How to cite this WIREs title:
WIREs Comp Stat

Minimum volume ellipsoid

Advanced Review

Stefan Van Aelst, Peter Rousseeuw

Published Online: Jul 13 2009

DOI: 10.1002/wics.19

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Abstract The minimum volume ellipsoid (MVE) estimator is based on the smallest volume ellipsoid that covers h of the n observations. It is an affine equivariant, high‐breakdown robust estimator of multivariate location and scatter. The MVE can be computed by a resampling algorithm. Its low bias makes the MVE very useful for outlier detection in multivariate data, often through the use of MVE‐based robust distances. We review the basic MVE definition as well as some useful extensions such as the one‐step reweighted MVE. We discuss the main properties of the MVE including its breakdown value, affine equivariance, and efficiency. We discuss the basic resampling algorithm to calculate the MVE and illustrate its use on two examples. An overview of applications is given, as well as some related classes of robust estimators of multivariate location and scatter. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Robust Methods

Distances of the observations in the pulp fiber dataset based on the four pulp fiber properties: (a) Mahalanobis distances based on sample mean and sample covariance matrix; (b) Robust distances based on MVE estimates of location and scatter. The horizontal cutoff line in both panels is at .

[ Normal View | Magnified View ]

MVE‐based robust distances of the observations in the Philips dataset. The horizontal cutoff line is at .

[ Normal View | Magnified View ]

Pairwise scatterplots of the four pulp fiber variables. The ellipses represent the 97.5% tolerance ellipsoid for the observations, based on the MVE estimates of location and scatter .

[ Normal View | Magnified View ]

Pairwise scatterplots of the four pulp fiber variables. The ellipses represent the 97.5% tolerance ellipsoid for the observations, based on the sample mean and sample covariance matrix.

[ Normal View | Magnified View ]

