## 3 Materials and Methods
### Feature Selection  
This paper addresses a classification problem where there exist multi-classes (\(C\) classes) for each pattern. Each pattern (patient) is represented by the vector formed by the set of \(M\) variables or features (as presented in Figure 5) and an additional value that indicates the class to which it belongs, that is, \(Patient_{k}=v_{i},c_{k}\) with \(i=1,\ldots,M\) and \(c_{k}=1,\ldots,C-1\)  
Figure 4: Operating mode of the 3D-DWT wavelet decomposition applied to a specific VOI. Figure 5: Feature matrix of 3D-DWT wavelet coefficients for each specific VOI in whole and gray matter. In this contribution the number of non-balanced classes is 5, represented by 1 = Control, 2 = SWEDD, 3 = Prodomal, 4 = PD and 5 = GenCohort. Mutual information is the most general measure of statistical independence between two random variables \(v_{i}\) and \(v_{j}\) and quantifies the information that \(v_{i}\) and \(v_{j}\) share (how knowledge of one of them affects the uncertainty about the other). Formally, the mutual information (\(I\)) between two random variables is defined from the marginal density functions \(p(v_{i})\) and \(p(v_{j})\), and the joint density function \(p(v_{i},v_{j})\) as follows:  
\[I(v_{i},v_{j})=\int\int p(v_{i},v_{j})log\frac{p(v_{i},v_{j})}{p(v_{i})p(v_{j} )}d(v_{i})d(v_{j}) \tag{6}\]  
If \(v_{i}\) and \(v_{j}\) are independent, the knowledge of \(v_{i}\) does not give any information about \(v_{j}\), being in this case \(I=0\). At the other extreme, if \(v_{i}\) and \(v_{j}\) are identically distributed, the knowledge of \(v_{i}\) implies full knowledge of \(v_{j}\) and vice versa. The main difficulty in calculating Equation (6) lies in estimating the densities \(p(v_{i},v_{j})\), \(p(v_{i})\) and \(p(v_{j})\) using only the data. For categorical variables, Equation (6) is reduced to a summation, and the calculation of mutual information is immediate by performing a simple count. Therefore, instead of the evaluation of the integral in Equation (6) for continuous variables, one possible solution (simplification) is to incorporate a discretization step in the data pre-processing. Determining the appropriate discretization is not an easy task for this specific problem. In this case, other estimation methods of density functions as maximum likelihood, Bayesian estimation or non-parametric techniques were used. The objective of a variable selection method based on mutual information is to find a subset \(S\) of \(m\) variables \(S=v_{1},\ldots,v_{m}\) that jointly have the maximum dependency (MaxDep) on the target variable \(c\):  
\[maxD(S,c),D=I(v_{i},i=1\ldots,m;c) \tag{7}\]  
Although MaxDep is an optimal theoretical criterion, the estimation of \(p(v_{i},\ldots,v_{m})\) and \(p(v_{1},\ldots,v_{m},c)\) is difficult to be computed due to the fact that the number of available examples is often insufficient and/or is computationally expensive. It is possible to use MaxDep when a small number of variables are wanted to be selected and the number of patterns is large. In this way, the lack of data for the estimation of the density functions would be solved, and their calculation would be carried out in a reduced feature space where the computational cost is affordable. As an alternative to MaxDep, the set of variables based on the dependency with the class (relevance) can be selected. The criterion of Maximum Relevance consists in finding the subset of variables \(S\) that maximize \(D(S,c)\), where \(D\) has been approximated as the mean value of the mutual information between each of the variables that make up the subset \(S\) and the class, that is:  
\[D=\frac{1}{|S|}\sum_{v_{i}\in S}I(v_{i};c) \tag{8}\]  
Therefore, the optimal subset selected by the algorithm will be the \(m\) variables with more mutual information with class \(c\). It is possible that the variables selected by MaxDep are quite redundant due to a great dependency between them. If the dependency between two variables is high, its discriminative power is not greatly affected by removing one of them.