Download Data Mining Classification: Alternative Techniques

Data Mining
Classification: Alternative Techniques
Lecture Notes for Chapter 5
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
1
Rule-Based Classifier
O
Classify records by using a collection of
“if…then…” rules
O
Rule:
(Condition) → y
– where
‹
Condition is a conjunctions of attributes
‹
y is the class label
– LHS: rule antecedent or condition
– RHS: rule consequent
– Examples of classification rules:
‹
(Blood Type=Warm) ∧ (Lay Eggs=Yes) → Birds
‹
(Taxable Income < 50K) ∧ (Refund=Yes) → Evade=No
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
2
Rule-based Classifier (Example)
Name
Blood Type
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
warm
cold
cold
warm
cold
cold
warm
warm
warm
cold
cold
warm
warm
cold
cold
cold
warm
warm
warm
warm
Give Birth
yes
no
no
yes
no
no
yes
no
yes
yes
no
no
yes
no
no
no
no
no
yes
no
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Live in Water
no
no
yes
yes
sometimes
no
no
no
no
yes
sometimes
sometimes
no
yes
sometimes
no
no
no
yes
no
Class
mammals
reptiles
fishes
mammals
amphibians
reptiles
mammals
birds
mammals
fishes
reptiles
birds
mammals
fishes
amphibians
reptiles
mammals
birds
mammals
birds
R1: (Give Birth = no) ∧ (Can Fly = yes) → Birds
R2: (Give Birth = no) ∧ (Live in Water = yes) → Fishes
R3: (Give Birth = yes) ∧ (Blood Type = warm) → Mammals
R4: (Give Birth = no) ∧ (Can Fly = no) → Reptiles
R5: (Live in Water = sometimes) → Amphibians
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
3
Application of Rule-Based Classifier
O
A rule r covers an instance x if the attributes of
the instance satisfy the condition of the rule
R1: (Give Birth = no) ∧ (Can Fly = yes) → Birds
R2: (Give Birth = no) ∧ (Live in Water = yes) → Fishes
R3: (Give Birth = yes) ∧ (Blood Type = warm) → Mammals
R4: (Give Birth = no) ∧ (Can Fly = no) → Reptiles
R5: (Live in Water = sometimes) → Amphibians
Name
hawk
grizzly bear
Blood Type
warm
warm
Give Birth
Can Fly
Live in Water
Class
no
yes
yes
no
no
no
?
?
The rule R1 covers a hawk => Bird
The rule R3 covers the grizzly bear => Mammal
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
4
Rule Coverage and Accuracy
Tid Refund Marital
Coverage of a rule:
Status
1
Yes
Single
– Fraction of records
2
No
Married
that satisfy the
3
No
Single
antecedent of a rule
4
Yes
Married
5
No
Divorced
O Accuracy of a rule:
6
No
Married
– Fraction of records
7
Yes
Divorced
that satisfy both the
8
No
Single
9
No
Married
antecedent and
10 No
Single
consequent of a
(Status=Single) → No
rule
O
Taxable
Income Class
125K
No
100K
No
70K
No
120K
No
95K
Yes
60K
No
220K
No
85K
Yes
75K
No
90K
Yes
10
Coverage = 40%, Accuracy = 50%
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
5
How does Rule-based Classifier Work?
R1: (Give Birth = no) ∧ (Can Fly = yes) → Birds
R2: (Give Birth = no) ∧ (Live in Water = yes) → Fishes
R3: (Give Birth = yes) ∧ (Blood Type = warm) → Mammals
R4: (Give Birth = no) ∧ (Can Fly = no) → Reptiles
R5: (Live in Water = sometimes) → Amphibians
Name
lemur
turtle
dogfish shark
Blood Type
warm
cold
cold
Give Birth
Can Fly
Live in Water
Class
yes
no
yes
no
no
no
no
sometimes
yes
?
?
?
A lemur triggers rule R3, so it is classified as a mammal
A turtle triggers both R4 and R5
A dogfish shark triggers none of the rules
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
6
Characteristics of Rule-Based Classifier
O
Mutually exclusive rules
– Classifier contains mutually exclusive rules if
the rules are independent of each other
– Every record is covered by at most one rule
O
Exhaustive rules
– Classifier has exhaustive coverage if it
accounts for every possible combination of
attribute values
– Each record is covered by at least one rule
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
7
From Decision Trees To Rules
Classification Rules
(Refund=Yes) ==> No
Refund
Yes
No
NO
{Single,
Divorced}
Marita l
Status
NO
{Married}
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
(Refund=No, Marital Status={Married}) ==> No
NO
Taxable
Income
< 80K
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
> 80K
YES
Rules are mutually exclusive and exhaustive
Rule set contains as much information as the
tree
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
8
Rules Can Be Simplified
Refund
Yes
No
NO
{Single,
Divorced}
Marita l
Status
NO
Taxable
Income
< 80K
{Married}
> 80K
NO
YES
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
6
No
Married
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
Yes
No
10
Initial Rule:
(Refund=No) ∧ (Status=Married) → No
Simplified Rule: (Status=Married) → No
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
9
Effect of Rule Simplification
O
Rules are no longer mutually exclusive
– A record may trigger more than one rule
– Solution?
