Decision Trees and Rule Induction

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Decision Trees and Rule Induction. Kurt Driessens with slides stolen from Evgueni Smirnov and Hendrik Blockeel. Overview. Concepts, Instances, Hypothesis space Decisions trees Decision Rules. Concepts - Classes. Instances & Representation. How to represent information about instances

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### Decision Trees and Rule Induction

Kurt Driessens

with slides stolen from Evgueni Smirnov and HendrikBlockeel

Overview
• Concepts, Instances, Hypothesis space
• Decisions trees
• Decision Rules
Instances & Representation

How to represent information about instances

• Attribute-Value

body = round

color = blue

legs = short

holding = balloon

smiling = false

Can be symbolic or numeric

body = square

color = red

legs = long

holding = knife

smiling = true

• Sequences
• dna, stock market, patient evolution
• Structures
• graphs: computer networks, Internet sites
• trees: html/xml documents, natural language
• Relational data-base
• molecules, complex problems

In this course: Attribute-Value

Induction of decision trees
• What are decision trees?
• How can they be induced automatically?
• top-down induction of decision trees
• avoiding overfitting
• a few extensions
What are decision trees?
• Cf. guessing a person using only yes/no questions:
• A decision tree
• Tells you which question to ask, depending on outcome of previous questions
• Gives you the answer in the end
• Usually not used for guessing an individual, but for predicting some property (e.g., classification)
Example decision tree 1
• Play tennis or not? (depending on weather conditions)

Each internal node tests an attribute

Outlook

Each branch corresponds to an attribute value

Sunny

Rainy

Overcast

Humidity

Yes

Wind

Normal

Strong

Weak

High

No

Yes

No

Yes

Each leaf assigns a classification

Example decision tree 2
• Tree for predicting whether C-section necessary
• Leaves are not pure here; ratio pos/neg is given

Fetal_Presentation

1

3

2

Previous_Csection

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0

[3+, 29-]

.11+ .89-

[8+, 22-]

.27+ .73-

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Primiparous

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[55+, 35-]

.61+ .39-

Representation power
• Trees can represent any Boolean function
• i.e., also disjunctive concepts (<-> VS: conjunctive concepts)
• E.g. A or B
• Trees can allow noise (non-pure leaves)
• posterior class probabilities

A

true

false

true

B

true

false

true

false

Classification, Regression and Clustering
• Classification trees represent function X -> C with C discrete (like the decision trees we just saw)
• Hence, can be used for concept learning
• Regression trees predict numbers in leaves
• can use a constant (e.g., mean), or linear regression model, or …
• Clustering trees just group examples in leaves

Most (but not all) decision tree research in data mining focuses on classification trees

Top-Down Induction of Decision Trees

Basic algorithm for TDIDT: (based on ID3; later more formal)

• find test that partitions examples as good as possible

= examples with same class, or otherwise similar, are put together

• for each outcome of test, create child node
• move examples to children according to outcome of test
• repeat procedure for each child that is not “pure”

Main questions:

• how to decide which test is “best”
• when to stop the procedure
Example problem

?

Is this drink going to

make me ill, or not?

?The decision tree

Shape

Colour

orange

Non-orange

Finding the best test (for classification)

Find test for which children are as “pure” as possible

• Purity measure borrowed from information theory: entropy
• measure of “missing information”; related to the minimum number of bits needed to represent the missing information

Given set S with instances belonging to class i with probability pi:

Entropy(S) = - pi log2 pi

Entropy

Entropy in function of p, for 2 classes:

Information gain
• Heuristic for choosing a test in a node:
• choose that test that on average provides most information about the class
• this is the test that, on average, reduces class entropy most
• entropy reduction differs according to outcome of test
• expected reduction of entropy = information gain
E = 0.940

E = 0.940

E = 0.985

E = 0.592

E = 0.811

E = 1.0

Gain(S, Humidity)

= .940 - (7/14).985 - (7/14).592

= 0.151

Gain(S, Wind)