References

Rousseeuw, PJ. Least median of squares regression. J Am Stat Assoc 1984, 79: 871–880.
Rousseeuw, PJ. Multivariate estimation with high breakdown point. In: Grossmann, W, Pflug, G, Vincze, I, Wertz, W, eds. Mathematical Statistics and Applications, Vol. B. Dordrecht: Reidel Publishing Company; 1985, 283–297.
Lee, J. Relationships between Properties of Pulp‐Fibre and Paper. PhD thesis, University of Toronto, 1992.
Rousseeuw, PJ, Van Aelst, S, Van Driessen, K, Agulló, J. Robust multivariate regression. Technometrics 2004, 46: 293–305.
Pison, G, Van Aelst, S. Diagnostic plots for robust multivariate methods. J Comput Graph Stat 2004, 13: 310–329.
Hampel, FR, Ronchetti, EM, Rousseeuw, PJ, Stahel, WA. Robust Statistics: The Approach Based on Influence Functions. New York, NY: Wiley, 1986.
Rousseeuw, PJ, Leroy, AM. Robust Regression and Outlier Detection. New York, NY: Wiley‐Interscience, 1987.
Maronna, RA, Martin, DR, Yohai, VJ. Robust Statistics: Theory and Methods. New York: Wiley, 2006.
Davies, L, Gather, U. The identification of multiple outliers. J Am Stat Assoc 1993, 88: 782–792.
Becker, C, Gather, U. The masking breakdown point of multivariate outlier identification rules. J Am Stat Assoc 1999, 94: 947–955.
Rousseeuw, PJ, Van Driessen, K. A fast algorithm for the minimum covariance determinant estimator. Technometrics 1999, 41: 212–223.
Lopuhaä, HP, Rousseeuw, PJ. Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann Stat 1991, 19: 229–248.
Davies, L. The asymptotics of Rousseeuw`s Minimum Volume Ellipsoid estimator. Ann Stat 1992, 20: 1828–1843.
Donoho,, DL, Huber,, PJ. The notion of breakdown point. In: Bickel, P, Doksum, K, Hodges, J, eds. A Festschrift for Erich Lehmann. Belmont, CA: Wadsworth; 1983, 157–184.
Davies, L. Asymptotic behavior of S‐estimators of multivariate location parameters and dispersion matrices. Ann Stat 1987, 15: 1269–1292.
Rousseeuw, PJ. Discussion of ‘Breakdown and Groups’. Ann Stat 2005, 33: 1004–1009.
Rousseeuw, PJ, van Zomeren, BC. Unmasking multivariate outliers and leverage points. J Am Stat Assoc 1990, 85: 633–651.
Lopuhaä, HP. Asymptotics of reweighted estimators of multivariate location and scatter. Ann Stat 1999, 27: 1638–1665.
Lopuhaä, HP. Highly efficient estimators of multivariate location with high breakdown point. Ann Stat 1992, 20: 398–413.
He, XM, Wang, G. Cross‐checking using the Minimum Volume Ellipsoid estimator. Stat Sin 1996, 6: 367–374.
Cook, RD, Hawkins, DM, Weisberg, S. Exact iterative computation of the robust multivariate Minimum Volume Ellipsoid estimator. Stat Probab Lett 1993, 16: 213–218.
Agulló, J. Exact iterative computation of the multivariate Minimum Volume Ellipsoid estimator with a branch and bound algorithm. In: Prat, A, ed. Compstat Proceedings in Computational Statistics. Heidelberg: Springer‐Verlag, 1996, 44–45.
Rousseeuw, PJ, Bassett, G. Robustness of the p‐subset algorithm for regression with high breakdown point. In: Stahel, W, Weisberg, S, eds. Directions in Robust Statistics and Diagnostics, Part II, The IMA Volumes in Mathematics and Its Applications. Springer verlag, New York, NY, 1991, 185–194.
Rousseeuw, PJ, Hubert,, M. Recent developments in progress. In: Dodge, Y, ed. L1‐Statistical Procedures and Related Topics, IMS Lecture Notes‐Monograph Series, Vol. 31. Hayward, CA: Institute of Mathematical Statistics, 1997, 201–214.
Croux, C, Haesbroeck, G. An easy way to increase the finite‐sample efficiency of the resampled Minimum Volume Ellipsoid estimator. Comput Stat Data Anal 1997, 25: 125–141.
Croux, C, Haesbroeck, G. A note on finite‐sample efficiencies of estimators for the Minimum Volume Ellipsoid. J Stat Comput Simulation 2002, 72: 585–596.
Croux, C, Haesbroeck, G, Rousseeuw, PJ. Location adjustment for the Minimum Volume Ellipsoid estimator. Stat Comput 2002, 12: 191–200.
Hawkins, DM. A feasible solution algorithm for the Minimum Volume Ellipsoid estimator in multivariate data. Comput Stat 1993, 8: 95–107.
Woodruff, DL, Rocke, DM. Heuristic search algorithms for the Minimum Volume Ellipsoid. J Comput Graph Stat 1993, 2: 69–95.
