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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

Classify records by using a collection of “if…then…”
rules

Rule: (Condition)  y
– where
Condition=(A1 op v1) & (A2 op v2)…(Ak op vk)

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
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
Blood Type
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

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
Coverage of a rule:
– Fraction of records
that satisfy the
antecedent of a rule
 Accuracy of a rule:
– Fraction of records
that satisfy both the
antecedent and
consequent of a rule

Tid Refund Marital
Status
Taxable
Income Class
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
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
No
10
Ex: (Status=Single)  No
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

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

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
Marita l
Status
{Single,
Divorced}
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
NO
{Married}
> 80K
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

Rules are no longer mutually exclusive
– A record may trigger more than one rule
– Solution?
Ordered rule set
 Unordered rule set – use voting schemes


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

Rules are rank ordered according to their priority
– An ordered rule set is known as a decision list

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
© Tan,Steinbach, Kumar
Blood Type
cold
Give Birth
Can Fly
Live in Water
Class
no
no
sometimes
?
Introduction to Data Mining
4/18/2004
11
Rule Ordering Schemes

Rule-based ordering
– Individual rules are ranked based on their quality

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
Holte’s 1R Algorithm:


Direct Method:
–
Extract rules directly from data
–
e.g.: RIPPER, CN2, Holte’s 1R
for each attribute
for each value of that attribute
find class distribution for attr/value
conc = most frequent class
make rule: attribute=value -> conc
calculate error rate of rule set
select rule set with lowest error rate
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
(i) Original Data
© Tan,Steinbach, Kumar
(ii) Step 1
Introduction to Data Mining
4/18/2004
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Example of Sequential Covering…
R1
R1
R2
(iii) Step 2
© Tan,Steinbach, Kumar
(iv) Step 3
Introduction to Data Mining
4/18/2004
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Aspects of Sequential Covering

Rule Growing

Instance Elimination

Rule Evaluation

Stopping Criterion

Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
17
Rule Growing

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
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Rule Growing (Examples)

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

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
Aspects of Sequential Covering

Rule Growing

Instance Elimination

Rule Evaluation

Stopping Criterion

Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Instance Elimination

Why do we need to
eliminate instances?
R3
– Otherwise, the next rule is
identical to previous rule

Why do we remove positive
instances?
R1
+
class = +
– Ensure that the next rule is
different

Why do we remove negative
instances?
-
class = -
– Prevent underestimating
accuracy of rule
– Compare rules R2 and R3 in
the diagram
© Tan,Steinbach, Kumar
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Introduction to Data Mining
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Aspects of Sequential Covering

Rule Growing

Instance Elimination

Rule Evaluation

Stopping Criterion

Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
22
Rule Evaluation

Metrics:
nc
– Accuracy 
n
nc  1
– Laplace 
nk
nc  kp
– M-estimate 
nk
n : Number of instances
covered by rule
nc : Number of instances
covered by rule
k : Number of classes
– Foil-gain
p : Prior probability
FOIL _ Gain  pos'(log 2
pos'
pos
 log 2
)
pos' neg '
pos  neg
Pos/neg are # of positive/negative tuples covered by R.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
23
Aspects of Sequential Covering

Rule Growing

Instance Elimination

Rule Evaluation

Stopping Criterion

Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
24
Stopping Criterion and Rule Pruning


Stopping criterion
– Compute the gain
– If gain is not significant, discard the new rule
FOIL _ Prune( R) 
Rule Pruning
– Similar to post-pruning of decision trees
– Reduced Error Pruning:
pos  neg
pos  neg
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
25
Summary of Direct Method

Grow a single rule

Remove Instances from rule

Prune the rule (if necessary)

Add rule to Current Rule Set

Repeat
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
26
Direct Method: RIPPER


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
27
Direct Method: RIPPER

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
28
Direct Method: RIPPER

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
29
Direct Method: RIPPER

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
30
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
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Indirect Method: C4.5rules
Extract rules from an unpruned decision tree
 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

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
32
Indirect Method: C4.5rules

