Download Data Mining Classification

Data Mining Classification
This lecture node is modified based on Lecture Notes for Chapter 4/5
of Introduction to Data Mining by Tan, Steinbach, Kumar, and slides
from Jiawei Han for the book of Data Mining – Concepts and
Techniqies by Jiawei Han and Micheline Kamber.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
1
Classification: Definition

Given a collection of records (training set )
– Each record contains a set of attributes, one of the
attributes is the class.


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.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Classification Techniques
Decision Tree
 Naïve Bayes
 kNN Classification

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Classification Task
Tid
Attrib1
Attrib2
Attrib3
1
Yes
Large
125K
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
Tree
Induction
algorithm
Class
Induction
Learn
Model
Model
10
Training Set
Tid
Attrib1
Attrib2
Attrib3
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Class
Decision
Tree
Deduction
10
Test Set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example of a Decision Tree
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
10
Model: Decision Tree
Training Data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Another Example of Decision Tree
MarSt
Tid Refund
Married
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
NO
Single,
Divorced
Refund
No
Yes
NO
TaxInc
< 80K
> 80K
NO
YES
There could be more than one tree that
fits the same data!
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Classification Task
Tid
Attrib1
Attrib2
Attrib3
1
Yes
Large
125K
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
Tree
Induction
algorithm
Class
Induction
Learn
Model
Model
10
Training Set
Tid
Attrib1
Attrib2
Attrib3
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Class
Decision
Tree
Deduction
10
Test Set
© 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
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
Assign Cheat to “No”
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Performance Metrics
PREDICTED CLASS
Class=Yes
ACTUAL
CLASS
Class=No
Class=Yes
a
(TP)
b
(FN)
Class=No
c
(FP)
d
(TN)
a+d
TP + TN
=
Accuracy =
a + b + c + d TP + TN + FP + FN
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
General Structure of Hunt’s Algorithm


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
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
Dt
?
4/18/2004
‹#›
Hunt’s Algorithm
Don’t
Cheat
Refund
Yes
No
Don’t
Cheat
Don’t
Cheat
Refund
Refund
Yes
Yes
No
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
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
Don’t
Cheat
Don’t
Cheat
Marital
Status
Single,
Divorced
Cheat
Married
Single,
Divorced
Don’t
Cheat
© Tan,Steinbach, Kumar
Marital
Status
Married
Don’t
Cheat
Taxable
Income
< 80K
>= 80K
Don’t
Cheat
Cheat
Introduction to Data Mining
4/18/2004
‹#›
Tree Induction

Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.

Issues
– Determine how to split the records
How
to specify the attribute test condition?
How to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
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
Own
Car?
Yes
Car
Type?
No
Family
Student
ID?
Luxury
c1
Sports
C0: 6
C1: 4
C0: 4
C1: 6
C0: 1
C1: 3
C0: 8
C1: 0
C0: 1
C1: 7
C0: 1
C1: 0
...
c10
c11
C0: 1
C1: 0
C0: 0
C1: 1
c20
...
C0: 0
C1: 1
Which test condition is the best?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to determine the Best Split
Greedy approach:
– Nodes with homogeneous class distribution
are preferred
 Need a measure of node impurity:

C0: 5
C1: 5
C0: 9
C1: 1
Non-homogeneous,
Homogeneous,
High degree of impurity
Low degree of impurity
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Find the Best Split
Before Splitting:
C0
C1
N00
N01
M0
A?
B?
Yes
No
Node N1
C0
C1
Node N2
N10
N11
C0
C1
N20
N21
M2
M1
Yes
No
Node N3
C0
C1
Node N4
N30
N31
C0
C1
M3
M12
N40
N41
M4
M34
Gain = M0 – M12 vs M0 – M34
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Measure of Impurity: GINI

Gini Index for a given node t :
GINI (t ) = 1   [ p( j | t )]2
j
(NOTE: p( j | t) is the relative frequency of class j at node t).
– Maximum 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
C1
C2
0
6
Gini=0.000
© Tan,Steinbach, Kumar
C1
C2
1
5
Gini=0.278
C1
C2
2
4
Gini=0.444
Introduction to Data Mining
C1
C2
3
3
Gini=0.500
4/18/2004
‹#›
Examples for computing GINI
GINI (t ) = 1   [ p( j | t )]2
j
C1
C2
0
6
P(C1) = 0/6 = 0
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C2) = 6/6 = 1
Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0
P(C2) = 5/6
Gini = 1 – (1/6)2 – (5/6)2 = 0.278
P(C2) = 4/6
Gini = 1 – (2/6)2 – (4/6)2 = 0.444
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on GINI

When a node p is split into k partitions (children), the
quality of split is computed as,
k
ni
GINI split =  GINI (i )
i =1 n
where,
© Tan,Steinbach, Kumar
ni = number of records at child i,
n = number of records at node p.
Introduction to Data Mining
4/18/2004
‹#›
Binary Attributes: Computing GINI Index