Ordered rule set
‹ Unordered rule set – use voting schemes
‹
O
Rules are no longer exhaustive
– A record may not trigger any rules
– Solution?
‹
Use a default class
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
10
Ordered Rule Set
O
Rules are rank ordered according to their priority
– An ordered rule set is known as a decision list
O
When a test record is presented to the classifier
– It is assigned to the class label of the highest ranked rule it has
triggered
– If none of the rules fired, it is assigned to the default class
R1: (Give Birth = no) ∧ (Can Fly = yes) → Birds
R2: (Give Birth = no) ∧ (Live in Water = yes) → Fishes
R3: (Give Birth = yes) ∧ (Blood Type = warm) → Mammals
R4: (Give Birth = no) ∧ (Can Fly = no) → Reptiles
R5: (Live in Water = sometimes) → Amphibians
Name
turtle
Blood Type
Give Birth
Can Fly
Live in Water
Class
no
no
sometimes
?
cold
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
11
Rule Ordering Schemes
O
Rule-based ordering
– Individual rules are ranked based on their quality
O
Class-based ordering
– Rules that belong to the same class appear together
Rule-based Ordering
Class-based Ordering
(Refund=Yes) ==> No
(Refund=Yes) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
(Refund=No, Marital Status={Married}) ==> No
(Refund=No, Marital Status={Married}) ==> No
© Tan,Steinbach, Kumar
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
Introduction to Data Mining
4/18/2004
12
Building Classification Rules
O
Direct Method:
Extract rules directly from data
‹ e.g.: RIPPER, CN2, Holte’s 1R
‹
O
Indirect Method:
Extract rules from other classification models (e.g.
decision trees, neural networks, etc).
‹ e.g: C4.5rules
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
13
Direct Method: Sequential Covering
1.
2.
3.
4.
Start from an empty rule
Grow a rule using the Learn-One-Rule function
Remove training records covered by the rule
Repeat Step (2) and (3) until stopping criterion
is met
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
14
Example of Sequential Covering
(ii) Step 1
(i) Original Data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
15
Example of Sequential Covering…
R1
R1
R2
(iii) Step 2
© Tan,Steinbach, Kumar
(iv) Step 3
Introduction to Data Mining
4/18/2004
16
Aspects of Sequential Covering
O
Rule Growing
O
Instance Elimination
O
Rule Evaluation
O
Stopping Criterion
O
Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
17
Rule Growing
O
Two common strategies
{}
Yes: 3
No: 4
Refund=No,
Status=Single,
Income=85K
(Class=Yes)
Refund=
No
Status =
Single
Status =
Divorced
Status =
Married
Yes: 3
No: 4
Yes: 2
No: 1
Yes: 1
No: 0
Yes: 0
No: 3
...
Income
> 80K
Yes: 3
No: 1
(a) General-to-specific
© Tan,Steinbach, Kumar
Introduction to Data Mining
Refund=No,
Status=Single,
Income=90K
(Class=Yes)
Refund=No,
Status = Single
(Class = Yes)
(b) Specific-to-general
4/18/2004
18
Rule Growing (Examples)
O
CN2 Algorithm:
– Start from an empty conjunct: {}
– Add conjuncts that minimizes the entropy measure: {A}, {A,B}, …
– Determine the rule consequent by taking majority class of instances
covered by the rule
O
RIPPER Algorithm:
– Start from an empty rule: {} => class
– Add conjuncts that maximizes FOIL’s information gain measure:
R0: {} => class (initial rule)
R1: {A} => class (rule after adding conjunct)
‹ Gain(R0, R1) = t [ log (p1/(p1+n1)) – log (p0/(p0 + n0)) ]
‹ where t: number of positive instances covered by both R0 and R1
p0: number of positive instances covered by R0
n0: number of negative instances covered by R0
p1: number of positive instances covered by R1
n1: number of negative instances covered by R1
‹
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
19
Instance Elimination
O
Why do we need to
eliminate instances?
