Distance Based Ranking Models

Abstract

SUMMARY A class of ranking models is proposed for which the probability of a ranking decreases with increasing distance from a modal ranking. Some special distances, namely those associated with Kendall and Cayley, decompose into a sum of independent components under the uniform distribution. These distances lead to multiparameter generalizations whose parameters may be interpreted as information at various stages in a ranking process. Estimation of model parameters is described, and the results are applied to an example of word associations. A censoring argument motivates simple extensions of these models to include partial rankings. The generalized Cayley distance model is illustrated for random arrangements arising from mechanisms other than ranking.

A Recursive Partitioning Method for the Prediction of Preference Rankings Based Upon Kemeny Distances

“…A serious candidate to be used as impurity measure is the well-known Kendall distance (Fligner & Verducci, 1986). Given two different rankings in the permutation polytope of n objects, the Kendall distance is equal to the total number of steps to migrate from the first to the second ranking by reversing adjacent pairs of objects (cf.…”

“…Probabilistic methods that model the ranking process include the so-called Thurstonian models, as well as distance-based and multistage models (Thurstone, 1927;Bradley & Terry, 1952;Mallows, 1957;Luce, 1957;Fligner & Verducci, 1986, 1988Critchlow, Fligner, & Verducci, For example, if n = 5 and λ = (1, 1, 1, 1, 1) then λ is expression of a full ranking in which the vector a could be equal to (1,2,3,4,5). If λ = (2, 1, 2) then λ represents a partial ranking in which two objects are tied at the first position, one object is ranked alone at the second position and two objects are tied at the last position.…”

Generating Nearly Sorted Sequences – The use of measures of disorder

“…If we want to test correlation or agreement of more than two rankings (because the objects are ranked independently by boards of judges), the corresponding techniques must be developed [19,20,21]. Random generation with bounded disorder may provide the answer for tabulating values of the distributions.…”

“…Kendall's τ , the most popular coefficient of correlation, is defined as τ = 1 − 4Inv(σ, π)/n(n − 1), where Inv(π, σ) is the minimum number of pairwise adjacent transpositions required to bring π −1 into the order σ −1 . Fligner and Verducci [21] use ri-metrics to generalize Mallow's [41] ranking models. They have studied ranking models based on Cayley's measure and the Hamming distance.…”

Prior distributions on symmetric groups

“…The Mallows model is unimodal with the probability of a ranking π decreasing as the distance in a certain metric between π and the mode increases. See Fligner and Verducci (1986, 1988, 1990, and Critchlow, Fligner, and verducci (1991) for a detailed discussion on this and other distance-based ranking models. In this paper, a different approach to this problem is developed based on defining a class of prior distributions -the Binary Tree prior obtained by partitioning the space of permutations.…”