Voronoi Homogeneity Analysis (2026-05-09)
In Homogeneity Analysis (HA), better known as Multiple Correpondence Analysis (MCA), there are n objects and m variables that measure the objects. By “measure” we mean that each variable j maps the n objects into a set of kj possible values, the categories of the variable. The number of categories can be finite or infinite and the categories can be nominal, ordinal, or numerical. In actual data analysis the number of categories that are actually present in the data will necessarily be finite.
HA maps objects and categories into a p-dimensional Euclidean space. The numerical output is an n x p matrix X of object scores and m matrices Yj, of dimension kj x p, called category quantifications for nominal and ordinal variables and category transformations for numerical variables. In HA, in the form in which it is implemented in Gifi (1990), we require that X is column-centered with X’X = nI. The category quantifications Yj are the centroids of the scores of the objects in the category (we also say that objects “belong to” a category or are “in” a category)
The most interesting graphical outputs of HA are the star plots, which are joint plots of objects scores and category quantifications/transformations. There is one star plot for each variable. In the star plot we draw stars by connecting each object with the categories the object belongs to. The resulting kj graphs are star-shaped, because the category point is the centroid of “its” object points.
In the paper we first reformulate HA as a form of non-metric multidimensional scaling, using Kruskal’s stress and requiring that the distances between object-points and the category-points the objects belong to are zero. We then relax this to the requirement that the distance of an object-point to the point of the category the object belongs to must be less than or equal to the distance to the other category points of the same variable. We adapt the monotone regression routines and the normalization of the solution to this requirement. The classical HA solution is used as an initial estimate of the smacof iterations.
Geometrically the category-points of a variable define a partition of the space into Voronoi regions. Our new requirement is that all object-points are in the Voronoi region of the category they belong to. For each variable the star graphs are replaced by regions partitioning the space. There is no centroid relation between category-points and object-points are more – the object points can be anywhere in the Voronoi region of the category they belong to.
The paper is at
https://jansweb.netlify.app/publication/deleeuw-e-26-f/
and the files are at
