2013-1 Machine Learning Lecture 02 - Decision Trees

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

Naï strategy of choosing attributes
ve
(i.e. choose the next available attribute)
Play=Yes Play=No
Outlook
3,4,5,7,9,10,11,12,13 (9) 1,2,6,8,14 (5)

Sunny
Rain
Overcast
Play=Yes Play=No Play=Yes Play=No

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

Temp Play=Yes Play=No
Temp
3,7,12,13 (4) (0)

Hot Cool Hot Cool
Play=Y
Mild Mild

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 (9+, 5-) Wind (9+, 5-)

High Norm Weak Strong

(3+, 4-) (6+, 1-) (3+, 3-)
(6+, 2-)

P(h, p) P(n, p) P( w, p) P ( s, p )
I h  P(h, p) log  P(n, p) log  I w  P( w, p) log  P( s, p) log 
P ( h) P ( p ) P ( n) P ( p ) P( w) P( p) P( s) P( p)
P(h, p) P(n, p) P( w, p) P( s, p)
P(h, p) log  P(n, p) log P( w, p) log  P( s, p) log
P(h) P(p) P(n) P(p) P( w) P(p) P( s ) P(p)
 0.151  0.048

Predication for Nodes

What is the predication for each node?

From Andrew Moore’s slides

Recursively Growing Trees
cylinders = 4

cylinders = 5

cylinders = 6

Original Partition it
Dataset according cylinders = 8
to the value of
the attribute
we split on

From Andrew Moore slides

Recursively Growing Trees

Build tree from Build tree from Build tree from Build tree from
These records.. These records.. These records.. These records..

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

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?

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

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
▫ Naï Bayes Tree
ve
▫ Attribute Value Taxonomy guided Decision Tree
▫ Recursive Naï Bayes
ve
▫ 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

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