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

Abstract

BackgroundSeriation is a procedure that orders stimuli according to structure within a proximity matrix. The method was developed within electrical engi-

Exploratory Approaches to Seriation by Asymmetric Multidimensional Scaling

“…Consequently, ordering (5, 3, 6, 1, 2) maximizes the sum of the elements of the upper triangle of Γ, which is a common criterion for unidimensional seriation of an asymmetric matrix (see Golledge 1981, Chapter 4 in Hubert et al 2001). On the contrary, ordering (2, 1, 6, 3, 5) maximizes the sum in the lower triangle of Γ, as observed in Rodgers and Thompson (1992). Ordering (5, 3, 6, 1, 2) can also be characterized as a result of applying to Γ the model…”

“…Hubert and Golledge 1981, Hubert et al 2001, Brusco andStahl 2001), some authors (e.g. Rodgers and Thompson 1992) argued that it can profitably interact with multidimensional scaling (MDS) in the analysis of asymmetric proximity matrices, for reasons such as MDS could add meaningful information regarding dependencies between objects. Following this line of thought, the next sections feature some procedures to combine seriation and asymmetric MDS, for the dichotomous and the quantitative case respectively, and two short possible applications will be provided.…”

“…Rodgers and Thompson (1992) argued that the two approaches can profitably interact in the analysis of asymmetric proximity matrices, and proposed a method that uses seriation to define an empirical ordering of the stimuli, and symmetric multidimensional scaling to scale the two separate triangles of the proximity matrix defined by this ordering (an approach anticipated by Method 3 in Gower (1977)). Following a similar concept, this paper proposes some procedures to explore seriation through asymmetric multidimensional scaling.…”

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.…”