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Data Mining
Classification: Basic Concepts, Decision
Trees, and Model Evaluation
Classification: Definition
l
– Each record contains a set of attributes, one of the
attributes is the class.
Lecture Notes for Chapter 4
l
Introduction to Data Mining
l
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
Attrib1
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125K
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Medium
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100K
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Medium
120K
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Large
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No
Medium
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Yes
Large
220K
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No
Small
85K
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No
Medium
75K
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No
Small
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Yes
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© Tan,Steinbach, Kumar
Attrib1
No
Small
55K
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12
Yes
Medium
Attrib2
80K
Attrib3
?
Class
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Yes
Large
110K
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No
Small
95K
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No
Large
67K
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Predicting tumor cells as benign or malignant
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Classifying credit card transactions
as legitimate or fraudulent
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Classifying secondary structures of protein
as alpha-helix, beta-sheet, or random
coil
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Categorizing news stories as finance,
weather, entertainment, sports, etc
Learn
Model
Apply
Model
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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10
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Introduction to Data Mining
2
Examples of Classification Task
Class
Tid
Find a model for class attribute as a function
of the values of other attributes.
Goal: previously unseen records should be
assigned a class as accurately as possible.
– A test set is used to determine the accuracy of the
model. Usually, the given data set is divided into
training and test sets, with training set used to build
the model and test set used to validate it.
Illustrating Classification Task
Tid
Given a collection of records (training set )
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© Tan,Steinbach, Kumar
Classification Techniques
Introduction to Data Mining
4/18/2004
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Example of a Decision Tree
al
al
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or
or
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in
eg
eg
nt
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ca
ca
co
Logistic Regression
l Decision Tree based Methods
l Rule-based Methods
l Memory based reasoning
l Neural Networks
l Naïve Bayes and Bayesian Belief Networks
l Support Vector Machines
l
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
Splitting Attributes
Refund
Yes
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
Married
NO
>=80K
YES
1
0
Model: Decision Tree
Training Data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
5
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
6
Another Example of Decision Tree
al
al
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ic
ic
uo
or
or
s
tin
eg
eg
t
t
n
as
cl
ca
ca
co
Tid Refund Marital
Status
1
Yes
Single
Taxable
Income Cheat
125K
Married
NO
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
60K
MarSt
Decision Tree Classification Task
Single,
Divorced
Refund
No
Yes
NO
TaxInc
< 80K
>= 80K
NO
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
Attrib1
1
Yes
Large
125K
No
2
No
Medium
Attrib2
100K
Attrib3
No
Class
3
No
Small
70K
No
4
Yes
Medium
120K
No
5
No
Large
95K
Yes
6
No
Medium
60K
No
7
Yes
Large
220K
No
8
No
Small
85K
Yes
9
No
Medium
75K
No
10
No
Small
90K
Yes
Tid
Attrib1
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Learn
Model
10
YES
No
8
Tid
There could be more than one tree that
fits the same data!
1
0
Attrib2
Attrib3
Apply
Model
Class
Decision
Tree
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
7
Apply Model to Test Data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
Apply Model to Test Data
Test Data
Start from the root of tree.
Refund
Yes
Test Data
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Status
Taxable
Income Cheat
No
80K
Married
?
Refund
1
0
No
NO
Yes
TaxInc
< 80K
© Tan,Steinbach, Kumar
TaxInc
NO
< 80K
4/18/2004
9
Apply Model to Test Data
Refund
NO
YES
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
Test Data
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
Refund
1
0
Yes
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
Introduction to Data Mining
TaxInc
< 80K
NO
4/18/2004
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No
80K
Married
?
MarSt
NO
YES
Taxable
Income Cheat
0
1
Single, Divorced
>= 80K
Refund Marital
Status
No
NO
MarSt
Single, Divorced
10
Apply Model to Test Data
No
NO
?
Married
Test Data
Yes
80K
Married
>= 80K
NO
Introduction to Data Mining
No
0
1
Single, Divorced
YES
NO
Taxable
Income Cheat
MarSt
Married
>= 80K
Refund Marital
Status
No
NO
MarSt
Single, Divorced
8
© Tan,Steinbach, Kumar
Married
NO
>= 80K
YES
Introduction to Data Mining
4/18/2004
12
Apply Model to Test Data
Apply Model to Test Data
Test Data
Refund
Test Data
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
Refund
1
0
Yes
No
NO
< 80K
TaxInc
NO
< 80K
YES
NO
© Tan,Steinbach, Kumar
4/18/2004
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Decision Tree Classification Task
Tid
Attrib1
1
Yes
Large
Attrib2
125K
Attrib3
No
2
No
Medium
100K
No
3
No
Small
70K
No
4
Yes
Medium
120K
No
5
No
Large
95K
Yes
6
No
Medium
60K
No
7
Yes
Large
220K
No
8
No
Small
85K
Yes
9
No
Medium
75K
No
10
No
Small
90K
Yes
NO
>= 80K
YES
NO
Introduction to Data Mining
?
