Difference Between Lasso and Ridge Regression

Both Lasso Regression and Ridge Regression are regularization techniques used to prevent overfitting in linear regression models by adding a penalty term to the loss function. However, they differ in how they apply this penalty.

1?? Ridge Regression (L2 Regularization)

• Ridge regression adds a penalty equal to the sum of the squared values of the coefficients.
• It shrinks the regression coefficients but does not force them to zero.
• Helps when we have multicollinearity (correlated independent variables).

Formula:

Loss=∑(yi?y^i)^2+λ∑βj^2

where:

• ∑(yi?y^i)^2 is the ordinary least squares (OLS) loss.
• λ∑βj^2 is the L2 penalty (sum of squared coefficients).
• λ is the regularization parameter (higher λ\lambda means stronger shrinkage).

Key Properties:

? Does not eliminate features, just shrinks coefficients.

? Works well when all features are useful but need to reduce their impact.

? Used in multicollinearity problems where independent variables are highly correlated.

?? Example Use Cases:

• Used in stock price prediction where multiple economic indicators are correlated.
• Applied in healthcare models predicting disease risk with multiple interrelated factors.

2?? Lasso Regression (L1 Regularization)

• Lasso regression adds a penalty equal to the absolute value of the coefficients.
• It shrinks some coefficients to exactly zero, effectively selecting a subset of features.
• Helps in feature selection (eliminating irrelevant variables).

Formula:

Loss=∑(yi?y^i)^2+λ∑∣βIj

where:

• λ∑∣βj∣ is the L1 penalty (sum of absolute values of coefficients).

Key Properties:

? Feature selection capability (removes irrelevant features by setting their coefficients to zero).

? Useful when we expect that only a few variables contribute significantly.

? Works well when the dataset has many irrelevant features.

?? Example Use Cases:

• Genomics: Selecting the most important genes affecting disease risk.
• Marketing analytics: Choosing the most relevant factors affecting sales.

3?? Other Types of Regularized Regression

Elastic Net Regression (L1 + L2 Regularization)

• Combination of Ridge and Lasso: Uses both L1 (Lasso) and L2 (Ridge) penalties.
• Helps when we have highly correlated variables (like Ridge) and also want feature selection (like Lasso).
• Formula: Loss=∑(yi?y^i)^2+λ1∑∣βj∣+λ2∑βj^2
• Best of both worlds: Can select features while keeping stability in coefficients.
• Used in large-scale datasets where feature selection and handling multicollinearity are both important.

4?? Other Types of Regression Beyond Lasso & Ridge

?? Key Takeaways

1?? Use Ridge Regression when all features are important, but you want to reduce their impact.

2?? Use Lasso Regression when you want feature selection and need to eliminate irrelevant variables.

3?? Use Elastic Net when both feature selection and multicollinearity handling are needed.

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