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Asymmetric multidimensional scaling (AMDS) is a visualization method that can be applied to asymmetric (dis)similarity data, which are adopted in several areas such as marketing research, psychometrics, and information science. Several researchers within these fields have examined and applied AMDS. However, the combination of the number of modes and ways remains fixed across these AMDS. Thus, the choice of model is largely contingent on the number of modes and ways. To overcome this problem, we propose an AMDS using a log-linear model with an m-way frequency table. Using the log-linear model, we apply AMDS to (dis)similarity data without a fixed number of modes and ways. In addition, we were able to simultaneously visualize two types of circles.

Asymmetric multidimensional scaling is a visualization method that reveals relations between objects using their asymmetric dissimilarity data as input. Such visualization is important in areas such as marketing science since it reveals consumer dynamics and brand competition. Large amounts of asymmetric dissimilarity data can be processed by modern information technology; nevertheless, it remains difficult to interpret the asymmetries between many objects. To overcome this problem, we propose a new visualization method called discriminant coordinates for asymmetric dissimilarity data, which interprets the relations between and within a set of object classes, given asymmetric dissimilarity data. The method easily interprets the estimated asymmetries between and within classes, even when the asymmetric dissimilarity data contain noise if some assumptions for asymmetric dissimilarity data is satisfied.

We propose a Bayesian approach to asymmetric multidimensional scaling (MDS), which incorporates an asymmetric data structure. The asymmetry is represented by the hill-climbing model, which introduces a slope vector that measures the extent of the difficulty in going from one point to another instead of vice versa, in the MDS space. By using Bayesian estimation with Markov chain Monte Carlo algorithm, both point and interval estimation of the parameters become possible, in addition to the many advantages of Bayesian estimation. The asymmetry is evaluated on the basis of the posterior credibility region of the slope vector. A numerical simulation demonstrates that the proposed method is effective for recovering the true parameter values. The proposed method is demonstrated by the analysis of brand-switching data.

We propose to use the mixed effect trend vector model for modeling of repeated multinomial choice data in the form of a square contingency table. Such data often shows asymmetries where more people change from category a to b than the other way around. In many cases an investigator has, besides the actual choices of the participants, auxiliary variables that pertain to the subjects under study. Most methodologies for asymmetric data do not take into account such variables. We will show how to incorporate these auxiliary variables into the mixed effects trend vector model and how they can be used to study differential change. The models are illustrated in detail with data from the Dutch parliamentary election studies 2006.

In test data analysis, it is very important to clarify the underlying inter-item dependency structure. Once the structure is identified, we can reorder the units in the relevant textbook and gain a deeper understanding of the errors made by students. The asymmetric triangulation scaling technique proposed in this paper is a kind of asymmetric multidimensional scaling technique used for visualizing the inter-item dependency structure in a 3D hemisphere by analyzing the conditional correct response rate matrix.

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.

A Bayesian nonmetric successive categories multidimensional scaling (MDS) method is proposed. The proposed method can be seen as a Bayesian alternative to the maximum likelihood multidimensional successive scaling method proposed by Takane (1981), or as a nonmetric extension of Bayesian metric MDS by Oh and Raftery (2001). The model has a graded-response type measurement model part and a latent metric MDS part. All the parameters are jointly estimated using a Markov chain Monte Carlo (MCMC) estimation technique. Moreover, WinBUGS/OpenBUGS code for the proposed methodology is also given to aid applied researchers. The proposed method is illustrated through the analysis of empirical two-mode three-way similarity data.

It is well known that health and social factors are closely related. The aim of this study is to examine subjective health through the analysis of cross-national comparative surveys to clarify cultural characteristics of subjective health (self-rated health). I examined countries/areas using the results of our past surveys (Table 1. Yoshino, Nikaido & Fujita, 2009, Behaviormetrica, 36, 2, 89–113). In this analysis, I examined self-rated health symptoms, health satisfaction, life satisfaction, and sense of anxiety as well as other social factors. Multidimensional data analyses of all items on subjective health and on social factors showed a similar pattern between countries/areas. Furthermore, multidimensional scaling for Japanese emigrants and their offsprings and their related countries showed meaningful cultural links of comparison but not so large generational difference between 2-SEI and 3-SEI with respect to health and life satisfactions.

