Efficient Computation of Location Depth Contours by Methods of Computational Geometry

Authors

Kim Miller, Tufts University
Suneeta Ramaswami, Rutgers University–Camden
Peter Rousseeuw, Universiteit Antwerpen
J. Antoni Sellarès, Universitat de Girona
Diane Souvaine, Smith College
Ileana Streinu, Smith CollegeFollow
Anja Struyf, Universiteit Antwerpen

Document Type

Article

Publication Date

4-1-2003

Publication Title

Statistics and Computing

Abstract

The concept of location depth was introduced 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. The depth contours form a collection of nested polygons, and the center of the deepest contour 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 bivariate depth contours in O(n 2) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log 2 n) time with no additional preprocessing or in O(log n) time after O(n 2) preprocessing. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.

Keywords

Bagplot, Bivariate median, Graphical display, Robust estimation, Tukey depth

Volume

13

Issue

2

First Page

153

Last Page

162

DOI

10.1023/A:1023208625954

ISSN

09603174

Comments

Archived as published.

Recommended Citation

Miller, Kim; Ramaswami, Suneeta; Rousseeuw, Peter; Antoni Sellarès, J.; Souvaine, Diane; Streinu, Ileana; and Struyf, Anja, "Efficient Computation of Location Depth Contours by Methods of Computational Geometry" (2003). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/291