## 4 Conclusions  
The original motivation of the present study was the fact that there is no published paper applying machine learning algorithms to classify clinical data related to the potential development of prostate cancer and to assist in decision-making by medical specialists concerning the issue of whether a patient should proceed to biopsy or not. The lack of such studies prompted us to construct surrogate data, based on a lognormal probability distribution function that presents authentic clinical PSA data reported elsewhere, and based on that, to extend our dataset in order to include additional features based on biomarkers like PSAD, PSAV, PSAR, and the DRE. These five features, combined with patient age as a sixth feature and the binary classification of the cases into two groups, i.e., those who are considered positive to proceed to biopsy and those who are negative for biopsy, formed the dataset on which the performance of seven simple classification algorithms was tested. The simple classification algorithms that were evaluated in the present study include Decision Tree, Random Forest, Neural Networks, K-Nearest Neighbors, Logistic Regression, Naive Bayes, and Support Vector Machines. The performance of these algorithms was examined for different sizes of the training set and the test set. The obtained results, presented in Figures 3 and 4, respectively, showed that the most efficient algorithms are the Decision Tree, Random Forest, and Neural Network algorithms, which achieved a score of at least 98% average classification accuracy on the test set. However, these results revealed a strong relationship between the performance of these three algorithms (as well as the other four) and the size of the training set. This relationship was investigated in more detail (Figures 5 and 6) by conducting 100,000 independent experiments for the KNN algorithm, which is among the algorithms that present the poorest performance. This investigation revealed that even the KNN algorithm can achieve 100% classification accuracy depending on the cases that constructed the training set. These findings prompted us to develop three hybrid classification algorithms for a two-objective Genetic Algorithm search (a) of the optimal training set regarding its size S and the cases that it is constructed of and (b) that, at the same time, optimizes the classification performance of the underlying classification algorithm. The classification methods in these three hybrid algorithms, namely, GAKNN, GAKMEANS, and GACLUST, were, respectively, the simple KNN classification algorithm, the simple k-means clustering algorithm, and a new genetic clustering algorithm introduced \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Algorithm** & _alpha_ & _Accuracy on_ & _Accuracy on_ & \(S\) & **Biopsy** & **No Biopsy** \\  & & _Test Set 1_ & _Test Set 2_ & & & \\ \hline GAKMEANS & 0.55 & 1.00000 & 1.00 & 234 & 34 & 200 \\ \hline GAKMEANS & 0.60 & 1.00000 & 1.00 & 235 & 44 & 191 \\ \hline GAKMEANS & 0.65 & 1.00000 & 0.98 & 233 & 44 & 189 \\ \hline GAKMEANS & 0.70 & 0.99055 & 0.97 & 237 & 31 & 206 \\ \hline GAKNN & 0.15 & 1.00000 & 1.00 & 122 & 64 & 58 \\ \hline GAKNN & 0.20 & 1.00000 & 1.00 & 114 & 60 & 54 \\ \hline GAKNN & 0.25 & 1.00000 & 0.98 & 124 & 66 & 58 \\ \hline GAKNN & 0.30 & 1.00000 & 0.97 & 126 & 60 & 50 \\ \hline GACLUST & 0.50 & 1.00000 & 0.96 & 125 & 66 & 60 \\ \hline GACLUST & 0.55 & 0.99813 & 0.98 & 126 & 38 & 88 \\ \hline GACLUST & 0.60 & 1.00000 & 0.99 & 123 & 39 & 84 \\ \hline GACLUST & 0.65 & 0.98168 & 0.98 & 114 & 40 & 74 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Classification results of four independent computer experiments for each one of the three hybrid algorithms regarding classification performance when a training–validation–test splitting is used on dataset 1 and a second independent test set of surrogate data is used (set 2). in the present work.