last updated 08-Sep-2020
Training errors (apparent errors)--Errors committed on the training set
Test errors -- Errors committed on the test set
Generalization errors -- Expected error of a model over random selection of records from same distribution
10 % of the data used for training and 90% of the data used for testing
Comparison between. Which is better?
Underfitting: when model is too simple, both training and test errors are large
Overfitting: when model is too complex, training error is small but test error is large
i.e. the model fits the training well but fails in general.
If training data is under-representative, testing errors increase and training errors decrease on increasing number of nodes
Increasing the size of training data reduces the difference between training and testing errors at a given number of nodes
Limited Training Size
High Complexity of the Model
Many algorithms employ a greedy strategy of trying different altermative models and tracking the best model
Consider the task of predicting whether stock market will rise/fall in the next 10 trading days
Random guessing: P(correct) = 0.5
Make 10 random guesses in a row:
Showing that you likely will find a great fit, use the following procedure
- Get 50 analysts
- Each analyst makes 10 random guesses
- Choose the analyst that makes the most number of correct predictions
Probability that at least one analyst makes at least 8 correct predictions: P(#correct≥8) = 1 - (1 - 0.0547)^{50} = 0.9333
Below on the left is fairly simple data set that should have a simple decision tree with just x and y attributes.
But if there are lots of noisy variables added, we can see fitting improves as more variables are added but the testing shows no improvement as shown in the upper left graph below.
Overfitting results in decision trees that are more complex than necessary.
Training error does not provide a good estimate of how well the tree will perform on previously test or future records.
Need ways for estimating generalization errors.
Model selection methodology can be used for estimating general errors.
Performed during model building
Purpose is to ensure that model is not overly complex (to avoid overfitting)
Need to estimate generalization error:
Approach is to split the training set into two partitions:
Rational: Occam's Razor: mathematics principle of given two models of similar generalization errors, one should prefer the simpler model over the more complex model
A complex model has a greater chance of being fitted accidentally by errors in data
Therefore, one should include model complexity when evaluating a model
Gen.Error(Model) = Train.Error(Model, Train.Data) + α * Complexity(Model)
Pessimistic Error Estimate of decision tree T with k leaf nodes:
Where:
Example:
Left tree has 4 wrong classifications, but with 7 nodes --> err()= 0.458
Right tree has 6 wrong classifications, with only 4 nodes --> err() = 0.417
So the right tree would be preferred.
There are other, similar, model selection approaches that we won't consider:
Pre-Pruning (Early Stopping Rule)
Stop the algorithm before it becomes a fully-grown tree
Typical stopping conditions for a node:
- Stop if all instances belong to the same class
- Stop if all the attribute values are the same
More restrictive conditions:
- Stop if number of instances is less than some user-specified threshold
- Stop if expanding the current node does not improve impurity measures (e.g., Gini or information gain).
- Stop if estimated generalization error falls below certain threshold
Post-pruning
Grow decision tree to its entirety, then apply the following:
Subtree replacement
- Trim the nodes of the decision tree in a bottom-up fashion
- If generalization error improves after trimming, replace sub-tree by a leaf node
- Class label of leaf node is determined by the largest class of instances in the sub-tree
Subtree raising
- Replace subtree with most frequently used branch
Example, with Ω = 0.5
Purpose: To estimate performance of classifier on previously unseen data (test set)
Holdout
Cross validation
Divide the data set into three or more even sized partitions (k partitions)
Cycle through each partition as the test set with the other k-1 partitions used as training.
The average error is used as the generalization error computation for the model
k=3 is shown below.