1. Decision Tree
R. Akerkar
TMRF, Kolhapur, India
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2. Introduction
 A classification scheme which generates a tree and
g
a set of rules from given data set.
 The t f
Th set of records available f d
d il bl for developing
l i
classification methods is divided into two disjoint
subsets – a training set and a test set.
g
 The attributes of the records are categorise into two
types:
 Attributes hose
Attrib tes whose domain is n merical are called n merical
numerical numerical
attributes.
 Attributes whose domain is not numerical are called the
categorical attributes
attributes.
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3. Introduction
 A decision tree is a tree with the following p p
g properties:
 An inner node represents an attribute.
 An edge represents a test on the attribute of the father
node.
node
 A leaf represents one of the classes.
 Construction of a decision tree
 Based on the training data
 Top Down strategy
Top-Down
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4. Decision Tree
Example
 The data set has five attributes.
 There is a special attribute: the attribute class is the class label.
 The attributes, temp (temperature) and humidity are numerical
attributes
 Other attributes are categorical, that is, they cannot be ordered.
 Based on the training data set, we want to find a set of rules to
know what values of outlook, temperature, humidity and wind,
determine whether or not to play golf.
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5. Decision Tree
Example
 We have five leaf nodes.
 In a decision tree, each leaf node represents a rule.
 We have the following rules corresponding to the tree given in
Figure.
 RULE 1 If it is sunny and the humidity is not above 75% then play
75%, play.
 RULE 2 If it is sunny and the humidity is above 75%, then do not play.
 RULE 3 If it is overcast, then play.
 RULE 4 If it is rainy and not windy, then play.
 RULE 5 If it is rainy and windy, then don't play.
i i d i d th d 't l
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6. Classification
 The classification of an unknown input vector is done by
p y
traversing the tree from the root node to a leaf node.
 A record enters the tree at the root node.
 At the root, a test is applied to determine which child
root
node the record will encounter next.
 This process is repeated until the record arrives at a leaf
node.
d
 All the records that end up at a given leaf of the tree are
classified in the same way. y
 There is a unique path from the root to each leaf.
 The path is a rule which is used to classify the records.
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7.  In our tree, we can carry out the classification
for a u
o an unknown record as follows.
o eco d o o s
 Let us assume, for the record, that we know
the values of the first four attributes (but we
do not know the value of class attribute) as
 outlook= rain; temp = 70; humidity = 65; and
windy true.
windy= true
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8.  We start from the root node to check the value of the attribute
associated at the root node.
 This attribute is the splitting attribute at this node.
 For a decision tree, at every node there is an attribute associated
with the node called the splitting attribute.
 In our example, outlook is the splitting attribute at root.
 Since for the given record, outlook = rain, we move to the right-
most child node of the root.
 At this node, the splitting attribute is windy and we find that for
the record we want classify, windy = true.
 Hence, we move to the left child node to conclude that the class
label Is "no l "
l b l I " play".
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9.  The accuracy of the classifier is determined by the percentage of the
test d t
t t data set that is correctly classified.
t th t i tl l ifi d
 We can see that for Rule 1 there are two records of the test data set
satisfying outlook= sunny and humidity < 75, and only one of these
is correctly classified as play.
 Thus, the accuracy of this rule is 0.5 (or 50%). Similarly, the
accuracy of Rule 2 is also 0.5 (or 50%). The accuracy of Rule 3 is
0.66.
RULE 1
If it is sunny and the humidity
is not above 75%, then play.
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10. Concept of Categorical Attributes
 Consider the following training
data set.
 There are three attributes,
namely, age, pincode and class.
 The attribute class is used for
class label.
The attribute age is a numeric attribute, whereas pincode is a categorical
one.
Though th d
Th h the domain of pincode i numeric, no ordering can b d fi d
i f i d is i d i be defined
among pincode values.
You cannot derive any useful information if one pin-code is greater than
another pincode
pincode.
