Efficient Algorithms for Maximum Regression Depth

  • Authors:

  • Marc van Kreveld;Joseph S. B. Mitchell;Peter Rousseeuw;Micha Sharir;Jack Snoeyink;Bettina Speckmann

  • Affiliations:

  • Utrecht University, Department of Information and Computing Sciences, Utrecht, The Netherlands;SUNY Stony Brook, Department of Applied Mathematics and Statistics, Stony Brook, USA;Universitaire Instelling Antwerpen, Department of Mathematics and Computer Science, Antwerpen, Belgium;Tel Aviv University, School of Computer Science, Tel Aviv, Israel;UNC Chapel Hill, Department of Computer Science, Chapel Hill, USA;TU Eindhoven, Department of Mathematics and Computer Science, Eindhoven, The Netherlands

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2008

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Abstract

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.