Decision Trees - RITamt/eXtremeT/decisiontreeGeneral.pdf · Hypothesis Space Search in ID3 [email protected] Dr. Ankur M. Teredesai P17 The hypothesis space is the set of all decision

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Slide 1

Decision Trees

AMT

OverviewDecision treesAppropriate problems for decision treesEntropy and Information GainThe ID3 algorithmAvoiding Overfitting via PruningHandling Continuous-Valued AttributesHandling Missing Attribute ValuesAlternative Measures for Selecting Attributes

[email protected] Dr. Ankur M. Teredesai P2

Time to look at the classification model

The Decision tree works a lot like playing twenty questions

The tree on the right decides if its possible to go play tennis outdoors

Eg. Its overcast, but its reasonably warm (55F). Answer: Go out and play!!

+

Outlook

Temp

Sunny Overcast

< 70F< 35F

- +



Outlook

sunny overcast rainy

Humidity Windy

high normal

no

false true

yes

yes yes no

Decision TreesDefinition: A decision tree is a tree s.t.:Each internal node tests an attributeEach branch corresponds to attribute valueEach leaf node assigns a classification

Data Set for Playing Tennis


Outlook Temp. Humidity Windy Play

Sunny Hot High False No

Sunny Hot High True No

Overcast Hot High False Yes

Rainy Mild High False Yes

Rainy Cool Normal False Yes

Rainy Cool Normal True No

Overcast Cool Normal True Yes

Outlook Temp. Humidity Windy play

Sunny Mild High False No

Sunny Cool Normal False Yes

Rainy Mild Normal False Yes

Sunny Mild Normal True Yes

Overcast Mild High True Yes

Overcast Hot Normal False Yes

Rainy Mild High True No

Decision Tree For Playing Tennis

Outlook


Humidity Windy

high normal

no

false true

yes

yes yes no


When to Consider Decision TreesEach instance consists of an attribute with discrete values (e.g. outlook/sunny, etc..)The classification is over discrete values (e.g. yes/no )It is okay to have disjunctive descriptions – each path in the tree represents a disjunction of attribute combinations. Any Boolean function can be represented!It is okay for the training data to contain errors – decision trees are robust to classification errors in the training data.It is okay for the training data to contain missing values – decision trees can be used even if instances have missing attributes.


Decision Tree InductionBasic Algorithm:1. A ← the “best" decision attribute for a node N.2. Assign A as decision attribute for the node N.3. For each value of A, create new descendant of the node N.4. Sort training examples to leaf nodes.5. IF training examples perfectly classified, THEN STOP.

ELSE iterate over new leaf nodes


How do we pick the splitting attribute?

Determine the attribute that contributes the most information, for example, If we knew the Outlook was Sunny, its more likely that we would go out and play than just knowing its not humid outside!

The measure we need is termed as the Information Gain for the attribute

Once we know the splitting attribute, we branch the the tree in the direction of all the unique values for that attribute. For example, for 3 unique values, a 3 way branch is necessary

?

? ?


Decision Tree Induction


_____________________________________Outlook Temp Hum Wind Play ---------------------------------------------------------Rain Mild High Weak YesRain Cool Normal Weak YesRain Cool Normal Strong NoRain Mild Normal Weak YesRain Mild High Strong No

Outlook

____________________________________Outlook Temp Hum Wind Play -------------------------------------------------------Sunny Hot High Weak NoSunny Hot High Strong NoSunny Mild High Weak NoSunny Cool Normal Weak YesSunny Mild Normal Strong Yes

_____________________________________Outlook Temp Hum Wind Play ---------------------------------------------------------Overcast Hot High Weak YesOvercast Cool Normal Strong Yes

SunnyOvercast

Rain

EntropyLet S be a sample of training examples, andp+ is the proportion of positive examples in S andp- is the proportion of negative examples in S. Then: entropy measures the impurity of S: E( S) = - p+ log2 p+ – p- log2 p-


Entropy

The Entropy for an attribute a, given a data set, s is:

E(a) = Σv(((|sk1|+|sk2|+…|skn|) / |s|) * (Ik(a)); for all ci

Ik(a) is the expected information to classify a sample, for ak|ski| is the number of training samples in s, for ci & ak

