C4.5 - pruning decision trees
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1 C4.5 - pruning decision trees
2 Quiz 1
3 Quiz 1 Q: Is a tree with only pure leafs always the best classifier you can have? A: No.
4 Quiz 1 Q: Is a tree with only pure leafs always the best classifier you can have? A: No. This tree is the best classifier on the training set, but possibly not on new and unseen data. Because of overfitting, the tree may not generalize very well.
5
6 Pruning Goal: Prevent overfitting to noise in the data Two strategies for pruning the decision tree: Postpruning - take a fully-grown decision tree and discard unreliable parts Prepruning - stop growing a branch when information becomes unreliable
7 Prepruning Based on statistical significance test Stop growing the tree when there is no statistically significant association between any attribute and the class at a particular node Most popular test: chi-squared test ID3 used chi-squared test in addition to information gain Only statistically significant attributes were allowed to be selected by information gain procedure
8 Early stopping a b class Pre-pruning may stop the growth process prematurely: early stopping Classic example: XOR/Parity-problem No individual attribute exhibits any significant association to the class Structure is only visible in fully expanded tree Pre-pruning won t expand the root node But: XOR-type problems rare in practice And: pre-pruning faster than post-pruning
9 Post-pruning First, build full tree Then, prune it Fully-grown tree shows all attribute interactions Problem: some subtrees might be due to chance effects Two pruning operations: 1. Subtree replacement 2. Subtree raising
10 Subtree replacement Bottom-up Consider replacing a tree only after considering all its subtrees
11 Subtree replacement Bottom-up Consider replacing a tree only after considering all its subtrees
12 Subtree replacement Bottom-up Consider replacing a tree only after considering all its subtrees
13 Subtree replacement Bottom-up Consider replacing a tree only after considering all its subtrees
14 Estimating error rates Prune only if it reduces the estimated error Error on the training data is NOT a useful estimator Use hold-out set for pruning ( reduced-error pruning ) C4.5 s method Derive confidence interval from training data Use a heuristic limit, derived from this, for pruning Standard Bernoulli-process-based method Shaky statistical assumptions (based on training data)
15 Estimating Error Rates
16 Estimating Error Rates Q: what is the error rate on the training set?
17 Estimating Error Rates Q: what is the error rate on the training set? A: 0.33 (2 out of 6)
18 Estimating Error Rates Q: what is the error rate on the training set? A: 0.33 (2 out of 6)
19 Estimating Error Rates Q: what is the error rate on the training set? A: 0.33 (2 out of 6) Q: will the error on the test set be bigger, smaller or equal?
20 Estimating Error Rates Q: what is the error rate on the training set? A: 0.33 (2 out of 6) Q: will the error on the test set be bigger, smaller or equal? A: bigger
21 Estimating the error Assume making an error is Bernoulli trial with probability p p is unknown (true error rate) We observe f, the success rate f = S/N For large enough N, f follows a Normal distribution Mean and variance for f : p, p (1 p)/n p -σ p p +σ
22 Estimating the error c% confidence interval [ z X z] for random variable with 0 mean is given by: With a symmetric distribution: c
23 z-transforming f Transformed value for f : (i.e. subtract the mean and divide by the standard deviation) Resulting equation: Solving for p:
24 C4.5 s method Error estimate for subtree is weighted sum of error estimates for all its leaves Error estimate for a node (upper bound): If c = 25% then z = 0.69 (from normal distribution) Pr[X z] z 1% % % % % % 0.25
25 C4.5 s method
26 C4.5 s method f is the observed error
27 C4.5 s method f is the observed error z = 0.69
28 C4.5 s method f is the observed error z = 0.69 e > f e = (f + ε 1 )/(1+ ε 2 )
29 C4.5 s method f is the observed error z = 0.69 e > f e = (f + ε 1 )/(1+ ε 2 ) N, e = f
30 Example f=0.33 e=0.47 f=0.5 e=0.72 f=0.33 e=0.47
31 Example f=0.33 e=0.47 f=0.5 e=0.72 f=0.33 e=0.47 Combined using ratios 6:2:6 gives 0.51
32 Example f = 5/14=0.36 e = 0.46 e < 0.51 so prune! f=0.33 e=0.47 f=0.5 e=0.72 f=0.33 e=0.47 Combined using ratios 6:2:6 gives 0.51
33 Summary Decision Trees splits binary, multi-way split criteria information gain, gain ratio, pruning No method is always superior experiment!
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