Shrinkage Methods: Lasso and Ridge Regression – Theory and Comparison

Project Overview

This project explores the application of Lasso and Ridge regression, two widely-used shrinkage methods. Both techniques address overfitting by penalizing large coefficients. The goal is to compare their performance in feature selection and coefficient shrinkage using a dataset containing various credit-related factors.

Key Topics Covered:

• Ridge Regression (L2 regularization): Shrinks all coefficients towards zero without eliminating any, making it useful when all features contribute to the prediction.
• Lasso Regression (L1 regularization): Can shrink some coefficients to zero, which is beneficial for feature selection.
• Comparison: A side-by-side comparison of how Ridge and Lasso affect coefficient estimates and their suitability for different use cases.

Dataset

The dataset used for this analysis is the Credit dataset, which contains various credit-related variables like Income, Limit, Rating, and Balance (the target variable). The aim is to predict the Balance using the other features.

Project Structure

1. Data Loading and Preprocessing

• Libraries: pandas, numpy, seaborn, matplotlib, sklearn
• Steps:
  • Load the dataset and display its summary statistics.
  • Visualize the relationships between variables using a pairplot.
  • Preprocess the dataset by one-hot encoding categorical variables and standardizing the features.

2. Ridge Regression

• Theory: Ridge Regression applies L2 regularization, adding a penalty term proportional to the square of the coefficients. This helps shrink the coefficients while retaining all variables.
• Implementation: Ridge regression is applied to the dataset using a range of alpha values (regularization strength). The solution path for Ridge is visualized, showing how coefficients shrink as alpha increases.

3. Lasso Regression

• Theory: Lasso Regression applies L1 regularization, which can set some coefficients exactly to zero, effectively performing feature selection.
• Implementation: Similar to Ridge, Lasso regression is applied with varying alpha values. The Lasso solution path is visualized, demonstrating how coefficients behave as the regularization strength increases.

4. Comparison of Ridge and Lasso

• A side-by-side comparison of the coefficients obtained using Ridge and Lasso at a fixed alpha value is performed. The comparison highlights:
  • Ridge shrinks coefficients uniformly, while Lasso sets some coefficients to zero.
  • Lasso performs feature selection, making it a good choice when interpretability and selecting important features are priorities.

5. Bias-Variance Tradeoff

• Ridge: By shrinking coefficients, Ridge reduces model variance at the cost of introducing some bias. It is especially useful when preventing overfitting is important.
• Lasso: Lasso also introduces bias by shrinking coefficients, but because it can set coefficients to zero, it reduces model complexity more aggressively than Ridge. This makes Lasso better suited for high-dimensional datasets with many irrelevant features.

6. Observations and Conclusions

• Ridge regression tends to shrink all coefficients gradually, with no coefficients being eliminated, even with high alpha values.
• Lasso regression sets some coefficients to zero, effectively removing less important features as alpha increases.
• Key Insight: Lasso is useful for feature selection, while Ridge is better suited when retaining all features is more important, but controlling their magnitude.