# Documentation

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## Introduction to Ridge Regression

Coefficient estimates for the models described in Linear Regression rely on the independence of the model terms. When terms are correlated and the columns of the design matrix X have an approximate linear dependence, the matrix (XTX)–1 becomes close to singular. As a result, the least-squares estimate

$\stackrel{^}{\beta }={\left({X}^{T}X\right)}^{-1}{X}^{T}y$

becomes highly sensitive to random errors in the observed response y, producing a large variance. This situation of multicollinearity can arise, for example, when data are collected without an experimental design.

Ridge regression addresses the problem by estimating regression coefficients using

$\stackrel{^}{\beta }={\left({X}^{T}X+kI\right)}^{-1}{X}^{T}y$

where k is the ridge parameter and I is the identity matrix. Small positive values of k improve the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of ridge estimates often result in a smaller mean square error when compared to least-squares estimates.

The Statistics and Machine Learning Toolbox™ function ridge carries out ridge regression.