## 3 Proposed Methodology
### Optimization of Support Vector Machine  
The position and orientation of the separating hyperplane influence the optimization problems that greatly influence the computation of the threshold and the classification of test and validation data and unknown data points. In the optimization phase, \(\theta\)\(\in\)\(\mathcal{O}\) is an important attribute to balance between margin maximization and error toleration. A large value of \(\theta\) leads to the least training errors while small values produce a larger margin, leading to more errors and more training points placed inside the margin. From this, it isnot possible to select a suitable parameter value since the number of training errors cannot be interpreted as an estimate for the absolute problem formulation. Since we consider unbalanced data, it is sensible to weigh incorrect categorizations of positive and negative points differently to obtain sensitive hyperplanes, thus, replacing the single parameter \(\theta\) with two values:  
\[\theta_{x}=\left\{\begin{array}{l}\theta^{+}if\ B_{x}=1,\\ \theta^{-}\ otherwise\end{array}\right.\ \ \ \ x=1,\ldots.l \tag{4}\]  
Likewise, the standard kernel of single parameter Z does not consider different feature scaling, thus it needs to be replaced with a multi-parameter Gaussian kernel.  
\[K^{\mathrm{G}}\left(A_{x},A_{y}\right)=\exp\left(-\sum_{k=1}^{n}\frac{\left(A _{x}^{k},A_{y}^{k}\right)^{2}}{2\theta_{k}^{2}}\right) \tag{5}\]  
The learning parameters are also kernel parameters and should be selected carefully to obtain a good classifier. The selection of appropriate learning parameters is a crucial step in obtaining well-tuned Support Vector Machines (SVMs) as this controls the behavior and performance of the SVM model. Grid search is used to find the appropriate parameter, where a finite number of possible parameter values are predefined, and all combinations of these values are evaluated to find the best-performing combination. However, the complexity of grid search grows exponentially with the number of parameters, limiting its practicality when dealing with a large number of parameters. These parameters are fine-tuned using numerical optimizer to define an objective function, a so-called quality measure to evaluate cross validation results, and are defined as \(\Pi\):P\(\rightarrow\)\(\emptyset\) for parameter vectors p in the parameter space P. It introduces a fitness relation:  
\[p^{j}>p^{j}\Leftrightarrow\Pi\left(p^{j}\right)>\Pi\left(p^{j}\right) \tag{6}\]  
\[p^{j}\sim p^{j}\Leftrightarrow\Pi\left(p^{j}\right)=\Pi\left(p^{j}\right) \tag{7}\]  
for all p\({}^{\mathrm{i}}\),p\({}^{\mathrm{j}}\)\(\in\)P and this is used in to ensure the good selection of parameter values. The percentage of simplest quality measures is validated for errors d and the error count measure is only the \(1+1\) discrete values 0, \(1/1\),..., 1. Good generalization abilities are required for SVM model in classification function that determines the data that was used during the training. The unbalanced datasets of cost for a false negative classification is significantly higher than for a false positive. Then F-measure  
\[F=\frac{2.pr.se}{pr+se}\in[0,1] \tag{8}\]  
Weighting between sensitivity is denoted by single measure defined as \(se\) and precision \(pr\), i.e.,  
\[se=\frac{tp}{p}\in[0,\ 1]\ \&\ pr=\frac{tp}{tp+fp}\in[0,\ 1] \tag{9}\]  
where, for validation of test set of \(p\), \(tp\), and \(fp\), denote the number of positive, true positive, and false positive points, respectively. Suppose se = 0 and pr = 0 is defined as F to be zero. Ideally, the F-measure shall be close or equal to 1.