Ridge

Overview

Ridge regression (L2-regularized linear regression) fits a linear model to the trial data and uses the regression coefficients as a surrogate for the response surface. |β| measures each parameter's contribution.

Formula

$$\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_p x_p$$

Objective (L2-regularized least squares):

$$\min_\beta |y - X\beta|^2 + \alpha|\beta|^2 \quad (\alpha = 1.0)$$

Closed-form solution:

$$\beta = (X^\top X + \alpha I)^{-1} X^\top y$$

Preprocessing

Z-score standardization applied to each column of X before fitting:

$$\tilde{x}_j = \frac{x_j - \mu_j}{\sigma_j}$$