Data Mining and Knowledge Discovery

, Volume 12, Issue 1, pp 29–45 | Cite as

Computing LTS Regression for Large Data Sets

Article

Abstract

Data mining aims to extract previously unknown patterns or substructures from large databases. In statistics, this is what methods of robust estimation and outlier detection were constructed for, see e.g. Rousseeuw and Leroy (1987). Here we will focus on least trimmed squares (LTS) regression, which is based on the subset of h cases (out of n) whose least squares fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The computation time of existing LTS algorithms grows too much with the size of the data set, precluding their use for data mining. In this paper we develop a new algorithm called FAST-LTS. The basic ideas are an inequality involving order statistics and sums of squared residuals, and techniques which we call ‘selective iteration’ and ‘nested extensions’. We also use an intercept adjustment technique to improve the precision. For small data sets FAST-LTS typically finds the exact LTS, whereas for larger data sets it gives more accurate results than existing algorithms for LTS and is faster by orders of magnitude. This allows us to apply FAST-LTS to large databases.

Keywords

breakdown value linear model outlier detection regression robust estimation 

References

  1. Agulló, J. 1997a. Computación de estimadores con alto punto de ruptura. Ph.D. Thesis, University of Alicante, Spain.Google Scholar
  2. Agulló, J. 1997b. Exact algorithms to compute the least median of squares estimate in multiple linear regression. In L1-Statistical Procedures and Related Topics, Y. Dodge (ed.), The IMS Lecture Notes – Monograph Series, Volume 31, pp. 133–146.Google Scholar
  3. Chork, C.J. 1990. Unmasking multivariate anomalous observations in exploration geochemical data from sheeted-vein tin mineralization near Emmaville, N.S.W., Australia. Journal of Geochemical Exploration, 37:191–203.Google Scholar
  4. Coakley, C.W. and Hettmansperger, T.P. 1993. A bounded influence, high breakdown, efficient regression estimator. Journal of the American Statistical Association, 88:872–880.Google Scholar
  5. Hawkins, D.M. 1994. The feasible solution algorithm for least trimmed squares regression. Computational Statistics and Data Analysis, 17:185–196.Google Scholar
  6. Hawkins, D.M. and Olive, D.J. 1999. Improved feasible solution algorithms for high breakdown estimation. Computational Statistics and Data Analysis, 30:1–11.Google Scholar
  7. Hössjer, O. 1994. Rank-based estimates in the linear model with high breakdown point. Journal of the American Statistical Association, 89:149–158.Google Scholar
  8. Huang, Z. 1998. Extensions of the k-means algorithm for clustering large data sets with categorical values. Data Mining and Knowledge Discovery, 2:283–304.Google Scholar
  9. Kaufman, L. and Rousseeuw, P.J. 1986. Clustering large data sets. In Pattern Recognition in Practice II, E.S. Gelsema and L.N. Kanal (eds.) Elsevier/North-Holland, pp. 425–437.Google Scholar
  10. Kaufman, L. and Rousseeuw, P.J. 1990. Finding Groups in Data, New York: John Wiley.Google Scholar
  11. Meer, P., Mintz, D., Rosenfeld, A., and Kim, D. 1991. Robust regression methods in computer vision: a review. International Journal of Computer Vision, 6:59–70.Google Scholar
  12. Mili, L., Phaniraj, V., and Rousseeuw, P.J. 1991. Least median of squares estimation in power systems (with discussion). IEEE Trans. on Power Systems, 6:511–523.Google Scholar
  13. Mili, L., Cheniae, N.S., and Rousseeuw, P.J. 1996. Robust state estimation based on projection statistics (with discussion). IEEE Trans. on Power Systems, 11:1118–1127.Google Scholar
  14. Ng, R.T. and Han, J., 1994. Efficient and effective clustering methods for spatial data mining. Proceedings of the International Conference on Very Large Data Bases (VLDB ’94), Santiago, Chile, September 1994, pp. 144–155.Google Scholar
  15. Odewahn, S.C., Djorgovski, S.G., Brunner, R.J., and Gal, R. 1998. Data From the Digitized Palomar Sky Survey. Technical Report, California Institute of Technology.Google Scholar
  16. Rousseeuw, P.J. 1984. Least median of squares regression. Journal of the American Statistical Association, 79:871–880.Google Scholar
  17. Rousseeuw, P.J. 1985. Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications, Vol B, W. Grossmann, G. Pflug, I. Vincze and W. Wertz (eds.) Dordrecht: Reidel, pp. 283–297.Google Scholar
  18. Rousseeuw, P.J. 1997. Introduction to positive-breakdown methods. In Handbook of Statistics, Vol. 15: Robust Inference, G.S. Maddala and C.R. Rao (eds.) Amsterdam: Elsevier, pp. 101–121.Google Scholar
  19. Rousseeuw, P.J. and Hubert, M. 1997. Recent developments in PROGRESS. In \({\rm L}_1\)-Statistical Procedures and Related Topics, Y. Dodge (ed.) The IMS Lecture Notes – Monograph Series, Vol. 31, pp. 201–214.Google Scholar
  20. Rousseeuw, P.J. and Leroy, A.M. 1987. Robust Regression and Outlier Detection, New York: John Wiley.Google Scholar
  21. Rousseeuw, P.J. and Van Driessen, K. 1999. A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41:212–223.Google Scholar
  22. Rousseeuw, P.J. and van Zomeren, B.C., 1990. Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association, 85:633–639.Google Scholar
  23. Steele, J.M. and Steiger, W.L. 1986. Algorithms and complexity for least median of squares regression. Discrete Applied Mathematics, 14:93–100.Google Scholar
  24. Stromberg, A.J. 1993. Computing the exact least median of squares estimate and stability diagnostics in multiple linear regression. SIAM Journal of Scientific Computing, 14:1289–1299.Google Scholar
  25. Simpson, D.G., Ruppert, D., and Carroll, R.J. 1992. On one-step GM-estimates and stability of inferences in linear regression. Journal of the American Statistical Association, 87:439–450.Google Scholar
  26. Wang, C.M., Vecchia, D.F., Young, M. and Brilliant, N.A. 1997. Robust regression applied to optical fiber dimensional quality control. Technometrics, 39:25–33.Google Scholar
  27. Woodruff, D.L. and Rocke, D.M. 1994. Computable robust estimation of multivariate location and shape in high dimension using compound estimators. Journal of the American Statistical Association, 89:888–896.Google Scholar
  28. Yohai, V.J. 1987. High breakdown point and high efficiency robust estimates for regression. Annals of Statistics, 15:642–656.Google Scholar
  29. Zhang, T., Ramakrishnan, R., and Livny, M. 1997. BIRCH: a new data clustering algorithm and its applications. Data Mining and Knowledge Discovery, 1:141–182.Google Scholar

Copyright information

© Springer Science + Business Media, Inc 2005

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversiteit AntwerpenAntwerpenBelgium
  2. 2.Faculty of Applied EconomicsUniversiteit AntwerpenAntwerpenBelgium

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