Lecture Notes: Ridge Regression and Regularization
Introduction
- Regularization: Technique to reduce model complexity and prevent overfitting by adding a penalty term to the loss function.
- Ridge Regression: A type of linear regression that includes a regularization term.
- Part of a series on regularization techniques.
- Assumes knowledge of regularization, bias & variance, linear models, and cross-validation.
Main Goals
- Understanding Ridge Regression: Introduction to concepts and how it works.
- Details of Ridge Regression: Mathematical formulation and functioning.
- Applications: How it works in various scenarios.
- Capabilities: Solving high-dimensional problems.
Ridge Regression Basics
- Model weight and size of mice using linear regression (least squares).
- Least Squares: Minimizes sum of squared residuals.
- Works well with large datasets.
- Risk of overfitting with small datasets.
Overfitting and Variance
- Overfitting occurs when a model fits training data too closely, leading to high variance on testing data.
- Ridge Regression introduces bias to reduce variance.
- Better long-term predictions by starting with a slightly worse fit.
Mathematical Formulation
- Ridge Regression Equation: Minimizes sum of squared residuals plus lambda times slope squared.
- Lambda (λ): Regularization parameter controlling penalty severity.
- When λ = 0, ridge regression is equivalent to least squares.
- Increasing λ reduces slope, decreasing sensitivity of predictions to changes in independent variables.
Effect of Ridge Regression
- Slope penalty reduces sensitivity of predictions to changes in independent variables.
- Ridge regression produces less sensitive predictions, reducing overfitting.
Choosing Lambda (λ)
- Determine optimal λ using cross-validation (e.g., 10-fold cross-validation).
Ridge Regression with Discrete Variables
- Also applies when predictors are discrete (e.g., diet types).
- Penalizes differences between group means.
- Effective for small datasets, improving prediction accuracy.
Ridge Regression in Logistic Regression
- Applies to binary outcomes (e.g., obesity prediction).
- Optimizes likelihoods rather than squared residuals.
High-Dimensional Problems
- Effective in situations with more parameters than data points.
- E.g., gene expression data.
- Solves for parameters in underdetermined systems using penalty term.
Summary
- Ridge regression adds a penalty term to reduce variance and improve predictions on new data.
- Useful for small sample sizes and high-dimensional datasets.
- Cross-validation used to determine optimal λ.
- Allows estimation of parameters when traditional least squares cannot.
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