A weighted localization of halfspace depth and its properties

A depth-based modification of the k-nearest neighbour method

“…There are two ways how to localise the depthbased procedures. The first possibility is to use some local depth, as suggested in, e. g., [1,13,16], or [23]. The second possibility is to plug-in some local classification procedure like k-nearest neighbours.…”

“…On the other hand, if the level sets of the probability density function are not convex, then the approximation (9) needs not be sufficiently good. This problem may be solved by using some version of a localised depth, see [13,16] or [1], rather than a usual global depth function such as the halfspace depth or the projection depth.…”

A local depth measure for general data

“…It is well known that the spatial median is not affine invariant, hence, transformation and retransformation methods have been designed to construct affine equivariant multivariate medians Chaudhuri 1996, 1998)). IDLD can be modified following the ideas of Kotík and Hlubinka (2017) to attain this property.…”

Weighted lens depth: Some applications to supervised classification

“…To overcome this issue, several modifications of the classical notions of depths have been proposed. For instance, in Kotík & Hlubinka (2017) a weighted Tukey's depth is considered. As shown in Li, Cuesta‐Albertos & Liu (2012), the depth‐depth approach behaves better than other learning procedures when the data have some nonstandard patterns.…”

“…Generality : Lens depth (and ) easily extends to general metric spaces and in particular to Riemannian manifolds. Ability to capture the level‐set structure : Some depths (under restrictive conditions like unimodality or symmetry) characterize the distribution of the data (Kong & Zuo, 2010; Kotík & Hlubinka, 2017) and have convex level sets. However, if those very restrictive conditions are not fulfilled, the level sets are not convex, see Dutta, Ghosh & Chaudhuri (2011).…”