Efficient Algorithms for Maximum Regression Depth

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(nd ) time and space algorithm computes directed depth of all points in d dimensions. Properties of undirected depth lead to an O(n log2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(n log n) time algorithm by Langerman and Steiger [17]. Furthermore, we describe the structure of depth in the plane and higher dimensions and also give approximation algorithms for hyperplane arrangements and degenerate line arrangements.

• Type: Journal
• Year: 2008
• Where: DCG
• Authors: Marc J. van Kreveld, Joseph S. B. Mitchell, Peter Rousseeuw, Micha Sharir, Jack Snoeyink, Bettina Speckmann
• Keywords: DCG 2008, Hyperplane, Maximum Regression Depth, Space Algorithm