Generalized Permutation Polytopes and Exploratory Graphical Methods for Ranked Data

Abstract

Exploratory graphical methods for fully and partiallyranked data are proposed. In fully ranked data, n itemsare ranked in order of preference by a group of judges.In partially ranked data, the judges do not completelyspecify their ranking of the n items. The resulting setof frequencies is a function on the symmetric group ofpermutations if the data is fully ranked, and afunction on a coset space of the symmetric group if thedata is partially ranked. Because neither the symmetricgroup nor its coset spaces have a natural linearordering, traditional graphical methods such ashistograms and bar graphs are inappropriate fordisplaying fully or partially ranked data. For fullyranked data, frequencies can be plotted naturally onthe vertices of a permutation polytope. A permutationpolytope is the convex hull of the n! points in Rnwhose coordinates are the permutations of n distinctnumbers. The metrics Spearman's {Ï} and Kendall's{Ï}„ are easily interpreted on permutationpolytopes. For partially ranked data, the concept of apermutation polytope must be generalized to includepermutations of nondistinct values. Thus, a generalizedpermutation polytope is defined as the convex hull ofthe points in Rn whose coordinates are permutations ofn not necessarily distinct values. The frequencies withwhich partial rankings are chosen can be plotted in anatural way on the vertices of a generalizedpermutation polytope. Generalized permutation polytopesinduce a new extension of Kendall's {Ï}„ forpartially ranked data. Also, the fixed vector versionof Spearman's {Ï} for partially ranked data iseasily interpreted on generalized permutationpolytopes. The problem of visualizing data plotted onpolytopes in Rn is addressed by developing the theoryneeded to define all the faces, especially the threeand four dimensional faces, of any generalizedpermutation polytope. This requires writing ageneralized permutation polytope as the intersection ofa system of linear equations, and extending results forpermutation polytopes to generalized permutationpolytopes. The proposed graphical methods isillustrated on five different data sets.