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Classification with Decision Trees I

Instructor: Qiang YangHong Kong University of Science and Technology

Qyang@cs.ust.hk

Thanks: Eibe Frank and Jiawei Han

Given a set of pre-classified examples, build a model or classifier to classify new cases.

Supervised learning in that classes are known for the examples used to build the classifier.

A classifier can be a set of rules, a decision tree, a neural network, etc.

Typical Applications: credit approval, target marketing, fraud detection, medical diagnosis, treatment effectiveness analysis, …..

INTRODUCTION

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Constructing a Classifier The goal is to maximize the accuracy on new

cases that have similar class distribution. Since new cases are not available at the time

of construction, the given examples are divided into the testing set and the training set. The classifier is built using the training set and is evaluated using the testing set.

The goal is to be accurate on the testing set. It is essential to capture the “structure” shared by both sets.

Must prune overfitting rules that work well on the training set, but poorly on the testing set.

Example

TrainingData

NAME RANK YEARS TENUREDMike Assistant Prof 3 noMary Assistant Prof 7 yesBill Professor 2 yesJim Associate Prof 7 yesDave Assistant Prof 6 noAnne Associate Prof 3 no

ClassificationAlgorithms

IF rank = ‘professor’OR years > 6THEN tenured = ‘yes’

Classifier(Model)

Example (Conted)

Classifier

TestingData

NAME RANK YEARS TENUREDTom Assistant Prof 2 noMerlisa Associate Prof 7 noGeorge Professor 5 yesJoseph Assistant Prof 7 yes

Unseen Data

(Jeff, Professor, 4)

Tenured?

Evaluation Criteria Accuracy on test set

the rate of correct classification on the testing set. E.g., if 90 are classified correctly out of the 100 testing cases, accuracy is 90%.

Error Rate on test set The percentage of wrong

predictions on test set Confusion Matrix

For binary class values, “yes” and “no”, a matrix showing true positive, true negative, false positive and false negative rates

Speed and scalability the time to build the classifier

and to classify new cases, and the scalability with respect to the data size.

Robustness: handling noise and missing values

Predicted class

Yes No

Actual class

Yes

True positive

False negative

No False positive

True negative

Evaluation Techniques

Holdout: the training set/testing set. Good for a large set of data.

k-fold Cross-validation: divide the data set into k sub-samples. In each run, use one distinct sub-sample as testing

set and the remaining k-1 sub-samples as training set.

Evaluate the method using the average of the k runs.

This method reduces the randomness of training set/testing set.

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Cross Validation: Holdout Method

— Break up data into groups of the same size — —

— Hold aside one group for testing and use the rest to build model

—

— Repeat

Testiteration

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Continuous Classes

Sometimes, classes are continuous in that they come from a continuous domain,

e.g., temperature or stock price. Regression is well suited in this case:

Linear and multiple regression Non-Linear regression

We shall focus on categorical classes, e.g., colors or Yes/No binary decisions.

We will deal with continuous class values later in CART

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DECISION TREE [Quinlan93]

An internal node represents a test on an attribute.

A branch represents an outcome of the test, e.g., Color=red.

A leaf node represents a class label or class label distribution.

At each node, one attribute is chosen to split training examples into distinct classes as much as possible

A new case is classified by following a matching path to a leaf node.

Outlook Tempreature Humidity Windy Classsunny hot high false Nsunny hot high true Novercast hot high false Prain mild high false Prain cool normal false Prain cool normal true Novercast cool normal true Psunny mild high false Nsunny cool normal false Prain mild normal false Psunny mild normal true Povercast mild high true Povercast hot normal false Prain mild high true N

Training Set

Outlook

overcast

humidity windy

high normal falsetrue

sunny rain

N NP P

P

overcast

Example

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Building Decision Tree [Q93]

Top-down tree construction At start, all training examples are at the root. Partition the examples recursively by choosing one

attribute each time. Bottom-up tree pruning

Remove subtrees or branches, in a bottom-up manner, to improve the estimated accuracy on new cases.

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Choosing the Splitting Attribute

At each node, available attributes are evaluated on the basis of separating the classes of the training examples. A Goodness function is used for this purpose.

Typical goodness functions: information gain (ID3/C4.5) information gain ratio gini index

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Which attribute to select?

