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For supervised classification we propose to use restricted multidimensional unfolding in a multinomial logistic framework. Where previous research proposed similar models based on squared distances, we propose to use usual (i.e., not squared) Euclidean distances. This change in functional form results in several interpretational advantages of the resulting biplot, a graphical representation of the classification model. First, the conditional probability of any class peaks at the location of the class in the Euclidean space. Second, the interpretation of the biplot is in terms of distances towards the class points, whereas in the squared distance model the interpretation is in terms of the distance towards the decision boundary. Third, the distance between two class points represents an upper bound for the estimated log-odds of choosing one of these classes over the other. For our multinomial restricted unfolding, we develop and test a Majorization Minimization algorithm that monotonically decreases the negative log-likelihood. With two empirical applications we point out the advantages of the distance model and show how to apply multinomial restricted unfolding in practice, including model selection. [ABSTRACT FROM AUTHOR]

Abstract: The fundamentals of preference mapping are revisited in the context of a new restricted unfolding method that has potential for wide application to product optimisation for consumers. Since more of an attribute is not necessarily preferred, the unfolding distance model provides estimates of ideal points for consumers and therefore provides a more adequate representation of the preference relationships, compared to conventional internal preference mapping. Compared to other ideal point methods, the new unfolding technique offers advantages in terms of allowing for the ordinal nature of the ratings, rather than implicitly assuming that ratings are linear. The proposed restricted unfolding model incorporates property fitting, both passive, as a separate, second step, and active, as a restriction on the product locations. This is also available as a restriction on the respondents’ locations and as such establishing a link between internal and external preference mapping. [Copyright &y& Elsevier]

It has long been thought that degeneracy in unfolding only concerned non-metric unfolding. Recently, Busing, Groenen, and Heiser have established that degeneracy occurs for all transformations that include estimation of an intercept and a slope. Consequently, degeneracy also plagues metric unfolding, since one member of the metric transformation family, the interval transformation, includes estimation of both an intercept and a slope. In this paper, a simple solution is proposed to the degeneracy problem for metric unfolding by penalizing for an undesirable intercept. An application of this approach will illustrate its potential. [ABSTRACT FROM AUTHOR]

In this exploratory study, a multidimensional unfolding technique is used to visualise goal salience. A total of 1,733 students from secondary vocational education reported on the importance of 16 goals, using the GIFI, which is based on the 24 goal taxonomy proposed by Ford and Nichols (1991). Results show that students make a clear distinction between ego and non-ego goals and that non-ego goals are reported as most important by most students. Also, differences within the group of boys and the group of girls, and differences between boys and girls, are shown. The results and possibilities of multidimensional unfolding for future research on goals are discussed. [ABSTRACT FROM AUTHOR]

Abstract: Nonmetric unfolding is a powerful (nonparametric) analytical tool generating a preference-based joint display of subjects (e.g., consumers) and objects (e.g., brands or products). Unfortunately, nonmetric unfolding frequently produces degenerate unfolding solutions (i.e., unfolding solutions showing close-to-perfect model fit irrespective of the data analyzed). Moreover, there are no methods to assess the quality of the unfolding solution in terms of stability and accuracy. In this paper, we resolve these important issues simultaneously by using a bootstrapped penalized nonmetric unfolding approach. In line with the explorative and visual nature of nonmetric unfolding, we introduce methods for visualizing the stability of unfolding solutions. In addition, we propose numerical measures for stability and validity of nonmetric unfolding solutions that can be used to identify accurate, nondegenerate and stable solutions. Finally, for large scale data, as often encountered in marketing, we propose a greedy search algorithm. We illustrate our methodology using three applications in the food domain. [Copyright &y& Elsevier]