= .940 - (8/14).811 - (6/14)1.0

= 0.048

Example
• Assume S has 9 + and 5 - examples; partition according to Wind or Humidity attribute

S: [9+,5-]

S: [9+,5-]

Humidity

Wind

Normal

Strong

Weak

High

S: [3+,4-]

S: [6+,1-]

S: [6+,2-]

S: [3+,3-]

Hypothesis space search in TDIDT
• Hypothesis space H = set of all trees
• H is searched in a hill-climbing fashion, from simple to complex
• maintain a single tree
• no backtracking
Inductive bias in TDIDT

Note: for e.g. Boolean attributes, H is complete: each concept can be represented!

• given n attributes, we can keep on adding tests until all attributes tested

• Clearly no “restriction bias”
• Preference bias: some hypotheses in H are preferred over others

In this case: preference for short trees with informative attributes at the top

Occam’s Razor
• Preference for simple models over complex models is quite generally used in data mining
• Similar principle in science: Occam’s Razor
• roughly: do not make things more complicated than necessary
• Reasoning, in the case of decision trees: more complex trees have higher probability of overfitting the data set
Avoiding Overfitting

Phenomenon of overfitting:

• keep improving a model, making it better and better on training set by making it more complicated …
• increases risk of modeling noise and coincidences in the data set
• may actually harm predictive power of theory on unseen cases

Cf. fitting a curve with too many parameters

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area with probably

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Overfitting: example

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Overfitting: effect on predictive accuracy
• Typical phenomenon when overfitting:
• training accuracy keeps increasing
• accuracy on unseen validation set starts decreasing

accuracy on training data

accuracy on unseen data

accuracy

size of tree

How to avoid overfitting?
• Option 1:
• stop adding nodes to tree when overfitting starts occurring
• need stopping criterion
• Option 2:
• don’t bother about overfitting when growing the tree
• after the tree has been built, start pruning it again
Stopping criteria
• How do we know when overfitting starts?
• use a validation set

= data not considered for choosing the best test

 when accuracy goes down on validation set: stop adding nodes to this branch

• use a statistical test
• significance test: is the change in class distribution significant? (2-test) [in other words: does the test yield a clearly better situation?]
• MDL: minimal description length principle
• entirely correct theory = tree + corrections for misclassifications
• minimize size(theory) = size(tree) + size(misclassifications(tree))
• Cf. Occam’s razor
Post-pruning trees

After learning the tree: start pruning branches away

• For all nodes in tree:
• Estimate effect of pruning tree at this node on predictive accuracy, e.g. on validation set
• Prune node that gives greatest improvement
• Continue until no improvements

Constitutes a second search in the hypothesis space

Reduced Error Pruning

accuracy

accuracy on training data

accuracy on unseen data

effect of pruning

size of tree

Turning trees into rules
• From a tree a rule set can be derived
• Path from root to leaf in a tree = 1 if-then rule
• Advantage of such rule sets
• may increase comprehensibility
• Disjunctive concept definition
• can be pruned more flexibly
• in 1 rule, 1 single condition can be removed
• vs. tree: when removing a node, the whole subtree is removed
• 1 rule can be removed entirely
Rules from trees: example

Outlook

Sunny

Rainy

Overcast

Humidity

Yes

Wind

Normal

Strong

Weak

High

No

Yes

No

Yes

if Outlook = Sunny and Humidity = High then No

if Outlook = Sunny and Humidity = Normal then Yes

Pruning rules

Possible method:

• convert tree to rules
• prune each rule independently
• remove conditions that do not harm accuracy of rule
• sort rules (e.g., most accurate rule first)
• more on this later
Handling missing values
• What if result of test is unknown for example?
• e.g. because value of attribute unknown
• Some possible solutions, when training:
• guess value: just take most common value (among all examples, among examples in this node / class, …)
• assign example partially to different branches
• e.g. counts for 0.7 in yes subtree, 0.3 in no subtree
• When using tree for prediction:
• assign example partially to different branches
• combine predictions of different branches
High Branching Factors
• Attributes with continuous domains (numbers)
• cannot different branch for each possible outcome
• allow, e.g., binary test of the form Temperature < 20
• sameevaluation as before, but need to generate value (e.g. 20)
• For instance, just try all reasonable values
• Attributes with many discrete values
• unfair advantage over attributes with few values

