# Comparative Analysis of Major Machine-Learning-Based Path Loss Models for Enclosed Indoor Channels
## 4 Machine-Learning-Based Models
### Support Vector Regression Model  
[PERSON] [80] proposed SVM algorithms for the binary classification problem. Later on, they worked on both multiclassification and regression problems, known as support vector classification (SVC) and SVR algorithms. The SVR applies a similar concept as the SVC algorithm with some changes. A few changes include that the target values are real numbers, the infinite possibility of which became challenging to predict using the same SVC. However, the SVR selects a boundary distance \(\{-\epsilon,\epsilon\}\) from the original hyperplane to predict the real numbers. This boundary distance is the margin of tolerance that takes only data points within this boundary. Therefore, the main goal is always similar: minimizing the prediction error and individualizing the hyperplane to maximize the margin [81].  
As aforementioned, the SVR applies a similar concept to SVC, but a target variable is a real number \(Y\in\mathbb{R}\). As stated by [PERSON] and [PERSON] [82] and [PERSON] et al. [83], the SVR seeks the linear regression function as an alternative to finding the hyperplane in the SVC by Equation (10). This can be achieved by selecting a threshold error \(\epsilon\), which is defined to minimize the expression in Equation (11). This expression is called the \(\epsilon\)-insensitivity loss error function. The SVR regression process, therefore, seeks to minimize \(\epsilon\) in Equation (11) and \(||W||^{2}\) in the expression of \(R\). The target value of the SVR method is given by:  
\[Y=W^{T}X+b, \tag{10}\]  
where \(Y\) is the target, \(W\) is the coefficient, \(X\) is the input feature, and \(b\) is a constant. We define:\[|\hat{Y}-Y|_{\epsilon}=\begin{cases}Zero,&|\hat{Y}-Y|\leq\epsilon\\ |\hat{Y}-Y|-\epsilon,&\text{otherwise}\end{cases} \tag{11}\]  
\[R=\frac{1}{\epsilon}||W||^{2}+C\Bigg{(}\sum_{i=1}^{N}|Y_{i}-\hat{Y}_{i}|_{ \epsilon}\Bigg{)}. \tag{12}\]  
Tolerance variables are also introduced, defined as the value in excess of \(\epsilon\) and \(\xi\) to limit the value to the regression target. Thus, the minimization of Equation (12) is updated to Equation (13), under the conditions of Equations (14) and (15) for \(\xi_{i}\) and \(\xi_{i}^{*}\leq 0\) and \(i=1,2,3,\ldots,N\). That is:  
\[R=\frac{1}{\epsilon}||W||^{2}+C\sum_{i=1}^{N}|\xi_{i}-\hat{\xi}_{i}^{*}|_{ \epsilon}, \tag{13}\]  
\[(W^{T}X_{i}+b)-Y_{i}\leq\epsilon-\xi_{i}, \tag{14}\]  
\[Y_{i}-(W^{T}X_{i}+b)\leq\epsilon+\xi_{i}^{*}. \tag{15}\]  
The standard kernel functions are considered in this study, given explicitly by the linear, radial, and polynomial functions in Equations (16)-(18), respectively.  
\[K(X_{i},X_{j})=X_{i}^{T}X_{j}, \tag{16}\]  
\[K(X_{i},X_{j})=e^{-\lambda||X_{i}-X_{j}||},\lambda>0, \tag{17}\]  
\[K(X_{i},X_{j})=(X_{i}^{T}X_{j}+1)^{d}. \tag{18}\]  
Note that the shape of the kernel function directly influences the values obtained by the SVR regression. Similarly, the constant \(C\) in Equation (12) and the parameters \(\lambda\) and \(d\) in Equations (17) and (18) should be optimized. For this purpose, the Optuna technique is applied to choose the optimal parameters for \(C,\lambda\), and \(d\), considering the lowest RMSE. According to our developed SVR model, the values of the primary hyperparameters obtained from the Optuna hyperparameter tuning technique are: (1) The kernel adopted was the radial basis function (RBF). (2) The kernel coefficient for RBF equals 0.001. (3) \(C=995.2783\). Figure 2 shows the principle of using the SVR in two dimensions.  
Figure 2: The principle of the SVR in two dimensions.