On the Use of M-Quantiles for Outlier Detection in Multidimensional Data

Defining the notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of Chakraborty and Chaudhuri [8] and Koltchinskii [26] leading to M-quantiles, is very appealing for its mathematical elegance, combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of Rᵈ and the unit ball in Rᵈ . By analogy with the $$d=1$$ case where the K-transform is a cumulative distribution function like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d≥ 1 . The resulting M-quantiles have seen applications in a variety of fields, with outlier detection featuring prominently. We show that for odd d≥ 3 it is not the first derivative, but a poly-Laplacian of the K-function that is proportional to the density function. (For even values of d no relationship is apparent.) Two-dimensional examples from non-standard distributions reinforce this point, and illustrate a feature of the K-transform whereby high density regions of the support are mapped to larger volumes, thereby producing a magnification effect that moves inliers closer to the boundary than outliers. In conclusion, the K-transform should not be viewed as a distribution function in higher dimensions, and its inverse does not lead to a ‘‘useful’’ quantile function, in the sense that its level set contours do not plausibly delineate outlying regions.