## 2 Background
### State-of-the-Art Algorithms  
CFD strategies have existed since the late 1990s and initially relied on expert driven technologies. In recent times, ML has predominated, as these new techniques offer better results in terms of accuracy [1; 2; 3; 6]. Good examples of these techniques are linear classifiers, which allow us to transform the input space into the weights space. This strategy links the contributions of the input variables to the result in a traceable way, offering direct interpretability of the final outcome [13]. In the following paragraphs, we summarize the linear ML algorithms we evaluated in this work, but let us first introduce the notation to be used throughout the paper. Let \(\mathbf{X}\in\mathbf{R}^{N\times L}\) be the input data matrix containing the input set of vectors in rows, with \(N\) observations of \(L\) features, where \(\mathbf{x}_{n}\) is a column vector with \(L\) features for \(n=1,\ldots,N\). We consider a classification problem with a binary output variable grouped in vector \(\mathbf{y}\in R^{N}\), such that \(\mathbf{y}_{i}e\{-1,+1\}\) for \(n=1,\ldots,N\). The relationship among each input feature and the output class is represented by the feature weights in vector \(\mathbf{w}\) learned by a linear classifier method so that \(\mathbf{y}_{n}=\mathbf{w}^{T}\mathbf{x}_{n}+b\) represents the discriminant function, and the sign of \(\mathbf{y}_{l}\) can be used as the decision output. Linear regression (LR)models the relationship between multiple input variables by fitting them through a linear equation. One variable is considered a dependent variable (\(\mathbf{y}_{n}\)), and the other variables \(\mathbf{x}_{n}\) are considered to be explanatory variables [26]. In general terms, this relationship may not hold for the whole sample base, and so a noise term or error terms need to be added to the mathematical formulation (\(\varepsilon\)). We can describe the underlying relationship with the following equation:  
\[\mathbf{y}_{l}=\mathbf{w}^{T}\mathbf{x}_{n}+b+\varepsilon_{l} \tag{1}\]  
Linear discriminant analysis (LDA)is a generalization of [PERSON]'s linear discriminant [27]. LDAis able to find a linear combination of features characterizing two or more sets with classification purposes, and for dimensionality reduction previous to subsequent classification. The main difference between LR and LDAis that LR analysis deals with a continuous dependent variable, whereas LDA must have a discrete dependent variable. LDAims to represent a single dependent variable as a linear combination of others. Given two classes of multidimensional observations, with means \(\mathbf{m}_{0},\mathbf{m}_{1}\) and covariances \(\mathbf{\Sigma}_{0},\mathbf{\Sigma}_{1}\), the linear combination of features \(\mathbf{w}^{T}\mathbf{x}\) has mean \(\mathbf{w}^{T}\mathbf{m}_{i}\) and variance \(\mathbf{w}^{T}\mathbf{\Sigma}_{i}\mathbf{w}\), for \(i=0,1\). The separation between these two distributions can be defined as the ratio of the variance between the classes to the variance within the classes, i.e.,  
\[S=\frac{\sigma_{b}^{2}}{\sigma_{w}^{2}}=\frac{\left(\mathbf{w}^{T}(\mathbf{m}_ {1}-\mathbf{m}_{0})\right)^{2}}{\mathbf{w}^{T}(\mathbf{\Sigma}_{1}+\mathbf{ \Sigma}_{0})\mathbf{w}} \tag{2}\]  
where \(\sigma_{b}^{2}\) represent the variances between the classes and \(\sigma_{w}^{2}\) represent the variances within the classes. It can be shown that the maximum separation happens when  
\[\mathbf{w}=c(\mathbf{\Sigma}_{0}+\mathbf{\Sigma}_{1})^{-1}(\mathbf{m}_{1}- \mathbf{m}_{0}) \tag{3}\]  
where \(c\) is a constant. When LDA assumptions are fulfilled, this is equivalent to LDA equations. For high-dimensional input feature spaces with highly correlated covariates, this algorithm also can exhibit some instability; hence, we often regularize the matrix inversion and use the following equation:  
\[\mathbf{w}=c(\mathbf{\Sigma}_{0}+\mathbf{\Sigma}_{1}+\lambda\mathbf{I})^{-1}( \mathbf{m}_{1}-\mathbf{m}_{0}) \tag{4}\]  
where \(\lambda\) is the regularization parameter, and \(\mathbf{I}\) is the identity matrix.