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11.  Figure gives a decision tree for the
training data
data.
 The splitting attribute at the root is
pincode and the splitting criterion
here is pincode = 500 046.
 Similarly, for the left child node, the
splitting criterion is age < 48 (the
p g g (
splitting attribute is age).
At root level, we have 9 records.
 Although the right child node has The associated splitting criterion is
p g
the same attribute as the splitting pincode = 500 046.
attribute, the splitting criterion is
different. As a result, we split the records
into two subsets. Records 1, 2, 4, 8,
and 9 are to the left child note and
remaining to the right node.
The process is repeated at every
node.
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12. Advantages and Shortcomings of Decision
Tree Classifications
 A decision tree construction process is concerned with identifying
the splitting attributes and splitting criterion at every l
h li i ib d li i i i level of the tree.
l f h
 Major strengths are:
 Decision tree able to generate understandable rules.
 They are able to handle both numerical and categorical attributes.
 They provide clear indication of which fields are most important for
prediction or classification
classification.
 Weaknesses are:
 The process of growing a decision tree is computationally expensive At
expensive.
each node, each candidate splitting field is examined before its best split
can be found.
 Some decision tree can only deal with binary-valued target classes.
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13. Iterative Dichotomizer (ID3)
 Quinlan (1986)
 Each node corresponds to a splitting attribute
 Each arc is a possible value of that attribute.
 At each node the splitting attribute is selected to be the most
informative among the attributes not yet considered in the path from
the root.
 Entropy is used to measure how informative is a node.
 The algorithm uses the criterion of information gain to determine the
goodness of a split.
d f lit
 The attribute with the greatest information gain is taken as
the splitting attribute, and the data set is split for all distinct
values of the attribute
attribute.
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14. Training Dataset
This follows an example from Quinlan’s ID3
The class label attribute,
buys_computer, has two distinct
values. age income student credit_rating buys_computer
<=30 high
g no fair no
Thus there are two distinct <=30 high no excellent no
classes. (m =2) 31…40 high no fair yes
Class C1 corresponds to yes >40 medium no fair yes
and class C2 corresponds to no
no. >40 low yes fair yes
>40 low yes excellent no
There are 9 samples of class yes 31…40 low yes excellent yes
and 5 samples of class no. <=30 medium no fair no
<=30 low yes fair yes
>40
40 medium
di yes f i
fair yes
<=30 medium yes excellent yes
31…40 medium no excellent yes
31…40 high yes fair yes
>40 medium no excellent no
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15. Extracting Classification Rules from Trees
g
 Represent the knowledge in
the form of IF-THEN rules
 One rule is created for each
path from the root to a leaf
 Each attribute-value pair
along a path forms a
conjunction
 The leaf node holds the class
p
prediction
What are the rules?
 Rules are easier for humans
to understand
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16. Solution (Rules)
IF age = “<=30” AND student = “no” THEN buys_computer = “no”
IF age = “<=30” AND student = “yes” THEN buys_computer = “yes”
IF age = “31…40” THEN buys_computer = “yes”
IF age = “>40” AND credit_rating = “excellent” THEN
buys_computer = “yes”
IF age = “<=30” AND credit_rating = “fair” THEN buys_computer =
“no”
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17. Algorithm for Decision Tree Induction
 Basic algorithm (a greedy algorithm)
 Tree is constructed in a top-down recursive divide-and-conquer
manner
 At start, all the training examples are at the root
 Attributes are categorical ( continuous-valued, they are
g (if y
discretized in advance)
 Examples are partitioned recursively based on selected attributes
 Test attributes are selected on the basis of a heuristic or
statistical measure (e.g., information gain)
 Conditions for stopping partitioning
 All samples for a g
p given node belong to the same class
g
 There are no remaining attributes for further partitioning –
majority voting is employed for classifying the leaf
 There are no samples left
p
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18. Attribute Selection Measure: Information
Gain (ID3/C4.5)
(ID3/C4 5)
 Select the attribute with the highest information gain
 S contains si tuples of class Ci for i = {1, …, m}
 information measures info required to classify any
q y y
arbitrary tuple m
si si
I( s1,s 2,...,s m )    log 2

i 1 s s ….information is encoded in bits.
