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

Unfolding applies multidimensional scaling to an off-diagonal nxm matrix, representing the scores (or the rank) assigned to a set of m items by n individuals or judges. The goal is to obtain two configurations of points representing the position of the judges and the items in a reduced geometrical space. Each point, representing each individual, is considered as an ideal point so that its distances to the object points correspond to the preference scores. Unfolding can be seen as a special case of multidimensional scaling because the off-diagonal matrix is considered as a block of an ideal distance matrix in which both the within judges and the within items dissimilarities are missing. The presence of blocks of missing data causes the phenomenon of the so-called degenerate solutions. To tackle the problem, several methods have been proposed. By following a previous approach, we adopt the strategy of augmenting the data matrix, trying to build a complete dissimilarity matrix, and then applying any MDS algorithms. In order to augment the data matrix, we use copulas-based association. Both experimental evaluations and applications to well-known real data sets show that the proposed strategy produces non-degenerate non-metric unfolding solutions.