R3
– Otherwise, the next rule is
identical to previous rule
O
Why do we remove
positive instances?
R1
+
class = +
– Ensure that the next rule is
different
O
Why do we remove
negative instances?
-
class = -
– Prevent underestimating
accuracy of rule
– Compare rules R2 and R3
in the diagram
© Tan,Steinbach, Kumar
+
Introduction to Data Mining
-
+ +
+
+
+
+
++
+
+
+++
+ + +
+ +
-
-
R2
-
-
+
+
+
+
+ +
-
+
-
-
+
+
-
-
-
-
4/18/2004
20
Rule Evaluation
O
Metrics:
– Accuracy
– Laplace
–
=
=
nc
n
nc + 1
n+k
n + kp
M-estimate = c
n+k
© Tan,Steinbach, Kumar
Introduction to Data Mining
n : Number of instances
covered by rule
nc : Number of instances
covered by rule
k : Number of classes
p : Prior probability
4/18/2004
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Stopping Criterion and Rule Pruning
O
Stopping criterion
– Compute the gain
– If gain is not significant, discard the new rule
O
Rule Pruning
– Similar to post-pruning of decision trees
– Reduced Error Pruning:
Remove one of the conjuncts in the rule
‹ Compare error rate on validation set before and
after pruning
‹ If error improves, prune the conjunct
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
22
Summary of Direct Method
O
Grow a single rule
O
Remove Instances from rule
O
Prune the rule (if necessary)
O
Add rule to Current Rule Set
O
Repeat
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
23
Direct Method: RIPPER
O
O
For 2-class problem, choose one of the classes as
positive class, and the other as negative class
– Learn rules for positive class
– Negative class will be default class
For multi-class problem
– Order the classes according to increasing class
prevalence (fraction of instances that belong to a
particular class)
– Learn the rule set for smallest class first, treat the rest
as negative class
– Repeat with next smallest class as positive class
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
24
Direct Method: RIPPER
O
Growing a rule:
– Start from empty rule
– Add conjuncts as long as they improve FOIL’s
information gain
– Stop when rule no longer covers negative examples
– Prune the rule immediately using incremental reduced
error pruning
– Measure for pruning: v = (p-n)/(p+n)
p: number of positive examples covered by the rule in
the validation set
‹ n: number of negative examples covered by the rule in
the validation set
‹
– Pruning method: delete any final sequence of
conditions that maximizes v
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
25
Direct Method: RIPPER
O
Building a Rule Set:
– Use sequential covering algorithm
Finds the best rule that covers the current set of
positive examples
‹ Eliminate both positive and negative examples
covered by the rule
‹
– Each time a rule is added to the rule set,
compute the new description length
stop adding new rules when the new description
length is d bits longer than the smallest description
length obtained so far
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
26
Direct Method: RIPPER
O
Optimize the rule set:
– For each rule r in the rule set R
‹
Consider 2 alternative rules:
– Replacement rule (r*): grow new rule from scratch
– Revised rule(r’): add conjuncts to extend the rule r
Compare the rule set for r against the rule set for r*
and r’
‹ Choose rule set that minimizes MDL principle
‹
– Repeat rule generation and rule optimization
for the remaining positive examples
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
27
Indirect Methods
P
No
Yes
Q
No
-
Rule Set
R
Yes
No
Yes
+
+
Q
No
-
© Tan,Steinbach, Kumar
Yes
r1: (P=No,Q=No) ==> r2: (P=No,Q=Yes) ==> +
r3: (P=Yes,R=No) ==> +
r4: (P=Yes,R=Yes,Q=No) ==> r5: (P=Yes,R=Yes,Q=Yes) ==> +
+
Introduction to Data Mining
4/18/2004
28
Indirect Method: C4.5rules
Extract rules from an unpruned decision tree
O For each rule, r: A → y,
– consider an alternative rule r’: A’ → y where A’
is obtained by removing one of the conjuncts
in A
– Compare the pessimistic error rate for r
against all r’s
– Prune if one of the r’s has lower pessimistic
error rate
– Repeat until we can no longer improve
generalization error
O
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
29
Indirect Method: C4.5rules
O
Instead of ordering the rules, order subsets of
rules (class ordering)
– Each subset is a collection of rules with the
same rule consequent (class)
– Compute description length of each subset
Description length = L(error) + g L(model)
‹ g is a parameter that takes into account the
presence of redundant attributes in a rule set
(default value = 0.5)
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
30
Example
Name
Give Birth
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
yes
no
no
yes
no
no
yes
no
yes
yes
no
no
yes
no
no
no
no
no
yes
no
Lay Eggs
no
yes
yes
no
yes
yes
no
yes
no
no
yes
yes
no
yes
yes
yes
yes
yes
no
yes
© Tan,Steinbach, Kumar
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Live in Water Have Legs
no
no
yes
yes
sometimes
no
no
no
no
yes
sometimes
sometimes
no
yes
sometimes
no
no
no
yes
no
yes
no
no
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
yes
yes
yes
yes
no
yes
Introduction to Data Mining
Class
mammals
reptiles
fishes
mammals
amphibians
reptiles
mammals
birds
mammals
fishes
reptiles
birds
mammals
fishes
amphibians
reptiles
mammals
birds
mammals
birds
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31
C4.5 versus C4.5rules versus RIPPER
C4.5rules:
Give
Birth?
