Repeated median regression
In robust statistics, repeated median regression, also known as the repeated median estimator, is a robust linear regression algorithm. The estimator has a breakdown point of 50%.[1] Although it is equivariant under scaling, or under linear transformations of either its explanatory variable or its response variable, it is not under affine transformations that combine both variables.[1] It can be calculated in [math]\displaystyle{ O(n^2) }[/math] time by brute force, in [math]\displaystyle{ O(n \log^2 n) }[/math] time using more sophisticated techniques,[2] or in [math]\displaystyle{ O(n\log n) }[/math] randomized expected time.[3] It may also be calculated using an on-line algorithm with [math]\displaystyle{ O(n) }[/math] update time.[4]
Method
The repeated median method estimates the slope of the regression line [math]\displaystyle{ y = A + Bx }[/math] for a set of points [math]\displaystyle{ (X_i, Y_i) }[/math] as
- [math]\displaystyle{ \widehat B = \underset{i}{\operatorname{median}} \ \underset{j\,\ne\,i}{\operatorname{median}} \ \operatorname{slope}(i, j) }[/math]
where [math]\displaystyle{ \operatorname{slope}(i,j) }[/math] is defined as [math]\displaystyle{ (Y_j - Y_i) / (X_j - X_i) }[/math].[5]
The estimated Y-axis intercept is defined as
- [math]\displaystyle{ \widehat A = \underset{i}{\operatorname{median}} \ \underset{j\,\ne\,i}{\operatorname{median}} \ \operatorname{intercept}(i, j) }[/math]
where [math]\displaystyle{ \operatorname{intercept}(i, j) }[/math] is defined as [math]\displaystyle{ (X_j Y_i - X_i Y_j ) / (X_j - X_i) }[/math].[5]
See also
References
- ↑ 1.0 1.1 Peter J. Rousseeuw, Nathan S. Netanyahu, and David M. Mount, "New Statistical and Computational Results on the Repeated Median Regression Estimator", in New Directions in Statistical Data Analysis and Robustness, edited by Stephan Morgenthaler, Elvezio Ronchetti, and Werner A. Stahel, Birkhauser Verlag, Basel, 1993, pp. 177-194.
- ↑ Stein, Andrew; Werman, Michael (1992). "Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '92)". Philadelphia, PA, USA: Society for Industrial and Applied Mathematics. pp. 409–413. ISBN 0-89791-466-X.
- ↑ "Efficient randomized algorithms for the repeated median line estimator", Algorithmica 20 (2): 136–150, 1998, doi:10.1007/PL00009190
- ↑ Bernholt, Thorsten; Fried, Roland (2003). "Computing the update of the repeated median regression line in linear time". Information Processing Letters 88 (3): 111–117. doi:10.1016/s0020-0190(03)00350-8.
- ↑ 5.0 5.1 Siegel, Andrew (September 1980). "Technical Report No. 172, Series 2 By Department of Statistics Princeton University: Robust Regression Using Repeated Medians". http://apps.dtic.mil/dtic/tr/fulltext/u2/a092660.pdf.
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