 entropy of attribute A with values {a1,a2,…,av}
f b h l {
v
s1 j  ... smj
E(A)  I ( s1 j ,...,smj )
j 1 s
 information gained by branching on attribute A
Gain(A)  I(s 1, s 2 ,..., sm)  E(A)
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19. Entropy
 Entropy measures the homogeneity (purity) of a set of examples.
 It gives the information content of the set in terms of the class labels of
the examples.
 Consider that you have a set of examples, S with two classes, P and N. Let
the set have p instances for the class P and n instances for the class N.
 So the total number of instances we have is t = p + n. The view [p, n] can
be seen as a class distribution of S.
The entropy for S is defined as
 Entropy(S) = - (p/t).log2(p/t) - (n/t).log2(n/t)
 Example: Let a set of examples consists of 9 instances for class positive,
and 5 instances for class negative.
 Answer: p = 9 and n = 5.
 So Entropy(S) = - (9/14).log2(9/14) - (5/14).log2(5/14)
 = -(0.64286)(-0.6375) - (0.35714)(-1.48557)
 = (0.40982) + (0.53056)
 = 0.940
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20. Entropy
y
The entropy for a completely pure set is 0 and is 1 for a set with
equal occurrences f both th classes.
l for b th the l
i.e. Entropy[14,0] = - (14/14).log2(14/14) - (0/14).log2(0/14)
= -1.log2(1) - 0 l 2(0)
1 l 2(1) 0.log2(0)
= -1.0 - 0
=0
i.e. Entropy[7,7] = - (7/14).log2(7/14) - (7/14).log2(7/14)
= - (0.5).log2(0.5) - (0.5).log2(0.5)
= - (0.5).(-1) - (0.5).(-1)
= 0.5 + 0.5
=1
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21. Attribute Selection by Information Gain
Computation
5 4
 Class P: buys_computer = “yes” E ( age )  I ( 2 ,3)  I ( 4,0 )
14 14
 Class N: buys_computer = “no”
5
 I(p, n) = I(9, 5) =0.940  I (3, 2 )  0 .694
 Compute the entropy for age:
14
age pi ni I(pi, ni) 5
<=30 2 3 0.971 I ( 2,3) means ““age <=30” h 5
30” has
14
30…40 4 0 0 out of 14 samples, with 2 yes's
>40
40 3 2 0.971 and 3 no’s. Hence
no s.
age income student credit_rating buys_computer
<=30
<=30
high
high
no
no
fair
excellent
no
no
Gain ( age )  I ( p , n )  E ( age )  0.246
31…40 high no fair yes
>40
>40
medium
low
no
yes
fair
fair
yes
yes
Similarly Gain(income)  0.029
Similarly,
>40 low yes excellent no Gain( student )  0.151
31…40 low yes excellent yes
<=30 medium no fair no Gain(credit _ rating )  0.048
<=30 low yes fair yes
>40 medium yes fair yes
<=30 medium yes excellent yes Since, age has the highest information gain
31…40 medium no excellent yes among the attributes, it is selected as the
31…40 high yes fair yes
>40 medium no excellent no R. Akerkar
test attribute. 21

22. Exercise 1
 The following table consists of training data from an employee
database.
database
 Let status be the class attribute. Use the ID3 algorithm to construct a
decision tree from the given data.
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23. Solution 1
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24. Other Attribute Selection Measures
 Gini index (CART IBM IntelligentMiner)
(CART,
 All attributes are assumed continuous-valued
 Assume there exist several possible split values for each
attribute
 May need other tools, such as clustering, to get the
possible split values
 Can be modified for categorical attributes
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25. Gini Index (IBM IntelligentMiner)
 If a data set T contains examples from n classes, gini index, gini(T) is
n
defined as gini (T )  1   p 2
i i j
j 1
where pj is the relative frequency of class j in T.
