Variations on least squares

TL;DR

This paper compares three least squares methods for line fitting, highlighting differences in their regression lines, invariance properties, and providing concise proofs for each method.

Contribution

It introduces and compares a less known distance-to-line least squares method, demonstrating its advantages and invariance under rotations.

Findings

Distance to line method is generally favorable.

Vertical and horizontal methods produce different slopes.

Distance to line method is rotation invariant.

Abstract

Three methods of least squares are examined for fitting a line to points in the plane. Two well known methods are to minimize sums of squares of vertical or horizontal distances to the line. Less known is to minimize sums of squares of distances to the line. Concise proofs are given for each method using a combination of the first derivative test for functions of two variables and completing the square. The three methods are compared and the distances to the line method appears to be favourable in most circumstances. They generally draw different regression lines. The method of vertical displacements typically gives a slope of too small magnitude while the method of horizontal displacements typically gives a slope of too large magnitude. An inequality involving the three slopes is proved. Rotating all the data points in the same way with these two methods does not result in the regression line being rotated the same way. However, the distance to the line method is invariant under rotations.