A practical approximation algorithm for the LMS line estimator
The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are widely used because of their lack of sensitivity to outlying data points. The least median-of-squares (LMS) regression line estimator is among...
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| Published in: | Computational statistics & data analysis Vol. 51; no. 5; pp. 2461 - 2486 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
01.02.2007
Elsevier Science Elsevier |
| Series: | Computational Statistics & Data Analysis |
| Subjects: | |
| ISSN: | 0167-9473, 1872-7352 |
| Online Access: | Get full text |
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| Abstract | The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are widely used because of their lack of sensitivity to outlying data points. The least median-of-squares (LMS) regression line estimator is among the best known robust estimators. Given a set of
n points in the plane, it is defined to be the line that minimizes the median squared residual or, more generally, the line that minimizes the residual of any given quantile
q, where
0
<
q
⩽
1
. This problem is equivalent to finding the strip defined by two parallel lines of minimum vertical separation that encloses at least half of the points.
The best known exact algorithm for this problem runs in
O
(
n
2
)
time. We consider two types of approximations, a
residual approximation, which approximates the vertical height of the strip to within a given error bound
ε
r
⩾
0
, and a
quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound
ε
q
⩾
0
. We present two randomized approximation algorithms for the LMS line estimator. The first is a conceptually simple quantile approximation algorithm, which given fixed
q and
ε
q
>
0
runs in
O
(
n
log
n
)
time. The second is a practical algorithm, which can solve both types of approximation problems or be used as an exact algorithm. We prove that when used as a quantile approximation, this algorithm's expected running time is
O
(
n
log
2
n
)
. We present empirical evidence that the latter algorithm is quite efficient for a wide variety of input distributions, even when used as an exact algorithm. |
|---|---|
| AbstractList | The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are widely used because of their lack of sensitivity to outlying data points. The least median-of-squares (LMS) regression line estimator is among the best known robust estimators. Given a set of
n points in the plane, it is defined to be the line that minimizes the median squared residual or, more generally, the line that minimizes the residual of any given quantile
q, where
0
<
q
⩽
1
. This problem is equivalent to finding the strip defined by two parallel lines of minimum vertical separation that encloses at least half of the points.
The best known exact algorithm for this problem runs in
O
(
n
2
)
time. We consider two types of approximations, a
residual approximation, which approximates the vertical height of the strip to within a given error bound
ε
r
⩾
0
, and a
quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound
ε
q
⩾
0
. We present two randomized approximation algorithms for the LMS line estimator. The first is a conceptually simple quantile approximation algorithm, which given fixed
q and
ε
q
>
0
runs in
O
(
n
log
n
)
time. The second is a practical algorithm, which can solve both types of approximation problems or be used as an exact algorithm. We prove that when used as a quantile approximation, this algorithm's expected running time is
O
(
n
log
2
n
)
. We present empirical evidence that the latter algorithm is quite efficient for a wide variety of input distributions, even when used as an exact algorithm. |
| Author | Romanik, Kathleen Silverman, Ruth Wu, Angela Y. Mount, David M. Netanyahu, Nathan S. |
| Author_xml | – sequence: 1 givenname: David M. surname: Mount fullname: Mount, David M. email: mount@cs.umd.edu organization: Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA – sequence: 2 givenname: Nathan S. surname: Netanyahu fullname: Netanyahu, Nathan S. email: nathan@cs.biu.ac.il, nathan@cfar.umd.edu organization: Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel – sequence: 3 givenname: Kathleen surname: Romanik fullname: Romanik, Kathleen email: kromanik@comcast.net organization: White Oak Technologies, Inc., Silver Spring, MD, USA – sequence: 4 givenname: Ruth surname: Silverman fullname: Silverman, Ruth email: ruthsilverman@verizon.net organization: Center for Automation Research, University of Maryland, College Park, MD, USA – sequence: 5 givenname: Angela Y. surname: Wu fullname: Wu, Angela Y. email: awu@american.edu organization: Department of Computer Science, American University, Washington, DC, USA |
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| CitedBy_id | crossref_primary_10_1016_j_ejor_2015_04_024 crossref_primary_10_1080_03610918_2016_1236954 crossref_primary_10_1016_j_jmva_2016_01_005 crossref_primary_10_1088_0031_9155_59_6_1367 crossref_primary_10_1080_00949655_2016_1249482 crossref_primary_10_1016_j_jvcir_2013_09_007 crossref_primary_10_1007_s00453_012_9721_8 crossref_primary_10_1007_s00180_013_0459_9 crossref_primary_10_1214_14_AOS1223 crossref_primary_10_1080_00949655_2015_1014371 crossref_primary_10_1109_LSP_2025_3542330 crossref_primary_10_1016_j_csda_2016_01_016 crossref_primary_10_3724_SP_J_1004_2010_01073 |
| Cites_doi | 10.1142/S0218195992000020 10.1142/S0218195900000334 10.1145/997817.997859 10.1214/aos/1176350366 10.1016/S0020-0190(97)00132-4 10.1007/BF00127126 10.1137/0914076 10.1016/S0031-3203(96)00189-6 10.1145/77635.77639 10.2307/2289532 10.1016/0022-0000(89)90038-X 10.1007/BF02574697 10.1016/0020-0190(91)90177-J 10.1016/0167-9473(93)90246-P 10.1007/BF02293051 10.1214/aoms/1177729330 10.1145/378583.378662 10.1137/0218055 10.1016/0020-0190(90)90214-I 10.1214/lnms/1215454133 10.1007/PL00009190 10.1109/34.464558 10.2307/2288718 10.1145/777792.777798 10.2307/2288788 |
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| Issue | 5 |
| Keywords | Approximation algorithms Randomized algorithms Robust estimation Line fitting Line arrangements Least median-of-squares regression Data analysis Approximation Error estimation Fitting Estimator robustness Median Statistical estimation Approximation algorithm Statistical regression Statistical computation Least squares method Least square median Distribution function Quantile |
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| SubjectTerms | Approximation algorithms Calculus of variations and optimal control Exact sciences and technology General topics Least median-of-squares regression Line arrangements Line fitting Mathematical analysis Mathematics Multivariate analysis Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Probability and statistics Randomized algorithms Robust estimation Sciences and techniques of general use Statistics |
| Title | A practical approximation algorithm for the LMS line estimator |
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