This paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their Euclidean nature. We present several particular transformations which preserve Euclideanarity and we complete, through the study of a one-parameter family, the current knowledge of the metric and Euclidean structure of coefficients based on binary data. These results are directly deduced from two theorems which prove the positive semi-definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used.

One of the main problems in cluster analysis is that of determining the number of groups in the data. In general, the approach taken depends on the cluster method used. For K-means, some of the most widely employed criteria are formulated in terms of the decomposition of the total point scatter, regarding a two-mode data set of N points in p dimensions, which are optimally arranged into K classes. This paper addresses the formulation of criteria to determine the number of clusters, in the general situation in which the available information for clustering is a one-mode $N\times N$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N\times N$$\end{document} dissimilarity matrix describing the objects. In this framework, p and the coordinates of points are usually unknown, and the application of criteria originally formulated for two-mode data sets is dependent on their possible reformulation in the one-mode situation. The decomposition of the variability of the clustered objects is proposed in terms of the corresponding block-shaped partition of the dissimilarity matrix. Within-block and between-block dispersion values for the partitioned dissimilarity matrix are derived, and variance-based criteria are subsequently formulated in order to determine the number of groups in the data. A Monte Carlo experiment was carried out to study the performance of the proposed criteria. For simulated clustered points in p dimensions, greater efficiency in recovering the number of clusters is obtained when the criteria are calculated from the related Euclidean distances instead of the known two-mode data set, in general, for unequal-sized clusters and for low dimensionality situations. For simulated dissimilarity data sets, the proposed criteria always outperform the results obtained when these criteria are calculated from their original formulation, using dissimilarities instead of distances.

Arrangements of feature sets that have been proposed to represent qualitative and quantitative variation among objects are shown to generate identical sets of set-symmetric distances. The set-symmetric distances for these feature arrangements can be represented by path lengths in an additive linear tree. Imperfect versions of these feature arrangements are proposed, which also are indistinguishable by the set-symmetric distance model. The distances for the imperfect versions can be represented by path lengths in an additive imperfectly linear tree. When dissimilarities are defined by the more general contrast model and a constant may be added to proximity data, then for both the perfect and imperfect arrangements an additive tree analysis obtains a perfect fit with an imperfectly linear tree. However, in the case of the contrast model also the distinction between the perfect and imperfect arrangements disappears in that also for the perfect arrangements the resulting tree need no longer be linear.

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or MDS and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.

In this article, we analyse the usefulness of multidimensional scaling in relation to performing K-means clustering on a dissimilarity matrix, when the dimensionality of the objects is unknown. In this situation, traditional algorithms cannot be used, and so K-means clustering procedures are being performed directly on the basis of the observed dissimilarity matrix. Furthermore, the application of criteria originally formulated for two-mode data sets to determine the number of clusters depends on their possible reformulation in a one-mode situation. The linear invariance property in K-means clustering for squared dissimilarities, together with the use of multidimensional scaling, is investigated to determine the cluster membership of the observations and to address the problem of selecting the number of clusters in K-means for a dissimilarity matrix. In particular, we analyse the performance of K-means clustering on the full dimensional scaling configuration and on the equivalently partitioned configuration related to a suitable translation of the squared dissimilarities. A Monte Carlo experiment is conducted in which the methodology examined is compared with the results obtained by procedures directly applicable to a dissimilarity matrix.

The representativeness heuristic suggests that similarity judgments provide a basis for judgments of likelihood. We use Tversky’s (1977) contrast model of similarity to design tests of this underlying mechanism. If similarity is used to judge likelihood, factors that are known to affect similarity should also affect judgments of likelihood. In two experiments, we manipulated two such factors described in the contrast model of similarity: the nature of the task and context effects. In a between-subject design, respondents assessed either similarity of fictive citizens of 15th century Florence, or the likelihood that they belonged to the same family. The factors that affected similarity also affected the likelihood judgments. These results support the assumption that similarity is an important contributor to judgments of likelihood.

The purpose of this study was to explore the relation between openness to diversity in terms of linguist, visible, value and informational heterogeneity and group processes depicted by group trust and group conflict. We collected data using a web-based survey among 498 academics in multicultural departments. The departments were located in four different Danish universities. We found openness to value diversity and informational diversity to have a positive association with group trust and a negative association with group conflict. Openness to linguistic diversity had positive relationships with group trust. The findings of this study suggest that openness to diversity should be highly prioritized by human resource managers in demographically diverse organizations. Possible interventions include organizational culture change programs, diversity training at individual and team levels as well as rewarding group oriented behavior.

Observers viewed two side-by-side arrays each of which contained three yellow Munsell papers, three blue, and one neutral Munsell. Each array was illuminated uniformly and independently of the other. The neutral light source intensities were 1380, 125, or 20 lux. All six possible combinations of light intensities were set as illumination conditions. On each trial, observers were asked to rate the dissimilarity between each chip in one array and each chip in the other by using a 30-point scale. Each pair of surfaces in each illumination condition was judged five times. We analyzed this data using non-metric multi-dimensional scaling to determine how light intensity and surface chroma contributed to dissimilarity and how they interacted. Dissimilarities were captured by a three-dimensional configuration in which one dimension corresponded to differences in light intensity.

Our concern here, is the characterization of dissimilarity indexes defined over finite sets, whose spatial representation is spherical. Consequently, we propose a methodology (NormedMultiDimensional Scaling) to determine the spherical euclidean representation of a set ofitems best accounting for the initial dissimilarity between items. Thismethodology has the advantage of being graphically readable on individual qualitiesof projection like the normed PCA, of which it constitutes a generalization. Moreover, it avoids the arbitrary character of spherical encoding which the use of similitude functions currently used in MDS, implies.