III.1 - The Best Least-Squares Line Fit

Abstract

This chapter discusses the best least-squares line fit. Traditional approaches for fitting least-squares lines to a set of two-dimensional data points involve minimizing the sum of the squares of the minimum vertical distances between the data points and the fitted line. That is, the fit is against a set of independent observations in the range y. This chapter presents a numerically stable algorithm that fits a line to a set of ordered pairs (x, y) by minimizing its least-squared distance to each point without regard to orientation. This is a true 2D point-fitting method exhibiting rotational invariance. A least-squares line-fitting method that is insensitive to coordinate system orientation can be constructed by minimizing instead the sum of the squares of the perpendicular distances between the data points and their nearest points on the target line. Such an algorithm has been presented in the literature, but the algorithm is based on a slope-intercept form of the line resulting in solution degeneracy and numerical inaccuracies. The algorithm presented in this chapter uses a θ–p (line angle, distance from the origin) parameterization of the line that results in no degenerate cases and provides a unique solution.