New approaches for choosing the ridge parameters

Authors : J. AL-JARARHA

Pages : 1625-1633

View : 15 | Download : 7

Publication Date : 2018-12-12

Article Type : Research Paper

Abstract :Consider the standard multiple linear regression model $y=x\beta+\varepsilon.$ If the correlation matrix $x^tx$  is ill-conditioned, the ordinary least squared estimate insert ignore into journalissuearticles values(ols); $\hat{\beta}$ of $\beta$ is not the best choice. In this paper, multiple regularization  parameters for different coefficients in ridge regression are proposed. The Mean Squared Error insert ignore into journalissuearticles values(MSE); of a ridge estimate  based on the multiple regularization parameters is less than or equal to the MSE of the ridge estimate based on Hoerl and Kennard, 1970. The proposed approach, depending on the condition numbers, leave`s zero for the largest eigenvalue of $x^tx$ and gives the largest value  for the smallest eigenvalue of $x^tx.$ Furthermore, if $x^tx$ is nearly a unit matrix, $x^tx$  is not an ill-conditioned one. The proposed approach gives approximately the same results as the ols estimates. The proposed approach can also be modified to give other new ridge parameters. The modified approach depends on the eigenvalues of  $x^tx$ and  differ from the ridge parameter proposed by Khalaf and Shukur by a factor. The body fat data set has severe multicollinearity and is used to compare different approaches.
Keywords : Ridge regression, multicollinearity, condition number, mean squared error, squared bias