a simulation comparison of Ridge regression estimators with Lars

Authors
Roshanak Alimohammadi
Jaleh Bahari
Alzahra University
Abstract
Introduction

Regression analysis is a common method for modeling relationships between variables. Usually Ordinary Least Squares method is applied to estimate regression model parameters. These estimators are unbiased and appropriate when design matrix is nonsingular.

In presence of multicollinearity, design matrix is singular and Ordinary Least Squares estimates cannot be obtained. In this situation, other methods‎, ‎such as Lasso‎, ‎Ridge‎ and ‎Lars may be considered.

Other hand, in many fields such as medicine, number of variables is greater than the number of observations‎ and usual methods such as Ordinary Least Squares are not proper and shrinkage methods‎, ‎such as Lasso‎, ‎Ridge‎ ‎and‎ ... ‎have a better performance to estimate regression model coefficients‎. ‎In the shrinkage methods‎, ‎tuning parameter plays an essential role in selecting variables and estimating parameters‎. ‎Bridge shrinkage estimators is an estimator that can be obtained by changing its tuning parameter‎. In other words, Bridge method is the extension of Ridge and Lasso regression methods. Selecting the appropriate amount of tuning parameter is important. There are many studies ‎on each of these methods under the assumed conditions. In this paper‎, performance of Bridge shrinkage estimators‎, ‎such as Lasso and Ridge‎ ‎are compared with Lars and Ordinary Least Squares estimators in a simulation study.

Material and Methods

A simulation study is applied to compare performance of the regression methods Ridge, Lasso, Lars and Ordinary Least Squares. MSE criterion is applied to evaluate the method performance.

Statistical software R is applied for simulation and comparing the regression methods.

Results and discussion

In the presence of collinearity, Bridge regression estimators will result in appropriate estimators. These estimators are biased but their performance is better than unbiased estimators such as Ordinary Least Squares. Indeed, Bridge estimators have the best performance in the class of biased estimators.

Conclusion

In this article, Ridge and Lasso estimators as special cases of Bridge estimators are compared with Lasso and Ordinary Least Squares in a simulation study.

This study shows that under the supposed conditions, Ridge, Lasso and Lars have better action than Ordinary Least Squares method.

Lars method has the best performance and Ridge estimators is better than Lasso Regression.
Keywords
Ridge Regression
Bridge Regression
Lasso Regression
LARS Regression
Tuning Parameter
Ordinary Least Squares Error Regression
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Mathematical Research
Volume 8, Issue 2 - Serial Number 21
Spring 2022
Pages 152-164