Modern Nonparametric, Robust and Multivariate Methods by Klaus Nordhausen & Sara Taskinen

Author:Klaus Nordhausen & Sara Taskinen
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

15.2 Classifier Based on Distribution of Multivariate Outlyingness

Suppose X is a d-dimensional random vector (or, feature vector) having a distribution F, which is assumed to be absolutely continuous with respect to the Lebesgue measure . The spatial rank function (Möttönen and Oja 1995) of any point with respect to F is defined as

(15.1)

Here is the usual Euclidian norm. It follows immediately from the definition that rank F (x) = 0 implies that x is the spatial median of the multivariate distribution F. Koltchinskii (1997) established that this spatial rank function is a one-to-one function of the distribution function F and hence it characterizes the distribution.

Define and then define the distribution function of r F (X) as

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