A practical approximation algorithm for the LMS line estimator

  • Authors:

  • David M. Mount;Nathan S. Netanyahu;Kathleen Romanik;Ruth Silverman;Angela Y. Wu

  • Affiliations:

  • Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA;Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel and Center for Automation Research, University of Maryland, College Park, MD, USA;White Oak Technologies, Inc., Silver Spring, MD, USA;Center for Automation Research, University of Maryland, College Park, MD, USA;Department of Computer Science, American University, Washington, DC, USA

  • Venue:
  • Computational Statistics & Data Analysis
  • Year:
  • 2007

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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=0, and a quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound @e"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 @e"q0 runs in O(nlogn) 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(nlog^2n). 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.