For attrib. a, there are v distinct values {a1,a2…ak,…av}

Entropy is a measure of how much we know about a particular class

• The more we know, lower the entropy


Entropy Example from the DatasetIn the Play Tennis dataset we had two target classes: yes and noOut of 14 instances, 9 classified yes, rest no

2

2

9 9log 0.4114 14

5 5log 0.5314 14

( ) 0.94

yes

no

yes no

p

p

E S p p

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=− =

=− =

= + =


Outlook Temp. Humidity Windy Play

Sunny Hot High False No

Sunny Hot High True No

Overcast Hot High False Yes

Rainy Mild High False Yes

Rainy Cool Normal False Yes

Rainy Cool Normal True No

Overcast Cool Normal True Yes

Outlook Temp. Humidity Windy play

Sunny Mild High False No

Sunny Cool Normal False Yes

Rainy Mild Normal False Yes

Sunny Mild Normal True Yes

Overcast Mild High True Yes

Overcast Hot Normal False Yes

Rainy Mild High True No

Information GainInformation Gain is the expected reduction in entropy caused by partitioning the instances according

to a given attribute.

Gain(S, A) = E(S) -

where SV = { s ∈ S | A(s) = V}

)(||||

)(v

AValuesv

v SESS∑

∈


Example

_____________________________________Outlook Temp Hum Wind Play ---------------------------------------------------------Rain Mild High Weak YesRain Cool Normal Weak YesRain Cool Normal Strong NoRain Mild Normal Weak YesRain Mild High Strong No

Outlook

____________________________________Outlook Temp Hum Wind Play -------------------------------------------------------Sunny Hot High Weak NoSunny Hot High Strong NoSunny Mild High Weak NoSunny Cool Normal Weak YesSunny Mild Normal Strong Yes

_____________________________________Outlook Temp Hum Wind Play ---------------------------------------------------------Overcast Hot High Weak YesOvercast Cool Normal Strong Yes

SunnyOvercast

Rain


Which attribute should be tested here?

Gain (Ssunny , Humidity) = = .970 - (3/5) 0.0 - (2/5) 0.0 = .970

Gain (Ssunny , Temperature) = .970 - (2/5) 0.0 - (2/5) 1.0 - (1/5) 0.0 = .570

Gain (Ssunny , Wind) = .970 - (2/5) 1.0 - (3/5) .918 = .019

ID3 AlgorithmInformally:

• Determine the attribute with the highest information gain on the training set.

• Use this attribute as the root, create a branch for each of the values the attribute can have.

• For each branch, repeat the process with subset of the training set that is classified by that branch.


Hypothesis Space Search in ID3


The hypothesis space is the set of all decision trees defined over the given set of attributes. ID3’s hypothesis space is a compete space; i.e., the target description is there!ID3 performs a simple-to-complex, hill climbing search through this space.


Hypothesis Space Search in ID3The evaluation function is the information gain.ID3 maintains only a single current decision tree.ID3 performs no backtracking in its search.ID3 uses all training instances at each step of the search.

Inductive Bias in ID3


Preference for short treesPreference for trees with high information gain attributes near the root.Bias is a preference to some hypotheses, not a restriction on the hypothesis space


OverfittingDefinition: Given a hypothesis space H, a hypothesis h ∈ H is said to overfit

the training data if there exists some hypothesis h’ ∈ H, such that h has smaller error than h’ over the training instances, but h’ has a smaller error than h over the entire distribution of instances.


Reasons for Reasons for OverfittingOverfitting


• Noisy training instances. Consider an noisy training example: Outlook = Sunny;Temperature = Hot; Humidity = Normal; Wind = Strong; PlayTennis = No

Outlook


Humidity Windy

high normal

no

false true

yes

yes yes no

add new test

Reasons for Overfitting

• Small number of instances are associated with leaf nodes. In this case it is possible that for coincidental regularities to occur that are unrelated to the actual target concept.

-++ + -+


area with probablywrong predictions

--

++ + --

-

- +-- -

- - - -- --

-

Approaches to Avoiding OverfittingPre-pruning: stop growing the tree earlier, before it reaches the point where it perfectly classifies the training data

Post-pruning: Allow the tree to overfit the data, and then post-prune the tree.


Criteria for PruningUse a separate set of instances, distinct from the training instances, to evaluate the utility of nodes in the tree. This requires the data to be split into a training set and a validation set which is then used for pruning. The reason is that the validation set is unlikely to suffer from same errors or fluctuations as the training set. Use all the available data for training, but apply a statistical test to estimate whether expanding/pruning a particular node is likely to produce improvement beyond the training set.


