Curious Pivot Points of Least-Squares Regression

TL;DR

This paper investigates how the least-squares regression line pivots around fixed points when multiple data points are repeated, exploring geometric regions of possible pivot points and raising new questions.

Contribution

It extends previous work on regression line pivots to multiple repetitions, providing a geometric description of pivot regions and introducing new open-ended questions.

Findings

Identifies regions where pivot points may lie under all repetitions

Generalizes the pivoting behavior to multiple repeated points

Raises new questions about the geometric properties of these regions

Abstract

It has been shown that for a given set of points in a plane, the least-squares regression line pivots about a fixed point when any single point in the set is repeated. We consider what happens when more than one point is repeated. Geometrically, we describe the regions where pivot points may lie under all possible combinations of repetitions. The underlying framework of this pivoting is explored, yielding new open-ended questions.