Reduced $k$-means clustering with MCA in a low-dimensional space

Abstract

In the two-step sequential approach called tandem analysis, we focus on applying a clustering algorithm on estimated object scores after dimensional reduction of variables. In this approach, reduction may obscure or mask taxonomic information (Arabie and Hubert in Handbook of marketing research. Blackwell, Oxford, 1994). As an alternative to tandem analysis, an approach combining two methods for categorical data is proposed by Hwang et al. (Psychometrika 71:161–171, 2006); however, this method does not consider the removal of object scores estimated as a vector of $1$ that has no meaning in the first dimension. In this study, we propose a method for clustering objects consisting of categorical variables in a low-dimensional space. Our proposed method uses simultaneous analysis of multi-dimensional nonmetric principal component analysis and $k$-means clustering for categorical data; that is, we reduce dimensions with category quantifications, thus clustering object scores. We display object scores and variable categories, and therefore, every relationship between objects and categories can be interpreted for each cluster. Using simulated data, this method has been compared with tandem clustering and applied to real world data.