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 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 (n²) 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{lunate}_r ≥ 0, and a quantile approximation, which approximates the fraction of points that lie within the strip to within a given error bound ε{lunate}_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{lunate}_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 log² 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.