## 3 Data Analysis  
The penetration of the projectile can be divided into three categories, namely, deep penetration (\(X\)/\(d\geq 5.0\)), intermediate penetration (\(0.5\leq X\)/\(d\) < 5.0), and narrow penetration (\(X\)/\(d\) < 0.5), where \(X\) is the penetration depth and d is the projectile's diameter [10]. Experimental data in dimensionless form, shown in Table 1, is used for the symbolic regression analysis to obtain a mathematical model [2; 8; 10; 13; 14; 15; 16; 19]. The data's dimensionless parameters include the impact factor \(I=\left(\frac{M\
u_{c}^{2}}{f_{c}d^{5.5}}\right)\), where \(M\) is mass of the rigid projectile, \(V_{o}\) is impacting projectile velocity, \(fc\) is the compressive strength of the concrete, \(d\) is the diameter of the rigid projectile, and \(S=72f_{c}^{-0.5}\) is the empirical constant for concrete strength. The geometry function of the projectile is \(N=\left(\frac{M}{Ld^{3}N}\right)\), where \(M\) is the mass of the rigid projectile, \(N^{*}\) is nose shape factor projectile, \(l_{c}\) is the density of concrete, and \(d\) is the diameter of the rigid projectile. The slenderness of the projectile is \(\lambda=\left(\frac{M}{Ld^{3}}\right)\), where \(M\) is the mass of the rigid projectile, \(l_{c}\) is the density of concrete, and \(d\) is the diameter of the rigid projectile. The empirical constant for concrete strength is \(S=72f_{c}^{-0.5}\), and \(X\)/\(d\) refers to the dimensionless penetration depth ratio-to-projectile diameter. The dimensionless penetration \(X\)/\(d\) is taken as a dependent target variable, whereas \(I\), \(N\), \(\lambda\), and \(S\) are independent variables or predictors. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\lambda\) & \(S\) & \(N\) & \(I\) & (\(X\)/\(d\))\({}_{\text{test}}\) & \(\lambda\) & \(S\) & \(N\) & \(I\) & (\(X\)/\(d\))\({}_{\text{test}}\) \\ \hline  
& 58.05 & 12.97 & 11.26 & 121.79 & 6.55 & 5.91 \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensionless data with penetration depths for regression analysis. In order to explore the distribution of predictors, a boxplot (or a box-and-whisker plot) of each predictor for the different penetration ranges is shown in Figure 5. The median is shown by the vertical line inside the box, whereas the left and right sides of the box are the first and third quartiles, respectively. Most of the data lie in the first and third quartiles, and the lines that are referred to as whiskers extend on the right and left sides of the box to indicate the range. There is a clear separation of \(I\), \(N\), and \(\lambda\) in the different penetration ranges. The median of \(S\) is the same for different penetration ranges, while there is more deep penetration data. More outliers in all predictors in the medium penetration are observed, as seen by the points plotted beyond the whiskers. Figures 6-8 show pair plots of the predictors for narrow, medium, and dense penetration ranges, respectively. The diagonals are density plots calculated from the data through a kernel density estimate (KDE) [20,21]. The KDE plots can be considered as smoothed histograms. The plots above the diagonal are scattered plots with bivariate KDE contours overlapping the scatter points. The bivariate KDE plot estimates the probability density of two variables. The shaded contours represent different density levels. The plots below the diagonal are scattered plots with linear regressions among predictors. In the narrow penetration range, the distribution of \(I\), \(N\), and \(S\) is skewed, with a high correlation between \(N\) and \(S\). In the medium penetration range, the distribution of all predictors is skewed with a high correlation of \(N\) with \(I\) and \(\lambda\), respectively. The I and S distributions are skewed in the deep penetration range, whereas \(N\) and \(\lambda\) have two modes. Furthermore, all predictors are weakly correlated in the deep penetration range.