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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Veröffentlicht in:Computational statistics & data analysis Jg. 51; H. 5; S. 2461 - 2486
Hauptverfasser: Mount, David M., Netanyahu, Nathan S., Romanik, Kathleen, Silverman, Ruth, Wu, Angela Y.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 01.02.2007
Elsevier Science
Elsevier
Schriftenreihe:Computational Statistics & Data Analysis
Schlagworte:
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
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
ISSN:0167-9473, 1872-7352
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Beschreibung
Zusammenfassung: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.
ISSN:0167-9473
1872-7352
DOI:10.1016/j.csda.2006.08.033