LEAST-SQUARES FITTING OF A STRAIGHT LINE

Concepts

Numerical AnalysisGeometric ModelingGeometric InterpolationBest Straight LineEngineeringBest SlopeNatural SciencesLeast-squares FittingInverse ProblemsCurve FittingCurve ModelingComputational GeometryApproximation TheoryLeast SquaresRegressionLinear Optimization

TL;DR

The paper discusses computing the best‑fit straight line using the least‑squares method. The authors illustrate the least‑squares fitting procedure with a concrete example. They derive a general solution for the least‑squares line, identify conditions for special cases, show that the optimal slope follows from a Least‑Squares Cubic, and demonstrate that this slope can lie outside the bounds of simple x‑on‑y and y‑on‑x regressions.

Abstract

A detailed discussion of the calculation of the "best straight line" by the method of least squares is given. The most general solution is found and the conditions under which certain previously derived special solutions are valid are clearly stated. The "best" slope is shown to be given by the solution of the "Least-Squares Cubic". An example is given to illustrate the method. It is shown that the best slope is not necessarily bounded by values found from the regressions of x on y and y on x.