DS 352 Syllabus

last updated 08-Sep-2020

Kinds of errors

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

Example: two class problem:

10 % of the data used for training and 90% of the data used for testing






Decision tree with 5 nodes

and with 50 nodes

Comparison between. Which is better?

Model Overfitting

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

Reasons for overfitting

Limited Training Size

High Complexity of the Model

Many algorithms employ a greedy strategy of trying different altermative models and tracking the best model


Multiple Comparison Problem

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

Probability that at least one analyst makes at least 8 correct predictions: P(#correct≥8) = 1 - (1 - 0.0547)50 = 0.9333


Effect of Multiple Comparisons.

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.

Notes on overfitting

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.


Model Selection

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:

Consider the complexity of the model

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:



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:

Model Selection for Decision Trees

Pre-Pruning (Early Stopping Rule)

Stop the algorithm before it becomes a fully-grown tree

Typical stopping conditions for a node:

More restrictive conditions:


Grow decision tree to its entirety, then apply the following:

Subtree replacement

Subtree raising

Example, with Ω = 0.5



Model Evaluation

Purpose: To estimate performance of classifier on previously unseen data (test set)


Cross validation

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.


Handling interactions


Limitations of single attribute-based decision boundaries