Clustering N Objects into K Groups under Optimal Scaling of Variables

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Abstract
We propose a method to reduce many categorical variables to one variable with k categories, or stated otherwise, to classify n objects into k groups. Objects are measured on a set of nominal, ordinal or numerical variables or any mix of these, and they are represented as n points in p-dimensional Euclidean space. Starting from homogeneity analysis, also called multiple correspondence analysis, the essential feature of our approach is that these object points are restricted to lie at only one of k locations. It follows that these k locations must be equal to the centroids of all objects belonging to the same group, which corresponds to a sum of squared distances clustering criterion. The problem is not only to estimate the group allocation, but also to obtain an optimal transformation of the data matrix. An alternating least squares algorithm and an example are given.