Nonmetric Multidimensional Scaling of Asymmetric Proximities

Self Cite

Brand switching data among 12 margarine brands were analyzed by the asymmetric multidimensional scaling based on the singular value decomposition. A two-dimensional result was adopted as the solution. A configuration based on the left and right singular vectors is given along each dimension. The left singular vector represents an outward tendency of switching from the corresponding brand to the other brands, and the right singular vector represents an inward tendency of being switched to the corresponding brand from the other brands. The configuration along Dimension 1 shows that the three brands with the larger market share compete vigorously with each other. The configuration along Dimension 2 classifies 12 brands into two groups; the brand switching between two groups is small, while that within each group is large.

Section: The Methodsmentioning

confidence: 62%

Section: The Datamentioning

confidence: 99%

Asymmetric Multidimensional Scaling of Brand Switching Among Margarine Brands

Okada

Tsurumi

2012

Self Cite

“…One short-coming of GIPSCAL is its employment of a metric MDS. Considering the nature of data in social and behavioral sciences, it will be better to develop a nonmetric asymmetric MDS as Okada and Imaizumi (1987) have done.…”

Section: Discussionmentioning

confidence: 99%

A Generalized Inner Product Model for the Analysis of Asymmetry

Chino

1990

A least squares procedure called GIPSCAL (a Generalized Inner Product multidimensional SCALing) is proposed which extends Chino's ASYMSCAL into higher dimensions than three. GIPSCAL fits the inner product of two vectors and the area of the parallelogram spanned by these vectors, respectively, for the symmetric and skew-symmetric parts of observed similarity judgements.It is shown that GIPSCAL has a very desirable property that the geometrical interpretation of asymmetric parts in similarity judgements is reducible to that of the area of the parallelogram spanned by vectors in two dimensions.It is also shown that GIPSCAL permits a social psychological justification for the cause of asymmetry.Relation to distance model is discussed. Examples of application are given to demonstrate the feasibility of the model.

“…This linear model is the simplest form of skew-symmetry, which has been studied by many authors (e.g. Gower 1977, 2000, 2008, Constantine and Gower 1978, Okada and Imaizumi 1987, Bove and Critchley 1993, Saito and Yadohisa 2005. A least squares solution for the r i 's in (2.2) in the general case of an n × n skew-symmetric matrix, provided by Mosteller (1951) in the context of Thurstone case V scaling, iŝ…”

Section: Dichotomous Casementioning

confidence: 99%

Exploratory Approaches to Seriation by Asymmetric Multidimensional Scaling

Bove

2012

Seriation and multidimensional scaling are two techniques aimed at exploring relationships in dominance or proximity data matrices. 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. The paper focuses on skew-symmetric components of a particular class of asymmetric matrices (including e.g. tournament or paired comparison matrices). Two short examples of possible applications are provided to illustrate the procedures, both in the dichotomous and quantitative case.