## 2 Materials and Methods
### Information Technologies Used in the Proposed Model
#### 2.2.2 Fundamentals of the \"Logistic Regression\" Algorithm  
Logistic Regression, also known in the literature as \"logit regression\", \"maximum entropy classification (MaxEnt)\", or \"log-linear classifier\", consists of a Machine Learning algorithm for classification problems and predictive analysis based on the probability concepts [36]. Logistic Regression is used to assign observations to a discrete set of classes. Logistic Regression is one of the most popular Machine Learning algorithms for binary classification, given a set of independent variables, and is used to predict a binary result (1 or 0, Yes or No, True or False) [32]. It has been applied successfully in various areas, such as Medicine [37], Finance [38], and Economics [39]. In this model, the probabilities describing the possible outcomes of a single trial are modeled using a logistic function, namely the logical relationship between a dichotomous response variable and a series of numerical (continuous, discrete) or categorical explanatory variables. [37; 38; 39; 40]. Logistic Regression is a type of linear regression in which the probability log is used as a dependent variable. It predicts the probability of an event occurring by adjusting data to a logit function [41]. The algorithm makes several assumptions, such as independence, answers (logits) at all subpopulation levels, where the explanatory variable is normally distributed. The variance is constant between reactions and all the values of the explanatory variable. Intuitively, a transformation for the response variable is applied to produce a continuous probability distribution over the output classes bounded between 0 and 1; this transformation is called a \"logistic\" or \"sigmoid\" function [42]. A logistic function or logistic curve is an S, sigmoid curve, generated by the Equation (1) [43]:  
\[f(x)=\frac{L}{1+e^{-k(x-x_{0})}} \tag{1}\]  
where:  
\(x_{0}=\) value of the midpoint of the sigmoid curve,  
\(L=\) maximum value of the curve,  
\(k=\) the logistic growth rate or slope of the curve. For values of \(x\) in the real-number domain from \(-\infty\) to \(+\infty\), the S curve, standard logistic sigmoid function, i.e., \(L=1\), \(k=1\), \(x_{0}=0\), as shown in Figure 1, is obtained in the graph with \(f\) approaching \(L\) when \(x\) approaches \(+\infty\) and with \(f\) approaching 0 (zero) when \(x\) approaches \(-\infty\). Logistic regression is more of a classifier than a regression technique, despite its name. This statement is especially true in medicine or the psychosocial sciences, focusing on predicting and explaining [44]. Moreover, for Logistic Regression, the response variable is quantitative; that is, the log of the chance of being classified in the i-th group of a binary or multiclass answer [45]. The probability of a positive event or diagnosis can be estimated directly in Logistic Regression. In the case of a dependent variable \(Y\) assuming only two possible responses or binary states of interest (True or False), there is also a set of \(p\) independent variables, \(X_{1},X_{2},\ldots,X_{p}\). The logistic regression model can be written as follows:  
\[P(Y=1|X_{1},\ldots,X_{p})=\frac{\exp(\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{p}X_ {p})}{1+\exp(\beta_{0}+\beta_{1}X_{1}+\ldots+\beta_{p}X_{p})^{\prime}} \tag{2}\]  
The coefficients \(\beta_{0},\beta_{1},\ldots,\beta_{p}\) are the regression coefficients that are estimated by the maximum likelihood of the data set. The probability of \(Y=1\) for a new instance is then estimated by replacing \(\beta\) with its estimated counterparts and \(X\) with its realizations for the new instance considered in Equation (2). The new instance is then assigned to class \(Y=1\) if \(P\left(Y=1\right)>c\), where c is a fixed limit, and class \(Y=0\) otherwise. The commonly used limit c = 0.5 produces a classifier called Bayes [46; 47]. For discrimination of two groups, the classification rule is as follows:  
* if \(P(Y=1)>0.5\), then \(Y=1\) is classified. * if \(P(Y=1)<0.5\), then \(Y=0\) is classified.