Rank-order principal components. A separation algorithm for ordinal data exploration.
Résumé
In most research studies, much of the information gathered is of qualitative nature. This paper concentrates on items for which multiple rankings exist that shall be combined optimally. This work presents a unsupervised deterministic approach that can be applied to rank-order data for decomposing them into composite signals. On the opposite to other techniques, it does not requite any prior knowledge about the distribution of the signal components. Typically the variables are necessarily dependent because of the inequality relations among them. The mathematical relationships found are of interest in themselves and in the theory of blind source separation (BSS).Multiple rankings of an item set are decomposed into other rankings of these items, each ranking being orthogonal to the others, hence the name of rank-order principal components. We transform the original ranking data into matrices that linearize the optimization problem for its resolution by linear programming.The resolution is sensitive to the distance between the items that is used. Several distances have been tested: The Euclidean distance (Spearman), the rank absolute deviation distance, the Hölder distance, the χ2 distance, etc. The results are concordant.