2013-1 machine learning lecture 02 - decision trees

Decision Tree Dae-Ki Kang

Definition

• Definition #1

▫ A hierarchy of if-then’s

▫ Node – test

▫ Edge – direction of control

• Definition #2

▫ A tree that represents compression of data based on class

• Manually generated decision tree is not interesting at all!

Decision tree for mushroom data

Algorithms

• ID3

▫ Information gain

• C4.5 (=J48 in WEKA) (and See5/C5.0)

▫ Information gain ratio

• Classification and regression tree (CART)

▫ Gini gain

• Chi-squared automatic interaction detection (CHAID)

Example from Tom Mitchell’s book

Naïve strategy of choosing attributes

(i.e. choose the next available attribute)

Outlook Play=Yes Play=No

3,4,5,7,9,10,11,12,13 (9) 1,2,6,8,14 (5)

Sunny

Overcast Rain

Play=Yes Play=No

9,11 (2) 1,2,8 (3)

Play=Yes Play=No

3,7,12,13 (4) (0)

Play=Yes Play=No

4,5,10 (3) 6,14 (2)

Temp Temp

Play=Y

Hot

Mild

Cool Hot

Mild

Cool

How to generate decision trees?

• Optimal one ▫ Equal to (or harder than) NP-Hard

• Greedy one Greedy means big questions first Strategy – divide and conquer

▫ Choose an easy-to-understand test such that divided sub-data sets by the chosen test are the easiest to deal with Usually choose an attribute as a test Usually adopt impurity measure to see how easy to deal

with the sub-data sets

• Are there any other approaches? – there are many and open

Impurity criteria

• Entropy Information Gain, Information Gain Ratio ▫ Most popular ▫ Entropy – Sum of -plogp ▫ IG – Entropy(S) - Sum of Entropy(sub-data t) * |t|/|S| ▫ IG favors Social Security Number or ID ▫ Information Gain Ratio

• Gini index Gini Gain (used in CART) ▫ Related with Area Under the Curve ▫ GG – 1 - Sum of fractions^2

• Misclassification rate ▫ (misclassified instances)/(all instances) ▫ Problematic – lead to many indistinguishable splits (where

other splits are more desirable)

Zero Occurrence

• When a feature is never occurred in the training set zero frequency PANIC: makes all terms zero

• Smoothing the distribution

▫ Laplacian Smoothing

▫ Dirichlet Priors Smoothing

▫ and many more (Absolute Discouting, Jelinek-Mercer smoothing, Katz smoothing, Good-Turing smoothing, etc.)

Overfitting • Training set error

▫ Error of the classifier on the training data ▫ It is a bad idea to use up all data for training. You will be out of data to

evaluate the learning algorithm. • Test set error

▫ Error of the classifier on the test data ▫ Jackknife – Use n-1 examples to learn and 1 to test. Repeat n times. ▫ x-folds stratified cross-validation – Divide data into x-folds with the

same proportion of class. x-1 folds to train and 1 fold to test. Repeat x times.

• Overfitting ▫ The input data is incomplete (Quine) ▫ The input data do not reflect all possible cases. ▫ The input data can include noise. ▫ I.e. fit the classifier tightly to the input data is a bad idea.

• Occam’s razor ▫ Old axiom used to prove the existence of God. ▫ “plurality should not be posited without necessity”

Razors and Canon

• Occam's razor (Ockham's razor) ▫ "Plurality is not to be posited without necessity" ▫ Similar to a principle of parsimony ▫ If two hypothesis have almost equal prediction power, we

prefer more concise one. • Hanlon's razor

▫ Never attribute to malice that which is adequately explained by stupidity.

• Morgan's Canon ▫ In no case is an animal activity to be interpreted in terms of

higher psychological processes if it can be fairly interpreted in terms of processes which stand lower in the scale of psychological evolution and development.

Example: Playing Tennis

(taken from Andrew Moore’s)

Humidity

High Norm

(9+, 5-)

(3+, 4-) (6+, 1-)

( , ) ( , )( , ) log ( , ) log

( ) ( ) ( ) ( )

( , ) ( , )( , ) log ( , ) log

( ) ( ) ( ) ( )

0.151

h

P h p P n pI P h p P n p

P h P p P n P p

P h p P n pP h p P n p

P h P p P n P p

Wind

Weak Strong

(9+, 5-)

(6+, 2-) (3+, 3-)

( , ) ( , )( , ) log ( , ) log

( ) ( ) ( ) ( )

( , ) ( , )( , ) log ( , ) log

( ) ( ) ( ) ( )

0.048

w

P w p P s pI P w p P s p

P w P p P s P p

P w p P s pP w p P s p

P w P p P s P p

Predication for Nodes

From Andrew Moore’s slides

What is the predication for each node?

