Explain ridge regression, Applied Statistics

Assignment Help:

Using log(x1), log(x2) and log(x3) as the predictors, do pair wise scatterplots of all pairs of variables (including the response) and comment (use the pairs function). Do you think that multi collinearity might be a problem with these data?

Plot the ridge trace for a grid of 50 values for the shrinkage parameter  over the range [0; 1]. Based on this plot suggest a reasonable value for . Find the estimates of the coecients for a ridge re gression with your chosen value of  (using centred and scaled predictors).

(The following question is based on Exercise 8.5 of Myers (1990), Classical and Modern Regression with Applications (Second Edition)," Duxbury).

With centred and scaled predictor variables, the ridge regression estimator for the coecients of the predictors is where y is the vector of responses, X is the design matrix for the centred and scaled predictors, is

1709_basic linear models.png

the shirnkage parameter and I denotes the identity matrix. We write n for the number of observations and k for the number of predictors. Writing biR for the ith component of bR, we will prove in this question that where 2 is the variance of the responses, and vi, i = 1,.......k are the eigenvalues of XTX. The di erent parts of the question below lead you through the proof.

735_basic linear models1.png

(a) Write XTX = QDQT for the eigenvalue decomposition of XTX, where D = diag(v1,........vk) is the diagonal matrix of eigenvalues and Q is an orthogonal matrix (QTQ = I) where the columns are the eigenvectors of XTX. Show that XTX +I = Q(D+I)QT .

2344_basic linear models2.png

where V ar(bR) denotes the covariance matrix of bR. (Hint: recall the result from basic linear models that if Y is a k  1 random vector with V ar(Y ) = V and if A is a k  k matrix and Z = AY then V ar(Z) = AV AT ).


Related Discussions:- Explain ridge regression

Test for equality of proportions, Test for Equality of Proportions For...

Test for Equality of Proportions For example, we may want to test whether the percentage of smokers (p 1 ) among the males equals the percentage of female smokers (p 2 ). W

Good measure of quality, Education seems to be a very difficult field in wh...

Education seems to be a very difficult field in which to use quality methods. One possible outcome measures for colleges is the graduation rate (the percentage of the students matr

Determine how the ordinary least squares, Question Following the general...

Question Following the general methodology used by econometricians as explained in the session for week 1 (eight steps), explain how you would proceed to determine if a good com

Confidence interval, a) List down several measures of central tendency and ...

a) List down several measures of central tendency and define the difference among them? b) What do you mean by confidence interval, and why it is useful? What is a confidence lev

Correlation, Correlation The board of directors of Bata Company is face...

Correlation The board of directors of Bata Company is faced with the problem of estimating what the annual sales might be in a shop to be opened in Bagpur where Bata has not op

Melissa Bakery, Melissa Bakery is preparing for the coming thanksgiving fes...

Melissa Bakery is preparing for the coming thanksgiving festival. The bakery plans to bake and sell its favourite cookies; butter cookies, chocolate cookies and almond cookies. A k

Determine the effects of stopping smoking on weight gain, Determine the Eff...

Determine the Effects of Stopping Smoking On Weight Gain As part of a study to determine the effects of stopping smoking on weight gain, nine females were weighed on the day t

X-bar charts when the mean and standard deviation not known , Charts when t...

Charts when the Mean and the Standard Deviation are not known We consider the data corresponding to the example of Piston India Limited. Since we do not know population mean a

Standard deviation , Standard Deviation  The concept of standard deviat...

Standard Deviation  The concept of standard deviation was first introduced by Karl Pearson in 1893. The standard deviation is the most important and the popular measure of disp

Dominant strategy equilibrium, Consider the following game: (a) If ...

Consider the following game: (a) If (top, left) is a Weakly Dominant Strategy Equilibrium, then what inequalities must hold among (a, ..., h)? (b) If (top, left) is a Na

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd