Pearson Distance is not a Distance

The Pearson distance between a pair of random variables X,Y with correlation ρxy, namely, 1-ρxy, has gained widespread use, particularly for clustering, in areas such as gene expression analysis, brain imaging and cyber security. In all these applications it is implicitly assumed/required that the distance measures be metrics, thus satisfying the triangle inequality. We show however, that Pearson distance is not a metric. We go on to show that this can be repaired by recalling the result, (well known in other literature) that √1-ρxy is a metric. We similarly show that a related measure of interest, 1-|ρxy|, which is invariant to the sign of ρxy, is not a metric but that √1-ρxy2 is. We also give generalizations of these results.