Assign Cheat to “No”
Married
Single, Divorced
>= 80K
80K
Married
MarSt
Married
TaxInc
No
No
NO
MarSt
Taxable
Income Cheat
0
1
Yes
Single, Divorced
Refund Marital
Status
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
14
Decision Tree Induction
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Class
Learn
Model
Many Algorithms:
– Hunt’s Algorithm (one of the earliest)
– CART
– ID3, C4.5
– SLIQ,SPRINT
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Tid
Attrib1
11
No
Small
Attrib2
55K
Attrib3
?
Class
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Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Decision
Tree
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
15
General Structure of Hunt’s Algorithm
l
l
Let Dt be the set of training records
that reach a node t
General Procedure:
– If Dt contains records that
belong the same class yt, then t
is a leaf node labeled as yt
– If Dt is an empty set, then t is a
leaf node labeled by the default
class, yd
– If Dt contains records that
belong to more than one class,
use an attribute test to split the
data into smaller subsets.
Recursively apply the
procedure to each subset.
© Tan,Steinbach, Kumar
Introduction to Data Mining
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
Hunt’s Algorithm
16
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
No
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
6
No
Married
5
No
Divorced 95K
Yes
7
Yes
Divorced 220K
No
6
No
Married
No
8
No
Single
85K
Yes
7
Yes
Divorced 220K
No
9
No
Married
75K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
10
No
Single
90K
Yes
60K
Refund
Don’t
Cheat
No
Don’t
Cheat
Don’t
Cheat
Refund
Refund
Yes
Yes
No
No
60K
1
0
Don’t
Cheat
Single,
Divorced
10
Dt
?
Cheat
4/18/2004
Yes
17
Marital
Status
Married
Don’t
Cheat
Single,
Divorced
Married
Don’t
Cheat
Taxable
Income
Don’t
Cheat
© Tan,Steinbach, Kumar
Marital
Status
< 80K
>= 80K
Don’t
Cheat
Cheat
Introduction to Data Mining
4/18/2004
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Tree Induction
Tree Induction
l
Greedy strategy.
– Split the records based on an attribute test
that optimizes a certain criterion.
l
Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.
l
Issues
– Determine how to split the records
l
Issues
– Determine how to split the records
uHow
uHow
to specify the attribute test condition?
to determine the best split?
uHow
uHow
– Determine when to stop splitting
© Tan,Steinbach, Kumar
– Determine when to stop splitting
Introduction to Data Mining
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How to Specify Test Condition?
l
l
l
Depends on number of ways to split
– 2-way split
– Multi-way split
l
CarType
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Large
l
{Medium,
Large}
l
What about this split?
© Tan,Steinbach, Kumar
Introduction to Data Mining
{Family,
Luxury}
Introduction to Data Mining
CarType
{Sports}
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Static – discretize once at the beginning
Dynamic – ranges can be found by equal interval
bucketing, equal frequency bucketing
(percentiles), or clustering.
Size
– Binary Decision: (A < v) or (A ≥ v)
{Small}
u
{Small,
Large}
OR
Different ways of handling
– Discretization to form an ordinal categorical
attribute
u
OR
{Family}
Splitting Based on Continuous Attributes
Binary split: Divides values into two subsets.
Need to find optimal partitioning.
{Large}
CarType
© Tan,Steinbach, Kumar
u
Size
Luxury
Binary split: Divides values into two subsets.
Need to find optimal partitioning.
{Sports,
Luxury}
Multi-way split: Use as many partitions as distinct
values.
{Small,
Medium}
20
Sports
Size
l
4/18/2004
Multi-way split: Use as many partitions as distinct
values.
Family
Introduction to Data Mining
Small
Medium
Introduction to Data Mining
Splitting Based on Nominal Attributes
Splitting Based on Ordinal Attributes
l
© Tan,Steinbach, Kumar
Depends on attribute types
– Nominal
– Ordinal
– Continuous
© Tan,Steinbach, Kumar
to specify the attribute test condition?
to determine the best split?
Size
u
{Medium}
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consider all possible splits and finds the best cut
can be more compute intensive
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Splitting Based on Continuous Attributes
Tree Induction
l
Greedy strategy.
– Split the records based on an attribute test
that optimizes a certain criterion.
l
Issues
– Determine how to split the records
uHow
uHow
to specify the attribute test condition?
to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
25
How to determine the Best Split
© Tan,Steinbach, Kumar
Greedy approach:
– Nodes with homogeneous class distribution
are preferred
l Need a measure of node impurity:
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Impurity Measures
Non-homogeneous,
Homogeneous,
High degree of impurity
Low degree of impurity
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Measures of Node Impurity
l
Should be at minimum when all examples belong
to a single class.
l
Should be at maximum when all classes have the
same relative frequency.
l
26
l
Which test condition is the best?
