Computational Statistics

, Volume 21, Issue 1, pp 33–51 | Cite as

Online signal extraction by robust linear regression

  • Ursula Gather
  • Karen Schettlinger
  • Roland Fried


In intensive care, time series of vital parameters have to be analysed online, i.e. without any time delay, since there may be serious consequences for the patient otherwise. Such time series show trends, slope changes and sudden level shifts, and they are overlaid by strong noise and many measurement artefacts. The development of update algorithms and the resulting increase in computational speed allows to apply robust regression techniques to moving time windows for online signal extraction. By simulations and applications we compare the performance of least median of squares, least trimmed squares, repeated median and deepest regression for online signal extraction.


Robust filtering least median of squares least trimmed squares repeated median deepest regression breakdown point 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bai, Z.D. and He, X. (1999). Asymptotic Distributions of the Maximal Depth Estimators for Regression and Multivariate Location, Ann. Stat. 27 (5), 1616–1637.zbMATHMathSciNetCrossRefGoogle Scholar
  2. Bernholt, T. (2004). Exact Algorithms for the Repeated Median, LMS, LTS and Deepest Regression, Personal Communication.Google Scholar
  3. Bernholt, T. and Fried, R. (2003). Computing the Update of the Repeated Median Regression Line in Linear Time, Inf. Process. Lett. 88 (1), 111–117.MathSciNetCrossRefGoogle Scholar
  4. Bernholt, T., Fried, R., Gather, U. and Wegener I. (2004). Modified Repeated Median Filters, Technical Report 46, SFB 475, University of Dortmund, Germany.Google Scholar
  5. Chang, W.H., McKean, J.W., Naranjo, J.D. and Sheather, S.J. (1999). High-Breakdown Rank Regression, J. Am. Stat. Assoc. 94, No. 445, 205–219.zbMATHMathSciNetCrossRefGoogle Scholar
  6. Croux, C., Rousseeuw, P.J. and Hössjer, O. (1994). Generalized S-Estimators, J. Am. Stat. Assoc. 89, No. 428, 1271–1281.zbMATHCrossRefGoogle Scholar
  7. Davies, P.L. (1993). Aspects of Robust Linear Regression, Ann. Stat. 21 (4), 1843–1899.zbMATHGoogle Scholar
  8. Davies, P.L., Fried, R. and Gather, U. (2004). Robust Signal Extraction for On-line Monitoring Data, J. Stat. Plann. Inference 122 (1–2), 65–78.zbMATHMathSciNetCrossRefGoogle Scholar
  9. Edelsbrunner, H. and Souvaine, D.L. (1990). Computing Least Median of Squares Regression Lines and Guided Topological Sweep, J. Am. Stat. Assoc. 85, No. 409, 115–119.zbMATHCrossRefGoogle Scholar
  10. Einbeck, J. and Kauermann, G. (2003). Online Monitoring with Local Smoothing Methods and Adaptive Ridging, J. Statist. Comput. Simul. 73, 913–929.zbMATHMathSciNetCrossRefGoogle Scholar
  11. Fried, R. (2004). Robust Filtering of Time Series with Trends, J. Nonparametric Statistics 16, 313–328.zbMATHMathSciNetCrossRefGoogle Scholar
  12. Hössjer, O., Rousseeuw, P.J. and Ruts, I. (1995). The Repeated Median Intercept Estimator: Influence Function and Asymptotic Normality, J. Multivariate Anal. 52, 45–72.zbMATHMathSciNetCrossRefGoogle Scholar
  13. Hössjer, O., Rousseeuw, P.J. and Croux, C. (1994). Asymptotics of the Repeated Median Slope Estimator, Ann. Stat. 22 (3), 1478–1501.zbMATHGoogle Scholar
  14. Imhoff, M., Bauer, M., Gather, U. and Fried, R. (2002). Pattern Detection in Intensive Care Monitoring Time Series with Autoregressive Models: Influence of the AR-Model Order, Biom. J. 44, 746–761.MathSciNetCrossRefGoogle Scholar
  15. Rousseeuw, P.J. (1983). Multivariate Estimation with High Breakdown Point, in W. Grossmann, G. Pflug, I. Vincze, W. Wertz (eds.) Proceedings of the 4th Pannonian Symposium on Mathematical Statistics and Probability, Vol. B, D. Reidel Publishing Company, Dordrecht (The Netherlands).Google Scholar
  16. Rousseeuw, P.J. (1984). Least Median of Squares Regression, J. Am. Stat. Assoc. 79, No. 388, 871–880.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Rousseeuw, P.J. and Hubert, M. (1999). Regression Depth, J. Am. Stat. Assoc. 94, No. 446, 388–402.zbMATHMathSciNetCrossRefGoogle Scholar
  18. Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection, Wiley, New York (USA).zbMATHCrossRefGoogle Scholar
  19. Rousseeuw, P.J., Van Aelst, S. and Hubert, M. (1999). Rejoinder to ‘Regression Depth’, J. Am. Stat. Assoc. 94, No. 446, 419–433.CrossRefGoogle Scholar
  20. Sheather, S.J., McKean, J.W. and Hettmansperger, T.P. (1997). Finite Sample Stability Properties of the Least Median of Squares Estimator, J. Stat. Comput. Simul. 58, 371–383.zbMATHMathSciNetGoogle Scholar
  21. Siegel, A.F. (1982). Robust Regression Using Repeated Medians, Biometrika 69, 242–244.zbMATHCrossRefGoogle Scholar
  22. Stromberg, A.J., Hössjer, O., Hawkins, D.M. (2000). The Least Trimmed Differences Regression Estimator and Alternatives, J. Am. Stat. Assoc. 95, No. 451, 853–864.zbMATHCrossRefGoogle Scholar
  23. Van Aelst, S., Rousseeuw, P.J., Hubert, M. and Struyf, A. (2002). The Deepest Regression Method, J. Multivariate Anal. 81, 138–166.zbMATHMathSciNetCrossRefGoogle Scholar
  24. Van Kreveld, M., Mitchell, J.S.B., Rousseeuw, P.J., Sharir, M., Snoeyink, J. and Speckmann, B. (1999). Efficient Algorithms for Maximum Regression Depth, Proceedings of the 15th Annual ACM Symposium of Computational Geometry, ACM Press, New York (NJ), 31–40.Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Ursula Gather
    • 1
  • Karen Schettlinger
    • 1
  • Roland Fried
    • 2
  1. 1.Department of StatisticsUniversity of DortmundDortmundGermany
  2. 2.Department of StatisticsUniversity Carlos IIIMadridSpain

Personalised recommendations