## 2 Materials and Methods
### Dataset Description
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Variable Name** & **GPSC Variable Representation** & **Count** & **Mean** & **Std** & **Range** \\ \hline Age & \(X_{0}\) & & 47.41766 & 9.931334 & 23–77 \\ \hline Sex & \(X_{1}\) & & 0.616299 & 0.4867 & 0–1 \\ \hline ALB & \(X_{2}\) & & 41.62428 & 5.761794 & 14.9–82.2 \\ \hline ALP & \(X_{3}\) & & 68.12309 & 25.92107 & 11.3–416.6 \\ \hline ALT & \(X_{4}\) & & 26.57538 & 20.86312 & 0.9–325.3 \\ \hline AST & \(X_{5}\) & & 33.77284 & 32.86687 & 10.6–324 \\ \hline BIL & \(X_{6}\) & & 11.01817 & 17.40657 & 0.8–209 \\ \hline CHE & \(X_{7}\) & & 8.203633 & 2.191073 & 1.42–16.41 \\ \hline CHOL & \(X_{8}\) & & 5.391341 & 1.128954 & 1.43–9.67 \\ \hline CREA & \(X_{9}\) & & 81.6691 & 50.69699 & 8–1079.1 \\ \hline GCT & \(X_{10}\) & & 38.19847 & 54.30241 & 4.5–650.9 \\ \hline PROT & \(X_{11}\) & & 71.89015 & 5.348883 & 44.8–86.5 \\ \hline Category Binary & \(y\) & & 0.095 & 0.293 & 0–1 \\ \hline Category Multivariate & & & 0.196944 & 0.666439 & 0-3 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of statistical analysis of the dataset with definition of input and output dataset variables. It can be seen from Figure 2 that the highest correlation is achieved between Category (target variable) and the input variables AST (0.63) and BIL (0.55). However, in this investigation, all input variables will be used in GPSC to develop symbolic expressions. As already stated in this paper two different cases were investigated i.e., the binary and the multi-class problem. Initially, the dataset contains four classes \"0--Blood Test\" (renamed to healthy), \"1--Hepatitis C\", \"2--Fibrosis\", and \"3--Cirrhosis\". The classes were renamed with numbers 0, 1, 2, and 3, respectively. In the case of the binary problem classes, 2 and 3 are joined with class 1. The number of samples per class and for binary/multiclass problems is shown in Figure 3.  
As seen from both subfigures in Figure 3 both cases have under-sampled classes. In the binary case, Hepatitis C is the under-sampled class while in the multiclass case the Hepatitis C, Fibrosis, and Cirrhosis class are all undersampled. To show the distribution of class samples the number of dimensions had to be reduced. The number of input variables in this investigation is 12 (12-dimensional space) and to visualize all samples for each class in 2-dimensional space the number of dimensions had to be reduced from 12 to 2-D.  
Figure 2: The Pearson s correlation heatmap of original dataset variables. The Kernel Principal Component Analysis (Kernel PCA), according to [38] is the nonlinear form of PCA that achieves non-linear dimensionality reduction through the use of kernels. In this paper, the Kernel PCA was used to reduce the dataset which has 12 input variables (12-dimensional space), and using the kernel function reduces it to 2-dimensional space. The reason for using Kernel PCA was to graphically show all class samples in 2-dimensional space and to visualize synthetically generated dataset samples with the application of random oversampling, SMOTE, BorderlineSMOTE, and ADASYN oversampling methods. Through trial and error, the best graphical representation was achieved using the radial basis function (RBF) kernel function. The Kernel PCA method [38] can be summarized in 4 steps i.e., construct the kernel matrix from the initial dataset, compute Gram matrix, use the Gram matrix to calculate the vectors \(\mathbf{a}_{i}\) and using them  
Figure 3: The number of samples per class in binary dataset and multiclass dataset. (**a**) Binary case; (**b**) Multiclass case. compute the kernel principal components. In Figure 4 the results of Kernel PCA applied on the original binary and multiclass dataset are shown in form of scatter plots with two kernel principal components (\(KPCA_{1}\), \(KPCA_{2}\)). As seen from Figure 4 the Hepatitis C, fibrosis, and cirrhosis patients overlap with Healthy patients.