(Give Birth=No, Can Fly=Yes) → Birds
(Give Birth=No, Live in Water=Yes) → Fishes
No
Yes
(Give Birth=Yes) → Mammals
(Give Birth=No, Can Fly=No, Live in Water=No) → Reptiles
Live In
Water?
Mammals
Yes
( ) → Amphibians
RIPPER:
No
(Live in Water=Yes) → Fishes
(Have Legs=No) → Reptiles
Sometimes
Fishes
Yes
Birds
© Tan,Steinbach, Kumar
(Give Birth=No, Can Fly=No, Live In Water=No)
→ Reptiles
Can
Fly?
Amphibians
(Can Fly=Yes,Give Birth=No) → Birds
No
() → Mammals
Reptiles
Introduction to Data Mining
4/18/2004
32
C4.5 versus C4.5rules versus RIPPER
C4.5 and C4.5rules:
PREDICTED CLASS
Amphibians Fishes Reptiles Birds
2
0
0
ACTUAL Amphibians
0
2
0
CLASS Fishes
1
0
3
Reptiles
1
0
0
Birds
0
0
1
Mammals
0
0
0
3
0
Mammals
0
1
0
0
6
0
0
0
2
0
Mammals
2
0
1
1
4
RIPPER:
PREDICTED CLASS
Amphibians Fishes Reptiles Birds
0
0
0
ACTUAL Amphibians
0
3
0
CLASS Fishes
0
0
3
Reptiles
0
0
1
Birds
0
2
1
Mammals
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
33
Advantages of Rule-Based Classifiers
As highly expressive as decision trees
O Easy to interpret
O Easy to generate
O Can classify new instances rapidly
O Performance comparable to decision trees
O
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
34
Instance-Based Classifiers
Set of Stored Cases
Atr1
……...
AtrN
Class
A
• Store the training records
• Use training records to
predict the class label of
unseen cases
B
B
C
A
Unseen Case
Atr1
……...
AtrN
C
B
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
35
Instance Based Classifiers
O
Examples:
– Rote-learner
Memorizes entire training data and performs
classification only if attributes of record match one of
the training examples exactly
‹
– Nearest neighbor
Uses k “closest” points (nearest neighbors) for
performing classification
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
36
Nearest Neighbor Classifiers
O
Basic idea:
– If it walks like a duck, quacks like a duck, then
it’s probably a duck
Compute
Distance
Training
Records
Test
Record
Choose k of the
“nearest” records
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
37
Nearest-Neighbor Classifiers
Unknown record
O
Requires three things
– The set of stored records
– Distance Metric to compute
distance between records
– The value of k, the number of
nearest neighbors to retrieve
O
To classify an unknown record:
– Compute distance to other
training records
– Identify k nearest neighbors
– Use class labels of nearest
neighbors to determine the
class label of unknown record
(e.g., by taking majority vote)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
38
Definition of Nearest Neighbor
X
X
(b) 2-nearest neighbor
(c) 3-nearest neighbor
X
(a) 1-nearest neighbor
K-nearest neighbors of a record x are data points
that have the k smallest distance to x
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
39
4/18/2004
40
1 nearest-neighbor
Voronoi Diagram
© Tan,Steinbach, Kumar
Introduction to Data Mining
Nearest Neighbor Classification
O
Compute distance between two points:
– Euclidean distance
d ( p, q ) =
O
−q )
∑ ( pi
2
i
i
Determine the class from nearest neighbor list
– take the majority vote of class labels among
the k-nearest neighbors
– Weigh the vote according to distance
‹
weight factor, w = 1/d2
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
41
Nearest Neighbor Classification…
O
Choosing the value of k:
– If k is too small, sensitive to noise points
– If k is too large, neighborhood may include points from
other classes
X
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
42
Nearest Neighbor Classification…
O
Scaling issues
– Attributes may have to be scaled to prevent
distance measures from being dominated by
one of the attributes
– Example:
height of a person may vary from 1.5m to 1.8m
‹ weight of a person may vary from 90lb to 300lb
‹ income of a person may vary from $10K to $1M
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
43
