Tadashi Imaizumi

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.

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.

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.

A Gravity-Based Multidimensional Unfolding Model for Preference Data

A new model for analyzing two-way, two-mode preference data is proposed. MultiDimensional Unfolding models (MDU) have been used widely. In these model, the observed preference value is related to the distance between the ideal point and object point only. The market share of each brand is ignored or assumed to be be the same for all objects. The attraction of each object, such as the market share of that object, must be incorporated in the analysis of marketing data. A gravity-based multidimensional unfolding model will be proposed. One specific characteristic of preference data of N subjects is that observed preference values of individuals are often not compatible between individuals. The de-generated configuration problem on applying the non-metric MDU method to a real data set will be caused by the week condition on the data matrix. A linearly constrained non-metric approach is also proposed to try to rescue from obtaining the de-generated configuration.

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.

A Hyperbolic Fuzzt k-Means Clustering and Algorithm for Neural Networks

A new fuzzy k-means clustering algorithm is proposed by introducing crisp regions of clusters. Boundaries of the regions are determined by hyperbolas and membership values are given by one or zero in each region. The area between crisp regions is a fuzzy region, where membership values are proportional to distances to crisp regions. Though the traditional hard k-means is a limit of the usual fuzzy k-means, results of the latter are fuzzy and then are not the same as results of the former. On the other hand a new method can produce the same results as those by the hard k-means. An algorithm for neural networks is given and a numerical example is illustrated.

Multi-Dimensional Scaling of Sparse Block Diagonal Similarity Matrix

Similarity matrix represents the relationship of n objects and gives us a useful information about these objects. Several models for analyzing this data assume that each object of n objects is embedded as a point or a vector in t dimensional “common space” of n objects in general. However, these models are not appropriate for analyzing a sparse block diagonal similarity matrix as each block diagonal matrix indicates us that each member of the set of objects in a block is represented as a point or a vector in not “common space,” but, “sub-space.” And a model is proposed to analyze this type of a sparse block diagonal similarity matrix. And application to a real data set will be shown.

An Unfolding Scaling Model for Aggregated Preferential Choice Data

The unfolding model has been widely used as a model for preferential choice data. This model is treated as the special case of multidimensional scaling with so-called “ideal” points. In this model, the distance between an “ideal” point and object points are related to the degree of individual preferential choice data for objects. However, the unfolding model has some difficulties, degeneracies, indeterminacies and multidimensionality problems in application to real data. In this paper, we propose a parametric unfolding model for aggregated choice data by introducing the attractiveness of objects.