The Rank Minrelation Coefficient

Bivariate (or pairwise) information measures such as mutual information or correlation are heavily used in variable selection and network inference algorithms mainly because they are faster and require fewer samples than multivariate (or multidimensional) strategies. This paper proposes a new relevance measure that aims at improving the detection of relevant variables based on pairwise measures. The new measure is called the rank minrelation coefficient because of its connection to the rank correlation coefficient. However, on the contrary to correlation, the minrelation is not symmetric. More explicitly, if a variable X exhibits a minrelation to Y then, as X increases, Y is likely to increases too, but, if X decreases, little can be said on Y values (except that the uncertainty on Y actually increases). In this paper, we introduce a new rank coefficient and an associated relevance measure that targets the detection of a characteristic dependency, connected to the concept of probabilistic implication. Finally, we show through several key examples and experiments that this new coefficient is competitive in order to select relevant variables, in particular when compared to correlations and mutual information.