Generalized Mahalanobis depth in the reproducing kernel Hilbert space

In this paper, Mahalanobis depth (MHD) in the Reproducing Kernel Hilbert Space (RKHS) is proposed. First, we extend the notion of MHD to a generalized version, i.e., the generalized MHD (GHMD), to make it suitable for the small sample with singular covariance matrix. We prove that GMHD is consistent with MHD when the sample has a full-rank covariance matrix. Second, we further extend GMHD to RKHS, i.e, the kernel mapped GMHD (kmGMHD), and discuss its main properties. Numeric results show that kmGMHD can give a better depth interpretation for the sample with special shape, such as a non-convex sample set. Our proposed kmGMHD can be potentially used as a robust tool for outliers detection and data classification. In addition, we also discuss the influence of parameters on the shape of the central regions.