC Hyperparameters (Population_Size,** \\  & **Number_of_Generations, Tournament_Size, Initial_Depth,** \\  & **Crossover, Subtree_Muation, Hoist_Mutation, Point_Mutation,** \\  & **Stopping_Criteria, Max_Samples, Constant_Range,** \\  & **Parsimony_Coefficient)** \\ \hline \multirow{3}{*}{Random oversampling} & 463, 114, 34, (6, 12), \\  & 0.16, 0.13, 0.63, 0.069, \(1\times 10^{-5}\), \\  & 0.64, (\(-8792.5\), 4309.56), \(3.78\times 10^{-5}\) \\ \hline \multirow{3}{*}{ADASYN} & 493, 121, 46, (7, 12), \\  & 0.057, 0.39, 0.013, 0.53, \(1\times 10^{-5}\), \\  & 0.67, (\(-4979.57\), 4518.54), \(5.02\times 10^{-5}\) \\ \hline \multirow{3}{*}{SMOTE} & 428, 183, 25, \\  & (7, 12), 0.2, 0.41, 0.35, 0.029, 0.00084, \\  & 0.64, (\(-7790.44\), 1461.71), \(9.19\times 10^{-6}\) \\ \hline \multirow{3}{*}{Borderline SMOTE} & 384, 170, 34, (4, 11), \\  & 0.1, 0.26, 0.32, 0.31, \(4\times 10^{-6}\), \\  & 0.67, (\(-4520.81\), 8562.5), \(8.9\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The randomly chosen hyperparameters with which the symbolic expressions with highest classification accuracy were obtained. Figure 10: The mean and standard deviation (error bars) values of ACC, AUC, Precision, Recall for binary case.