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-   -   Exercise 3.4 (http://book.caltech.edu/bookforum/showthread.php?t=4353)

 xuewei4d 06-14-2013 10:08 AM

Exercise 3.4

:clueless:

I didn't get correct answer to Exercise 3.4 (c).

Exercise 3.4(b), I think the answer would be

Exercise 3.4(c), by independence between different , I have .

Where am I wrong?

 htlin 06-17-2013 08:27 PM

Re: Exercise 3.4

You can consider double-checking your answer of 3.4(b). Hope this helps.

 i_need_some_help 10-06-2013 02:16 PM

Re: Exercise 3.4

I am not sure how to approach part (a). Are we supposed to explain why that in-sample estimate intuitively makes sense, or (algebraically) manipulate expressions given earlier into it?

 magdon 10-06-2013 09:18 PM

Re: Exercise 3.4

Algebraically manipulate earlier expressions and you should get 3.4(a). It is essentially a restatement of .

 Sweater Monkey 10-07-2013 12:00 AM

Re: Exercise 3.4

I'm not sure if I'm going about part (e) correctly.

I'm under the impression that

where as derived earlier
and

This lead me to

I carried out the expansion of this expression and then simplified into the relevant terms but my final answer is because the N term cancels out.

Am I starting out correctly up until this expansion or is my thought process off from the start? And if I am heading in the right direction is there any obvious reason that I may be expanding the expression incorrectly? Any help would be greatly appreciated.

 ddas2 10-07-2013 01:46 AM

Re: Exercise 3.4

1. I got $y^{\prime}=y-\epsilon+\epsilon^{\prime}$.
and $\hat{y}-y^{\prime}=H\epsilon +\epsilon^{\prime}$.

 magdon 10-07-2013 05:53 AM

Re: Exercise 3.4

You got it mostly right. Your error is assuming both term, the H term and the one without the H give an N to cancel the N in the denominator. One term gives an N and the other gives a (d+1).

Quote:
 Originally Posted by Sweater Monkey (Post 11541) I'm not sure if I'm going about part (e) correctly. I'm under the impression that where as derived earlier and This lead me to I carried out the expansion of this expression and then simplified into the relevant terms but my final answer is because the N term cancels out. Am I starting out correctly up until this expansion or is my thought process off from the start? And if I am heading in the right direction is there any obvious reason that I may be expanding the expression incorrectly? Any help would be greatly appreciated.

 Sweater Monkey 10-07-2013 09:09 AM

Re: Exercise 3.4

Quote:
 Originally Posted by magdon (Post 11544) You got it mostly right. Your error is assuming both term, the H term and the one without the H give an N to cancel the N in the denominator. One term gives an N and the other gives a (d+1).
Yes I realize that only one term should have the N so the issue must be in how I'm expanding the expression.

I think my problem is how I'm looking at the trace of the matrix.

I'm under the impression that produces an NxN matrix with a diagonal of all values and 0 elsewhere. I come to this conclusion because the are all independent so when multiplied together the covariance of any two should be zero while the covariance of any should be the variance of . So then the trace of this matrix should have a sum along the diagonal of , shouldn't it? :clueless:

 aaoam 10-07-2013 09:18 AM

Re: Exercise 3.4

I'm having a bit of difficulty with 3.4b. I take \hat(y) - y and multiply by (XX^T)^{-1}XX^T, which ends up reducing the expression to just H\epsilon. However, then I can't use 3.3c in simplifying 3.3c, which makes me think I did something wrong. Can somebody give me a pointer?

Also, it'd be great if there was instructions somewhere about how to post in math mode. Perhaps I just missed them?

 magdon 10-07-2013 09:19 AM

Re: Exercise 3.4

Yes, that is right. You have to be more careful but use similar reasoning with

Quote:
 Originally Posted by Sweater Monkey (Post 11545) Yes I realize that only one term should have the N so the issue must be in how I'm expanding the expression. I think my problem is how I'm looking at the trace of the matrix. I'm under the impression that produces an NxN matrix with a diagonal of all values and 0 elsewhere. I come to this conclusion because the are all independent so when multiplied together the covariance of any two should be zero while the covariance of any should be the variance of . So then the trace of this matrix should have a sum along the diagonal of , shouldn't it? :clueless:

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