 If a data set T is split into two subsets T1 and T2 with sizes N1 and N2
respectively, the gini index of the split data contains examples from n
classes, the gini index gini(T) is defined as
gini split
(T )  N 1 gini (T 1)  N 2 gini (T 2 )
N N
 The attribute provides the smallest ginisplit(T) is chosen to split the node
(need to enumerate all possible splitting points for each attribute).
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26. Exercise 2
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27. Solution 2
 SPLIT: Age <= 50
 ----------------------
 | High | Low | Total
 --------------------
 S1 (left) | 8 | 11 | 19
 S2 (right) | 11 | 10 | 21
 --------------------
 For S1: P(high) = 8/19 = 0.42 and P(low) = 11/19 = 0.58
 For S2: P(high) = 11/21 = 0.52 and P(low) = 10/21 = 0.48
 Gini(S1) = 1-[0.42x0.42 + 0.58x0.58] = 1-[0.18+0.34] = 1-0.52 = 0.48
 Gini(S2) = 1-[0.52x0.52 + 0.48x0.48] = 1-[0.27+0.23] = 1-0.5 = 0.5
 Gini-Split(Age<=50) = 19/40 x 0.48 + 21/40 x 0.5 = 0.23 + 0.26 = 0.49
 SPLIT: Salary <= 65K
<
 ----------------------
 | High | Low | Total
 --------------------
 S1 (top) | 18 | 5 | 23
 S2 (bottom) | 1 | 16 | 17
 --------------------
 For S1: P(high) = 18/23 = 0.78 and P(low) = 5/23 = 0.22
 For S2: P(high) = 1/17 = 0.06 and P(low) = 16/17 = 0.94
 Gini(S1) = 1-[0.78x0.78 + 0.22x0.22] = 1-[0.61+0.05] = 1-0.66 = 0.34
 Gini(S2) = 1-[0.06x0.06 + 0 94x0 94] = 1-[0 004+0 884] = 1-0 89 = 0 11
1-[0 06x0 06 0.94x0.94] 1-[0.004+0.884] 1-0.89 0.11
 Gini-Split(Age<=50) = 23/40 x 0.34 + 17/40 x 0.11 = 0.20 + 0.05 = 0.25
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28. Exercise 3
 In previous exercise which is a better split of
exercise,
the data among the two split points? Why?
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29. Solution 3
 Intuitively Salary <= 65K is a better split point since it produces
relatively ``pure'' partitions as opposed to Age <= 50 which
pure'' 50,
results in more mixed partitions (i.e., just look at the distribution
of Highs and Lows in S1 and S2).
 More formally, let us consider the properties of the Gini index.
If a partition is totally pure, i.e., has all elements from the same
class, then gini(S) = 1-[1x1+0x0] = 1-1 = 0 (for two classes).
On the other hand if the classes are totally mixed, i.e., both
classes have equal probability then
gini(S) = 1 - [0 5x0 5 + 0 5x0 5] = 1 [0 25+0 25] = 0 5
[0.5x0.5 0.5x0.5] 1-[0.25+0.25] 0.5.
In other words the closer the gini value is to 0, the better the
partition is. Since Salary has lower gini it is a better split.
is split
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30. Avoid Overfitting in Classification
vo d v g C ss c o
 Overfitting: An induced tree may overfit the training data
 Too many branches, some may reflect anomalies due to noise
or outliers
 Poor accuracy for unseen samples
y p
 Two approaches to avoid overfitting
 Prepruning: Halt tree construction early—do not split a node if
this would result in the goodness measure falling below a
threshold
 Difficult to choose an appropriate threshold
 Postpruning: Remove branches from a “fully grown” tree—get a
sequence of progressively pruned t
f i l d trees
 Use a set of data different from the training data to decide
which is the “best pruned tree”
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