Split into two partitions
Effect of weighing partitions:
– Larger and Purer Partitions are sought for.
Parent
B?
Yes
No
C1
6
C2
6
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
Node N2
N1 N2
C1
5
1
C2
2
4
Gini=0.371
Introduction to Data Mining
Gini(Children)
= 7/12 * 0.408 +
5/12 * 0.320
= 0.371
4/18/2004
‹#›
Categorical Attributes: Computing Gini Index


For each distinct value, gather counts for each class in
the dataset
Use the count matrix to make decisions
Multi-way split
Two-way split
(find best partition of values)
CarType
Family Sports Luxury
C1
C2
Gini
1
4
2
1
0.393
© Tan,Steinbach, Kumar
1
1
C1
C2
Gini
CarType
{Sports,
{Family}
Luxury}
3
1
2
4
0.400
Introduction to Data Mining
C1
C2
Gini
CarType
{Family,
{Sports}
Luxury}
2
2
1
5
0.419
4/18/2004
‹#›
Continuous Attributes: Computing Gini Index




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
Introduction to Data Mining
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
Taxable
Income
> 80K?
Yes
4/18/2004
No
‹#›
Alternative Splitting Criteria based on INFO

Entropy at a given node t:
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.
 Maximum
when records are equally distributed among
all classes implying least information
 Minimum (0.0) when all records belong to one class,
implying most information
– Entropy based computations are similar to the
GINI index computations
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples for computing Entropy
Entropy(t ) =  p( j | t ) log p( j | t )
j
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C1) = 0/6 = 0
2
P(C2) = 6/6 = 1
Entropy = – 0 log 0 – 1 log 1 = – 0 – 0 = 0
P(C2) = 5/6
Entropy = – (1/6) log2 (1/6) – (5/6) log2 (1/6) = 0.65
P(C2) = 4/6
Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on INFO...

Information Gain:
n


GAIN = Entropy ( p)    Entropy (i ) 
 n

k
split
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)
– 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
‹#›
Splitting Based on INFO...

Gain Ratio:
GAIN
n
n
GainRATIO =
SplitINFO =  log
SplitINFO
n
n
Split
split
k
i
i
i =1
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!
– Designed to overcome the disadvantage of Information
Gain
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Splitting Criteria based on Classification Error

Classification error at a node t :
Error (t ) = 1  max P(i | t )
i

Measures misclassification error made by a node.
 Maximum
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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples for Computing Error
Error (t ) = 1  max P(i | t )
i
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C1) = 0/6 = 0
P(C2) = 6/6 = 1
Error = 1 – max (0, 1) = 1 – 1 = 0
P(C2) = 5/6
Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6
P(C2) = 4/6
Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3
Introduction to Data Mining
4/18/2004
‹#›
Comparison among Splitting Criteria
For a 2-class problem:
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Stopping Criteria for Tree Induction

Stop expanding a node when all the records
belong to the same class

Stop expanding a node when all the records have
similar attribute values

Early termination
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Training Dataset
This
follows
an
example
from
Quinlan’s
ID3
© Tan,Steinbach, Kumar
age
<=30
<=30
30…40
>40
>40
>40
31…40
<=30
<=30
>40
<=30
31…40
31…40
>40
income student credit_rating
high
no fair
high
no excellent
high
no fair
medium
no fair
low
yes fair
low
yes excellent
low
yes excellent
medium
no fair
low
yes fair
medium
yes fair
medium
yes excellent
medium
no excellent
high
yes fair
medium
no excellent
Introduction to Data Mining
buys_computer
no
no
yes
yes
yes
no
yes
no
yes
yes
yes
yes
yes
no
4/18/2004
‹#›
More on Information Gain
Entropy(t ) =  p( j | t ) log p( j | t )
j
2

S contains si tuples of class Ci for i = {1, …, m}
information measures info required to classify any
m
arbitrary tuple
si
si
I(s1 ,s2 ,...,sm ) = -å log2
s
i=1 s
entropy of attribute A with values {a1,a2,…,av}

s1 j +... + smj
E(A) = å
I(s1 j ,..., smj )
s
j=1
information gained by branching on attribute A


v
Gain(A) = I(s1,s2,...,sm )- E(A)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Attribute Selection by Information Gain

Class P: buys_computer = “yes”

Class N: buys_computer = “no”