Woodruff, DL, Rocke, DM. Computable robust estimation of multivariate location and shape in high dimension using compound estimators. J Am Stat Assoc 1994, 89: 888–896.
Rocke, DM, Woodruff, DL. Robust estimation of multivariate location and shape. J Stat Plan Inference 1997, 57: 245–255.
Poston, WL, Wegman, EJ, Priebe, CE, Solka, JL. A deterministic method for robust estimation of multivariate location and shape. J Comput Graph Stat 1997, 6: 300–313.
Poston, WL, Wegman, EJ, Solka, JL. D‐optimal design methods for robust estimation of multivariate location and scatter. J Stat Plan Inference 1998, 73: 205–213.
Hawkins, DM, Olive, DJ. Improved feasible solution algorithms for high breakdown estimation. Comput Stat Data Anal 1999, 30: 1–11.
Mount, DM, Netanyahu, NS, Piatko, CD, Silverman, R, Wu, AY. Quantile approximation for robust statistical estimation and kappa‐enclosing problems. Int J Comput Geometry Appl 2000, 10: 593–608.
Croux, C, Haesbroeck, G. Maxbias curves of location estimators based on subranges. J Nonparametr Stat 2002, 14: 295–306.
Croux, C, Haesbroeck, G. Maxbias curves of robust scale estimators based on subranges. Metrika 2001, 53: 101–122.
Adrover, J, Yohai, V. Projection estimates of multivariate location. Ann Stat 2002, 30: 1760–1781.
Adrover, J, Yohai, V. Bias behaviour of the Minimum Volume Ellipsoid estimate. In: Hubert, M, Pison, G, Struyf, A, Van Aelst, S, eds. Theory and Applications of Recent Robust Methods. Basel: Statistics for Industry and Technology, Birkhäuser, 2004, 1–12.
Stahel, WA. Robuste Schätzungen: infinitesimale Optimalität und Schätzungen von Kovarianzmatrizen. PhD thesis, ETH Zürich, 1981.
Donoho, DL. Breakdown properties of multivariate location estimators, Qualifying paper. Boston: Harvard University, 1982.
Maronna, RA, Yohai, VJ. The behavior of the Stahel‐Donoho robust multivariate estimator. J Am Stat Assoc 1995 90 330 341.
Rousseeuw, PJ, van Zomeren, BC. Robust distances: simulations and cutoff values. In: Stahel, W, Weisberg, S, eds. Directions in Robust Statistics and Diagnostics, Part II, The IMA Volumes in Mathematics and Its Applications. New York: Springer Verlag, 1991, 195–203.
Cho, JH, Gemperline, PJ. Pattern‐recognition analysis of near‐infrared spectra by robust distance method. J Chemom 1995, 9: 169–178.
Seaver, BL, Triantis, KP. The impact of outliers and leverage points for technical efficiency measurement using high breakdown procedures. Manage Sci 1995, 41: 937–956.
Plets, H, Vynckier, C. An analysis of the incidence of the vega phenomenon among main‐sequence and post main‐sequence stars. Astron Astrophys 1999, 343: 496–506.
Simpson, DG, Ruppert, D, Carroll, RJ. On one‐step GM‐estimates and stability of inferences in linear regression. J Am Stat Assoc 1992, 87: 439–450.
Coakley, CW, Hettmansperger, TP. A bounded influence, high breakdown, efficient regression estimator. J Am Stat Assoc 1993, 88: 872–880.
Du, ZY, Wiens, DP. Jackknifing, weighting, diagnostics and variance estimation in generalized M‐estimation. Stat Probab Lett 2000, 46: 287–299.
Simpson, DG, Chang, YI. Reweighting approximate GM estimators: asymptotics and residual‐based graphics. J Stat Plan Inference 1997, 57: 273–293.
Naranjo, JD, Hettmansperger, TP. Bounded influence rank regression. J R Stat Soc B 1994, 56: 209–220.
Chang, WH, McKean, JW, Naranjo, JD, Sheather, SJ. High‐breakdown rank regression. J Am Stat Assoc 1999, 94: 205–219.
Bianco, A, Boente, G, di Rienzo, J. Some results for robust GM‐based estimators in heteroscedastic regression models. J Stat Plan Inference 2000, 89: 215–242.
Bianco, A, Boente, G. On the asymptotic behavior of one‐step estimates in heteroscedastic regression models. Stat Probab Lett 2002, 60: 33–47.
Fried, R. Robust location estimation under dependence. J Stat Comput Simulation 2007, 77: 131–147.
Jackson, DA, Chen, Y. Robust principal component analysis and outlier detection with ecological data. Environmetrics 2004, 15: 129–139.
Chen, Y, Chen, X, Xu, L. Developing a size indicator for fish populations. Sci Mar 2008, 72: 221–229.
Chork, CY, Rousseeuw, PJ. Integrating a high‐breakdown option into discriminant analysis in exploration geochemistry. J Geochem Explor 1992, 43: 191–203.
