 LFD Book Forum Exercise 3.4
 User Name Remember Me? Password
 Register FAQ Calendar Mark Forums Read Thread Tools Display Modes
#1
 tomaci_necmi Junior Member Join Date: May 2014 Posts: 3 Exercise 3.4

I asked the question at math stack exchange,

http://math.stackexchange.com/questi...to-a-dataset-d

Can anyone explain me how it works ?
#2
 tomaci_necmi Junior Member Join Date: May 2014 Posts: 3 Re: Exercise 3.4

In my textbook, there is a statement mentioned on the topic of linear regression/machine learning, and a question, which is simply quoted as,

Consider a noisy target, , for generating the data, where is a noise term with zero mean and variance, independently generated for every example . The expected error of the best possible linear fit to this target is thus .

For the data , denote the noise in as , and let ; assume that is invertible. By following the steps below, ***show that the expected in-sample error of linear regression with respect to is given by***, Below is my methodology,

Book says that,

In-sample error vector, , can be expressed as , which is simply, hat matrix, , times, error vector, .

So, I calculated in-sample error, , as, Since it is given by the book that, , and also is symetric, I got the following simplified expression, Here, I see that, And, also, the sum formed by , gives the following sum, I undestand that, However, I don't understand why,  should be equal to zero in order to satisfy the equation, ***Can any one mind to explain me why leads to a zero result ?***
#3
 tomaci_necmi Junior Member Join Date: May 2014 Posts: 3 Re: Exercise 3.4

Well all fits now, just my mind played a game to me.

off course #4 yaser Caltech Join Date: Aug 2009 Location: Pasadena, California, USA Posts: 1,478 Re: Exercise 3.4

Thank you for the question and the answer.
__________________
Where everyone thinks alike, no one thinks very much
#5
 yongxien Junior Member Join Date: Jun 2015 Posts: 8 Re: Exercise 3.4

Why the last statement is 0? I don't quite understand. Does the mean being zero imply E(e_i) and E(e_j) = 0? I find it weird if that is the case. Because that will mean E(e_i) = 0 but E(e_i^2) = \sigma^2. I understand E(e_i^2) = \sigma^2 from statistics but not the first part.

If it is not the case, then what is the reason for the last statement to be 0?
#6 htlin NTU Join Date: Aug 2009 Location: Taipei, Taiwan Posts: 604 Re: Exercise 3.4

In the problem statement, I think "zero mean of the noise" is a given condition? __________________
When one teaches, two learn.
#7
 zhout2 Junior Member Join Date: Oct 2016 Posts: 2 Re: Exercise 3.4

#8
 zhout2 Junior Member Join Date: Oct 2016 Posts: 2 Re: Exercise 3.4

Quote:
 Originally Posted by zhout2 Never mind. It's just a typo in the original post. The answer is still correct.
#9
 johnwang Junior Member Join Date: Oct 2017 Posts: 2 Re: Exercise 3.4

I still don't understand why "eq1" leads to zero. I know that e_i and e_j are zero mean independent variables. However, H_ij is dependent on both e_i and e_j,, so I don't know how to prove that the sum of H_ij*e_i*e_j has an expected value of zero.

Quote:
 Originally Posted by tomaci_necmi In my textbook, there is a statement mentioned on the topic of linear regression/machine learning, and a question, which is simply quoted as, Consider a noisy target, , for generating the data, where is a noise term with zero mean and variance, independently generated for every example . The expected error of the best possible linear fit to this target is thus . For the data , denote the noise in as , and let ; assume that is invertible. By following the steps below, ***show that the expected in-sample error of linear regression with respect to is given by***, Below is my methodology, Book says that, In-sample error vector, , can be expressed as , which is simply, hat matrix, , times, error vector, . So, I calculated in-sample error, , as, Since it is given by the book that, , and also is symetric, I got the following simplified expression, Here, I see that, And, also, the sum formed by , gives the following sum, I undestand that, However, I don't understand why,  should be equal to zero in order to satisfy the equation, ***Can any one mind to explain me why leads to a zero result ?***
#10
 johnwang Junior Member Join Date: Oct 2017 Posts: 2 Re: Exercise 3.4

Is it because the noise is generated independently for each datapoint? Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home General     General Discussion of Machine Learning     Free Additional Material         Dynamic e-Chapters         Dynamic e-Appendices Course Discussions     Online LFD course         General comments on the course         Homework 1         Homework 2         Homework 3         Homework 4         Homework 5         Homework 6         Homework 7         Homework 8         The Final         Create New Homework Problems Book Feedback - Learning From Data     General comments on the book     Chapter 1 - The Learning Problem     Chapter 2 - Training versus Testing     Chapter 3 - The Linear Model     Chapter 4 - Overfitting     Chapter 5 - Three Learning Principles     e-Chapter 6 - Similarity Based Methods     e-Chapter 7 - Neural Networks     e-Chapter 8 - Support Vector Machines     e-Chapter 9 - Learning Aides     Appendix and Notation     e-Appendices

All times are GMT -7. The time now is 09:41 AM.

 Contact Us - LFD Book - Top

Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd. The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.