Nearest Neighbor Classification…
O
Problem with Euclidean measure:
– High dimensional data
‹
curse of dimensionality
– Can produce counter-intuitive results
111111111110
vs
100000000000
011111111111
000000000001
d = 1.4142
d = 1.4142
‹
Solution: Normalize the vectors to unit length
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
44
Nearest neighbor Classification…
O
k-NN classifiers are lazy learners
– It does not build models explicitly
– Unlike eager learners such as decision tree
induction and rule-based systems
– Classifying unknown records are relatively
expensive
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
45
Example: PEBLS
O
PEBLS: Parallel Examplar-Based Learning
System (Cost & Salzberg)
– Works with both continuous and nominal
features
‹For
nominal features, distance between two
nominal values is computed using modified value
difference metric (MVDM)
– Each record is assigned a weight factor
– Number of nearest neighbor, k = 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
46
Example: PEBLS
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
d(Single,Married)
2
No
Married
100K
No
= | 2/4 – 0/4 | + | 2/4 – 4/4 | = 1
3
No
Single
70K
No
4
Yes
Married
120K
No
d(Single,Divorced)
5
No
Divorced 95K
Yes
6
No
Married
No
d(Married,Divorced)
7
Yes
Divorced 220K
No
= | 0/4 – 1/2 | + | 4/4 – 1/2 | = 1
8
No
Single
85K
Yes
d(Refund=Yes,Refund=No)
9
No
Married
75K
No
10
No
Single
90K
Yes
= | 0/3 – 3/7 | + | 3/3 – 4/7 | = 6/7
60K
Distance between nominal attribute values:
= | 2/4 – 1/2 | + | 2/4 – 1/2 | = 0
10
Refund
Marital Status
Class
Single
Married
Divorced
Yes
2
0
1
No
2
4
1
© Tan,Steinbach, Kumar
Class
Yes
No
Yes
0
3
No
3
4
d (V1 ,V2 ) =
Introduction to Data Mining
∑
i
n1i n 2 i
−
n1 n 2
4/18/2004
47
Example: PEBLS
Tid Refund Marital
Status
Taxable
Income Cheat
X
Yes
Single
125K
No
Y
No
Married
100K
No
10
Distance between record X and record Y:
d
∆ ( X , Y ) = w X wY ∑ d ( X i , Yi ) 2
i =1
where:
wX =
Number of times X is used for prediction
Number of times X predicts correctly
wX ≅ 1 if X makes accurate prediction most of the time
wX > 1 if X is not reliable for making predictions
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
48
Bayes Classifier
A probabilistic framework for solving classification
problems
O Conditional Probability:
P( A, C )
P (C | A) =
P ( A)
O
P( A | C ) =
O
P( A, C )
P (C )
Bayes theorem:
P (C | A) =
© Tan,Steinbach, Kumar
P ( A | C ) P (C )
P ( A)
Introduction to Data Mining
4/18/2004
49
Example of Bayes Theorem
O
Given:
– A doctor knows that meningitis causes stiff neck 50% of the
time
– Prior probability of any patient having meningitis is 1/50,000
– Prior probability of any patient having stiff neck is 1/20
O
If a patient has stiff neck, what’s the probability
he/she has meningitis?
P( M | S ) =
© Tan,Steinbach, Kumar
P ( S | M ) P ( M ) 0.5 ×1 / 50000
=
= 0.0002
P( S )
1 / 20
Introduction to Data Mining
4/18/2004
50
Bayesian Classifiers
O
Consider each attribute and class label as random
variables
O
Given a record with attributes (A1, A2,…,An)
– Goal is to predict class C
– Specifically, we want to find the value of C that
maximizes P(C| A1, A2,…,An )
O
Can we estimate P(C| A1, A2,…,An ) directly from
data?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
51
Bayesian Classifiers
O
Approach:
– compute the posterior probability P(C | A1, A2, …, An) for
all values of C using the Bayes theorem
P (C | A A K A ) =
1
2
n
P ( A A K A | C ) P (C )
P(A A K A )
1
2
n
1
2
n
– Choose value of C that maximizes
P(C | A1, A2, …, An)
– Equivalent to choosing value of C that maximizes
P(A1, A2, …, An|C) P(C)
O
How to estimate P(A1, A2, …, An | C )?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
52
Naïve Bayes Classifier
O
Assume independence among attributes Ai when class is
given:
– P(A1, A2, …, An |C) = P(A1| Cj) P(A2| Cj)… P(An| Cj)
– Can estimate P(Ai| Cj) for all Ai and Cj.
