Ridge Regression

What is Ridge Regression? Ridge Regression Explained

Ridge regression is a regularization technique used in linear regression models to mitigate the problem of multicollinearity and overfitting. It adds a penalty term to the standard linear regression objective function to control the model’s complexity and reduce the impact of highly correlated predictors.

In this regression technique, the objective is to minimize the sum of squared residuals between the observed target values and the predicted values, while also penalizing the magnitude of the regression coefficients. The penalty term is proportional to the squared L2 norm (Euclidean norm) of the coefficient vector, multiplied by a regularization parameter, often denoted as lambda or alpha.

The ridge regression objective function can be expressed as:

minimize ||Y – Xβ||^2 + lambda * ||β||^2

where:

Y is the vector of observed target values,
X is the matrix of predictor variables,
β is the vector of regression coefficients,
lambda is the regularization parameter.

The lambda parameter controls the amount of regularization applied. A higher lambda value increases the penalty on the coefficient magnitudes, resulting in a simpler and more constrained model. Conversely, a lower lambda value reduces the regularization effect, allowing the model to fit the data more closely.

Ridge regression has several benefits and applications:

Multicollinearity Reduction: It is effective in dealing with multicollinearity, which occurs when predictor variables are highly correlated with each other. The penalty term in ridge regression shrinks the coefficients towards zero, reducing their impact and making the model more stable and robust in the presence of multicollinearity.

Model Stability: By reducing the coefficient magnitudes, this technique improves the stability of the model. It reduces the sensitivity to small changes in the input data and helps avoid overfitting by discouraging extreme coefficient values.

Feature Selection: It does not drive coefficients to exactly zero, but it can push them close to zero. As a result, it can be used for feature selection by identifying less important predictors that have near-zero coefficients.

Regularization Tuning: The choice of the regularization parameter lambda is crucial in ridge regression. It can be determined using techniques like cross-validation or by optimizing performance metrics such as mean squared error (MSE) or adjusted R-squared on a validation set.

Ridge regression is a widely used technique for linear regression problems, particularly when dealing with multicollinearity and overfitting. It provides a balance between model simplicity and performance by incorporating regularization. However, it should be noted that ridge regression assumes a linear relationship between predictors and the target variable, making it less suitable for capturing nonlinear relationships without appropriate transformations or extensions.