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YuhangZhou88 / ESL_Solution

https://yuhangzhou88.github.io/ESL_Solution/
MIT License
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Ex 9.5a #3

Closed qichun-dai closed 2 years ago

qichun-dai commented 3 years ago

Hi, Yuhang: Thanks for developing this site for ESL!

I learned how to solve 9.5 (a) from your solution. You can probably generalize the proof as follows.

Screen Shot 2021-04-19 at 21 39 04

Best, Qichun

YuhangZhou88 commented 3 years ago

Thank you Qichun!

Yes I agree with you. I was a bit lazy and only wrote "Similar arguments apply to the general m" for brevity :(

qichun-dai commented 3 years ago

Haha, I see :)

On Tue, Apr 20, 2021 at 12:14 AM Yuhang @.***> wrote:

Thank you Qichun!

Yes I agree with you. I was a bit lazy and only wrote "Similar arguments apply to the general m" for brevity :(

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-- Best, Qichun

szcf-weiya commented 3 years ago

Hello, recently I am also working on this exercise (https://github.com/szcf-weiya/ESL-CN/issues/236). It seems that the partition region R_m depends on y_i, so the covariance you calculated is just a conditional one. (I am struggling in the closed-form without fixing R_m, and by googling related materials, I found your great repo here!

Your code indeed treats R_m as random, since you fit a new regression tree for each column of y, but there is a typo that might cause the estimation is close to the number of terminal nodes, image the summation is over each observation, i.e., along the row.

YuhangZhou88 commented 3 years ago

@szcf-weiya Thank you! you are correct, that should be 0.

For (a), I feel what the authors ask is the conditional one, from the context. Dof is a complicated concept for models like trees.

szcf-weiya commented 3 years ago

@YuhangZhou88 To my best knowledge, I had not heard of "conditioanl dof" in any related papers. If you find one, I am appreciated if you can share its link.

YuhangZhou88 commented 3 years ago

@szcf-weiya I mean the variance, as you pointed out, is "a conditional one". My second comment on dof is a general comment, so for the "closed-form without fixing R_m" that you are looking for, I can imagine it's difficult.