– New point is classified to Cj if P(Cj) Π P(Ai| Cj) is
maximal.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
53
How to Estimate Probabilities from Data?
c
c
O
c
Tid
Refund
Marital
Status
Taxable
Income
Evade
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
60K
No
7
Yes
Divorced
220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
Class: P(C) = Nc/N
– e.g., P(No) = 7/10,
P(Yes) = 3/10
O
For discrete attributes:
P(Ai | Ck) = |Aik|/ Nc k
– where |Aik| is number of
instances having attribute
Ai and belongs to class Ck
– Examples:
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
P(Status=Married|No) = 4/7
P(Refund=Yes|Yes)=0
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54
How to Estimate Probabilities from Data?
O
For continuous attributes:
– Discretize the range into bins
one ordinal attribute per bin
‹ violates independence assumption
‹
k
– Two-way split: (A < v) or (A > v)
‹
choose only one of the two splits as new attribute
– Probability density estimation:
Assume attribute follows a normal distribution
‹ Use data to estimate parameters of distribution
(e.g., mean and standard deviation)
‹ Once probability distribution is known, can use it to
estimate the conditional probability P(Ai|c)
‹
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
55
How to Estimate Probabilities from Data?
Tid
1
Refund
Yes
Marital
Status
Single
Taxable
Income
125K
Evade
O
Normal distribution:
1
P( A | c ) =
e
2πσ
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced
95K
Yes
6
No
Married
60K
No
7
Yes
Divorced
220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
i
j
−
( Ai − µ ij ) 2
2 σ ij2
2
ij
– One for each (Ai,ci) pair
O
For (Income, Class=No):
– If Class=No
‹
sample mean = 110
‹
sample variance = 2975
10
1
P ( Income = 120 | No) =
e
2π (54.54)
© Tan,Steinbach, Kumar
Introduction to Data Mining
−
( 120 −110 ) 2
2 ( 2975 )
= 0.0072
4/18/2004
56
Example of Naïve Bayes Classifier
Given a Test Record:
X = (Refund = No, Married, Income = 120K)
naive Bayes Classifier:
P(Refund=Yes|No) = 3/7
P(Refund=No|No) = 4/7
P(Refund=Yes|Yes) = 0
P(Refund=No|Yes) = 1
P(Marital Status=Single|No) = 2/7
P(Marital Status=Divorced|No)=1/7
P(Marital Status=Married|No) = 4/7
P(Marital Status=Single|Yes) = 2/7
P(Marital Status=Divorced|Yes)=1/7
P(Marital Status=Married|Yes) = 0
For taxable income:
If class=No:
sample mean=110
sample variance=2975
If class=Yes: sample mean=90
sample variance=25
© Tan,Steinbach, Kumar
O
P(X|Class=No) = P(Refund=No|Class=No)
× P(Married| Class=No)
× P(Income=120K| Class=No)
= 4/7 × 4/7 × 0.0072 = 0.0024
O
P(X|Class=Yes) = P(Refund=No| Class=Yes)
× P(Married| Class=Yes)
× P(Income=120K| Class=Yes)
= 1 × 0 × 1.2 × 10-9 = 0
Since P(X|No)P(No) > P(X|Yes)P(Yes)
Therefore P(No|X) > P(Yes|X)
=> Class = No
Introduction to Data Mining
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57
Naïve Bayes Classifier
If one of the conditional probability is zero, then
the entire expression becomes zero
O Probability estimation:
O
Original : P ( Ai | C ) =
N ic
Nc
N +1
Laplace : P( Ai | C ) = ic
Nc + c
N + mp
m - estimate : P( Ai | C ) = ic
Nc + m
© Tan,Steinbach, Kumar
Introduction to Data Mining
c: number of classes
p: prior probability
m: parameter
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Example of Naïve Bayes Classifier
Name
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
Give Birth
yes
Give Birth
yes
no
no
yes
no
no
yes
no
yes
yes
no
no
yes
no
no
no
no
no
yes
no
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Can Fly
no
© Tan,Steinbach, Kumar
Live in Water Have Legs
no
no
yes
yes
sometimes
no
no
no
no
yes
sometimes
sometimes
no
yes
sometimes
no
no
no
yes
no
Class
yes
no
no
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
yes
yes
yes
yes
no
yes
mammals
non-mammals
non-mammals
mammals
non-mammals
non-mammals
mammals
non-mammals
mammals
non-mammals
non-mammals
non-mammals
mammals
non-mammals
non-mammals
non-mammals
mammals
non-mammals
mammals
non-mammals
Live in Water Have Legs
yes
no
A: attributes
M: mammals
N: non-mammals
6 6 2 2
P ( A | M ) = × × × = 0.06
7 7 7 7
1 10 3 4
P ( A | N ) = × × × = 0.0042
13 13 13 13
7
P ( A | M ) P( M ) = 0.06 × = 0.021
20
13
P ( A | N ) P( N ) = 0.004 × = 0.0027
20
Class
?
