Lecture notes on ridge regression

TL;DR

This paper reviews ridge regression, discussing its theoretical properties, practical applications, and extensions, including its Bayesian interpretation, generalizations, and adaptation to logistic regression, with illustrations on simulation and omics data.

Contribution

It provides a comprehensive review of ridge regression, including new generalizations, extensions to logistic regression, and comparisons with lasso, enhancing understanding of penalized regression methods.

Findings

Ridge regression reduces variance and improves stability in high-dimensional data.

Its Bayesian interpretation links it to prior distributions.

Extensions to logistic regression preserve key properties.

Abstract

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapters introduce its lasso counterpart and generalizations thereof.