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 O(nd−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2n) 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.

[1]  Micha Sharir.  k-sets and random hulls , 1993, Comb..

[2]  Mia Hubert,et al.  The Catline for Deep Regression , 1998 .

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

[4]  P. Rousseeuw,et al.  High-dimensional computation of the deepest location , 2000 .

[5]  Leonidas J. Guibas,et al.  The Exact Fitting Problem in Higher Dimensions , 1996, Comput. Geom..

[6]  M. Hubert,et al.  Regression depth - Rejoinder , 1999 .

[7]  Mia Hubert,et al.  Depth in an Arrangement of Hyperplanes , 1999, Discret. Comput. Geom..

[8]  P. Rousseeuw,et al.  Robustness of Deepest Regression , 2000 .

[9]  Jan van Leeuwen,et al.  Maintenance of Configurations in the Plane , 1981, J. Comput. Syst. Sci..

[10]  Bernard Chazelle,et al.  Halfspace range search: An algorithmic application ofk-sets , 1986, Discret. Comput. Geom..

[11]  Leonidas J. Guibas,et al.  The exact fitting problem for points , 1991 .

[12]  Raimund Seidel,et al.  Constructing arrangements of lines and hyperplanes with applications , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[13]  G. W. Peck.  On 'k-sets' in the plane , 1985, Discret. Math..

[14]  M. Keil,et al.  A Simple Algorithm for Determining the Envelope of a Set of Lines , 1991, Inf. Process. Lett..

[15]  Micha Sharir,et al.  Optimal Slope Selection Via Expanders , 1993, CCCG.

[16]  Géza Tóth,et al.  Point Sets with Many k-Sets , 2000, SCG '00.

[17]  Micha Sharir,et al.  On the Zone Theorem for Hyperplane Arrangements , 1991, SIAM J. Comput..

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

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

[20]  P. Rousseeuw,et al.  Halfspace Depth and Regression Depth Characterize the Empirical Distribution , 1999 .

[21]  Endre Szemerédi,et al.  An Optimal-Time Algorithm for Slope Selection , 1989, SIAM J. Comput..

[22]  David Eppstein,et al.  Regression Depth and Center Points , 1998, Discret. Comput. Geom..

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

[24]  Jirí Matousek,et al.  Randomized Optimal Algorithm for Slope Selection , 1991, Inf. Process. Lett..

[25]  Godfried T. Toussaint,et al.  On Envelopes of Arrangements of Lines , 1996, J. Algorithms.

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

[27]  Timothy M. Chan.  Dynamic planar convex hull operations in near-logarithmic amortized time , 2001, JACM.

[28]  Regina Y. Liu,et al.  Regression depth. Commentaries. Rejoinder , 1999 .

[29]  David M. Mount,et al.  A randomized algorithm for slope selection , 1992, Int. J. Comput. Geom. Appl..

[30]  Bernard Chazelle,et al.  Optimal Slope Selection Via Cuttings , 1994, CCCG.

[31]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[32]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

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