Introduction to Data Mining
4/18/2004
How to determine the Best Split
Before Splitting: 10 records of class 0,
10 records of class 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
l
Gini Index
l
Entropy
l
Misclassification error
Should be symmetrical in class frequencies.
© Tan,Steinbach, Kumar
Introduction to Data Mining
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© Tan,Steinbach, Kumar
Introduction to Data Mining
How to Find the Best Split
Before Splitting:
C0
C1
Measure of Impurity: GINI
N00
N01
M0
A?
Gini Index for a given node t :
GINI (t ) = 1 − ∑ [ p ( j | t )]2
B?
Yes
j
No
Node N1
Yes
Node N3
M2
M1
Node N4
N30
N31
C0
C1
N20
N21
C0
C1
No
(NOTE: p( j | t) is the relative frequency of class j at node t).
Node N2
N10
N11
C0
C1
l
M3
– Maximum (1 - 1/nc) when records are equally
distributed among all classes, implying least
interesting information
– Minimum (0.0) when all records belong to one class,
implying most interesting information
N40
N41
C0
C1
M4
C1
C2
M12
M34
0
6
C1
C2
Gini=0.000
1
5
C1
C2
Gini=0.278
2
4
Gini=0.444
C1
C2
3
3
Gini=0.500
Gain = M0 – M12 vs M0 – M34
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Examples for computing GINI
l
j
0
6
P(C1) = 0/6 = 0
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
Introduction to Data Mining
l
P(C2) = 6/6 = 1
k
GINI split = ∑
Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0
l
ni
GINI (i )
n
P(C2) = 5/6
where,
Gini = 1 – (1/6)2 – (5/6)2 = 0.278
ni = number of records at child i,
n = number of records at node t.
P(C2) = 4/6
Gini = 1 – (2/6)2 – (4/6)2 = 0.444
Introduction to Data Mining
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Binary Attributes: Computing GINI Index
l
32
Used in CART, SLIQ, SPRINT.
When a node t is split into k partitions (children), the
quality of split is computed as,
i =1
© Tan,Steinbach, Kumar
4/18/2004
Splitting Based on GINI
GINI (t ) = 1 − ∑ [ p ( j | t )]2
C1
C2
© Tan,Steinbach, Kumar
Splits into two partitions
Effect of Weighing partitions:
– Larger and Purer Partitions are sought for.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Categorical Attributes: Computing Gini Index
l
l
For each distinct value, gather counts for each class in
the dataset
Use the count matrix to make decisions
Parent
B?
Yes
No
C1
6
C2
6
Multi-way split
Gini = 0.500
Gini(N1)
= 1 – (5/7)2 – (2/7)2
= 0.408
Gini(N2)
= 1 – (1/5)2 – (4/5)2
= 0.320
© Tan,Steinbach, Kumar
Node N1
CarType
Node N2
N1 N2
C1
5
1
C2
2
4
Gini=0.371
Introduction to Data Mining
C1
C2
Gini
Gini(Children)
= 7/12 * 0.408 +
5/12 * 0.320
= 0.371
4/18/2004
Two-way split
(find best partition of values)
35
Family Sports Luxury
1
2
1
4
1
1
0.393
© Tan,Steinbach, Kumar
C1
C2
Gini
CarType
{Sports,
{Family}
Luxury}
3
1
2
4
0.400
Introduction to Data Mining
CarType
{Family,
Luxury}
2
2
1
5
0.419
{Sports}
C1
C2
Gini
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Continuous Attributes: Computing Gini Index
l
l
l
l
Use Binary Decisions based on one
value
Several Choices for the splitting value
– Number of possible splitting values
= Number of distinct values
Each splitting value has a count matrix
associated with it
– Class counts in each of the
partitions, A < v and A ≥ v
Simple method to choose best v
– For each v, scan the database to
gather count matrix and compute
its Gini index
– Computationally Inefficient!
Repetition of work.
© Tan,Steinbach, Kumar
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
For efficient computation: for each attribute,
– Sort the attribute on values
– Linearly scan these values, each time updating the count matrix
and computing gini index
– Choose the split position that has the least gini index
l
Cheat
Introduction to Data Mining
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No
No
No
Yes
60
70
75
85
Yes
Yes
No
No
No
No
100
120
125
220
Taxable Income
Sorted Values
Split Positions
55
65
72
90
80
95
87
92
97
110
122
172
230
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
Yes
0
3
0
3
0
3
0
3
1
2
2
1
3
0
3
0
3
0
3
0
3
0
No
0
7
1
6
2
5
3
4
3
4
3
4
3
4
4
3
5
2
6
1
7
0
Gini
0.420
0.400
© Tan,Steinbach, Kumar
0.375
0.343
0.417
0.400
0.300
0.343
Introduction to Data Mining
0.375
0.400
4/18/2004
0.420
38
Examples for computing Entropy
Alternative Splitting Criteria based on INFO
l
Continuous Attributes: Computing Gini Index...
