AS 307

Algorithm AS 307: Bivariate location depth. he half-space depth of a point θ relative to a bivariate data set {x 1 ,⋯,x n } is given by the smallest number of data points contained in a closed half-plane of which the boundary line passes through θ. A straightforward algorithm for the half-space depth needs O(n 2 ) steps. The simplicial depth of θ relative to {x 1 ,⋯,x n } is given by the number of data triangles Δ(x i ,x j ,x k ) that contain θ; this appears to require O(n 3 ) steps. The algorithm proposed here computes both depths in O(nlogn) time, by combining geometric properties with certain sorting and updating mechanisms. Both types of depth can be used for data description, bivariate confidence regions, p-values, quality indices and control charts. Moreover, the algorithm can be extended to the computation of depth contours and bivariate sign test statistics.

References in zbMATH (referenced in 46 articles , 1 standard article )

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  1. Gagolewski, Marek: Penalty-based aggregation of multidimensional data (2017)
  2. Liu, Xiaohui: Fast implementation of the Tukey depth (2017)
  3. Zasenko, Olga; Stephen, Tamon: Algorithms for colourful simplicial depth and medians in the plane (2016)
  4. Hubert, Mia; Rousseeuw, Peter J.; Segaert, Pieter: Multivariate functional outlier detection (2015)
  5. Liu, Xiaohui; Ren, Haiping; Wang, Guofu: Computing halfspace depth contours based on the idea of a circular sequence (2015)
  6. Li, Zhonghua; Dai, Yi; Wang, Zhaojun: Multivariate change point control chart based on data depth for phase I analysis (2014)
  7. Mustafa, Nabil H.; Tiwary, Hans Raj; Werner, Daniel: A proof of the Oja depth conjecture in the plane (2014)
  8. Chenouri, Shojaeddin; Small, Christopher G.: A nonparametric multivariate multisample test based on data depth (2012)
  9. Dutta, Subhajit; Ghosh, Anil K.: On robust classification using projection depth (2012)
  10. Elmasry, Amr: Enumerating trichromatic triangles containing the origin in linear time (2012)
  11. Elbassioni, Khaled; Elmasry, Amr; Makino, Kazuhisa: Finding simplices containing the origin in two and three dimensions (2011)
  12. Hallin, Marc; Paindaveine, Davy; Šiman, Miroslav: Multivariate quantiles and multiple-output regression quantiles: from $L_1$ optimization to halfspace depth (2010)
  13. Hlubinka, Daniel; Vencálek, Ondřej; Kotík, Lukáš: Weighted halfspace depth (2010)
  14. Hubert, Mia; Van der Veeken, Stephan: Robust classification for skewed data (2010)
  15. Massé, Jean-Claude: Multivariate trimmed means based on the Tukey depth (2009)
  16. Mosler, Karl; Lange, Tatjana; Bazovkin, Pavel: Computing zonoid trimmed regions of dimension $d>2$ (2009)
  17. Plante, Jean-François: Asymptotic properties of the MAMSE adaptive likelihood weights (2009)
  18. Messaoud, Amor; Weihs, Claus; Hering, Franz: Detection of chatter vibration in a drilling process using multivariate control charts (2008)
  19. Abellanas, Manuel; Claverol, Mercè; Hurtado, Ferran: Point set stratification and Delaunay depth (2007)
  20. Wilcox, Rand R.: Robust ANCOVA: some small-sample results when there are multiple groups and multiple covariates (2007)

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