Tukey depth: linear programming and applications

Determining the representativeness of a point within a data cloud has recently become a desirable task in multivariate analysis. The concept of statistical depth function, which reflects centrality of an arbitrary point, appears to be useful and has been studied intensively during the last decades. Here the issue of exact computation of the classical Tukey data depth is addressed. The paper suggests an algorithm that exploits connection between the Tukey depth and linear separability and is based on iterative application of linear programming. The algorithm further develops the idea of the cone segmentation of the Euclidean space and allows for efficient implementation due to the special search structure. The presentation is complemented by relationship to similar concepts and examples of application.

[1]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[2]  D. Donoho,et al.  Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness , 1992 .

[3]  Alicia Nieto-Reyes,et al.  The random Tukey depth , 2007, Comput. Stat. Data Anal..

[4]  Pat Morin,et al.  Absolute approximation of Tukey depth: Theory and experiments , 2013, Comput. Geom..

[5]  Pat Morin,et al.  Output-sensitive algorithms for Tukey depth and related problems , 2008, Stat. Comput..

[6]  Xiaohui Liu,et al.  Computing Halfspace Depth and Regression Depth , 2014, Commun. Stat. Simul. Comput..

[7]  Tatjana Lange,et al.  Fast nonparametric classification based on data depth , 2012, Statistical Papers.

[8]  Tatjana Lange,et al.  Classifying real-world data with the $${ DD}\alpha $$DDα-procedure , 2014, Adv. Data Anal. Classif..

[9]  Miroslav Siman,et al.  Computing multiple-output regression quantile regions , 2012, Comput. Stat. Data Anal..

[10]  Timothy M. Chan.  An optimal randomized algorithm for maximum Tukey depth , 2004, SODA '04.

[11]  Regina Y. Liu,et al.  DD-Classifier: Nonparametric Classification Procedure Based on DD-Plot , 2012 .

[12]  Mia Hubert,et al.  ROBPCA: A New Approach to Robust Principal Component Analysis , 2005, Technometrics.

[13]  Xiaohui Liu,et al.  Computing projection depth and its associated estimators , 2012, Statistics and Computing.

[14]  D. Paindaveine,et al.  Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth , 2010, 1002.4486.

[15]  M. Hubert,et al.  Multivariate Functional Halfspace Depth , 2012 .

[16]  Franco P. Preparata,et al.  The Densest Hemisphere Problem , 1978, Theor. Comput. Sci..

[17]  Tatjana Lange,et al.  Computing zonoid trimmed regions of dimension d>2 , 2009, Comput. Stat. Data Anal..

[18]  S. Majumdar.  Robust estimation of principal components from depth-based multivariate rank covariance matrix , 2015, 1502.07042.

[19]  R. Koenker,et al.  Regression Quantiles , 2007 .

[20]  K. Mosler,et al.  OF ECONOMIC AND SOCIAL STATISTICS UNIVERSITY OF COLOGNE No . 06 / 2010 An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension , 2010 .

[21]  J. Tukey.  Mathematics and the Picturing of Data , 1975 .

[22]  Davy Paindaveine,et al.  Computing multiple-output regression quantile regions from projection quantiles , 2011, Computational Statistics.

[23]  K. Mosler.  Depth Statistics , 2012, Encyclopedia of Image Processing.

[24]  P. Chaudhuri,et al.  On data depth and distribution-free discriminant analysis using separating surfaces , 2005 .

[25]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[26]  P. Rousseeuw.  Least Median of Squares Regression , 1984 .

[27]  R. Dyckerhoff.  Data depths satisfying the projection property , 2004 .

[28]  Pavlo Mozharovskyi,et al.  Exact computation of the halfspace depth , 2014, Comput. Stat. Data Anal..

[29]  Gleb A. Koshevoy,et al.  The Tukey Depth Characterizes the Atomic Measure , 2002 .