Ridge , Lasso, and Elastic Net Regression
Regression analysis is a cornerstone in both statistics and machine learning and it helps to model relationship between variables. Although linear regression is one of the most popular methods, But in data of high dimension or where predictive variable multicollinearity exists the results may not be very accurate. Ridge and Lasso regression are two techniques that have become increasingly popular in recent years, aiming to compensate for these drawbacks by introducing a type of regularization. Here, we explain the mathematical intuition behind both methods, how they function and when you should choose them.
Understanding Linear Regression:
Before diving into Ridge and Lasso regression, recap linear regression. click here for understand Linear Regression.
Introduction of Regularization
In the area of linear models, ordinary least squares (OLS) regression is one of the chief ways to estimate the relationship between variables. But OLS also has serious limitations: it tends to overfit, especially with multicollinearity and when the number of predictors is greater than observations. Regularization methods such as Ridge or Lasso regression, however, have been developed to tackle such problems.
1. Ridge Regression
Ridge regression, also known as L2 regularization, utilizes a penalty term to improved ordinary least squares modeling. This penalty comprises the tuning parameter multiplied by the squared sum of the coefficient values.
The key intuition behind Ridge regression is that by penalizing large coefficients, it reduces the model’s sensitivity to changes in the predictor variables and mitigates multicollinearity issues.
2. Lasso Regression
In Lasso regression, also known as L1 regularization, a penalty term is added to the coefficients that is proportional to their absolute values.
As Lasso regression puts some of these coefficient estimates exactly equal to zero it encourages sparsity. Because the coefficients of Lasso regression can be used to generate a sparse representation, Lasso regression is especially useful in feature selection: it automatically selects variables by directly setting some coefficients to zero.
3. Elastic Net Regression
Elastic Net regression combines the strengths of Ridge and Lasso regression, which involves both penalties 𝐿1 and 𝐿2. For Elastic Net regression, the objective function is:
Elastic Net regression is a versatile approach that effectively deals with both multicollinearity and feature selection and so is especially appropriate for datasets where both multicollinearity and sparse predictors feature simultaneously.
Comparing Ridge, Lasso Regression, and Elastic Net:
- Ridge regression can very effectively reduce the impact of multicollinearity and stabilize coefficient estimates, while it typically will not end up with exact zero coefficients that tallied-off each other in underdetermined cases.
- Lasso regression also actually includes a procedure for variable selection. It can set the tuned parameters down to zero thereof contents became more sparse in feature space.
- Elastic Net is a combination of Ridge and Lasso, giving us a generalization on the benefits both. It has been developed to take account of trade-offs between coefficient shrinkage and sparsity control.
Conclusion
Regularized regression techniques like Ridge, Lasso, and Elastic Net offer powerful solutions for addressing the limitations of conventional linear regression. By introducing penalty terms to the regression objective function, these methods strike a balance between model complexity and fit, leading to more robust and interpretable models. Understanding the mathematical foundations and practical implications of Ridge, Lasso, and Elastic Net regression enables practitioners to effectively apply these techniques to real-world datasets, improving predictive performance and model interpretability.
Ridge and Lasso Regression Practical Implementation in Python :-
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