Multivariate Spearman’sρfor Aggregating Ranks Using Copulas

Justin Bedo, Cheng Soon Ong

Abstract: We study the problem of rank aggregation: given a set of ranked lists, we want to form aconsensus ranking. Furthermore, we consider the case of extreme lists: i.e., only the rank ofthe best or worst elements are known. We impute missing ranks and generalise Spearman’sρto extreme ranks. Our main contribution is the derivation of a non-parametric estimatorfor rank aggregation based on multivariate extensions of Spearman’sρ, which measurescorrelation between a set of ranked lists. Multivariate Spearman’sρis defined using copulas,and we show that the geometric mean of normalised ranks maximises multivariate correlation.Motivated by this, we propose a weighted geometric mean approach for learning to rankwhich has a closed form least squares solution. When only the best (top-k) or worst (bottom-k) elements of a ranked list are known, we impute the missing ranks by the average value,allowing us to apply Spearman’sρ. We discuss an optimistic and pessimistic imputation ofmissing values, which respectively maximise and minimise correlation, and show its effecton aggregating university rankings. Finally, we demonstrate good performance on the rankaggregation benchmarks MQ2007 and MQ2008.