GENFOLD2: A set of models and algorithms for the general UnFOLDing analysis of preference/dominance data

A general set of multidimensional unfolding models and algorithms is presented to analyze preference or dominance data. This class of models termed GENFOLD2 (GENeral UnFOLDing Analysis-Version 2) allows one to perform internal or external analysis, constrained or unconstrained analysis, conditional or unconditional analysis, metric or nonmetric analysis, while providing the flexibility of specifying and/or testing a variety of different types of unfolding-type preference models mentioned in the literature including Caroll's (1972, 1980) simple, weighted, and general unfolding analysis. An alternating weighted least-squares algorithm is utilized and discussed in terms of preventing degenerate solutions in the estimation of the specified parameters. Finally, two applications of this new method are discussed concerning preference data for ten brands of pain relievers and twelve models of residential communication devices.     

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.     

Fitting a distance model to homogeneous subsets of variables: Points of view analysis of categorical data

An approach is presented for analyzing a heterogeneous set of categorical variables assumed to form a limited number of homogeneous subsets. The variables generate a particular set of proximities between the objects in the data matrix, and the objective of the analysis is to represent the objects in lowdimensional Euclidean spaces, where the distances approximate these proximities. A least squares loss function is minimized that involves three major components: a) the partitioning of the heterogeneous variables into homogeneous subsets; b) the optimal quantification of the categories of the variables, and c) the representation of the objects through multiple multidimensional scaling tasks performed simultaneously. An important aspect from an algorithmic point of view is in the use of majorization. The use of the procedure is demonstrated by a typical example of possible application, i.e., the analysis of categorical data obtained in a free-sort task. The results of points of view analysis are contrasted with a standard homogeneity analysis, and the stability is studied through a Jackknife analysis.     

A spherical representation of a correlation matrix

It is common practice to perform a principal component analysis (PCA) on a correlation matrix to represent graphically the relations among numerous variables. In such a situation, the variables may be considered as points on the unit hypersphere of an Euclidean space, and PCA provides a sort of best fit of these points within a subspace. Taking into account their particular position, this paper suggests to represent the variables on an optimal three-dimensional unit sphere.

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Résumé Il est classique d'utiliser une analyse en composantes principales pour représenter graphiquement une matrice de corrélation. Dans une telle situation, les variables peuvent être considérées comme des points sur l'hypersphère unité d'un espace Euclidien, et l'analyse en composantes principales permet d'obtenir une bonne approximation de ces points à l'aide d'un sous-espace Euclidien. Prenant en compte une telle situation géométrique, le présent article suggère de représenter les variables sur une sphère tri-dimensionelle optimale.