A Multidimensional Scaling Model for Three-Way Data Analysis

Abstract

This article proposes a model that facilitates the analysis of triadic relationships among three objects. Recently, interest in studies of three-way data models has increased, and many significant contributions have been made in this area. However, one-mode, three-way models have yet to be considered. This study focuses on a onemode, three-way model in which three-way distances are explained as the subtraction of the smallest squared distance among the three squared distances from the sum of these squared distances. This formulation is used to illustrate the idea that relationships with many differences carry more information than relationships with few differences. Moreover, the distance between two objects is weighted more heavily when two objects differ greatly, and this weight indicates the salience of their dyadic distance. Finally, the model and algorithm are applied to purchase data for convenience stores. The proposed model of multidimensional scaling clearly identifies differences among groups of objects.

A Brief Survey of Asymmetric MDS and Some Open Problems

“…Methods for analyzing this type of triadic distance data go back to Hayashi (1972). Representative methods in this category are , Daws (1996), De Rooij and, Gower and De Rooij (2003), Hayashi (1972Hayashi ( , 1989, Heiser and Bennani (1997), Joly and Le Calvé (1995), Nakayama (2005), and Nakayama and Okada (2011). These are descriptive MDS methods which assume certain symmetric, triadic distance models.…”

On multi-way metricity, minimality and diagonal planes

“…Examples of functions that satisfy inequality (c4) can be found in Joly and Le Calvé (1995), Bennani-Dosse (1993), Heiser and Bennani (1997) and Nakayama (2005). However, it should be noted that, although there are many cases in which this three-way metric inequality is valid, the three-way models can be used regardless of the validity of the three-way axiom.…”

“…Daws (1996) incorporated three-way information in the analysis of free-sorting data. Methods for multi-way multidimensional scaling can be found in Cox et al (1991), Heiser and Bennani (1997), Gower and De Rooij (2003) and Nakayama (2005). Cox et al (1991), Diatta (2006) and Daws (1996) convincingly showed that multi-way dissimilarities may be used to detect possible higher-order relations between the objects.…”