# ID3 Algorithm Decision Tree – Solved Example

## Problem Definition:

Build a decision tree using ID3 algorithm for the given training data in the table (Buy Computer data), and predict the class of the following new example: age<=30, income=medium, student=yes, credit-rating=fair

### Solution:

First, check which attribute provides the highest Information Gain in order to split the training set based on that attribute. We need to calculate the expected information to classify the set and the entropy of each attribute.

### Video Tutorial ID3 Algorithm Decision Tree – Solved Example

The information gain is this mutual information minus the entropy:

The mutual information of the two classes,

Entropy(S)= E(9,5)= -9/14 log2(9/14) – 5/14 log2(5/14)=0.94

Now Consider the Age attribute

For Age, we have three values age<=30 (2 yes and 3 no), age31..40 (4 yes and 0 no), and age>40 (3 yes and 2 no)

Entropy(age) = 5/14 (-2/5 log2(2/5)-3/5log2(3/5)) + 4/14 (0) + 5/14 (-3/5log2(3/5)-2/5log2(2/5))

= 5/14(0.9709) + 0 + 5/14(0.9709) = 0.6935

Gain(age) = 0.94 – 0.6935 = 0.2465

Next, consider Income Attribute

For Income, we have three values incomehigh (2 yes and 2 no), incomemedium (4 yes and 2 no), and incomelow (3 yes 1 no)

Entropy(income) = 4/14(-2/4log2(2/4)-2/4log2(2/4)) + 6/14 (-4/6log2(4/6)-2/6log2(2/6)) + 4/14 (-3/4log2(3/4)-1/4log2(1/4))

= 4/14 (1) + 6/14 (0.918) + 4/14 (0.811)

= 0.285714 + 0.393428 + 0.231714 = 0.9108

Gain(income) = 0.94 – 0.9108 = 0.0292

Next, consider Student Attribute

For Student, we have two values studentyes (6 yes and 1 no) and studentno (3 yes 4 no)

Entropy(student) = 7/14(-6/7log2(6/7)-1/7log2(1/7)) + 7/14(-3/7log2(3/7)-4/7log2(4/7)

= 7/14(0.5916) + 7/14(0.9852)

= 0.2958 + 0.4926 = 0.7884

Gain (student) = 0.94 – 0.7884 = 0.1516

Finally, consider Credit_Rating Attribute

For Credit_Rating we have two values credit_ratingfair (6 yes and 2 no) and credit_ratingexcellent (3 yes 3 no)

Entropy(credit_rating) = 8/14(-6/8log2(6/8)-2/8log2(2/8)) + 6/14(-3/6log2(3/6)-3/6log2(3/6))

= 8/14(0.8112) + 6/14(1)

= 0.4635 + 0.4285 = 0.8920

Gain(credit_rating) = 0.94 – 0.8920 = 0.479

Since Age has the highest Information Gain we start splitting the dataset using the age attribute.

Since all records under the branch age31..40 are all of the class, Yes, we can replace the leaf with Class=Yes

### Now build the decision tree for the left subtree

The same process of splitting has to happen for the two remaining branches.

For branch age<=30 we still have attributes income, student, and credit_rating. Which one should be used to split the partition?

The mutual information is E(Sage<=30)= E(2,3)= -2/5 log2(2/5) – 3/5 log2(3/5)=0.97

For Income, we have three values incomehigh (0 yes and 2 no), incomemedium (1 yes and 1 no) and incomelow (1 yes and 0 no)

Entropy(income) = 2/5(0) + 2/5 (-1/2log2(1/2)-1/2log2(1/2)) + 1/5 (0) = 2/5 (1) = 0.4

Gain(income) = 0.97 – 0.4 = 0.57

For Student, we have two values studentyes (2 yes and 0 no) and studentno (0 yes 3 no)

Entropy(student) = 2/5(0) + 3/5(0) = 0

Gain (student) = 0.97 – 0 = 0.97

We can then safely split on attribute student without checking the other attributes since the information gain is maximized.

Since these two new branches are from distinct classes, we make them into leaf nodes with their respective class as label:

### Now build the decision tree for right left subtree

The mutual information is Entropy(Sage>40)= I(3,2)= -3/5 log2(3/5) – 2/5 log2(2/5)=0.97

For Income, we have two values incomemedium (2 yes and 1 no) and incomelow (1 yes and 1 no)

Entropy(income) = 3/5(-2/3log2(2/3)-1/3log2(1/3)) + 2/5 (-1/2log2(1/2)-1/2log2(1/2))

= 3/5(0.9182)+2/5 (1) = 0.55+0. 4= 0.95

Gain(income) = 0.97 – 0.95 = 0.02

For Student, we have two values studentyes (2 yes and 1 no) and studentno (1 yes and 1 no)

Entropy(student) = 3/5(-2/3log2(2/3)-1/3log2(1/3)) + 2/5(-1/2log2(1/2)-1/2log2(1/2)) = 0.95

Gain (student) = 0.97 – 0.95 = 0.02

For Credit_Rating, we have two values credit_ratingfair (3 yes and 0 no) and credit_ratingexcellent (0 yes and 2 no)

Entropy(credit_rating) = 0

Gain(credit_rating) = 0.97 – 0 = 0.97

We then split based on credit_rating. These splits give partitions each with records from the same class. We just need to make these into leaf nodes with their class label attached:

New example: age<=30, income=medium, student=yes, credit-rating=fair

Follow branch(age<=30) then student=yes we predict Class=yes