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 marginal maximum likelihood (MML) estimation method is developed for the analysis of both categorical rating data and choice data for decision making in the context of uncertain outcomes. The proposed method fits the weighted additive models with interaction terms to the data, allowing for individual differences in weights, category boundaries, and thresholds. The present study demonstrates that the MML approach will be useful for dealing with individual differences in those variables as well as the subject points previously dealt with within the framework of multidimensional scaling. Bock and Aitkin’s EM algorithm is used for the MML estimation of the proposed models.

The attraction among a group of subjects was predicted by a procedure which had been originally introduced to predict the amount of purchase of a brand. The procedure, which is based on two multidimensional scaling methods; INDSCAL and PREFMAP, was applied to friendship tie data among members of a group of 17 students collected once a week for 16 weeks (Newcomb, 1961; Nordlie, 1958). Three subjects, who had received relatively unstable and dissimilar attractions among the 17 subjects, were selected. The attraction from the 17 subjects to the three subjects was analyzed by the procedure. The 17 subjects were cluster analyzed using the data at weeks 0 through 8, and were grouped into three clusters. The attraction from each of the three clusters to the three subjects at weeks 0 through 8 were analyzed to predict the attraction at weeks 9 through 15. The predicted attraction was compared with the actual attraction at weeks 10 through 15 (data at week 9 were missing). The comparison shows that five of the nine root mean squares of discrepancy between the predicted and actual attractions were smaller than unit rank, and that the prediction was successful.

A new multidimensional unfolding method is developed. The most important advantage of the proposed method is that the goodness of fit criterion is constructed on the basis of the response model of ranking behavior so that the analysis never fails to arrive at a solution space where data and distances correspond in their rank order as much as possible. This implies that the method guarantees elimination of such degeneracies as caused by minimizing the so-called stress measure at the expense of order correspondences between the data and the model as is often the case with the conventional nonmetric unfolding approach. The method is applied to two sets of real data for illustrative purposes.

The study proposes and empirically tests a new model for delineating product markets-MARKDEF. In MARKDEF product market definitions are analytically obtained using consumer perceptions. MARKDEF provides important diagnostic capability. It not only enables one to gain insight into the overall similarity/dissimilarity of products but also the attributes which account for these. In particular, MARKDEF gives a measure called the overall index of deviation which reflects the overall degree to which two products differ from each other. In addition, MARKDEF provides a measure of deviation on each attribute for the products under consideration, which aids in understanding the attributes responsible for differentiating the products. MARKDEF is supported in the product class investigated, namely candy bars. Attempts are made to validate the methodology by comparing the results obtained from MARKDEF with those of other standard procedures like multidimensional scaling and cluster analysis, and it is demonstrated that MARKDEF has good overall validity.

This paper proposes a mutidimensional scaling method for asymmetric proximity data whose diagonal entries are not all equal. For a suggested model, necessary and sufficient conditions and algebraic solutions are presented. The method consisting of two algorithms is developed, which gives multidimensional configuration, unidimensional scale of any psychological quantities and the degree of asymmetry of the data. The quantities should be interpreted through data analysis or examined with any hypothesis. A simulation study using artificial data demonstrates that the method works successfully, showing sufficient robustness and capability to recover the original structure. Illustrative applications are presented. Extensions of the model are argued, and properties of the model and the algorithms are discussed with another simulation study.

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.

The present paper provides two response models: one for binary ranking and the other for sorting. The former is a behavior of choosing, in a random order, only those comparison stimuli which are judged to be very similar to a standard stimulus, and the latter is that of selecting stimuli which are judged very similar to each other to form them into clusters. The key assumption of these models is that the subject perceives any two stimuli as very similar to each other when their dissimilarity, which varies over time, is below response thresholds that are associated with those stimuli. Maximum likelihood estimation procedures are used for the estimation of parameters of these models. The proposed models are applied, for illustrative purposes, to the similarity data collected by the binary ranking and sorting methods. We discuss some advantages of the binary ranking method to be used for collecting similarity data and a practical limitation of our response model for sorting.