Reduced-Error PruningSplit data into training and validation sets.

Pruning a decision node d consists of:removing the subtree rooted at d.making d a leaf node. assigning d the most common classification of the training instances associated with d.

Do until further pruning is harmful:Evaluate impact on validation set of pruning each possible node (plus those below it).Greedily remove the one that most improves validation set accuracy.

Outlook


Humidity Windy

high normal

no

false true

yes

yes yes no


Reduced Error Pruning Example


Rule Post-PruningConvert tree to equivalent set of rules.Prune each rule independently of others.Sort final rules by their estimated accuracy, and consider them in this sequence when classifying subsequent instances.

IF (Outlook = Sunny) & (Humidity = High)THEN PlayTennis = NoIF (Outlook = Sunny) & (Humidity = Normal)THEN PlayTennis = Yes……….

Outlook


Humidity Windy

normal

no

false true

yes

yes yes no


Continuous Valued Attributes

Create a set of discrete attributes to test continuous.Apply Information Gain in order to choose the best attribute.

Temperature: 40 48 60 72 80 90PlayTennis: No No Yes Yes Yes No

Temp>54 Tem>85


An Alternative Measure for Attribute Selection

),(),(),(

ASmationSplitInforASGainASGainRatio =

where:

||||log

||||),(

12 S

SSSASmatioSplitInfro i

c

i

i∑=

−=


Missing Attribute ValuesStrategies:

Assign most common value of A among other examples belonging to the same concept.If node n tests the attribute A, assign most common value of A among other examples sorted to node n.If node n tests the attribute A, assign a probability to each of possible values of A. These probabilities are estimated based on the observed frequencies of the values of A. These probabilities are used in the information gain measure.


Summary PointsDecision tree learning provides a practical method for concept learning. ID3-like algorithms search complete hypothesis space.The inductive bias of decision trees is preference (search) bias.Overfitting the training data is an important issue in decision tree learning.A large number of extensions of the ID3 algorithm have been proposed for overfitting avoidance, handling missing attributes, handling numerical attributes, etc.


ReferencesMitchell, Tom. M. 1997. Machine Learning. New York: McGraw-HillQuinlan, J. R. 1986. Induction of decision trees. Machine LearningStuart Russell, Peter Norvig, 1995. Artificial Intelligence: A Modern Approach. New Jersey: Prantice Hall


[email protected] Proprietary and Confidential

NOTICE: Proprietary and Confidential

This material is proprietary to A. Teredesai and GCCIS, RIT.

Slide 34

RainForestRainForest - A Framework for Fast Decision Tree Construction of Large Datasets

Paper By: J. Gehrke, R. Ramakrishnan, V. GantiDept. of Computer Sciences University of Wisconsin-Madison

Introduction to ClassificationAn important Data Mining ProblemInput: a database of training records

– Class label attributes– Predictor Attributes

Goal

• to build a concise model of the distribution of class label in terms of predictor attributes

Applications

• scientific experiments,medical diagnosis, fraud detection, etc.


Decision Tree:A Classification Model

It is one of the most attractive classification modelsThere are a large number of algorithms to construct decision trees

• E.g.: SLIQ, CART, C4.5 SPRINT

• Most are main memory algorithms

• Tradeoff between supporting large databases, performance and constructing more accurate decision trees


Motivation of RainForest

Developing a unifying framework that can be applied to most decision tree algorithms, and results in a scalable version of this algorithm without modifying the results.Separating the scalability aspects of these algorithms from the central features that determine the quality of the decision trees


Decision Tree Terms

Root,Leaf, Internal NodesEach leaf is labeled with one class labelEach internal node is labeled with one predictor attribute called the splitting attributeEach edge e from node n has a predicate q associated with it, q only involves splitting attributes.P : set of predicates on all outgoing edges of an internal node; Non-overlapping, ExhaustiveCrit(n): splitting criteria of n; combination of splitting attributes and predicates


Decision Tree Terms(Cont’d)

F(n) :Family of database(D) tuples of a node nDefinition: let E={e1,e2,…,ek}, Q={q1,q2,…,qk} be the edge set and predicate set for

a node n; p be the parent node of n If n=root, F(n) = DIf n≠root, let q(p→n) be the predicate on e(p→n),