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
33
Example
Name
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
© Tan,Steinbach, Kumar
Give Birth
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
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Introduction to Data Mining
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
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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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
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C4.5 versus C4.5rules versus RIPPER
Confusion Matrix
C4.5 and C4.5rules:
PREDICTED CLASS
Amphibians Fishes Reptiles Birds
ACTUAL Amphibians
2
0
0
CLASS Fishes
0
2
0
Reptiles
1
0
3
Birds
1
0
0
Mammals
0
0
1
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
ACTUAL Amphibians
0
0
0
CLASS Fishes
0
3
0
Reptiles
0
0
3
Birds
0
0
1
Mammals
0
2
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
36
Advantages of Rule-Based Classifiers
As highly expressive as decision trees
 Easy to interpret
 Easy to generate
 Can classify new instances rapidly
 Performance comparable to decision trees

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
37
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
38
Instance Based Classifiers

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
39
Nearest Neighbor Classifiers

Basic idea:
– If it walks like a duck, quacks like a duck, then it’s
probably a duck
Compute
Distance
Training
Records
© Tan,Steinbach, Kumar
Test
Record
Choose k of the
“nearest” records
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Nearest-Neighbor Classifiers
Unknown record

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

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
41
Definition of Nearest Neighbor
X
(a) 1-nearest neighbor
X
X
(b) 2-nearest neighbor
(c) 3-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
42
1 nearest-neighbor
Voronoi Diagram
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
43
Nearest Neighbor Classification

Compute distance between two points:
– Euclidean distance
d ( p, q ) 

 ( pi
i
q )
2
i
Determine the class from nearest neighbor list
– take the majority vote of class labels among the knearest neighbors
– Weigh the vote according to distance

weight factor, w = 1/d2
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
44
Nearest Neighbor Classification…

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
45
Nearest Neighbor Classification…

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
46
Nearest Neighbor Classification…

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
47
Nearest neighbor Classification…

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
48
Example: PEBLS

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
49
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
Marital Status
Class
Refund
Single
Married
Divorced
Yes
2
0
1
No
2
4
1
© Tan,Steinbach, Kumar
Class
Yes
No
Yes
0
3
No
3
4
Introduction to Data Mining
d (V1 ,V2 )  
i
4/18/2004
n1i n2i

n1 n2
50
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 )  wX wY  d ( X i , Yi )
2
i 1
where:
Number of times X is used for prediction
wX 
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
51
Naïve Bayes Classifier
Thomas Bayes
1702 - 1761
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
52
Grasshoppers
Katydids
Antenna Length
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Abdomen Length
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
53
With a lot of data, we can build a histogram. Let us
just build one for “Antenna Length” for now…
Antenna Length
10
Katydids
Grasshoppers 9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
54
We can leave the histograms as they
are, or we can summarize them with
two normal distributions.
Let us us two normal distributions for
ease of visualization in the following
slides…
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
55
p(cj | d) = probability of class cj, given that we have observed d
P(Grasshopper | 3 ) = 10 / (10 + 2)
= 0.833
P(Katydid | 3 )
= 0.166
= 2 / (10 + 2)
10
2
3
Antennae length is 3
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
56
p(cj | d) = probability of class cj, given that we have observed d
P(Grasshopper | 7 ) = 3 / (3 + 9)
= 0.250
P(Katydid | 7 )
= 0.750
= 9 / (3 + 9)
9
3
7
Antennae length is 7
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
57
p(cj | d) = probability of class cj, given that we have observed d
P(Grasshopper | 5 ) = 6 / (6 + 6)
= 0.500
P(Katydid | 5 )
= 0.500
= 6 / (6 + 6)
6 6
5
Antennae length is 5
© Tan,Steinbach, Kumar
Introduction to Data Mining
?
4/18/2004
58
Bayesian Theorem

Given training data X, posteriori probability of a hypothesis H,
P(H|X) follows the Bayes theorem
Grasshopper
P(H | X )  P( X | H )P(H )
P( X )

Katydid
Informally, this can be written as
posterior =likelihood x prior probability/ evidence


