## 3 Methods and Materials
### MCA for Categorical Feature Description  
MCA is a well-known method in multivariate statistical analysis which is being given increasing attention in Big Data applications using categorical variables. It can be seen as the counterpart of Principal Component Analysis (PCA) in metric features, since both are based on matrix eigendecompositions of the input data. The main difference between them is that the former represents a variance orthonormalization, whereas the later represents a probability-space orthonormalization. We follow some of the principles pointed out in previous works on MCA (see, e.g., [42]), but with emphasis here on the behavior and adaptability of the method to summarize categorical features together with the description of a number of cases, as is usually the case of the client view in CRMs in hospitality. The identification of relations among features and samples is not straightforward, specially in datasets consisting of categorical variables. Correspondence analysis (CA) is a well-known exploratory bivariate technique for analyzing contingency or count tables both visually and numerically [17]. In contrast, MCA can be seen as a generalization of CA, which allows us to study relationships among categorical variables [18; 42]. The method first transforms the categorical variables in order to construct a binary matrix \(\mathbf{X}\in\mathbb{R}^{I\times I}\), being \(I\) and \(J\) the total number of samples and categories in our dataset, respectively. Note that \(J\) is not the number of categorical features, but the sum of all of the different categories related to the categorical features. Then, MCA decomposes this \(\mathbf{X}\) matrix into two sets of projection matrices (also known as factor scores), one for the dimension of the samples (rows) and another for the dimension of the categories (columns), in order to be able to project the samples and features to points on a low-dimensional space,which allows us to establish relationships among them through distances in this projection space. In addition, let  
\[\mathbf{Z}=\frac{1}{M}\mathbf{X}^{\top}\mathbf{X} \tag{1}\]  
where denote the corresponding relative frequency matrix, known as the Burt matrix, being \(M=I\times J\) the number of elements of the \(\mathbf{X}\) matrix. Furthermore, let us define \(\mathbf{r}\) and \(\mathbf{c}\) as the vectors of the horizontal and vertical sum of \(\mathbf{Z}\), respectively, i.e., \(\mathbf{r}=\mathbf{1}^{\top}\mathbf{Z}\) and \(\mathbf{c}=\mathbf{Z}\mathbf{1}\), where \(\mathbf{1}\) is a column vector of ones. Lastly, let \(\mathbf{D_{c}}\) and \(\mathbf{D_{r}}\) represent diagonal matrices whose diagonal entries are the elements of \(\mathbf{c}\) and \(\mathbf{r}\), respectively. The projection matrices can be obtained through the following singular value decomposition of the normalized Burt matrix:  
\[\mathbf{D_{r}}^{-\frac{1}{2}}(\mathbf{Z}-\mathbf{r}\mathbf{c}^{T})\mathbf{D_ {c}}^{-\frac{1}{2}}=\mathbf{P}\mathbf{\Lambda}\mathbf{Q}^{T}, \tag{2}\]  
where being \(\mathbf{P}\) and \(\mathbf{Q}\) the left and right singular vector matrices, respectively, and \(\mathbf{\Lambda}\) a diagonal matrix with the singular values organized in decreasing order, where \(\mathbf{\Lambda}=\mathbf{\Lambda}^{2}\) is the matrix of eigenvalues. Thus, the factor scores corresponding to the samples and categories dimensions can be calculated as follows  
\[\mathbf{F} =\mathbf{D_{r}}^{-\frac{1}{2}}\mathbf{P}\mathbf{\Lambda},\] \[\mathbf{G} =\mathbf{D_{c}}^{-\frac{1}{2}}\mathbf{Q}\mathbf{\Lambda}, \tag{3}\]  
respectively. Note that, since \(\mathbf{Z}\) is a symmetric matrix, \(\mathbf{r}\) and \(\mathbf{c}\) are equal. Therefore, \(\mathbf{P}\) and \(\mathbf{Q}\) are the same matrices, being \(\mathbf{V}=\mathbf{P}=\mathbf{Q}\) the eigenvector matrix, and thus \(\mathbf{F}\) is also equal to \(\mathbf{G}\). Using these eigenvector and eigenvalue matrices, MCA results can be analyzed [42].