F(n) = {t: t€F(n),t €F(p), and q(p→ n= True}


Decision Tree Terms (Cont’d)P { q1, q2, … , qk }

n

ek { qk}e2

{ q2

}e1 { q1 }


RainForest Framework:Top-down Tree Induction Schema


Input: node n, partition D, classification algorithm CL Output: decision tree for D rooted at n Top-Down Decision Tree Induction Schema: BuildTree(Node n, datapartition D, algorithm CL) (1) Apply CL to D to find crit(n)(2) let k be the number of children of n(3) if (k > 0) (4) Create k children c1 ; ... ; ck of n (5) Use best split to partition D into D1 ; . . . ; Dk(6) for (i = 1; i <= k; i++) (7) BuildTree(ci , Di ) (8) endfor(9) endifRainForest Refinement: (1a) for each predictor attribute p (1b) Call CL.find_best_partitioning(AVC-set of p)(1c) endfor

(2a) k = CL:decide_splitting_criterion();

RainForest:Tree Induction Schema (Cont’d)

AVC stands for Attribute-Value, ClasslabelAVC-set: AVC-set of a predictor attribute a to be the projection of F(n) onto a and the class label whereby counts of the individual class labels are aggregatedAVC-group: the AVC-group of a node n to be the set of all AVC-sets at node n. Size of the AVC-set of a predictor attribute a at node n

• depends only on the number of distinct attribute values of a and the number of class labels in F(n).

AVC-group(r) is not equal to F( r ) : contains aggregated information that is sufficient for decision tree construction


AVC-groups and Main Memory

The memory size determines the implementation of RainForest Schema.Case 1: AVC-group of the root node fits in the M.M.

• RF-Write; RF-Read; RF-HybridCase 2: each individual AVC-set of the root node fits into M.M., but the AVC-group does not.

• RF-VerticalCase 3: Other than Case 1&2


Steps for Algorithms in RainForest Family1. AVC-group Construction2. Choose Splitting Attribute and Predicate

• This step uses the decision tree algorithm CL that is being scaled using the RainForest framework

3. Partition D Across the Children Nodes

• We must read the entire dataset and write out all records, partitioning them into child ``buckets'' according to the splitting criterion chosen in the previous step.


Algorithms: RF-Write/RF-ReadPrerequisite:AVC-group fits into M.M.RF-Write:

• For each level of the tree,it reads the entire database twice and writes the entire database once

RF-Read

• Makes an increasing number of scans of entire database

• Marks one end of the design spectrum in the RainForest framework


Algorithm:RF-HybridCombination of RF-Write and RF-ReadPerformance can be improved by concurrent construction of AVC-sets


Algorithm:RF-VerticalPrerequisite: individual AVC-set can fit into M.M.For very large sets, a temporary file is generated for each node, the large sets are constructed from this temporary file.For small sets, construct them in M.M.


Experiments: Datasets


Experiment Results: (1)When the overall maximum number of entries in the AVC-group of the root node is about 2.1 million, requiring a maximum memory size of 17MB.


Experiment Results (2)

The performance of RF-Write, RF-Read and RF-Hybrid as the input database increases:



How internal properties of the AVC-groups of the training database influence performance?Result: the AVC-group size and Main Memory size are the two factors which determine the performance.



How performance is affected as the number of attributes is increasing?Result: a roughly linear scaleup with the number of attributes.


ConclusionA scaling decision tree algorithm that is applicable to all decision tree algorithms at that time.AVC-group is the key idea.

Database scan at each level of the decision treeToo much dependence over the size of available main memory



Common Node Decision Trees

Plays PS2 ?

Watched ET ?

Yes

Plays PS2 ?

Watched ET ? Enjoys Sci Fi ?

NoNo

Yes No

Yes YesNo No

No

Yes

No

Star Trek Fan?

Yes

No Yes

No

Yes No

Yes

No

Plays PS2 ?

Watched ET ?

Plays PS2 ?

Watched ET ?

Proof in collaboration Gautam Das @ microsoft research.

Documents

Decision Trees - RITamt/eXtremeT/decisiontreeGeneral.pdf · Hypothesis Space Search in ID3 [email protected] Dr. Ankur M. Teredesai P17 The hypothesis space is the set of all decision