Fitting an extended INDSCAL model to three-way proximity data

The INDSCAL individual differences scaling model is extended by assuming dimensions specific to each stimulus or other object, as well as dimensions common to all stimuli or objects. An alternating maximum likelihood procedure is used to seek maximum likelihood estimates of all parameters of this EXSCAL (Extended INDSCAL) model, including parameters of monotone splines assumed in a quasi-nonmetric approach. The rationale for and numerical details of this approach are described and discussed, and the resulting EXSCAL method is illustrated on some data on perception of musical timbres.     

Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL

Carroll and Chang have derived the symmetric CANDECOMP model from the INDSCAL model, to fit symmetric matrices of approximate scalar products in the least squares sense. Typically, the CANDECOMP algorithm is used to estimate the parameters. In the present paper it is shown that negative weights may occur with CANDECOMP. This phenomenon can be suppressed by updating the weights by the Nonnegative Least Squares Algorithm. A potential drawback of the resulting procedure is that it may produce two different versions of the stimulus space matrix. To obviate this possibility, a symmetry preserving algorithm is offered, which can be monitored to produce non-negative weights as well. This work was partially supported by the Royal Netherlands Academy of Arts and Sciences.     

A generalization of GIPSCAL for the analysis of nonsymmetric data

Graphical representation of nonsymmetric relationships data has usually proceeded via separate displays for the symmetric and the skew-symmetric parts of a data matrix. DEDICOM avoids splitting the data into symmetric and skewsymmetric parts, but lacks a graphical representation of the results. Chino's GIPSCAL combines features of both models, but may have a poor goodness-of-fit compared to DEDICOM. We simplify and generalize Chino's method in such a way that it fits the data better. We develop an alternating least squares algorithm for the resulting method, called Generalized GIPSCAL, and adjust it to handle GIPSCAL as well. In addition, we show that Generalized GIPSCAL is a constrained variant of DEDICOM and derive necessary and sufficient conditions for equivalence of the two models. Because these conditions are rather mild, we expect that in many practical cases DEDICOM and Generalized GIPSCAL are (nearly) equivalent, and hence that the graphical representation from Generalized GIPSCAL can be used to display the DEDICOM results graphically. Such a representation is given for an illustration. Finally, we show Generalized GIPSCAL to be a generalization of another method for joint representation of the symmetric and skew-symmetric parts of a data matrix.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author, and by research grant number A6394 to the second author, from the Natural Sciences and Engineering Research Council of Canada. The authors are obliged to Jos ten Berge and Naohito Chino for stimulating comments.     

A graph-theoretic method for organizing overlapping clusters into trees,multiple trees,or extended trees

A clustering that consists of a nested set of clusters may be represented graphically by a tree. In contrast, a clustering that includes non-nested overlapping clusters (sometimes termed a “nonhierarchical” clustering) cannot be represented by a tree. Graphical representations of such non-nested overlapping clusterings are usually complex and difficult to interpret. Carroll and Pruzansky (1975, 1980) suggested representing non-nested clusterings with multiple ultrametric or additive trees. Corter and Tversky (1986) introduced the extended tree (EXTREE) model, which represents a non-nested structure as a tree plus overlapping clusters that are represented by marked segments in the tree. We show here that the problem of finding a nested (i.e., tree-structured) set of clusters in an overlapping clustering can be reformulated as the problem of finding a clique in a graph. Thus, clique-finding algorithms can be used to identify sets of clusters in the solution that can be represented by trees. This formulation provides a means of automatically constructing a multiple tree or extended tree representation of any non-nested clustering. The method, called “clustrees”, is applied to several non-nested overlapping clusterings derived using the MAPCLUS program (Arabie and Carroll 1980).