Applications of combinatorial programming to data analysis: Seriation using asymmetric proximity measures

Abstract

Based on a given Bsymmetric proximity function, a two-stage computetional heuristic for sequencing a set of objmts along a continuum is presented and illustrated with the type of example common in the paired-comparison literature. The first stage, defined by the pairwise interchange of objects, is intended to generate reasonably good orderinga from randomly chosen initial starts; the second stage c m be considered to be a refinement phase and depends on a general condition for m optimal solution suggested by Younger involving the interchange of consecutively placed groups of objects. An extension of the heuristic is also described that allows the selection and sequencing of a subset of all the possible objects.

An Interactive Multiobjective Programming Approach to Combinatorial Data Analysis

“…The ordering of n objects along a continuum has long been recognized as an important and practical problem (Baker & Hubert, 1977;Flueck & Korsh, 1974;Hubert, 1974;Hubert, 1976;Hubert & Golledge, 1981;Rodgers & Thompson, 1992). The general goal associated with asymmetric unidimensional seriation is to find a permutation of the rows and columns of an n x n asymmetric proximity matrix, such that the information in the reordered matrix is more easily interpreted.…”

“…First, seriation and scaling of asymmetric proximity matrices is an important and well-studied problem in the psychometric literature (Baker & Hubert, 1977;Constantine & Gower, 1978;De Soete, Hubert, &Arabie, 1988;Groenen & Heiser, 1996;Holman, 1979;Hubert, 1976;Hubert & Golledge, 1981;Hutchinson, 1989;Levin & Brown, 1979;Rodgers & Thompson, 1992;Weeks & Bentler, 1982;Zielman & Heiser, 1996). Second, a variety of solution methods, employing various objective criteria, have been suggested for asymmetric seriation (Baker & Hubert, 1977;Hubert, 1976;Hubert & Golledge, 1981;Rodgers & Thompson, 1992). Third, single objective problems--hereafter referred to as uniobjective problems--of a reasonable size can be solved to optimality using dynamic programming (Hubert & Golledge) and these approaches are readily extensible to weighted multiobjective problems.…”

A Branch-and-Bound Algorithm for Fitting Anti-Robinson Structures to Symmetric Dissimilarity Matrices

“…The seriation of large matrices (n > 25) typically requires the use of heuristic solution procedures that provide good (but not necessarily globally optimal) solutions. Perhaps the most common type of local-search operation deployed in heuristic procedures for seriation and unidimensional scaling is based on the pairwise interchange of objects in the sequence (Baker & Hubert, 1977;Groenen, 1993). Hubert and Arable (1994) suggested using object block reversals and object insertions as additional local-search strategies.…”

Seriation and Multidimensional Scaling: A Data Analysis Approach to Scaling Asymmetric Proximity Matrices

“…Young (1975) Cunningham (1978), DeSarbo (1982), Hutchinson (1989), and Klauer (1989 Hubert (1974Hubert ( , 1976Baker & Hubert, 1977). He asserted: &dquo;One of the basic problems of data analysis that has concerned applied researchers for many years deals with the sequencing of objects along a continuum&dquo; (Hubert, 1974, p. 9).…”

“…Once the seriation solution is found, the ordering of the stimuli is no longer arbitrary. With asymmetric data, the most popular seriation objective function has been to find the ordering of the stimuli maximizing the sum of the elements within one triangle (say, the lower triangle) which simultaneously minimizes the (Baker & Hubert, 1977), dynamic programming (Hubert & Arabie, 1986;Hubert & Golledge, 1981), and simulated annealing (De Soete et al, 1988) The seriation algorithm presented here is a slight adaptation of a nonmetric counting rule suggested by Hubert and Golledge (1981) and has relationships to early procedures proposed by Kendall (1955) and Flueck and Korsh (1974). From (Younger, 1963) (Hubert, 1976, pp.…”