Introduction to Data Mining
P(A|M)P(M) > P(A|N)P(N)
=> Mammals
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59
Naïve Bayes (Summary)
O
Robust to isolated noise points
O
Handle missing values by ignoring the instance
during probability estimate calculations
O
Robust to irrelevant attributes
O
Independence assumption may not hold for some
attributes
– Use other techniques such as Bayesian Belief
Networks (BBN)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
60
Artificial Neural Networks (ANN)
X1 X2 X3
1
1
1
1
0
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
1
0
Y
Input
0
1
1
1
0
0
1
0
X1
Black box
Output
X2
Y
X3
Output Y is 1 if at least two of the three inputs are equal to 1.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
61
Artificial Neural Networks (ANN)
X1 X2 X3
1
1
1
1
0
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
1
0
Y
0
1
1
1
0
0
1
0
Input
nodes
Black box
X1
Output
node
0.3
0.3
X2
X3
0.3
Y
Σ
t=0.4
Y = I ( 0 .3 X 1 + 0 .3 X 2 + 0 .3 X 3 − 0 .4 > 0 )
1
where I ( z ) = 
0
© Tan,Steinbach, Kumar
if z is true
otherwise
Introduction to Data Mining
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62
Artificial Neural Networks (ANN)
O
O
O
Model is an assembly of
inter-connected nodes
and weighted links
Output node sums up
each of its input value
according to the weights
of its links
Input
nodes
Black box
X1
Output
node
w1
w2
X2
Y
Σ
w3
X3
t
Perceptron Model
Y = I ( ∑ wi X i − t )
Compare output node
against some threshold t
or
i
Y = sign( ∑ wi X i − t )
i
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
63
General Structure of ANN
x1
x2
x3
Input
Layer
x4
x5
Input
I1
I2
Hidden
Layer
I3
Neuron i
wi1
wi2
wi3
Si
Activation
function
g(Si )
Output
Oi
Oi
threshold, t
Output
Layer
Training ANN means learning
the weights of the neurons
y
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
64
Algorithm for learning ANN
O
Initialize the weights (w0, w1, …, wk)
O
Adjust the weights in such a way that the output
of ANN is consistent with class labels of training
examples
2
– Objective function: E = ∑ [Yi − f ( wi , X i )]
i
– Find the weights wi’s that minimize the above
objective function
‹
e.g., backpropagation algorithm (see lecture notes)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
65
Support Vector Machines
O
Find a linear hyperplane (decision boundary) that will separate the data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
66
Support Vector Machines
B1
O
One Possible Solution
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Support Vector Machines
B2
O
Another possible solution
© Tan,Steinbach, Kumar
Introduction to Data Mining
Support Vector Machines
B2
O
Other possible solutions
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
69
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Support Vector Machines
B1
B2
O
O
Which one is better? B1 or B2?
How do you define better?
© Tan,Steinbach, Kumar
Introduction to Data Mining
Support Vector Machines
B1
B2
b21
b22
margin
b11
b12
O
Find hyperplane maximizes the margin => B1 is better than B2
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
71
Support Vector Machines
B1
r r
w• x + b = 0
r r
w • x + b = +1
r r
w • x + b = −1
b11
 1
r
f (x) = 
− 1
r r
if w • x + b ≥ 1
r r
if w • x + b ≤ − 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
b12
Margin =
4/18/2004
2
r 2
|| w ||
72
Support Vector Machines
O
2
We want to maximize: Margin = r 2
|| w ||
r
|| w ||2
– Which is equivalent to minimizing: L( w) =
2
– But subjected to the following constraints:
r r
if w • x i + b ≥ 1
 1
r
f ( xi ) = 
r r
 − 1 if w • x i + b ≤ − 1
‹
This is a constrained optimization problem
– Numerical approaches to solve it (e.g., quadratic programming)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
73
Support Vector Machines
O
What if the problem is not linearly separable?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
74
Support Vector Machines
O
What if the problem is not linearly separable?
