Graphical Techniques for Ranked Data

Abstract

Exploratory graphical methods are critically needed for displaying ranked data. Fully and partially ranked data are functions on the symmetric group of n elements, S _n , and on the related coset spaces. Because neither S _n nor its coset spaces have a natural linear ordering, graphical methods such as histograms and bar graphs are inappropriate for displaying ranked data. However, a very natural partial ordering on S _n and its coset spaces is induced by two reasonable measures of distance: Spearman’s p and Kendall’s τ. Graphical techniques that preserve this partial ordering can be developed to display ranked data and to illustrate related probability density functions by using permutation polytopes. A polytope is the convex hull of a finite set of points in ℜⁿ⁻¹, and a permutation polytope is the convex hull of the n! permutations of n elements when regarded as vectors in ℜⁿ (see, for example, Yemelichev, et. al.[9]). This concept is closely related to the observation by McCullagh [7] that the n! elements of S _n lie on the surface of a sphere in ℜⁿ⁻¹.