Nonmetric multidimensional scaling which could be applied to a square asymmetric inter-stimulus proximity matrix is presented. In the model each stimulus is represented as a point and a circle (sphere, hypersphere) whose center is at that point in a multidimensional Euclidean space. The radius of a circle (sphere, hypersphere) tells the skew-symmetry of the corresponding stimulus. In a sense the model is a nonmetric generalization of Weeks and Bentler (1982) ’s model. An algorithm to derive the coordinates of points and radii of circles (spheres, hypers-pheres) which minimize the discrepancy of the coordinates and radii from the monotone relationship with given interstimulus proximities is described. An application to car switching data among 16 car segments is represented.

This paper is concerned with a methodological study of Hayashi’s eej-type quantification method (EQ) and development of a new method (PAMSE) of multidimensional scaling (MDS). First we propose procedures to compare EQ with other methods of MDS in terms of the separation of eigenvalues. Secondly we examine a problem of the fit and dimensionality and point out a peculiar property inherent to EQ. Third we propose a new method of MDS, which is developed from EQ, intended principally for dissimilarity data measured on the medium level between ordinal and interval scale levels. Our method is conceptually similar to but fundamentally different from Johnson’s in that our index of fit is specially formulated in view of the scale level. The method has been found to work successfully, giving a high degree of the rank correlation between the dissimilarities and the derived distances.

The present paper is a review of recent theoretical and methodological developments in multidimensional scaling and its related methods in Japan. The term multidimensional scaling (MDS) is used in two senses. One is a narrow sense, and the other is a wider sense. The former is a technical term which is divided into metric MDS and non-metric MDS. The latter has many variations in multidimensional data analysis. In this review, MDS in a narrow sense and its closely related methods, which are a part of MDS in a wider sense, are taken into consideration. And Japanese works are introduced.

This paper proposes a model to analyze the structure and context effects involved in dissimilarity judgment. With some restrictions incorporated, the model is also interpreted as a kind of the distance-density model and further interpreted as a mixed distance and content model. Algorithms for multidimensional scaling based on the proposed model are implemented using nonlinear optimization methods. The algorithms are evaluated through a Monte Carlo study, and applications are demonstrated with real data. Problems regarding the model, the algorithms and possible applications are discussed.

A new algorithm of multidimensional scaling is proposed. It performs metric or nonmetric analyses of two- or three-way data. It is a simple, convergent algorithm analogous to ALSCAL, but it approximately optimizes the least-squares fit of distances (not squared). This paper mainly considers the nonmetric two-way case.

Dr. Hayashi developed one of methods for multidimensional scaling problem, named MDA. This method is a very useful one, but its computation is very difficult. In the present paper, the author shows a modification of its computation with some numerical examples in one dimensional case.

This paper proposes a model to analyze the symmetric dissimilarity matrix by an Euclidean distance model with additional terms. The proposed model is a restricted model of Weeks and Bentler’s model. The symmetric dissimilarity is assumed to be induced from asymmetric dissimilarities. A Monte Carlo simulation study conducted demonstrated that the proposed least squares algorithm was effective in recovering the true number of dimensions and true configuration. This algorithm was applied to the real data set. The local minimum problem is discussed.

An information processing model of pattern perception has been constructed and applied to computer recognition of hand-written numerals. Patterns are picked up in the form of 32 × 32 binary array and orthogonally transformed by mutual additions and substructions among the elements. The Hadamard power spectra are defined as the feature variates to make distribution memories of the patterns which are called engram. Discriminations are performed through template matching by Euclid distance minimization method. A clusteing method has been introduced as the learning process to rewrite the engram. It is possible to identify the patterns correctly, not only for their position shift but also for their changes of parts or inclination to some extent. Multidimensional scaling method is used for the appreciation of effectiveness of the model.

The weighted inner-product model is discussed for a class of techniques of multidimensional scaling incorporating individual differences. Data to be anlyzed are sets of arbitrary ones, from some individuals, expressing similarity between objects. The model discussed is such that an inner product of position vectors, in a space, corresponding to the objects represents given similarity between the objects. A set of parameters called weights is utilized for indicating individual differences. In order to determine the vectors and the parameters, a least-squares criterion to be minimized is defined. For minimization of the criterion, an iterative computation procedure is proposed considering quantitative data. The model and the procedure are further extended to treat ordinal data. Illustrative examples are attatched with using artificial data and real survey data.

