Ridge Regression
Introduction
Ridge regression is a technique used in statistical learning to analyze multiple regression data that suffer from multicollinearity. By adding a degree of bias to the regression estimates, ridge regression reduces the standard errors.
Background
Ridge regression was first introduced by Arthur E. Hoerl and Robert W. Kennard in 1970 as a response to the problem of multicollinearity in multiple linear regression models. Multicollinearity occurs when predictor variables in a regression model are highly correlated, which can lead to unstable and unreliable estimates of the regression coefficients.
Methodology
Ridge regression improves the ill-posed problem of multicollinearity by imposing a penalty on the size of coefficients. This penalty term, or 'shrinkage', is a tuning parameter that is multiplied by the sum of the squared coefficients. This has the effect of shrinking the estimates of the coefficients towards zero, hence the term 'ridge regression'.
Mathematical Formulation
The ridge regression estimator is defined as the matrix expression:
β̂ ridge = (X'X + λI)^-1 X'y
where: - β̂ ridge is the ridge regression estimator, - X is the matrix of predictors, - y is the response vector, - λ is the tuning parameter, - I is the identity matrix.
Applications
Ridge regression is widely used in areas where predictor variables are expected to be highly correlated, such as in genomics, econometrics, and chemometrics. It is also used in machine learning as a method to prevent overfitting.
Advantages and Disadvantages
Ridge regression has several advantages over ordinary least squares regression. It can provide more reliable estimates when the predictors are highly correlated, and it can help to prevent overfitting. However, it also has some disadvantages. The inclusion of the penalty term makes the estimates biased, and the choice of the tuning parameter can be somewhat arbitrary and requires validation.