Spherical Subfamily Models

A new method is presented for modeling low-dimensional representations of high-dimensional multinomial and compositional data. The data are t to subfamilies of the multinomial family which are deened using the multinomial information geometry. These collections of spherical subfamilies have a number of advantages over the aane subfamilies contructed by methods such as canonical and correspondence analysis, traditionally t to such data. First, they can describe more complex shapes in the data, and are particularly well-suited to modelling sparse data. Second, the subfamilies provide a convenient variance-stabilizing parametrization for the tted data. An algorithm which uses iterative Singular Value Decompositions is presented for tting the models. Two example applications are presented: one is to Latent Semantic Indexing, a method for the automatic indexing of text documents. The ability of the method to model sparse data is an advantage here. A second example is an analysis of compositional data from a geological study. This example shows the ability of the method to model curvature in the data, and illustrates its variance stabilization properties.