Some intriguing relationships among four methods of metric and nonmetric multidimensional scaling (MDS) are explicated. It is shown that all four methods of MDS considered here amount to solving, explicitly or implicitly, the stationary point of a matrix which can be generally represented as A’HA, where A is a difference matrix (to be defined), and where H depends on a particular criterion being optimized. H may be a matrix of fixed constants or of functions of unknown parameters (stimulus coordinates) of the representation model. A conceptual distinction is made as to the scale level of measurement in reference to MDS methods and solutions.

A new method is introduced which can explain individual differences in dissimilarity judgments by N subjects. The method is based on a weighted distance model which is a generalization of DeLeeuw and Pruzansky’s model to Minkowski r metric. A set of N dissimilarity matrices is analyzed nonmetrically to derive a group stimulus configuration and a subject configuration just like as INDSCAL. The method optimizes the goodness of fit measure which is based on the stress formula two by the method of steepest descent. This optimizing measure makes the computing process independent on Minkowski constant. The method was applied to schematic face data, and gave satisfactory results.

We are discussing the problem of impression fromations of others in person perception applying the nonmetric multidimensional scaling techniques, especially comparing the Smallest Space Analysis (SSA) with the Minimum Dimension Analysis (MDA), using the data from rating of the impression formations of a person measured by the Semantic Differential Method. The usefulness of nonmetric multidimensional scaling methods for the psychological phenomena is examined under several considerations from various points of view.

A minimization problem in metric multidimensional scaling is discussed. Each object is embeded as a point in t-dimensional Euclidean space such that the points minimize the sum of Euclidean norm between the mean of dissimilarity value and Euclidean distance of two points. The existence of a solution which attains the minimum value of the least squares criterion is shown.

This paper proposes a new representation method of multivariate data by face pattern -termed as the face method — considering the human visual characteristics of facial expressions. The visual characteristics of facial expressions illustrated by the face pattern are analyzed by using multidimensional scaling (MDS), and psychometrical distances of facial expressions are then obtained from the configuration formed by MDS. A mathematical formulation of the proposed representation method is derived introducing the obtained psychometrical distances. The verification of the method is performed by some psychometrical experiments, and its efficiency is also tested applying in the classification of multivariate fossil data.

A sensitivity analysis method for eigenvalue problems in multivariate analysis was proposed by De Sarbo, et al. (1982) and developed by Ueda (1988). Metric multidimensional scaling (MDS) is a spectral decomposition of symmetric matrices, where eigenvalues and eigenvectors play important roles. It is shown that the e -neighborhood vectors defined in Ueda (1988) give the 2e-neighborhood vectors for the goodness-of-fit index in a spectral decomposition. Considering this fact and using the same method as Ueda (1988), sensitivity for metric MDS is discussed.

In order to investigate perceptual dimensions of facial expressions, three-way multidimensional scaling methods were applied to the analysis of two types of judgment data. That is, dissimilarity judgments among facial expressions and rating judgments of each face on 20 emotion-term scales were obtained in experiments, and analyzed by the scaling methods which yielded uniquely oriented configurations. The faces used in the experiments consisted of seventeen schematic facial expressions constructed by combinations of different curvatures of the mouth and eyes. The results successfully exposed two perceptual dimensions, which were interpreted as friendliness-unfriendliness and high activation — low activation. And it was clarified that these dimensions were fairly consistent over the different data types and scaling methods used in the present study.

A fundamental idea of a new algorithm of minimum dimension analysis of ordered class belonging (MDA-OR) which is one of the methods for nonmetric multidimensional scaling (MDS) is described in the author’s previous paper (1978). But the previous paper is mainly concerned with one dimensional case. In the present paper, the author gives a full account of a new algorithm together with its computational techniques, and shows some numerical examples in multidimensional case.

"Multidimensional scaling technique was applied to the Brazilian migration matrix in order to identify the functional regions of the nation, using the 1970 census data for males. The results obtained from three types of proximity matrices confirmed the key roles of Minas Gerais and Mato Grosso as the core states in addition to Rio de Janeiro and Sao Paulo. Substantial circulation of migrants among the core states runs counter to the popular notion about the Northeastern states as the major source of labor in Rio de Janeiro and Sao Paulo. The Northeast [was] found to consist of three groups of states: Maranhao, Piaui and Ceara showed a stable clustering, but peripherally located, in all configurations; and, the other two groups failed to form persistent regions across configurations. Though limited in scope, the observed attraction of Espirito Santo and Goias violated the widely held negative effect of distance and deserves further investigation."