Non-metric unfolding on augmented data matrix: a copula-based approach

Unfolding applies multidimensional scaling (Cox & Cox, 2000) to an off-diagonaln×mmatrix, representing the scores (or the rank) assigned to a set of m items by n indi-viduals or judges (Borg & Groenen, 1997). The goal is to obtain two configurations of pointsrepresenting the position of the judges and the items in a reduced geometrical space. Eachpoint, representing each individual, is considered as an ideal point so that its distances tothe object points correspond to the preference scores (Coombs, 1964). Unfolding can be seenas a special case of multidimensional scaling because the off-diagonal matrix is consideredas a block of an ideal distance matrix in which both the within judges and the within itemsdissimilarities are missing. The presence of blocks of missing data causes the phenomenonof the so-called degenerate solutions, i.e., solutions that return excellent badness of fit mea-sures but not graphically interpretable at all. To tackle the problem of degenerate solutions,several methods have been proposed (Borg & Groenen, 1997). By following the approachintroduced by Van Deun et al. (2007), we adopt the strategy of augmenting the data matrix,trying to build a complete dissimilarity matrix, and then applying any MDS algorithms. Inorder to augment the data matrix, we use copulas-based association measures (Joe, 1997;Nelsen, 2013) among rankings (the individuals), and between rankings and objects (namely,a rank-order representation of the objects through tied rankings). Both experimental evalua-tions and applications to well-known real data sets show that the proposed strategy producesnon-degenerate non-metric unfolding solutions.