Statistics and Computing 1573-1375 0960-3174 76798a853d5fa4b3d9612da441dc51c14fb7572595214100ce1a3dcd16f0b2ac readcube_id Department of Mathematics and Computer Science, U.I.A., Universiteitsplein 1, B-2610, Antwerp, Belgium 193-203 false 2019-04-10T21:31 The location depth (Tukey 1975) of a point θ relative to a p-dimensional data set Z of size n is defined as the smallest number of data points in a closed halfspace with boundary through θ. For bivariate data, it can be computed in O(nlogn) time (Rousseeuw and Ruts 1996). In this paper we construct an exact algorithm to compute the location depth in three dimensions in O(n2logn) time. We also give an approximate algorithm to compute the location depth in p dimensions in O(mp3+mpn) time, where m is the number of p-subsets used. Recently, Rousseeuw and Hubert (1996) defined the depth of a regression fit. The depth of a hyperplane with coefficients (θ1,...,θp) is the smallest number of residuals that need to change sign to make (θ1,...,θp) a nonfit. For bivariate data (p=2) this depth can be computed in O(nlogn) time as well. We construct an algorithm to compute the regression depth of a plane relative to a three-dimensional data set in O(n2logn) time, and another that deals with p=4 in O(n3logn) time. For data sets with large n and/or p we propose an approximate algorithm that computes the depth of a regression fit in O(mp3+mpn+mnlogn) time. For all of these algorithms, actual implementations are made available. research_article articles 1998-08-01 http://link.springer.com/10.1023/A:1008945009397 https://scigraph.springernature.com/explorer/license/ Computing location depth and regression depth in higher dimensions 1998-08 en Peter J. Rousseeuw Artificial Intelligence and Image Processing 8 Department of Mathematics and Computer Science, U.I.A., Universiteitsplein 1, B-2610, Antwerp, Belgium 3 Information and Computing Sciences Struyf Anja doi 10.1023/a:1008945009397 pub.1024868624 dimensions_id Springer Nature - SN SciGraph project