The problem is:
Selecting best pruned tree from CCP with Cross-validation
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Professor　Ricco RAKOTOMALALA’s lecture[1] has described Cart-CCP process clearly.
And I have implemented it in my Github[2]:
What [1] have described :
is:
67% of all data,for growing a CART tree and get a sequence of pruned tree.
33% of all data,for selecting the best pruned tree.
This is only for when the data is large enough.
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according to[3]
(This author is not in this world any longer,so I can NOT contact him),

Two methods of estimation are discussed: Use of an independent test sample and cross-validation. Of the two, use of an independent test sample is computationally more efficient and is preferred when the learning sample contains a large number of cases. As a useful by-product it gives relatively unbiased estimates of the node misclassification costs. **Cross-validation is computationally more expensive,but makes more effective use of all cases and gives useful** information regarding the stability of the tree structure.

So,there’s the second way to choose the best pruned tree:cross-validation.I mean :
The difficuly lies in selecting best pruned tree with cross-validation,
NOT in Pruning algorithm.

-----------almost all material about CCP in Google-------------
in[4],is said in page 19th:

It is important to note that in the procedure described here we are effectively using cross-validation to select the best value of the complexity parameter from the set β1, . . . , βK. Once the best value has been determined, the corresponding tree from the original cost-complexity sequence is returned.

no details about how to select.

in[5],mlwikipedia only talk about the same method as [2].

in[6],no validation about selecting is in discussion

in[7],IBM files don’t discuss how to use cross-validation to select the best-tree.

in[8],no details about how to select with cross-validation
(I have contacted the author,but no reply).

Here’s a picture about cross-validation after CCP,but no details.

The goal of Cross-validation is:
Estimation of the prediction error(or MSE)of the trees in sequenceMain(use all the data) with the sequenceV:(remove the vth fold data before growing Tv1)

For main tree,we have pruned tree sequenceMain
[$T_1$,$T_2$,$T_3$,$T_4$]
For CV tree V,we have pruned tree sequenceV:
[$T_{v1}$,$T_{v2}$,$T_{v3}$,Tv4 ]
(because only 3 decision nodes,so it can only be pruned twice )
Then how could I estimate error rate of $T_4$ in　sequenceMain when no $T_{v4}$ in SequenceV?
In[3] part3.4.2,the formula to estimate $T_k$is:
$$R^{cv}(T_k)=R^{cv}(T(a_k^{{}^{\prime }}))$$,
now in SequenceV,$T_{v4}(=T(a_4^{{}^{\prime }}))$ does not exist ,
how to estimate$R^{cv}(T_4)$ in sequenceMain via the above equation ?

The other reference about CCP were:
in[9],only pruning algorithm ,no details about how to select the best pruned tree.
in[10]page 110th:" There is no theoretical justification for
this heuristic tree-matching process as it was mentioned by Esposito et al. (1997)."
Quotation of[10] means that:
sequenceMain and sequenceV can NOT be matched,no matter in shape of pruned-tree or in the length of sequence.
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So,selecting the best pruned tree(from CCP) with cross-validation is NOT rigrous in theory,it has limitations,it is available only in large and balalanced datasets which can ensure:
For different $L_v$,it can produces similar tress.

Reference:
[1]Decision tree learning algorithms
[2]https://github.com/appleyuchi/Decision_Tree_Prune
[3]《Classification and regresssion trees》“Chapter 3.4 THE BEST PRUNED SUBTREE: AN ESTIMATION PROBLEM”

[4]http://www.cs.uu.nl/docs/vakken/mdm/trees.pdf
[5]Cost-Complexity Pruning
[6]https://onlinecourses.science.psu.edu/stat857/node/60/
[7]Cost-Complexity Pruning Process-IBM
[8]Classification and Regression Trees
[9]Lecture 19: Decision trees
[10]Chapter4 Overfitting Avoidance in Regression Trees

---------------------appendix--------------------------------------
Why $a_k^{{}^{\prime }}=\sqrt{a_k{a}_{k+1}}$ in CCP and ECP?
Because the final best pruned tree is built based on all the data,the real cost complexity will larger than $a_k$,we need to give it a modification,so we use $a_k=\sqrt{a_k{a}_{k+1}}$,but it’s of course not rigorously modified,just by experience.

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