Generalized Permutation Polytopes and Exploratory Graphical Methods for Ranked Data

Abstract

Exploratory graphical methods for fully and partially ranked data are proposed. In fully ranked data, $n$ items are ranked in order of preference by a group of judges. In partially ranked data, the judges do not completely specify their ranking of the $n$ items. The resulting set of frequencies is a function on the symmetric group of permutations if the data is fully ranked, and a function on a coset space of the symmetric group if the data is partially ranked. Because neither the symmetric group nor its coset spaces have a natural linear ordering, traditional graphical methods such as histograms and bar graphs are inappropriate for displaying fully or partially ranked data. For fully ranked data, frequencies can be plotted naturally on the vertices of a permutation polytope. A permutation polytope is the convex hull of the $n$! points in $\mathbb{R}^n$ whose coordinates are the permutations of $n$ distinct numbers. The metrics Spearman's $\rho$ and Kendall's $\tau$ are easily interpreted on permutation polytopes. For partially ranked data, the concept of a permutation polytope must be generalized to include permutations of nondistinct values. Thus, a generalized permutation polytope is defined as the convex hull of the points in $\mathbb{R}^n$ whose coordinates are permutations of $n$ not necessarily distinct values. The frequencies with which partial rankings are chosen can be plotted in a natural way on the vertices of a generalized permutation polytope. Generalized permutation polytopes induce a new extension of Kendall's $\tau$ for partially ranked data. Also, the fixed vector version of Spearman's $\rho$ for partially ranked data is easily interpreted on generalized permutation polytopes. The problem of visualizing data plotted on polytopes in $\mathbb{R}^n$ is addressed by developing the theory needed to define all the faces, especially the three and four dimensional faces, of any generalized permutation polytope. This requires writing a generalized permutation polytope as the intersection of a system of linear equations, and extending results for permutation polytopes to generalized permutation polytopes. The proposed graphical methods is illustrated on five different data sets.