Multiple Linear Regression and Correlation: A Geometric Analysis

TL;DR

This paper offers a geometric perspective on multiple linear regression, illustrating how vector lengths and angles can describe regression outputs, correlation, and effects like multicollinearity and enhancement.

Contribution

It introduces a geometric framework for understanding linear regression, providing formulas and interpretations based on vector geometry that clarify complex phenomena.

Findings

Geometric interpretation of regression outputs using vector lengths and angles

A formula for multiple correlation based on geometric relationships

Analysis of multicollinearity and the enhancement phenomenon

Abstract

In this review article we consider linear regression analysis from a geometric perspective, looking at standard methods and outputs in terms of the lengths of the relevant vectors and the angles between these vectors. We show that standard regression output can be written in terms of the lengths and angles between the various input vectors, such that this geometric information is sufficient in linear regression problems. This allows us to obtain a standard formula for multiple correlation and give a geometric interpretation to this. We examine how multicollinearity affects the total explanatory power of the data, and we examine a counter-intuitive phenomena called "enhancement" where the total information from the explanatory vectors is greater than the sum of the marginal parts.