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A criterion for attribute selection

Which is the best attribute? The one which will result in the smallest tree Heuristic: choose the attribute that produces the

“purest” nodes Popular impurity criterion: information gain

Information gain increases with the average purity of the subsets that an attribute produces

Strategy: choose attribute that results in greatest information gain

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Computing information

Information is measured in bits Given a probability distribution, the info required to

predict an event is the distribution’s entropy Entropy gives the information required in bits (this

can involve fractions of bits!) Formula for computing the entropy:

nnn ppppppppp logloglog),,,entropy( 221121

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Example: attribute “Outlook”

“ Outlook” = “Sunny”:

“Outlook” = “Overcast”:

“Outlook” = “Rainy”:

Expected information for attribute:

bits 971.0)5/3log(5/3)5/2log(5/25,3/5)entropy(2/)info([2,3]

bits 0)0log(0)1log(10)entropy(1,)info([4,0]

bits 971.0)5/2log(5/2)5/3log(5/35,2/5)entropy(3/)info([3,2]

Note: this isnormally notdefined.

971.0)14/5(0)14/4(971.0)14/5([3,2])[4,0],,info([3,2] bits 693.0

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Computing the information gain

Information gain: information before splitting – information after splitting

Information gain for attributes from weather data:

0.693-0.940[3,2])[4,0],,info([2,3]-)info([9,5])Outlook"gain("

bits 247.0

bits 247.0)Outlook"gain(" bits 029.0)e"Temperaturgain("

bits 152.0)Humidity"gain(" bits 048.0)Windy"gain("

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Continuing to split

bits 571.0)e"Temperaturgain(" bits 971.0)Humidity"gain("

bits 020.0)Windy"gain("

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The final decision tree

Note: not all leaves need to be pure; sometimes identical instances have different classes Splitting stops when data can’t be split any further

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Highly-branching attributes

Problematic: attributes with a large number of values (extreme case: ID code)

Subsets are more likely to be pure if there is a large number of values

Information gain is biased towards choosing attributes with a large number of values

This may result in overfitting (selection of an attribute that is non-optimal for prediction)

Another problem: fragmentation

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The gain ratio

Gain ratio: a modification of the information gain that reduces its bias on high-branch attributes

Gain ratio takes number and size of branches into account when choosing an attribute

It corrects the information gain by taking the intrinsic information of a split into account

Also called split ratio Intrinsic information: entropy of distribution of

instances into branches (i.e. how much info do we need to tell which branch

an instance belongs to)

.||

||2

log||||

),(SiS

SiSASnfoIntrinsicI

.),(

),(),(ASnfoIntrinsicI

ASGainASGainRatio

Gain Ratio

Gain ratio should be Large when data is evenly spread Small when all data belong to one branch

Gain ratio (Quinlan’86) normalizes info gain by this reduction:

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Computing the gain ratio

Example: intrinsic information for ID code

Importance of attribute decreases as intrinsic information gets larger

Example of gain ratio:

Example:

bits 807.3)14/1log14/1(14),1[1,1,(info

)Attribute"info("intrinsic_)Attribute"gain("

)Attribute"("gain_ratio

246.0bits 3.807bits 0.940

)ID_code"("gain_ratio

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Gain ratios for weather data

Outlook Temperature

Info: 0.693 Info: 0.911

Gain: 0.940-0.693 0.247 Gain: 0.940-0.911 0.029

Split info: info([5,4,5]) 1.577 Split info: info([4,6,4]) 1.362

Gain ratio: 0.247/1.577 0.156 Gain ratio: 0.029/1.362 0.021

Humidity Windy

Info: 0.788 Info: 0.892

Gain: 0.940-0.788 0.152 Gain: 0.940-0.892 0.048

Split info: info([7,7]) 1.000 Split info: info([8,6]) 0.985

Gain ratio: 0.152/1 0.152 Gain ratio: 0.048/0.985 0.049

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More on the gain ratio

“ Outlook” still comes out top However: “ID code” has greater gain ratio

Standard fix: ad hoc test to prevent splitting on that type of attribute

Problem with gain ratio: it may overcompensate May choose an attribute just because its intrinsic

information is very low Standard fix:

First, only consider attributes with greater than average information gain

Then, compare them on gain ratio

n

jjpTgini

1

21)(

splitgini NT

NTT

Ngini

Ngini( ) ( ) ( ) 1

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2

Gini Index If a data set T contains examples from n classes, gini index,

gini(T) is defined as

where pj is the relative frequency of class j in T. gini(T) is

minimized if the classes in T are skewed. After splitting T into two subsets T1 and T2 with sizes N1 and

N2, the gini index of the split data is defined as

The attribute providing smallest ginisplit(T) is chosen to split the node.

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Discussion

Algorithm for top-down induction of decision trees (“ID3”) was developed by Ross Quinlan

Gain ratio just one modification of this basic algorithm

Led to development of C4.5, which can deal with numeric attributes, missing values, and noisy data

Similar approach: CART (linear regression tree, WF book, Chapter 6.5)

There are many other attribute selection criteria! (But almost no difference in accuracy of result.)