Repeated median regression explained
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%. 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
time by brute force, in
time using more sophisticated techniques,
[2] or in
randomized expected time. It may also be calculated using an
on-line algorithm with
update time.
[3] Method
The repeated median method estimates the slope of the regression line
for a set of points
as
\widehatB=\underset{i}{\operatorname{median}} \underset{j\nei}{\operatorname{median}} \operatorname{slope}(i,j)
where
\operatorname{slope}(i,j)
is defined as
.
[4] The estimated Y-axis intercept is defined as
\widehatA=\underset{i}{\operatorname{median}} \underset{j\nei}{\operatorname{median}} \operatorname{intercept}(i,j)
where
\operatorname{intercept}(i,j)
is defined as
.
[4] A simpler and faster alternative to estimate the intercept
is to use the value
just estimated, thus:
[4] \widehatA=\underset{i}{\operatorname{median}} (yi-\widehat{B}xi)
Note: The direct and hierarchical methods of estimating
give slightly different values, with the hierarchical method normally being the best estimate. This latter hierarchical approach is idential to the method of estimating
in
Theil–Sen estimator regression.
See also
Notes and References
- 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 . Finding the repeated median regression line . https://dl.acm.org/citation.cfm?id=139404.139485 . 0-89791-466-X . Philadelphia, PA, USA . 409–413 . Society for Industrial and Applied Mathematics . Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '92) . 1992.
- Bernholt. Thorsten. Fried. Roland. Computing the update of the repeated median regression line in linear time. Information Processing Letters. 88. 3. 111–117. 10.1016/s0020-0190(03)00350-8. 2003. 2003/5224. free.
- Web site: Technical Report No. 172, Series 2 By Department of Statistics Princeton University: Robust Regression Using Repeated Medians. https://web.archive.org/web/20180728183415/http://www.dtic.mil/dtic/tr/fulltext/u2/a092660.pdf. live. July 28, 2018. Siegel. Andrew. September 1980. 20 February 2018.
