Exact computation of angular halfspace depth

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Exact computation of angular halfspace depth

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Exact computation of angular halfspace depth

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10103 - Statistics and probability

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Statistics and Computing

ISSN

0960-3174

e-ISSN

1573-1375

Svazek periodika

35

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

29

Strana od-do

173

Kód UT WoS článku

001551714600001

EID výsledku v databázi Scopus

2-s2.0-105013649806