Advances in Data Analysis and Classification

, Volume 4, Issue 4, pp 301–334 | Cite as

Outlier detection and robust covariance estimation using mathematical programming

Regular Article

Abstract

The outlier detection problem and the robust covariance estimation problem are often interchangeable. Without outliers, the classical method of maximum likelihood estimation (MLE) can be used to estimate parameters of a known distribution from observational data. When outliers are present, they dominate the log likelihood function causing the MLE estimators to be pulled toward them. Many robust statistical methods have been developed to detect outliers and to produce estimators that are robust against deviation from model assumptions. However, the existing methods suffer either from computational complexity when problem size increases or from giving up desirable properties, such as affine equivariance. An alternative approach is to design a special mathematical programming model to find the optimal weights for all the observations, such that at the optimal solution, outliers are given smaller weights and can be detected. This method produces a covariance estimator that has the following properties: First, it is affine equivariant. Second, it is computationally efficient even for large problem sizes. Third, it easy to incorporate prior beliefs into the estimator by using semi-definite programming. The accuracy of this method is tested for different contamination models, including recently proposed ones. The method is not only faster than the Fast-MCD method for high dimensional data but also has reasonable accuracy for the tested cases.

Keywords

Covariance matrix estimation Robust statistics Outlier detection Optimization Semi-definite programming Newton–Raphson method 

Mathematics Subject Classification (2000)

62-07 (Statistics-Data analysis) 90-08 (Operations Research-Computational methods) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alqallaf F, Van Aelst S, Yohai VJ, Zamar RH (2009) Propagation of outliers in multivariate data. Ann Stat 37(1): 311–331MATHCrossRefMathSciNetGoogle Scholar
  2. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, BelmontMATHGoogle Scholar
  3. Chakraborty B, Chaudhuri P (2008) On an optimization problem in robust statistics. J Comput Graph Stat 17(3): 683–702CrossRefMathSciNetGoogle Scholar
  4. Chandola V, Banerjee A, Kumar V (2007) Outlier detection: a review. Technical Report, University of MinnesotaGoogle Scholar
  5. Critchley F, Schyns M, Haesbroeck G, Fauconnier C, Lu G, Atkinson RA, Wang DQ (2010) A relaxed approach to combinatorial problems in robustness and diagnostics. Stat Comput 20(1): 99–115CrossRefGoogle Scholar
  6. Critchley F, Schyns M, Haesbroeck G, Kinns D, Atkinson RA, Lu G (2004) The case sensitivity function approach to diagnostics and robust computation: a relaxation strategy. In: COMPSTAT: 2004 Proceedings in Computational Statistics, vol 36, pp 113–125Google Scholar
  7. Huber PJ (2004) Robust statistics. Wiley, New YorkGoogle Scholar
  8. Khan J, Van Aelst S, Zamar R (2007) Robust linear model selection based on least angle regression. J Am Stat Assoc 102: 1289–1299MATHCrossRefMathSciNetGoogle Scholar
  9. Kuhn HW, Tucker AW (1951) Nonlinear programming. In: Proceedings of second Berkeley symposium. University of California Press, Berkeley, pp 481–492Google Scholar
  10. Ledoit O, Wolf M (2004) A well-conditioned estimator for large-dimensional covariance matrices. J Multivar Anal 88(2):365–411Google Scholar
  11. Maronna RA, Martin RD, Yohai VJ (2004) Robust statistics: theory and methods. Wiley, New York (2006)Google Scholar
  12. Nguyen TD, Welsch R (2009) Outlier detection and least trimmed squares approximation using semi-definite programming. Comput Stat Data Anal (to appear)Google Scholar
  13. Rousseeuw PJ (1984) Least median of squares regression. J Am Stat Assoc 79: 871–880MATHCrossRefMathSciNetGoogle Scholar
  14. Rousseeuw PJ, van Driessen K (1999) A fast algorithm for the minimum covariance determinant estimator. Technometrics 41: 212–223CrossRefGoogle Scholar
  15. Schyns M, Haesbroeck G, Critchley F (2010) RelaxMCD: smooth optimisation for the minimum covariance determinant estimator. Comput Stat Data Anal 54(4): 843–857MATHCrossRefGoogle Scholar
  16. Toh KC, Todd MJ, Tutuncu RH (2006) Sdpt3 version 4.0 (beta)—a matlab software for semidefinite-quadratic-linear programmingGoogle Scholar
  17. Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38(1): 49–95MATHCrossRefMathSciNetGoogle Scholar
  18. Vandenberghe L, Boyd S (1999) Applications of semidefinite programming. Appl Numer Math Trans IMACS 29(3): 283–299MATHCrossRefMathSciNetGoogle Scholar
  19. Verboven S, Hubert M (2005) Libra: a MATLAB library for robust analysis. Chemom Intell Lab Syst 75(2): 127–136CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.3103 Newmark Civil Engineering Laboratory, UIUCUrbanaUSA
  2. 2.Center for Computational Research in Economics and Management ScienceSloan School of Management, MITCambridgeUSA

Personalised recommendations