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Decision Tree - Overfitting
Overfitting is a significant practical difficulty for decision tree models and many other predictive models. Overfitting happens when the learning algorithm continues to develop hypotheses that reduce training set error at the cost of an
increased test set error. There are several approaches to avoiding overfitting in building decision trees. 
  • Pre-pruning that stop growing the tree earlier, before it perfectly classifies the training set.
  • Post-pruning that allows the tree to perfectly classify the training set, and then post prune the tree. 
Practically, the second approach of post-pruning overfit trees is more successful because it is not easy to precisely estimate when to stop growing the tree. 
The important step of tree pruning is to define a criterion be used to determine the correct final tree size using one of the following methods:
  1. Use a distinct dataset from the training set (called validation set), to evaluate the effect of post-pruning nodes from the tree.
  2. Build the tree by using the training set, then apply a statistical test to estimate whether pruning or expanding a particular node is likely to produce an improvement beyond the training set.
    • Error estimation
    • Significance testing (e.g., Chi-square test)
  3. Minimum Description Length principle : Use an explicit measure of the complexity for encoding the training set and the decision tree, stopping growth of the tree when this encoding size (size(tree) + size(misclassifications(tree)) is minimized.
The first method is the most common approach. In this approach, the available data are separated into two sets of examples: a training set, which is used to build the decision tree, and a validation set, which is used to evaluate the impact of pruning the tree. The second method is also a common approach. Here, we explain the error estimation and Chi2 test. 
 

Post-pruning using Error estimation
Error estimate for a sub-tree is weighted sum of error estimates for all its leaves. The error estimate (e) for a node is:

In the following example we set Z to 0.69 which is equal to a confidence level of 75%.

The error rate at the parent node is 0.46 and since the error rate for its children (0.51) increases with the split, we do not want to keep the children.
 
Post-pruning using Chi2 test
In Chi2 test we construct the corresponding frequency table and calculate the Chi2 value and its probability.

 
  Bronze Silver Gold
Bad 4 1 4
Good 2 1 2

Chi2 = 0.21          Probability = 0.90         degree of freedom=2
 
If we require that the probability has to be less than a limit (e.g., 0.05), therefore we decide not to split the node.