No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Asymmetric CLUster analysis based on SKEW-symmetry: ACLUSKEW
Abstract
A procedure of cluster analysis to deal with asymmetric similarities is introduced, where the similarity from one object to the other object is not necessarily equal to the similarity from the latter to the former. The procedure analyzes one-mode two-way asymmetric similarities among objects to classify objects into clusters. Each cluster consists of a dominant (central) object and the other (noncentral) objects. The central object of a cluster represents the cluster and dominates the other objects in the cluster. In the present procedure, differences between two conjugate similarities (two times of skew-symmetries) are weighted by multiplying with the sum of the two corresponding similarities. Thus the larger the similarity between two objects is, the more prominently the difference is evaluated. The present procedure is applied to car switching data among car categories, and the result is compared with the result which was obtained by analyzing unweighted differences between two conjugate similarities. The comparison shows the weight is reasonable.
In this chapter, we present applications of asymmetric multidimensional scaling based on the models described in Chap. 3 to empirical data. These applications reveal features of objects which underlie asymmetric relationships between objects, and depict differences in revealing features of objects which determine asymmetric relationships between objects from one model to another.
This chapter presents remarks on performing an analysis by using mainly one-mode two-way asymmetric multidimensional scaling based on the radius-distance and ellipse-distance models.
Models and methods of cluster analysis for asymmetric data are presented by considering two main classes: hierarchical and non-hierarchical methods. They are presented and applied to the same small illustrative data set which allows to highlight their different features and capabilities by using, when appropriate, graphical representations of the results. Attention is also paid to the issues of model selection and evaluation which are critical in applications.
A nonhierarchical clustering model is proposed here which jointly fits the symmetric and skew-symmetric components of an asymmetric pairwise dissimilarity matrix. Two similar clustering structures are defined depending on two (generally different) partitions of the objects: a “complete” partition fitting the symmetries (where all objects belong to some cluster) and an “incomplete” partition fitting the skew-symmetries, where only a subset of objects is assigned to some cluster, while the remaining ones may remain non-assigned. The exchanges between clusters are accounted for by the model which is formalized in a least squares framework and an appropriate Alternating Least Squares algorithm is provided to fit the model to illustrative real data.
Asymmetric pairwise relationships are frequently observed in experimental and non-experimental studies. They can be analysed with different aims and approaches. A brief review of models and methods of multidimensional scaling and cluster analysis able to deal with asymmetric proximities is provided taking a ‘data-analytic’ approach and emphasizing data visualization.
Using a model of consumer variety-seeking, the authors study the long-term market share implications of changes in variety-seeking intensity, brand preferences, and pairwise similarities between brands. Those analytically derived guidelines are examined in three-brand and five-brand markets through simulation. The least preferred brand is found generally to gain market share as variety-seeking intensifies whereas the most preferred brand tends to lose share. If two brands are perceived as having become more similar without a change in overall preferences, the repositioned brands are likely to lose market share while uninvolved brands gain share. If two brands are perceived as having become more similar in a way that increases overall preference for those repositioned brands, they should gain market share while uninvolved brands lose it. A behavioral experiment provides preliminary empirical support for some of the findings.
Over the last decade, numerous methods for the multidimensional scaling (MDS) of perceptions and preferences have been applied by researchers in marketing. However, one notable gap in MDS methodology has been the lack of suitable models for analyzing inherently asymmetric data relationships. Recently, Harshman (Harshman, R. A. 1978. Models for analysis of asymmetrical relationships among N objects or stimuli. Paper presented at the First Joint Meeting of the Psychometric Society and the Society for Mathematical Psychology. McMaster University, Hamilton, Ontario, August; Harshman, R. A. 1982a. DEDICOM: A family of models generalizing factor analysis and multidimensional scaling for decomposition of asymmetric relationships. Unpublished manuscript, University of Western Ontario.) has proposed a new family of models—called DEDICOM (DEcomposition into Directional COMponents)—for analyzing data matrices that are intrinsically asymmetric.
In this article, the single-domain DEDICOM model is described and applied to two illustrative cases in marketing research. The examples demonstrate that DEDICOM solutions will sometimes make more substantive sense and provide significantly better fits to asymmetric data than solutions obtained by factor analysis or MDS. DEDICOM also provides a novel type of information—a description of asymmetric relations among dimensions or clusters. Such information will often have useful marketing implications.
Questions the metric and dimensional assumptions that underlie the geometric representation of similarity on both theoretical and empirical grounds. A new set-theoretical approach to similarity is developed in which objects are represented as collections of features and similarity is described as a feature-matching process. Specifically, a set of qualitative assumptions is shown to imply the contrast model, which expresses the similarity between objects as a linear combination of the measures of their common and distinctive features. Several predictions of the contrast model are tested in studies of similarity with both semantic and perceptual stimuli. The model is used to uncover, analyze, and explain a variety of empirical phenomena such as the role of common and distinctive features, the relations between judgments of similarity and difference, the presence of asymmetric similarities, and the effects of context on judgments of similarity. The contrast model generalizes standard representations of similarity data in terms of clusters and trees. It is also used to analyze the relations of prototypicality and family resemblance. (39 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The slide-vector scaling model attempts to account for the asymmetry of a proximity matrix by a uniform shift in a fixed direction imposed on a symmetric Euclidean representation of the scaled objects. Although no method for fitting the slide-vector model seems available in the literature, the model can be viewed as a constrained version of the unfolding model, which does suggest one possible algorithm. The slide-vector model is generalized to handle three-way data, and two examples from market structure analysis are presented.
