# Comparative Analysis of Major Machine-Learning-Based Path Loss Models for Enclosed Indoor Channels
## 4 Machine-Learning-Based Models
### Linear Regression Models  
Linear Regression Models are essential techniques for addressing the regression challenges in machine learning using data modeling. The linear regression models contain various types of models and depend upon several elements [70; 71; 72; 73; 74; 75; 76; 77; 78; 79]. These elements incorporate the type of target variable, the shape of the regression line, and the number of independent variables. This work adopted two types of linear regression models to predict the path loss for the selected enclosed indoor environment. These models are MLR and PR. The MLR model is a predictive model that considers more than one input to predict the target. This model identifies the correlation between the various features (dependent variables) and the target (independent variable) to find the best fit for the measured data. The MLR assumes that the inputs are \(X_{1},X_{2},X_{3},\ldots,X_{n}\), and seeks to predict the target real-value \(Y\). The MLR model has the form:  
\[Y=f(X)=\beta_{0}+\sum_{i=1}^{n}\beta_{i}X_{i}+\epsilon, \tag{6}\]  
where \(\beta_{i}\) are unknown coefficients (model coefficients) and \(\epsilon\) defines the errors or noise. The most famous estimator that is used to estimate the coefficients model is the LeastSquares (LS), in a way that it picks the coefficients \(\beta_{0},\beta_{1},\beta_{3},\dots,\beta_{n}\) to minimize the mean squares error (MSE) presented in the following equations:  
\[\text{MSE} =\sum_{j=1}^{M}(Y_{j}-\beta_{0}-f(X_{j}))^{2}, \tag{7}\] \[=\sum_{j=1}^{M}\Bigg{(}Y_{j}-\beta_{0}-\sum_{i=1}^{N}X_{ji}\beta_{ j}\Bigg{)}^{2}. \tag{8}\]  
However, if basic expansion is made to Equation (6) by substituting \(X_{2}=X_{1}^{2}\), \(X_{3}=X_{1}^{3}\),..., this leads to a new form of model called polynomial regression representation. A polynomial regression model can be defined as a new function that takes the form:  
\[Y=f(X)=\Gamma_{0}+\sum_{n=1}^{N}\Gamma_{n}X^{n}, \tag{9}\]  
where \(n\) is a polynomial degree of the PR model and \(\Gamma_{n}\) represents the model coefficients of PR, where the LS method estimation is applied to estimate the PL model coefficient using the measured data. The polynomial regression model attempts to generate a polynomial function that estimates the measured data points. It determines the best-fit curve that passes through the entire measured data to minimize the predicted error.  
The PR model is a modified version of the MLR model where the relationship between the independent and dependent variables is defined by the \(n\)-th degree. The best fit curve in polynomial regression passes through all the data points, depending on the power of \(X\) (or the value of \(n\)). It is recommended to analyze the turn towards the end as the higher polynomials can give undesired results.  
The number of iterations adopted using the Optuna technique was 100 for both the MLR and PR methods. Moreover, the number of degrees for the PR model was 6. Finally, it is worth noting that the datasets were normalized before training and testing these models to achieve the best performance.