Ridge Regression

In ridge regression, the goal is to find the best-fitting linear equation, similar to multiple regression. The difference lies in the objective function, which is modified to include the L2 regularization term.

The ridge regression objective function is given by,

Ridge Regression Objective Function: Objective = Σ(Yᵢ - Ŷᵢ)² + λ * Σ(βⱼ²)

The first term in the objective function represents the residual sum of squares (RSS), which measures the discrepancy between the actual and predicted dependent variable values. The second term is the regularization term, which penalizes large coefficient values. The coefficient values are shrunk towards zero, but not exactly to zero, as the regularization term ensures that they remain non-zero.

The ridge regression coefficients are obtained by minimizing the objective function. The formula for calculating the ridge regression coefficients β we use below formula

Ridge Regression Coefficients Calculation: β = (Xᵀ X + λ * I)⁻¹ Xᵀ Y

Where:
- Ŷᵢ is the predicted dependent variable using the ridge regression model.
- Yᵢ is the actual dependent variable (observed data).
- p is the number of independent variables (features).
- βⱼ is the coefficient for the j-th independent variable.
- λ is the regularization parameter (hyperparameter) controlling the penalty.
- Xᵀ is the transpose of the matrix of independent variables X.
- I is the identity matrix of appropriate dimensions.

Ridge regression allows for more stable and accurate coefficient estimates, especially when dealing with multicollinearity and high-dimensional data, making it a valuable tool in statistical modeling and machine learning.