Entropy (t ) = − ∑ p ( j | t ) log p ( j | t )
Entropy at a given node t:
2
j
Entropy (t ) = − ∑ p ( j | t ) log p ( j | t )
j
(NOTE: p( j | t) is the relative frequency of class j at node t).
– Measures homogeneity of a node.
(log nc) when records are equally distributed
among all classes implying least information (where nc
denotes the number of classes)
u Minimum (0.0) when all records belong to one class,
implying most information
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
P(C1) = 0/6 = 0
P(C2) = 6/6 = 1
Entropy = – 0 log 0 – 1 log 1 = – 0 – 0 = 0
u Maximum
– Entropy based computations are similar to the
GINI index computations
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
39
Splitting Based on INFO...
l
© Tan,Steinbach, Kumar
l
n
= Entropy( p ) −  ∑ Entropy(i ) 
 n

k
split
P(C2) = 4/6
Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92
Introduction to Data Mining
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Splitting Based on INFO...
Information Gain:
GAIN
P(C2) = 5/6
Entropy = – (1/6) log2 (1/6) – (5/6) log2 (5/6) = 0.65
i
Gain Ratio:
GainRATIO
i =1
split
=
GAIN
n
n
SplitINFO = − ∑ log
SplitINFO
n
n
Split
k
i
i
i =1
Parent Node, p is split into k partitions;
ni is number of records in partition i
– Measures Reduction in Entropy achieved because of
the split. Choose the split that achieves most reduction
(maximizes GAIN)
– Used in ID3 and C4.5
– Disadvantage: Tends to prefer splits that result in large
number of partitions, each being small but pure.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
41
Parent Node, p is split into k partitions
ni is the number of records in partition i
– Adjusts Information Gain by the entropy of the
partitioning (SplitINFO). Higher entropy partitioning
(large number of small partitions) is penalized!
– Used in C4.5
– Designed to overcome the disadvantage of Information
Gain
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
42
Splitting Criteria based on Classification Error
l
Examples for Computing Error
Error (t ) = 1 − max P (i | t )
Classification error at a node t :
i
Error (t ) = 1 − max P (i | t )
i
l
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
P(C1) = 0/6 = 0
P(C2) = 6/6 = 1
Error = 1 – max (0, 1) = 1 – 1 = 0
Measures misclassification error made by a node.
u Maximum
(1 - 1/nc) when records are equally distributed
among all classes, implying least interesting information
(where nc denotes the number of classes)
(0.0) when all records belong to one class, implying
most interesting information
P(C2) = 5/6
Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6
u Minimum
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
43
Comparison among Splitting Criteria
P(C2) = 4/6
Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3
© Tan,Steinbach, Kumar
Introduction to Data Mining
Parent
A?
Yes
Node N1
Gini(N1)
= 1 – (3/3)2 – (0/3)2
=0
Gini(N2)
= 1 – (4/7)2 – (3/7)2
= 0.489
Introduction to Data Mining
4/18/2004
45
Tree Induction
Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.
l
Issues
– Determine how to split the records
uHow
© Tan,Steinbach, Kumar
No
Node N2
N1 N2
C1
3
4
C2
0
3
Gini=0.342
C1
7
C2
3
Gini = 0.42
Gini(Children)
= 3/10 * 0
+ 7/10 * 0.489
= 0.342
Gini improves !!
Introduction to Data Mining
4/18/2004
46
Stopping Criteria for Tree Induction
l
uHow
44
Misclassification Error vs Gini
For a 2-class problem:
© Tan,Steinbach, Kumar
4/18/2004
to specify the attribute test condition?
to determine the best split?
l
Stop expanding a node when all the records
belong to the same class
l
Stop expanding a node when all the records have
the same attribute values
l
Early termination (to be discussed later)
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
47
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
48
Decision Tree Based Classification
l
Example: C4.5
Advantages:
– Inexpensive to construct
– Extremely fast at classifying unknown records
– Easy to interpret for small-sized trees
– Accuracy is comparable to other classification
techniques for many simple data sets
Simple depth-first construction.
Uses Information Gain
l Sorts Continuous Attributes at each node.
l Needs entire data to fit in memory.
l Unsuitable for Large Datasets.
– Needs out-of-core sorting.
l
l
l
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
49
Implemented in Weka as J48.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
50