Introduction

Ridge regression is a powerful technique used to address multicollinearity in linear regression models by adding a regularization term, specifically the L2 norm of the coefficient vector, to the loss function. One of the specifications available in certain implementations of Ridge regression, notably in earlier versions of `scikit-learn`, is the `normalize=True` parameter. This parameter was originally intended to standardize (mean-center and scale) the predictor variables before applying the regression. However, in later versions of libraries like `scikit-learn`, this parameter is deprecated due to the recommendation of preprocessing steps. This article will guide you through reproducing the effect of `Ridge(normalize=True)` using manual standardization of input features in Python.

Technical Explanation

Understanding Ridge Regression

Ridge regression modifies the ordinary least squares cost function by adding a penalty proportional to the square of the magnitude of coefficients: $$\min | \mathbf{y} - \mathbf{X} \beta |^2 + \alpha | \beta |^2$$ where: • $\mathbf{y}$ is the vector of observations. • $\mathbf{X}$ is the matrix of input features. • $\beta$ is the vector of coefficients. • $\alpha$ is the regularization parameter.

This method is effective at dealing with multicollinearity and preventing overfitting by constraining the coefficient size.

The Impact of Normalization

Normalization ensures that all features contribute equally to the regression analysis. This step involves:

1. Subtracting the mean from each feature.
2. Dividing each feature by its standard deviation.

The transformation can be expressed for each feature $x_i$ as: $$x\_{i,\text{norm}} = \frac{x\_i - \bar{x}\_i}{s\_i}$$ where $\bar{x}_i$ is the mean and $s_i$ is the standard deviation of the feature $x_i$.

Reproducing `normalize=True` Manually

Step 1: Standardize Features

Before fitting the Ridge model, manually standardize the features of your dataset:

• $\alpha = 0$: The model reduces to simple linear regression. • Large $\alpha$: The model becomes heavily regularized, possibly underfitting data.