# Ridge

Ridge regression modeling explained in detail.

**Ridge Regression Modeling**

Ridge regression is a type of linear regression modeling technique that introduces a regularization term, or penalty term, into the linear regression equation. The regularization term helps to prevent overfitting and improve the stability of the model by adding a constraint on the magnitude of the coefficients, parameters, of the regression model.

The ridge regression model modifies the linear regression equation by adding a penalty term to the sum of squared residuals (SSR) objective function. The objective function of ridge regression can be written as Objective function = SSR + α * Σ(βi^2) where:

- SSR is the sum of squared residuals, which measures the discrepancy between the observed values of the dependent variable and the predicted values by the regression model.
- α is the regularization parameter, or hyperparameter, that controls the strength of the penalty term. A higher value of α results in a stronger penalty, and a lower value of α results in a weaker penalty. It is a tuning parameter that we determine when constructing the model.
- Σ(βi^2) is the sum of the squared coefficients of the regression model, or the L2 norm of the coefficients. It represents the magnitude of the coefficients, and the penalty term is proportional to the square of the magnitude of the coefficients.

The addition of the penalty term in the objective function of ridge regression results in a different estimation approach compared to ordinary least squares (OLS) used in linear regression. The penalty term shrinks the estimated coefficients towards zero, which helps to reduce the risk of overfitting by discouraging the model from assigning excessively large values to the coefficients.

Ridge regression is particularly useful when dealing with multicollinearity, which occurs when there is a high correlation between the independent variables in the regression model. Multicollinearity can lead to unstable estimates of the coefficients in linear regression, but ridge regression helps to mitigate this issue by stabilizing the estimates and improving the model’s predictive performance.

We use ridge regression for prediction, estimation, feature selection, and model regularization. It is a powerful tool for improving the stability and performance of linear regression models, especially when dealing with multicollinearity or when there are a large number of correlated predictors in the model.