Consensus among preference rankings: a new weighted correlation coefficient for linear and weak orderings

Preference data are a particular type of ranking data where some subjects (voters, judges,...) express their preferences over a set of alternatives (items). In most real life cases, some items receive the same preference by a judge, thus giving rise to a ranking with ties. An important issue involving rankings concerns the aggregation of the preferences into a “consensus”. The purpose of this paper is to investigate the consensus between rankings with ties, taking into account the importance of swapping elements belonging to the top (or to the bottom) of the ordering (position weights). By combining the structure of $$\tau _x$$ τ x proposed by Emond and Mason (J Multi-Criteria Decis Anal 11(1):17–28, 2002) with the class of weighted Kemeny-Snell distances, a position weighted rank correlation coefficient is proposed for comparing rankings with ties. The one-to-one correspondence between the weighted distance and the rank correlation coefficient is proved, analytically speaking, using both equal and decreasing weights.

A New Weighted-learning Approach for Exploiting Data Sparsity in Tag-based Item Recommendation Systems

The tag-based recommendation systems that are built based on tensor models commonly suffer from the data sparsity problem. In recent years, various weighted-learning approaches have been proposed to tackle such a problem. The approaches can be categorized by how a weighting scheme is used for exploiting the data sparsity – like employing it to construct a weighted tensor used for weighing the tensor model during the learning process. In this paper, we propose a new weighted-learning approach for exploiting data sparsity in tag-based item recommendation system. We introduce a technique to represent the users’ tag preferences for leveraging the weighted-learning approach. The key idea of the proposed technique comes from the fact that users use different choices of tags to annotate the same item while the same tag may be used to annotate various items in tag-based systems. This points out that users’ tag usage likeliness is different and therefore their tag preferences are also different. We then present three novel weighting schemes that are varied in manners by how the ordinal weighting values are used for labelling the users’ tag preferences. As a result, three weighted tensors are generated based on each scheme. To implement the proposed schemes for generating item recommendations, we develop a novel weighted-learning method called as WRank (Weighted Rank). Our experiments show that considering the users' tag preferences in the tensor-based weightinglearning approach can solve the data sparsity problem as well as improve the quality of recommendation.