Multiple Linear Regression and Correlation: A Geometric Analysis

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.