## Section: New Results

### Dimension Reduction for categorical data

Participants : Marie Chavent, Vanessa Kuentz, Jérôme Saracco.

**Clustering of variables.**

Clustering of variables is studied as a way to arrange variables into homogeneous clusters, thereby organizing data into meaningful structures. Once the variables are clustered into groups such that variables are similar to the other variables belonging to their cluster, the selection of a subset of variables is possible. Several specific methods have been developed for the clustering of numerical variables. However concerning categorical variables, much less methods have been proposed. In [15] , we extend the criterion used by Vigneau and Qannari (2003) in their Clustering around Latent Variables approach for numerical variables to the case of categorical data. The homogeneity criterion of a cluster of categorical variables is defined as the sum of the correlation ratio between the categorical variables and a latent variable, which is in this case a numerical variable. We show that the latent variable maximizing the homogeneity of a cluster can be obtained with Multiple Correspondence Analysis. Different algorithms for the clustering of categorical variables are proposed: iterative relocation algorithm, ascendant and divisive hierarchical clustering. The proposed methodology is illustrated by a real data application to satisfaction of pleasure craft operators.

**Rotation in Multiple Correspondence Analysis**

Multiple Correspondence Analysis (MCA) is a well-known multivariate method for statistical description of categorical data. Similarly to what is done in Principal Component Analysis (PCA) and Factor Analysis, the MCA solution can be rotated to increase the components simplicity. The idea behind a rotation is to find subsets of variables which coincide more clearly with the rotated components. This implies that maximizing components simplicity can help in factor interpretation and in variables clustering. In [16] , we propose a two-dimensional analytic solution for rotation in MCA. Similarly to what is done by Kaiser (1958) for PCA, this planar solution is computed in a practical algorithm applying successive pairwise planar rotations for optimizing the rotation criterion. This criterion is a varimax-based one relying on the correlation ratio between the categorical variables and the MCA components. A simulation study is used to illustrate the proposed solution. An application on a real data set shows the possible benefits of using rotation in MCA.