• To compensate: divide gain by “max. potential gain” SI

Gain Ratio: GR(S,A) = Gain(S,A) / SI(S,A)

• Split-information SI(S,A) = -  |Si|/|S| log2 |Si|/|S|

with i ranging over different results of test A

Generic TDIDT algorithm
• Many different algorithms for top-down induction of decision trees exist
• What do they have in common, and where do they differ?
• We look at a generic algorithm
• General framework for TDIDT algorithms
• Several “parameter procedures”
• instantiating them yields a specific algorithm
• Summarizes previously discussed points and puts them into perspective
Generic TDIDT algorithm

function TDIDT(E: set of examples) returns tree;

T' := grow_tree(E);

T := prune(T');

returnT;

function grow_tree(E: set of examples) returns tree;

T := generate_tests(E);

t := best_test(T, E);

P := partition induced on E by t;

ifstop_criterion(E, P)

thenreturn leaf(info(E))

else

for allEjin P: tj := grow_tree(Ej);

return node(t, {(j,tj)};

For classification...
• prune: e.g. reduced-error pruning, ...
• generate_tests : Attr=val, Attr
• for numeric attributes: generate val
• best_test : Gain, Gainratio, ...
• stop_criterion : MDL, significance test (e.g. 2-test), ...
• info : most frequent class ("mode")

Popular systems: C4.5 (Quinlan 1993), C5.0

For regression...
• change
• best_test: e.g. minimize average variance
• info: mean
• stop_criterion: significance test (e.g., F-test), ...

{1,3,4,7,8,12}

{1,3,4,7,8,12}

A1

A2

{1,4,12}

{3,7,8}

{1,3,7}

{4,8,12}

Model trees
• Make predictions using linear regression models in the leaves
• info: regression model (y=ax1+bx2+c)
• best_test: ?
• variance: simple, not so good (M5 approach)
• residual variance after model construction: better, computationally expensive (RETIS approach)
• stop_criterion: significant reduction of variance

A

Summary
• Decision trees are a practical method for concept learning
• TDIDT = greedy search through complete hypothesis space
• search based bias only
• Overfitting is an important issue
• Large number of extensions of basic algorithm exist that handle overfitting, missing values, numerical values, etc.
Induction of Rule Sets
• What are decision rules?
• Induction of predictive rules
• Sequential covering approaches
• Learn-one-rule procedure
• Pruning
Decision Rules

Another popular representation for concept definitions:

if-then-rules

IF THEN belongs to concept

• Can be more compact and easier to interpret than trees

How can we learn such rules ?

• By learning trees and converting them to rules
• With specific rule-learning methods (“sequential covering”)
Decision Boundaries

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if A and B then pos

if C and D then pos

Sequential Covering Approaches
• Or: “separate-and-conquer” approach
• Versus trees: “divide-and-conquer”
• General principle: learn a rule set one rule at a time
• Learn one rule that has

High accuracy

• When it predicts something, it should be correct

Any coverage

• Does not make a prediction for all examples, just for some of them
• Mark covered examples

These have been taken care of; now focus on the rest

• Repeat this until all examples covered
Sequential Covering

function LearnRuleSet(Target, Attrs, Examples, Threshold):

LearnedRules := 

Rule := LearnOneRule(Target, Attrs, Examples)

while performance(Rule,Examples) > Threshold, do

LearnedRules := LearnedRules  {Rule}

Examples := Examples \ {examples classified correctly by Rule}

Rule := LearnOneRule(Target, Attrs, Examples)

sort LearnedRules according to performance

return LearnedRules

Learning One Rule

To learn one rule:

