Ex. 9.5

[szcf-weiya]

[szcf-weiya]

The following calculation assumes R_m is fixed, but I think it should not be fixed (see the following first several sentences). However, I failed to obtain a closed form when treating R_m depends on y.

[szcf-weiya]

binary greedy partition (without pruning)

for simplicity, I wrote a no-pruning version from scratch,

julia> includet("df_regtree.jl")

julia> [rep_calc_df(maxdepth=i) for i=0:4]
5-element Vector{Tuple{Float64, Float64}}:
 (1.080629443872466, 0.041187110826985264)
 (9.996914473623567, 0.11093327184178654)
 (21.34913882167176, 0.14589493792386243)
 (34.57260013395476, 0.2469943540268895)
 (47.90389232392168, 0.31056757408950914)

the number of terminal nodes M equal to 2^i, it can be shown that the empirical df is much larger than M except for M=1.

[szcf-weiya]

call tree::prune.tree

to return a tree with the given number of terminal nodes, I call tree::prune.tree, which has an argument best to specify the number of terminal nodes, but note that there might be situations that the specified best cannot be achieved, as mentioned in ?prune.tree

If there is no tree in the sequence of the requested size, the next largest is returned.

In my experiments, I indeed found such cases.

> source("df_regtree.R")
> mean(replicate(10, calc_df(m=1)))
[1] 1.137945
> mean(replicate(10, calc_df(m=5)))
[1] 32.18979
> mean(replicate(10, calc_df(m=10)))
[1] 50.46288

Again, the estimated dfs are much larger than m, except for m=1.

[szcf-weiya]

similar experiments in Ye (1998)

Ye, J. (1998). On Measuring and Correcting the Effects of Data Mining and Model Selection. Journal of the American Statistical Association, 93(441), 120–131. https://doi.org/10.2307/2669609 It also shows that the estimated (generalized) df are much larger than m.

And a close result is reported if I set m=19 in my code

> mean(replicate(10, calc_df(m=19)))
[1] 60.31616

[litsh]

Thanks for your great solution. May I ask why is the estimated degree of freedom so far from the one in theory?

[szcf-weiya]

@litsh what do you mean "the one in theory"? You mean the number of nodes? Actually, here I am trying to say that the number of nodes is not the theoretical degrees of freedom. There is a gap, and the gap is referred to as search cost.

If you are interested, you can check the paper on the excess part of degrees of freedom by comparing lasso and the best subset regression: Tibshirani, Ryan J. “Degrees of Freedom and Model Search.” Statistica Sinica 25, no. 3 (2015): 1265–96.

I also discussed the search cost of degrees of freedom for more methods (including the regression tree here) in my paper. https://arxiv.org/abs/2308.13630

[litsh]

Thank you for your reply! I will read the paper.

Thaison @.***

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Subject:Re: [szcf-weiya/ESL-CN] Ex. 9.5 (#236)

@litsh what do you mean "the one in theory"? You mean the number of nodes? Actually, here I am trying to say that the number of nodes is not the theoretical degrees of freedom. There is a gap, and the gap is referred to as search cost.

If you are interested, you can check the paper on the excess part of degrees of freedom by comparing lasso and the best subset regression: Tibshirani, Ryan J. “Degrees of Freedom and Model Search.” Statistica Sinica 25, no. 3 (2015): 1265–96.

I also discussed the search cost of degrees of freedom for more methods (including the regression tree here) in my paper. https://arxiv.org/abs/2308.13630

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