Exact computation of angular halfspace depth

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10506633" target="_blank" >RIV/00216208:11320/25:10506633 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OYine05jr-" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OYine05jr-</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s11222-025-10700-z" target="_blank" >10.1007/s11222-025-10700-z</a>

Result language

angličtina

Original language name

Exact computation of angular halfspace depth

Original language description

The angular halfspace depth (ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document}) was, already in 1987, the first depth function proposed for the nonparametric analysis of directional data. Mainly due to its presumed high computational cost and lack of efficient computational algorithms, it was never widely used in directional data analysis. We address the problem of the exact computation of ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document} in any dimension d. We proceed in two steps: (i) We express ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document} as a generalized (Euclidean) halfspace depth in dimension d-1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d-1$$end{document}, using a projection approach. That allows us to develop fast exact computational algorithms for ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document} in dimensions d=1,2,3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d=1, 2, 3$$end{document}. (ii) In spaces of dimension 3]]d 3 we design an inductive procedure that reduces the dimensionality d in the computation of ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document}, until the algorithms for d &lt;= 3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$d le 3$$end{document} can be used. Using our advances we develop a family of powerful algorithms for the computation of ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document} in any dimension d. Our procedures are implemented efficiently in C++ with an interface in R. A detailed analysis of the complexity of the novel algorithms is performed. Surprisingly, we show that computing ahDdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ahD$$end{document} of multiple points with respect to the same dataset is substantially faster than the same task for the classical (Euclidean) halfspace depth.

Czech name

—

Czech description

—

Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10103 - Statistics and probability

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Statistics and Computing

ISSN

0960-3174

e-ISSN

1573-1375

Volume of the periodical

35

Issue of the periodical within the volume

6

Country of publishing house

GB - UNITED KINGDOM

Number of pages

29

Pages from-to

173

UT code for WoS article

001551714600001

EID of the result in the Scopus database

2-s2.0-105013649806