Todorov, VK, Neykov, NM, Neytchev, PN. Robust selection of variables in the discriminant‐analysis based on MVE and MCD estimators. In: Momirovic, K, Mildner, V, eds. Compstat Proceedings in Computational Statistics. Heidelberg: Physica‐Verlag, 1990, 193–198.
Chork, CY, Salminen, R. Interpreting exploration geochemical data from Outokumpu, Finland ‐ a MVE‐robust factor‐analysis. J Geochem Explor 1993, 48: 1–20.
Pison, G, Rousseeuw, PJ, Filzmoser, P, Croux, C. Robust factor analysis. J Multivar Anal 2003, 84: 145–172.
Jolion, JM, Meer, P, Bataouche, S. Robust clustering with applications in computer vision. IEEE Trans Pattern Anal Mach Intell 1991, 13: 791–802.
Vargas, JA. Robust estimation in multivariate control charts for individual observations. J Qual Technol 2003, 35: 367–376.
Williams, JD, Woodal, WH, Birch, JB. Statistical monitoring of quality profiles of products and processes. Qual Reliab Eng Int 2007, 23: 925–941.
Jensen, WA, Birch, JB, Woodall, WH. High breakdown estimation methods for phase I multivariate control charts. Qual Reliab Eng Int 2007, 23: 615–629.
Everitt, B. An R and S‐PLUS^® Companion to Multivariate Analysis. London: Springer‐Verlag, 2007.
Rousseeuw, PJ, Yohai, V. Robust regression by means of S‐estimators. In: Franke, J, Härdle, W, Martin, D, eds. Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics No. 26. Berlin: Springer Verlag, 1984, 256–272.
Lopuhaä, HP. On the relation between S‐estimators and M‐estimators of multivariate location and covariance. Ann Stat 1989, 17: 1662–1683.
Roelant, E, Van Aelst, S, Croux, C. Multivariate generalized S‐estimators. J Multivar Analysis 2009, 100: 876–887.
Tatsuoka, KS, Tyler, DE. On the uniqueness of S‐functionals and M‐functionals under nonelliptical distributions. Ann Stat 2000, 28: 1219–1243.
Salibian‐Barrera, M, Van Aelst, S, Willems, G. PCA based on multivariate MM‐estimators with fast and robust bootstrap. J Am Stat Assoc 2006, 101: 1198–1211.
Kent, JT, Tyler, DE. Constrained M‐estimation for multivariate location and scatter. Ann Stat 1996, 24: 1346–1370.
Lopuhaä, HP. Multivariate τ‐estimators for location and scatter. Can J Stat 1991, 19: 307–321.
Martin, RD, Zamar, RH. Bias robust estimation of scale. Ann Stat 1993, 21: 991–1017.
Salibian‐Barrera, M, Van Aelst, S, Willems, G. Fast and robust bootstrap. Stat Methods Appl 2008, 17: 41–71.
Van Aelst, S, Willems, G. Multivariate regression S‐estimators for robust estimation and inference. Stat Sin 2005, 15: 981–1001.
Agulló, J, Croux, C, Van Aelst, S. The multivariate Least Trimmed Squares estimator. J Multivar Analysis 2008, 99: 311–338.
Butler, RW, Davies, PL, Jhun, M. Asymptotics for the Minimum Covariance Determinant estimator. Ann Stat 1993, 21: 1385–1400.
Croux, C, Haesbroeck, G. Influence function and efficiency of the Minimum Covariance Determinant scatter matrix estimator. J Multivar Analysis 1999, 71: 161–190.
Maronna, RA, Stahel, WA, Yohai, VJ. Bias‐robust estimators of multivariate scatter based on projections. J Multivar Analysis 1992, 42: 141–161.
Tyler, DE. Finite‐sample breakdown points of projection‐based multivariate location and scatter statistics. Ann Stat 1994, 22: 1024–1044.
Zuo, Y. Projection‐based depth functions and associated median. Ann Stat 2003, 31: 1460–1490.
Zuo, Y, Cui, H, He, X. On the Stahel‐Donoho estimator and depth‐weighted means of multivariate data. Ann Stat 2004, 32: 167–188.
Zuo, Y, Cui, H, Young, D. Influence function and maximum bias of projection depth based estimators. Ann Stat 2004, 32: 189–218.
Zuo, Y, Cui, H. Depth weighted sactter estimators. Ann Stat 2005, 33: 381–413.
Hadi, AS, Luceño, A. Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Comput Stat Data Anal 1997, 25: 251–272.
Hubert, M, Rousseeuw, PJ, Van Aelst, S. High‐breakdown robust multivariate methods. Stat Sci 2008, 23: 92–119.
Alqallaf, F, Van Aelst, S, Yohai, VJ, Zamar, RH. Propagation of outliers in multivariate data. Ann Stat 2009, 37: 311–331.

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Minimum volume ellipsoid

Related Articles

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox