## 2 Materials and Methods
### SVM Model and Its Variants  
One of the most prominent supervised machine learning (ML) techniques that apply statistical learning principles and the necessary risk minimization system is the SVM [91]. This technique revises the non-linear system into a linear format by creating a hyperplane and converting the aforementioned system into a simplistic and processable setup [92] as shown in Figure 2. The data transmutation is conducted utilizing an analytical and precise function recognized as the Kernel function. The SVM intends to obtain the best margin of division between the groups and creates a classification hyperplane within the middle of the most significant margin [93]. These couple classes are named as \"+1\" (positive samples), which indicates the circumstance over the hyperplane, and \"\(-1\)\" (negative samples) describes the circumstance under the hyperplane. The characteristics of new data afterward can forecast the assortment to which a new record should fit.  
The abovementioned step is executed for both classification and regression. With regards to classification, the aforementioned minimization is made assuming that all samples are entirely classified, while the regression analysis follows the provision that the \"\(y\)\" value of each example varies less than the demanded precision of \(\epsilon\) from \(f(x)\). For classification, the main aim is to find a function \(f(x)=wx+b\) where \(f(x)\geq 1\) for positive examples and \(f(x)\leq-1\) for negative examples. Under these conditions, we want to maximize the margin which is nothing more than minimizing the derivative of \(f^{\prime}=w\). For regression, the objective is to determine a function \(f(x)=wx+b\) (pale diagonal line) following the condition that \(f(x)\) is within a required accuracy \(\epsilon\) from the value \(y(x)\) (vertical bars) of every data point, namely \(\,\mid y(x)-f(x)\mid\,\leq\epsilon\) where epsilon is the distance between the dashed and the pale  
Figure 2: Representation of SVM.  
This research examined the SVM model with diverse kernels, including RBF, LIN, SIG, and POL to predict the PPV caused by quarry blasting. Typically, \"Kernel\" refers to implementing a linear classifier to resolve a non-linear problem. In ML technique, this Kernel is also called \"Kernel trick\". LIN Kernel is suitable for simple and linearly separated data. Otherwise, other functions should be employed. It is worth mentioning that the SIG are identical to the RBF for some parameters of SVM [94]. The kernel of LIN is the particular form of the RBF and in circumstances that RBF is adopted during processing, it is unnecessary to apply the kernel of LIN. With regards to precision, the RBF has a greater ability to interpolate compared to the SIG. This triggers RBF to produce additional consistent outcomes. Instead, the RBF is not able to create longer-range extrapolation. The SIG may have a great inconsistency since it is not severely positive certain which may cause incorrect calculation. In a study by [PERSON] et al. [95], it was asserted that the POL is able to produce better extrapolations. Figure 4 presents the kernels' formulas. This study utilized every type of kernels to investigate the efficiency of each kernel to predict the PPV induced by blasting. Figure 4 shows that there are some critical coefficients like \"\(\gamma\)\" and \"\(d\)\" for different kernels such as RBF and POL that need to be designed. In Figure 4, \"\(\gamma\)\" is the kernel width and \"\(d\)\" is degree of polynomial kernel. It is vital to discover the correct value of \"\(\gamma\)\" and \"\(d\)\" because \"\(\gamma\)\" regulates the level of nonlinearity of the SVM model and \"\(d\)\" determines the level of the polynomial kernel.