Fast implementation of depth contours using topological sweep

The concept of location depth was introduced in statistics as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. These require the computation of depth regions, which form a collection of nested polygons. The center of the deepest region is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all depth contours in &Ogr;(n2) time and space, using topological sweep of the dual arrangement of lines. Once the contours are known, the location depth of any point is computed in &Ogr;(log2 n) time. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.

[1]  Jirí Matousek.  Computing the Center of Planar Point Sets , 1990, Discrete and Computational Geometry.

[2]  P. Rousseeuw,et al.  Computing depth contours of bivariate point clouds , 1996 .

[3]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[4]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[5]  Trevor Hastie,et al.  Statistical Models in S , 1991 .

[6]  Stefan Langerman,et al.  An optimal algorithm for hyperplane depth in the plane , 2000, SODA '00.

[7]  D. Donoho,et al.  Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness , 1992 .

[8]  Peter Rousseeuw,et al.  Computing location depth and regression depth in higher dimensions , 1998, Stat. Comput..

[9]  M. Braga,et al.  Exploratory Data Analysis , 2018, Encyclopedia of Social Network Analysis and Mining. 2nd Ed..

[10]  Richard Cole,et al.  On k-hulls and related problems , 1984, STOC '84.

[11]  謙太郎 野間口,et al.  仮説に制約条件がある場合の Bivariate Sign Test , 1986 .

[12]  P. Rousseeuw,et al.  Constructing the bivariate Tukey median , 1998 .

[13]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[14]  J. Matou Sek,et al.  Computing the center of planar point sets , 1991 .

[15]  P. Rousseeuw,et al.  The Bagplot: A Bivariate Boxplot , 1999 .

[16]  Chee-Keng Yap,et al.  Algorithmic and geometric aspects of robotics , 1987 .

[17]  Regina Y. Liu,et al.  Notions of Limiting P Values Based on Data Depth and Bootstrap , 1997 .