GENERALIZED PERMUTATION POLYTOPES AND EXPLORATORY GRAPHICAL METHODS FOR RANKED DATA

Exploratory graphical methods for fully and partially ranked data are proposed. In fully ranked data, n items are ranked in order of prefere nce 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 th e data is fully ranked, and a function on a coset space of the symmetr ic group if the data is partially ranked. Because neither the symmetri c group nor its coset spaces have a natural linear ordering, tradition al graphical methods such as histograms and bar graphs are inappropria te for displaying fully or partially ranked data. For fully ranked dat a, frequencies can be plotted naturally on the vertices of a permutati on polytope. A permutation polytope is the convex hull of the n! point s in 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 no ndistinct values. Thus, a generalized permutation polytope is defined as the convex hull of the points in R(n) whose coordinates are permuta tions of n not necessarily distinct values. The frequencies with which partial rankings are chosen can be plotted in a natural way on the ve rtices 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 R(n) is addr essed by developing the theory needed to define all the faces, especia lly the three and four dimensional faces, of any generalized permutati on polytope. This requires writing a generalized permutation polytope as the intersection of a system of linear equations, and extending res ults for permutation polytopes to generalized permutation polytopes. T he proposed graphical methods is illustrated on five different data se ts.