Multivariate Spearman's $\rho$ for Aggregating Ranks Using Copulas

Justin Bedő, Cheng Soon Ong.

Year: 2016, Volume: 17, Issue: 201, Pages: 1−30

Abstract

We study the problem of rank aggregation: given a set of ranked lists, we want to form a consensus ranking. Furthermore, we consider the case of extreme lists: i.e., only the rank of the best or worst elements are known. We impute missing ranks and generalise Spearman's $\rho$ to extreme ranks. Our main contribution is the derivation of a non-parametric estimator for rank aggregation based on multivariate extensions of Spearman's $\rho$, which measures correlation between a set of ranked lists. Multivariate Spearman's $\rho$ is 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 rank which 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's $\rho$. We discuss an optimistic and pessimistic imputation of missing values, which respectively maximise and minimise correlation, and show its effect on aggregating university rankings. Finally, we demonstrate good performance on the rank aggregation benchmarks MQ2007 and MQ2008.