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Dyckerhoff, Rainer ORCID: 0000-0002-3631-8497 and Nagy, Stanislav ORCID: 0000-0002-8610-4227 (2025). Exact computation of angular halfspace depth. Statistics and Computing, 35 (6). pp. 1-29. Springer Nature. ISSN 0960-3174

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Abstract

[Artikel-Nr.: 173] The angular halfspace depth (ah D) 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 ah D in any dimension d. We proceed in two steps: (i) We express ah D as a generalized (Euclidean) halfspace depth in dimension d − 1, using a projection approach. That allows us to develop fast exact computational algorithms for ah D in dimensions d = 1, 2, 3. (ii) In spaces of dimension 3]]d 3 we design an inductive procedure that reduces the dimensionality d in the computation of ah D, until the algorithms for d ≤ 3 can be used. Using our advances we develop a family of powerful algorithms for the computation of ah D 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 ah D of multiple points with respect to the same dataset is substantially faster than the same task for the classical (Euclidean) halfspace depth.

Item Type:	Article
Creators:	Creators Email ORCID ORCID Put Code Dyckerhoff, Rainer UNSPECIFIED https://orcid.org/0000-0002-3631-8497 UNSPECIFIED Nagy, Stanislav UNSPECIFIED https://orcid.org/0000-0002-8610-4227 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-809782
Identification Number:	10.1007/s11222-025-10700-z
Journal or Publication Title:	Statistics and Computing
Volume:	35
Number:	6
Page Range:	pp. 1-29
Number of Pages:	29
Date:	18 December 2025
Publisher:	Springer Nature
ISSN:	0960-3174
Language:	English
Faculty:	Faculty of Management, Economy and Social Sciences
Divisions:	Faculty of Management, Economics and Social Sciences > Economics > Econometrics and Statistics > Professorship for Statistics and Econometrics
Subjects:	Social sciences General statistics Economics Mathematics
Uncontrolled Keywords:	Keywords Language Angular halfspace depth ; Exact computation ; Depth ; Directional data analysis UNSPECIFIED
['eprint_fieldname_oa_funders' not defined]:	Publikationsfonds UzK
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/80978