A practical approximation algorithm for the LMS line estimator

Computational Statistics and Data Analysis

Volumn 51, Issue 5, 2007, Pages 2461-2486

  Mount, David M ^a   Netanyahu, Nathan S ^b,c   Romanik, Kathleen ^d   Silverman, Ruth ^c   Wu, Angela Y ^e  

^a University of Maryland ^*  (United States)

^b BAR ILAN UNIVERSITY   (Israel)

^c UNIVERSITY OF MARYLAND   (United States)

^d White Oak Technologies   (United States)

^e AMERICAN UNIVERSITY   (United States)

Author keywords

Approximation algorithms; Least median of squares regression; Line arrangements; Line fitting; Randomized algorithms; Robust estimation

Indexed keywords

APPROXIMATION THEORY; ERROR ANALYSIS; LEAST SQUARES APPROXIMATIONS; OPTIMIZATION; PROBLEM SOLVING; REGRESSION ANALYSIS;

APPROXIMATION ALGORITHMS; LEAST MEDIAN-OF-SQUARES REGRESSION; LINE ARRANGEMENTS; LINE FITTING; RANDOMIZED ALGORITHMS; ROBUST ESTIMATION;

ALGORITHMS;

EID: 33751009905     PISSN: 01679473     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.csda.2006.08.033     Document Type: Article

References (37)

1. 1-Statistical Procedures and Related Topics, IMS Lecture Notes Monograph Series. Institute of Mathematical Statistics, vol. 31, pp. 133-146.
2. Chernoff H. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 (1952) 493-507
3. Cole R., Salowe J., Steiger W., and Szemerédi E. An optimal-time algorithm for slope selection. SIAM J. Comput. 18 4 (1989) 792-810
4. Cormen T.H., Leiserson C.E., and Rivest R.L. Introduction to Algorithms (1990), MIT Press, Cambridge, MA
5. de Berg M., van Kreveld M., Overmars M., and Schwarzkopf O. Computational Geometry: Algorithms and Applications. second ed (2000), Springer, Berlin
6. Dillencourt M.B., Mount D.M., and Netanyahu N.S. A randomized algorithm for slope selection. Internat. J. Comput. Geom. Appl. 2 (1992) 1-27
7. Edelsbrunner H. Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science vol. 10 (1987), Springer, Heidelberg, West Germany
8. Edelsbrunner H., and Guibas L.J. Topologically sweeping an arrangement. J. Comput. System Sci. 38 (1989) 165-194 (Corrigendum in 42 (1991) 249-251.)
9. Edelsbrunner H., and Mücke E.P. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9 (1990) 66-104
10. Edelsbrunner H., and Souvaine D.L. Computing median-of-squares regression lines and guided topological sweep. J. Amer. Statist. Assoc. 85 (1990) 115-119
11. Erickson, J., Har-Peled, S., Mount, D.M., 2004. On the least median square problem. In: Proceedings of the 20th Annual ACM Symposium on Computational Geometry, pp. 273-279; Discrete Comput. Geom., to appear.
12. Hagerup T., and Rüb C. A guided tour of Chernoff bounds. Inform. Process. Lett. 33 (1990) 305-308
13. Har-Peled, S., 2001. A practical approach for computing the diameter of a point-set. In: Proceedings of the 17th Annual ACM Symposium on Computational Geometry, pp. 177-186.
14. Har-Peled, S., Wang, Y., 2003. Shape fitting with outliers. In: Proceedings of the 19th Annual ACM Symposium on Computational Geometry, pp. 29-38.
15. Hawkins D.M. The feasible set algorithm for least median of squares regression. Comput. Statist. Data Anal. 16 (1993) 81-101
16. Langerman, S., Steiger, W., 2000. An optimal algorithm for hyperplane depth in the plane. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 54-59.
17. Matoušek J. Cutting hyperplane arrangements. Discrete Comput. Geom. 6 (1991) 385-406
18. Matoušek J. Randomized optimal algorithm for slope selection. Inform. Process. Lett. 39 (1991) 183-187
19. Matoušek J. Efficient partition trees. Discrete Comput. Geom. 8 (1992) 315-334
20. Matoušek J., Mount D.M., and Netanyahu N.S. Efficient randomized algorithms for the repeated median line estimator. Algorithmica 20 (1998) 136-150
21. Meer P., Mintz D., Rosenfeld A., and Kim D.Y. Robust regression methods for computer vision-a review. Internat. J. Comput. Vision 6 (1991) 59-70
22. Motwani R., and Raghavan P. Randomized Algorithms (1995), Cambridge University Press, Cambridge
23. Mount D.M., Netanyahu N.S., Piatko C., Silverman R., and Wu A.Y. Quantile approximation for robust statistical estimation and k-enclosing problems. Internat. J. Comput. Geom. Appl. 10 (2000) 593-608
24. Mount D.M., Netanyahu N.S., and Zuck E. Analyzing the number of samples required for an approximate Monte-Carlo LMS line estimator. In: Hubert M., Pison G., Struyf A., and Van Aelst S. (Eds). Theory and Applications of Recent Robust Methods, Statistics for Industry and Technology (2004), Birkhauser, Basel 207-219
25. Mulmuley K. Computational Geometry: An Introduction Through Randomized Algorithms (1994), Prentice-Hall, Englewood Cliffs, NJ
26. Netanyahu, N.S., Philomin, V., Rosenfeld, A., Stromberg, A.J., 1997. Robust detection of road segments in noisy aerial images. Pattern Recogn. 30, 1673-1686.
27. Olson C.F. An approximation algorithm for least median of squares regression. Inform. Process. Lett. 63 (1997) 237-241
28. Rafalin, E., Souvaine, D., Streinu, I., 2002. Topological sweep in degenerate cases. In: Proceedings of the Fourth International Workshop on Algorithm Engineering and Experiments, Lecture Notes in Computer Science, vol. 2409. Springer, Berlin, Germany, pp. 155-156.
29. Rousseeuw P.J. Least median-of-squares regression. J. Amer. Statist. Assoc. 79 (1984) 871-880
30. 1-Statistical Procedures and Related Topics. IMS Lecture Notes Monograph Series, vol. 31. Institute of Mathematical Statistics (1977) 201-214
31. Rousseeuw P.J., and Leroy A.M. Robust Regression and Outlier Detection (1987), Wiley, New York
32. Samet H. The Design and Analysis of Spatial Data Structures (1990), Addison-Wesley, Reading, MA
33. Souvaine D.L., and Steele J.M. Time- and space- efficient algorithms for least median of squares regression. J. Amer. Statist. Assoc. 82 (1987) 794-801
34. Stein A., and Werman M. Robust statistics in shape fitting. Proceedings of the IEEE Computer Society Conference Computer on Vision and Pattern Recognition (1992), Champaign, IL June
35. Stewart C.V. MINPRAN: a new robust estimator for computer vision. IEEE Trans. Pattern Anal. Mach. Intell. 17 (1996) 925-938
36. Stromberg A.J. Computing the exact least median of squares estimate and stability diagnostics in multiple linear regression. SIAM J. Sci. Comput. 14 6 (1993) 1289-1299
37. Yohai V.J. High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 (1987) 642-656