Predication for Nodes

Recursively Growing Trees

Original Dataset

Partition it according to the value of the attribute we split on

cylinders = 4

cylinders = 5

cylinders = 6

cylinders = 8

From Andrew Moore slides

Recursively Growing Trees

cylinders = 4 cylinders = 5 cylinders = 6 cylinders = 8

Build tree from These records..




From Andrew Moore slides

A Two Level Tree

Recursively growing trees

When should We Stop Growing Trees?

Should we split this node ?

Base Cases

• Base Case One: If all records in current data subset have the same output then don’t recurse

• Base Case Two: If all records have exactly the same set of input attributes then don’t recurse

Base Cases: An idea

• Base Case One: If all records in current data subset have the same output then don’t recurse

• Base Case Two: If all records have exactly the same set of input attributes then don’t recurse

Proposed Base Case 3:

If all attributes have zero information gain then don’t recurse

Is this a good idea?

Old Topic: Overfitting

Pruning

• Prepruning (=forward pruning)

• Postpruning (=backward pruning)

▫ Reduced error pruning

▫ Rule-post pruning

Pruning Decision Tree

• Stop growing trees in time

• Build the full decision tree as before.

• But when you can grow it no more, start to prune:

▫ Reduced error pruning

▫ Rule post-pruning

Reduced Error Pruning

• Split data into training and validation set

• Build a full decision tree over the training set

• Keep removing node that maximally increases validation set accuracy

Original Decision Tree

Pruned Decision Tree

Reduced Error Pruning

Rule Post-Pruning

• Convert tree into rules

• Prune rules by removing the preconditions

• Sort final rules by their estimated accuracy

Most widely used method (e.g., C4.5)

Other methods: statistical significance test (chi-square)

Real Value Inputs

• What should we do to deal with real value inputs?

mpg cylinders displacementhorsepower weight acceleration modelyear maker

good 4 97 75 2265 18.2 77 asia

bad 6 199 90 2648 15 70 america

bad 4 121 110 2600 12.8 77 europe

bad 8 350 175 4100 13 73 america

bad 6 198 95 3102 16.5 74 america

bad 4 108 94 2379 16.5 73 asia

bad 4 113 95 2228 14 71 asia

bad 8 302 139 3570 12.8 78 america

: : : : : : : :

: : : : : : : :

: : : : : : : :

good 4 120 79 2625 18.6 82 america

bad 8 455 225 4425 10 70 america

good 4 107 86 2464 15.5 76 europe

bad 5 131 103 2830 15.9 78 europe

Information Gain

• x: a real value input

• t: split value

• Find the split value t such that the mutual information I(x, y: t) between x and the class label y is maximized.

Pros and Cons

• Pros ▫ Easy to understand ▫ Fast learning algorithms (because they are greedy) ▫ Robust to noise ▫ Good accuracy

• Cons ▫ Unstable ▫ Hard to represent some functions (Parity, XOR, etc.) ▫ Duplication in subtrees ▫ Cannot be used to express all first order logic because

the test cannot refer to two or more different objects

Generation of data from a decision

tree (based on the definition #2) • Decision tree with support for each node

Rule set

▫ support = # of training instances assigned for a node

• Rule set Instances

• In this way, one can combine multiple decision trees by combining rule sets

• cf. Bayesian classifiers Fractional instances

Extensions and further considerations

• Extensions ▫ Alternating decision tree ▫ Naïve Bayes Tree ▫ Attribute Value Taxonomy guided Decision Tree ▫ Recursive Naïve Bayes ▫ and many more

• Further Researches ▫ Decision graph ▫ Bottom up generation of decision tree ▫ Evolutionary construction of decision tree ▫ Integrating two decision trees ▫ and many more

2013-1 machine learning lecture 02 - decision trees

Documents

p log p s

p log pn

p logp w

input data

training data

laplacian smoothing

data toevaluate

compression of data