– Introduce slack variables
r
‹ Need to minimize:
|| w ||2
 N k
+ C  ∑ ξi 
L( w ) =
2
 i =1 
‹
Subject to:
 1
r
f ( xi ) = 
− 1
© Tan,Steinbach, Kumar
r r
if w • x i + b ≥ 1 - ξ i
r r
if w • x i + b ≤ − 1 + ξ i
Introduction to Data Mining
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Nonlinear Support Vector Machines
O
What if decision boundary is not linear?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
76
Nonlinear Support Vector Machines
O
Transform data into higher dimensional space
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
77
Ensemble Methods
O
Construct a set of classifiers from the training
data
O
Predict class label of previously unseen records
by aggregating predictions made by multiple
classifiers
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
78
General Idea
D
Step 1:
Create Multiple
Data Sets
Step 2:
Build Multiple
Classifiers
D1
D2
C1
C2
Step 3:
Combine
Classifiers
....
Original
Training data
Dt-1
Dt
Ct -1
Ct
C*
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
79
Why does it work?
O
Suppose there are 25 base classifiers
– Each classifier has error rate, ε = 0.35
– Assume classifiers are independent
– Probability that the ensemble classifier makes
a wrong prediction:
 25  i
ε (1 − ε ) 25−i = 0.06
i =13 

25
∑  i
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
80
Examples of Ensemble Methods
O
How to generate an ensemble of classifiers?
– Bagging
– Boosting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
81
Bagging
O
Sampling with replacement
Original Data
Bagging (Round 1)
Bagging (Round 2)
Bagging (Round 3)
1
7
1
1
2
8
4
8
3
10
9
5
4
8
1
10
5
2
2
5
6
5
3
5
7
10
2
9
8
10
7
6
9
5
3
3
10
9
2
7
O
Build classifier on each bootstrap sample
O
Each sample has probability (1 – 1/n)n of being
selected
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
82
Boosting
O
An iterative procedure to adaptively change
distribution of training data by focusing more on
previously misclassified records
– Initially, all N records are assigned equal
weights
– Unlike bagging, weights may change at the
end of boosting round
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
83
Boosting
Records that are wrongly classified will have their
weights increased
O Records that are classified correctly will have
their weights decreased
O
Original Data
Boosting (Round 1)
Boosting (Round 2)
Boosting (Round 3)
1
7
5
4
2
3
4
4
3
2
9
8
4
8
4
10
5
7
2
4
6
9
5
5
7
4
1
4
8
10
7
6
9
6
4
3
10
3
2
4
• Example 4 is hard to classify
• Its weight is increased, therefore it is more
likely to be chosen again in subsequent rounds
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
84
Example: AdaBoost
O
Base classifiers: C1, C2, …, CT
O
Error rate:
1
εi =
N
O
∑ w δ (C ( x ) ≠ y )
N
j
i
j
j
j =1
Importance of a classifier:
1  1 − εi 

αi = ln
2  ε i 
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
85
Example: AdaBoost
O
Weight update:
exp −α j if C j ( xi ) = yi
wi

α
 exp j if C j ( xi ) ≠ yi
where Z j is the normalization factor
( j +1)
wi( j )
=
Zj
If any intermediate rounds produce error rate
higher than 50%, the weights are reverted back
to 1/n and the resampling procedure is repeated
O Classification:
T
O
C * ( x ) = arg max ∑ α jδ (C j ( x ) = y )
y
© Tan,Steinbach, Kumar
Introduction to Data Mining
j =1
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Illustrating AdaBoost
Initial weights for each data point
0.1
Original
Data
0.1
0.1
0.0094
0.4623
Data points
for training
- - - - - ++
+++
B1
Boosting
Round 1
0.0094
- - - - - - -
+++
© Tan,Steinbach, Kumar
Introduction to Data Mining
α = 1.9459
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Illustrating AdaBoost
B1
Boosting
Round 1
0.0094
0.0094
+++
0.4623
- - - - - - -
α = 1.9459
B2
Boosting
Round 2
0.0009
0.3037
- - -
0.0422
- - - - - ++
α = 2.9323
B3
0.0276
0.1819
0.0038
Boosting
Round 3
+++
++ ++ + ++
Overall
+++
- - - - - ++
© Tan,Steinbach, Kumar
Introduction to Data Mining
α = 3.8744
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88