I(p, n) = I(9, 5) =0.940

Compute the entropy for age:
age
<=30
30…40
>40
pi
2
4
3
age
income student
<=30
high
no
<=30
high
no
31…40 high
no
>40
medium
no
>40
low
yes
>40
low
yes
31…40 low
yes
<=30
medium
no
<=30
low
yes
>40
medium
yes
<=30
medium
yes
31…40 medium
no
© Tan,Steinbach,
Kumar
31…40
high
yes
>40
medium
no
ni I(pi, ni)
3 0.971
0 0
2 0.971
5
4
I (2,3) +
I ( 4,0)
14
14
5
+
I (3,2) = 0.694
14
E ( age) =
5
I (2,3) means “age <=30” has 5
14
out of 14 samples, with 2 yes’es
and 3 no’s. Hence
Gain(age) = I ( p, n)  E (age) = 0.246
credit_rating buys_computer
fair
no
excellent
no
fair
yes
fair
yes
fair
yes
excellent
no
excellent
yes
fair
no
fair
yes
fair
yes
excellent
yes
excellent
yes
Introduction
to Data Mining
fair
yes
excellent
no
Similarly,
Gain(income) = 0.029
Gain( student ) = 0.151
Gain(credit _ rating ) = 0.048
4/18/2004
‹#›
Output: A Decision Tree for “buys_computer”
age?
<=30
overcast
30..40
student?
yes
>40
credit rating?
no
yes
excellent
fair
no
yes
no
yes
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Classification Rules






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 prediction
Rules are easier for humans to understand
Example
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”
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Based Classification

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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Boundary
1
0.9
x < 0.43?
0.8
0.7
Yes
No
y
0.6
y < 0.33?
y < 0.47?
0.5
0.4
Yes
0.3
0.2
:4
:0
0.1
No
Yes
:0
:4
:0
:3
No
:4
:0
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
• Border line between two neighboring regions of different classes is
known as decision boundary
• Decision boundary is parallel to axes because test condition involves
a single attribute at-a-time
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Oblique Decision Trees
x+y<1
Class = +
Class =
• Test condition may involve multiple attributes
• More expressive representation
• Finding optimal test condition is computationally expensive
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Model Underfitting/Overfitting




The errors of a classification model are divided into two
types
– Training errors: the number of misclassification
errors committed on training records.
– Generalization errors: the expected error of the
model on previously unseen records.
A good model must have both errors low.
Model underfitting: both type of errors are large when
the decision tree is too small.
Model overfitting: training error is small but
generalization error is large, when the decision tree is
too large.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Underfitting and Overfitting (Example)
500 circular and 500
triangular data points.
Circular points:
0.5  sqrt(x12+x22)  1
Triangular points:
sqrt(x12+x22) > 0.5 or
sqrt(x12+x22) < 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Underfitting and Overfitting
Overfitting
Underfitting: when model is too simple, both training and test errors are large
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
An Example: Training Dataset
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
An Example: Testing Dataset
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
An Example: Two Models, M1 and M2
0% training error
30% testing error
20% training error
10% testing error
(human, warm-blooded, yes, no, no)
(dolphin, warm-blooded, yes, no, no)
(spiny anteater, warm-blooded, no, yes, yes)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Occam’s Razor

Given two models of similar generalization errors,
one should prefer the simpler model over the
more complex model

For complex models, there is a greater chance
that it was fitted accidentally by errors in data

Therefore, one should include model complexity
when evaluating a model
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Incorporating Model Complexity (2)



Training errors: error on training ( e(t))
Generalization errors: error on testing ( e’(t))
Methods for estimating generalization errors:
– Optimistic approach: e’(t) = e(t)
– Pessimistic approach:

For each leaf node: e’(t) = (e(t)+0.5)

Total errors: e’(T) = e(T) + N  0.5 (N: number of leaf nodes)

For a tree with 30 leaf nodes and 10 errors on training
(out of 1000 instances):
Training error = 10/1000 = 1%
Generalization error = (10 + 300.5)/1000 = 2.5%
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
An Example: Pessimistic Error Estimate


The generalization error of TL is (4 + 7 * 0.5)/24 = 0.3125, and the
generalization error of TR is (6 + 4 * 0.5)/24 = 0.3333,
where the penalty term is 0.5.
Based on pessimistic error estimate, the TL should be used.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Address Overfitting (1)

Pre-Pruning (Early Stopping Rule)
– Stop the algorithm before it becomes a fully-grown tree
– Typical stopping conditions for a node:

Stop if all instances belong to the same class

Stop if all the attribute values are the same
– More restrictive conditions:
Stop if number of instances is less than some user-specified
threshold

Stop if class distribution of instances are independent of the
available features (e.g., using chi-squared test)


Stop if expanding the current node does not improve impurity
measures (e.g., Gini or information gain).
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Address Overfitting (2)

Post-pruning
– Grow decision tree to its entirety
– Trim the nodes of the decision tree in a
bottom-up fashion
– If generalization error improves after trimming,
replace sub-tree by a leaf node.
– Class label of leaf node is determined from
majority class of instances in the sub-tree
– Can use MDL (Minimum Description Length)
for post-pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example of Post-Pruning
Training Error (Before splitting) = 10/30
Class = Yes
20
Pessimistic error = (10 + 0.5)/30 = 10.5/30
Class = No
10
Training Error (After splitting) = 9/30
Pessimistic error (After splitting)
Error = 10/30
= (9 + 4  0.5)/30 = 11/30
PRUNE!
A?
A1
A4
A3
A2
Class = Yes
8
Class = Yes
3
Class = Yes
4
Class = Yes
5
Class = No
4
Class = No
4
Class = No
1
Class = No
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›