• Perform greedy search
• Could be top-down or bottom-up
• Top-down:
• Start with maximally general rule (has maximal coverage but low accuracy)
• Add literals one by one
• Gradually maximize accuracy without sacrificing coverage (using some heuristic)
• Bottom-up:
• Start with maximally specific rule (has minimal coverage but maximal accuracy)
• Remove literals one by one
• Gradually maximize coverage without sacrificing accuracy (using some heuristic)
Learning One Rule

function LearnOneRule(Target, Attrs, Examples):

NewRule := “IF true THEN pos”

NewRuleNeg := Neg

while NewRuleNeg not empty, do

// add a new literal to the rule

Candidates := generate candidate literals

BestLit := argmaxLCandidates performance(Specialise(NewRule,L))

NewRule := Specialise(NewRule, BestLit)

NewRuleNeg := {xNeg | x covered by NewRule}

return NewRule

function Specialise(Rule, Lit):

let Rule = “IF conditions THEN pos”

return “IF conditions and Lit THEN pos”

IF true THEN pos

IF A THEN pos

IF A & B THEN pos

Illustration

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IF C & D THEN posIllustration

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IF A & B THEN pos

IF C THEN pos

IF true THEN pos

Bottom-up vs. Top-down

Bottom-up: typically more specific rules

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Top-down: typically more general rules

Heuristics
• Heuristics
• When is a rule “good”?
• High accuracy
• Somewhat less important: high coverage
• Possible evaluation functions:
• Accuracy: p / (p+n) (p=#positives, n=#negatives)
• A variant of accuracy: m-estimate: (p+mq) / (p+n+m)
• Weighted mean between accuracy on covered set of examples and a priori estimate of true accuracy q (m is weight)
• Entropy: more symmetry between pos and neg
Example-driven Top-down RuleInduction
• Example: AQ algorithms (Michalski et al.)
• for a given class C:
• as long as there are uncovered examples for C
• pick one such example e
• consider He = {rules that cover this example}
• search top-down in He to find best rule
• Much more efficient search
• Hypothesis spaces He much smaller than H (set of all rules)
• Less robust with respect to noise
• what if noisy example picked?
• some restarts may be necessary
Illustration: not example-driven

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Looking for a good rule in the format “IF A=... THEN pos”

Illustration: not example-driven

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If A=b then posIllustration: example-driven

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Try only rules that cover the seed “+” which has A=b.

Hence, A=b is a reasonable test, A=a is not.

We do not try all 4 alternatives in this case! Just one.

How to Arrange the Rules
• According to the order they have been learned.
• According to their accuracy.
• Unordered: devise a strategy how to apply the rules
• E.g., an instance covered by conflicting rules use the rule with higher training accuracy; if an instance is not covered by any rule, then it is assigned the majority class
Approaches to Avoiding Overfitting
• Pre-pruning: stop learning the decision rules before they reach the point where they perfectly classify the training data
• Post-pruning: allow the decision rules to overfit the training data, and then post-prune the rules.
Post-Pruning
• Split instances into Growing Set and Pruning Set;
• Learn set SR of rules using Growing Set;
• Find the best simplification BSR of SR.
• while (Accuracy(BSR, Pruning Set) >
• Accuracy(SR, Pruning Set) ) do
• 4.1 SR = BSR;
• 4.2 Find the best simplification BSR of SR.
• 5. returnBSR;
Incremental Reduced Error Pruning

Post-pruning

D1

D1

D21

D3

D2

D22

D3

Incremental Reduced Error Pruning
• Split Training Set into Growing Set and Validation Set;
• Learn rule R using Growing Set;
• Prune the rule R using Validation Set;
• if performance(R, Training Set) > Threshold
• 4.1 Add R to Set of Learned Rules
• 4.2 Remove in Training Set the instances covered by R;
• 4.2 go to 1;
• 5. else return Set of Learned Rules
Summary Points
• Decision rules are easier for human comprehension than decision trees.
• Decision rules have simpler decision boundaries than decision trees.
• Decision rules are learned by sequential covering of the training instances.