This paper is concerned with the hierachical clustering algorithms. The first half part of this paper presents eight clustering algorithms using asymmetric similarity measures, and the graph-theoretic structure of clusters obtained by these clustering algorithms are described.The second half part of this paper presents a clustering algorithm for ordering items using homogeneity coeffcient as similarity measure, and compares it with other clustering algorithms.
Nonmetric multidimensional scaling which could be applied to a square asymmetric interstimulus 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, hyperspheres) 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.
Car switching data among 16 car segments was analyzed by an asymmetric multidimensional scaling. The car switching data was formed by rescaling a 16 × 16 car switching matrix whose (j, k) element represents the frequency with which any car in car segment j was traded-in for any car in car segment k. The asymmetric multidimensional scaling utilized in the present study represents each car segment as a point and a circle (shpere, hypershpere) centered at the point representing the car segment in a multidimensional Euclidean space. The resulting three-dimensional solution revealed size or price dimension, imports-captive imports-domestic dimension, and specialty dimension. It seems that radii of shperes might represent the relative dominance or attractiveness of car segments.
In this paper, an asymmetric k-means clustering algorithm is presented. The asymmetric version of this algorithm is derived using the asymmetric coefficients, which convey the information provided by the asymmetry in analyzed data sets. The formulation of the asymmetric k-means algorithm is motivated by the fact that, when an analyzed data set has the asymmetric nature, a data analysis algorithm should properly adjust to this nature. The traditional k-means approach using the symmetric dissimilarities does not apply correctly to this kind of phenomenon in data. We propose the k-means algorithm using the asymmetric coefficients, which has the ability to reflect the asymmetric relationships between objects in analyzed data sets. The results of our experimental study on real data show that the asymmetric k-means approach outperforms its symmetric counterpart.
Fundamentals of MDS.- The Four Purposes of Multidimensional Scaling.- Constructing MDS Representations.- MDS Models and Measures of Fit.- Three Applications of MDS.- MDS and Facet Theory.- How to Obtain Proximities.- MDS Models and Solving MDS Problems.- Matrix Algebra for MDS.- A Majorization Algorithm for Solving MDS.- Metric and Nonmetric MDS.- Confirmatory MDS.- MDS Fit Measures, Their Relations, and Some Algorithms.- Classical Scaling.- Special Solutions, Degeneracies, and Local Minima.- Unfolding.- Unfolding.- Avoiding Trivial Solutions in Unfolding.- Special Unfolding Models.- MDS Geometry as a Substantive Model.- MDS as a Psychological Model.- Scalar Products and Euclidean Distances.- Euclidean Embeddings.- MDS and Related Methods.- Procrustes Procedures.- Three-Way Procrustean Models.- Three-Way MDS Models.- Modeling Asymmetric Data.- Methods Related to MDS.
This study examines the important and often underestimated role that switching barriers play in the propensity to stay with service providers. Three service types (with different structural characteristics) were studied across two diverse cultures—Australia (Western, individualistic culture) and Thailand (Eastern, collectivist culture). Six potential switching barriers are examined: search costs; loss of social bonds; setup costs; functional risk; attractiveness of alternatives; and loss of special treatment benefits. The results from a series of multiple regression analyses show switching costs capture a substantial amount of the explained variance in the dependent variable, propensity to stay with a focal service provider. Furthermore we demonstrate using interaction terms that these switching costs appear to be universal across west–east cultures. However, significant variations were found across industries. Next, using a hierarchical regression procedure, we add a satisfaction variable into each model. The incremental gain in R2 is significant in each industry. Nonetheless the significant impact of switching barriers gives rise to the identification of a new type of service loyalty, which we term “captive loyalty.”
In this paper, an asymmetric version of the k-means clustering algorithm is proposed. The asymmetry arises caused by the use of asymmetric dissimilarities in the k-means algorithm. Application of asymmetric measures of dissimilarity is motivated with a basic nature of the k-means algorithm, which uses dissimilarities in an asymmetric manner. Clusters centroids are treated as the dominance points governing the asymmetric relationships in the entire cluster analysis. The results of experimental study on the real data have shown the superiority of asymmetric dissimilarities employed for the k-means method over their symmetric counterparts.
The min and the max hierarchical clustering methods discussed by Johnson are extended to include the use of asymmetric similarity
values. The first part of the paper presents the basic min and max procedures but in the context of graph theory; this description
is then generalized to directed graphs as a way of introducing the less restrictive characterization of the original clustering
techniques.
This paper presents asymmetric
agglomerative hierarchical clustering algorithms in an extensive view point. First, we develop a new updating formula for
these algorithms, proposing a general framework to incorporate many algorithms. Next we propose measures to evaluate the fit
of asymmetric clustering results to data. Then we demonstrate numerical examples with real data, using the new updating formula
and the indices of fit. Discussing empirical findings, through the demonstrative examples, we show new insights into the asymmetric
clustering.
Hitaisho sokudo to toshitsuseikeisu o michiita kurasuta bunsekiho [Methods for cluster analysis using asymmetric measures and homogeneity coefficient]
- H Fujiwara
Jugyou no kategori bunseki ni okeru keiretsu no chushutsu [Abstraction of behavior sequence on categorical analysis of instruction]
- K Akahori
Hitaisho ruijido ni yoru kurasuta bunseki [Cluster analysis using asymmetric similarity measures
- H Fujiwara
- H Ozaki
Partitioning asymmetric dissimilarity data
- D Vicari