Akinori Okada

Multidimensional Scaling of Asymmetric Proximities with a Dominance Point

The purpose of the present study is to introduce a model and the associated nonmetric algorithm of multidimensional scaling for analyzing one-mode two-way (object × object) asymmetric proximities. In the model each object is represented as a point in a multidimensional Euclidean space, and a point, called the dominance point, is also embedded in the same multidimensional Euclidean space. The dominance point governs the asymmetry in the proximity relationships among objects, and represents the whole one-mode two-way asymmetric proximities dealt with in the analysis. An application to car switching data is presented.

An Asymmetric Cluster Analysis Study of Car Switching Data

An asymmetric cluster analysis was applied to car switching data among 16 car segments. The asymmetric cluster analysis, which is based on an algorithm corresponding to the mean method, can represent asymmetric relationships among car segments on a dendrogram and can deal with non switchers of the car switching data (diagonal elements of the car switching matrix). The obtained result showed that two features: (a) difference between import cars or domestic and captive import cars; and (b) size or price of cars, are playing important roles in car switching, and that the first feature played the primary factor in the car switching.

Joint Space Model for Multidimensional Scaling of Two-Mode Three-Way Asymmetric Proximities

A joint space model and an associated nonmetric algorithm to analyze two-mode three-way asymmetric proximities (object × object × source) are presented. Each object is represented as a point and a circle (sphere, hyper sphere) in the common joint configuration which is common to all sources. Each source is represented as a point in the common joint configuration. For each source, the radius of an object is stretched or shrunk according to the distance between the dominance point representing the source and the point representing the object. An application to intergenerational occupational mobility data is shown.

A Generalization of Asymmetric Multidimensional Scaling

An asymmetric multidimensional scaling model, which is derived by generalizing the predecessor, and the nonmetric algorithm to fit the present model to an asymmetric proximity matrix is presented. In the present model asymmetries in proximities are accounted for not only by the term related with stimuli or with dimensions but by the term related with stimuli and dimensions. Geometrically each stimulus is represented as a point and an ellipse (ellipsoid, hyperellipsoid) in a multidimensional Euclidean space. The present model is applied to car switching data among 12 car segments. The obtained two-dimensional solution is compatible with the previous analyses. And it seems that the length of the semiaxes of the ellipse might represent the relative dominance or attractiveness on the two dimensions respectively.

External Analysis of Two-mode Three-way Asymmetric Multidimensional Scaling

An external analysis of two-mode three-way (object×object×source) asymmetric multidimensional scaling is introduced, which is similar to the external analysis of INDSCAL. The present external analysis discloses the asymmetry of each object, and source differences in symmetric and in asymmetric proximity relationships among objects respectively for an externally given configuration of objects. The present external asymmetric multidimensional scaling is applied to the university enrollment flow among Japanese prefectures.

Two-Mode Three-Way Nonmetric Multidimensional Scaling with Different Directions of Asymmetry for Different Sources

A model and an associated nonmetric algorithm of multidimensional scaling (MDS) for analyzing two-mode three-way (object×object×source) asymmetric proximities are introduced. The model and the algorithm of the present two-mode three-way asymmetric MDS were extended from those of Okada and Imaizumi (1997) which can represent differences of symmetric and asymmetric relationships among sources. But the model assumes that the directions of asymmetry are the same for all sources. The present model allows different sources to have different directions of asymmetry. The present two-mode three-way asymmetric MDS was applied to analyze the data on the international trade.

Multidimensional Scaling with Different Orientations of Dimensions for Symmetric and Asymmetric Relationships

A model and an associated nonmetric algorithm for analyzing two-mode three-way asymmetric proximities are presented. The model represents proximity relationships among objects which are common to all sources, the salience of symmetric proximity relationships along dimensions for each source, and the salience of asymmetric proximity relationships along dimensions. The salience of asymmetric proximity relationships is represented by a set of dimensions, which have different orientations from that for the symmetric relationships.

Rescaling Proximity Matrix Using Entropy Analyzed by INDSCAL

[Yokoyama and Okada (2005, in press)] suggested a new method that rescales a brand-switching data matrix using entropy. These studies applied the rescaling method to car-switching data, and the configuration derived by Kruskal’s multidimensional scaling was interpreted as the circumplex. In the present paper, we apply that rescaling method to intergenerational occupational mobility data for four years, and analyzed the resluts by Krsukal’s multidimensional scaling. As a result, the configurations are also interpreted as the circumplex. Furthermore, we also find that the result of the analysis of these rescaled data by INDSCAL is interpreted as the circumplex.

Asymmetric Multidimensional Scaling of Relationships Among Managers of a Firm

Relationships among managers of a firm were analyzed. Each manager responded who goes to whom for help or advice for work-related problems. Resulting responses constitute a set of matrices, each comes from a manager. Each matrix is transformed to a matrix of geodesic distances from row to column managers. The set of geodesic distance matrices was analyzed by the asymmetric multidimensional scaling. The result represents the hierarchical structure of the firm. The dimensions for symmetric relationships represent differences among departments, and those for asymmetric relationships represent differences within and between supervisors.

Developing a Layout of a Supermarket Through Asymmetric Multidimensional Scaling and Cluster Analysis of Purchase Data

A layout of a supermarket, which increases the jointly purchasing items of different categories, was developed based on purchase data of more than 14,000 customers for half a year at a supermarket. Similarities among 24 categories of items sold at the supermarket were defined, for each of 12 conditions (the age of the customers, the place they live, and the day they purchased), by using (a) the proportion of the jointly purchasing items of any two categories and (b) the amount of items of categories purchased. Because differences among conditions were not significant, one layout was derived based on the similarities for all (unconditional) purchases.