Predicts X belongs to Ci iff the probability P(Ci|X) is the highest
among all the P(Ck|X) foi all the k classes
Practical difficulty:
require initial knowledge of many probabilities
=>significant computational cost
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
59
Example
Is Chris a Male or Female
Female?
?
Name
Sex
Chris
Male
Claudia
Female
Chris
Female
Chris
Female
Alberto
Male
Karin
Female
Nina
Female
Sergio
Male
Chris
Female
© Tan,Steinbach, Kumar
2010/03/26
P(H | X )  P( X | H )P(H )
P( X )
p(male
male |Chris) = 1/3 * 3/9
4/9
p(female
female |Chris) = 3/6 * 6/9
4/9
=>Chris is a Female
Introduction to Data Mining
4/18/2004
60
60
Bayes Classifier
A probabilistic framework for solving classification
problems
 Conditional Probability:
P( A, C )
P(C | A) 
P( A)

P( A, C )
P( A | C ) 
P(C )

Bayes theorem:
P ( A | C ) P(C )
P(C | A) 
P ( A)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
61
Bayesian Classifiers

Consider each attribute and class label as random
variables

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 )

Can we estimate P(C| A1, A2,…,An ) directly from data?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
63
Bayesian Classifiers

Approach:
– compute the posterior probability P(C | A1, A2, …, An) for all
values of C using the Bayes theorem
P (C | A A  A ) 
1
2
n
P ( A A  A | C ) P (C )
P( A A  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)

How to estimate P(A1, A2, …, An | C )?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
64
Naïve Bayes Classifier

Assume independence among attributes Ai when class is given:
– P(A1, A2, …, An |C) = P(A1| Cj) P(A2| Cj)… P(An| Cj) difficult
A1 (Height=170), A2 (Weight=50), …
– Can estimate P(Ai| Cj) for all Ai and Cj
much easier
– New point is classified to Cj if P(Cj) P(Ai| Cj) is maximal.
P (C | A A  A ) 
1
2
n
P ( A A  A | C ) P (C )
P( A A  A )
1
2
n
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
2
n
4/18/2004
65
How to Estimate
Probabilities
from
Data?
l
l
c
at
o
eg
a
c
i
r
c
at
o
eg
a
c
i
r
c
on
u
it n
s
u
o
s
s
a 
cl
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
For discrete attributes:
P(Ai | Ck) = |Aik|/ Nck
– 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
4/18/2004
66
Training dataset
Class:
C1:buys_computer=‘yes’
C2:buys_computer=‘no’
Data sample:
X (age<=30, Income=medium, Student=yes, Credit_rating=Fair)
age
<=30
<=30
30…40
>40
>40
>40
31…40
<=30
<=30
>40
<=30
31…40
31…40
>40
© Tan,Steinbach, Kumar
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
67
Naïve Bayesian Classifier: Example



X=(age<=30, income=medium, student=yes, credit_rating=fair)
P(buys_computer=“yes”)=9/14, P(buys_computer=“no”)=5/14
Compute P(X|Ci) for each class
P(age=“<30” | buys_computer=“yes”) = 2/9=0.222
P(income=“medium” | buys_computer=“yes”)= 4/9 =0.444
P(student=“yes” | buys_computer=“yes)= 6/9 =0.667
P(credit_rating=“fair” | buys_computer=“yes”)=6/9=0.667
∵ X=(age<=30 ,income =medium, student=yes,credit_rating=fair)
 P(X|Ci) : P(X|buys_computer=“yes”)= 0.222 x 0.444 x 0.667 x 0.667 =0.044
P(credit_rating=“fair” | buys_computer=“no”)=2/5=0.4
P(age=“<30” | buys_computer=“no”) = 3/5 =0.6
P(income=“medium” | buys_computer=“no”) = 2/5 = 0.4
P(student=“yes” | buys_computer=“no”)= 1/5=0.2
=> P(X|buys_computer=“no”)= 0.6 x 0.4 x 0.2 x 0.4 =0.019
P(X|Ci)*P(Ci ) :
P(X|buys_computer=“yes”) * P(buys_computer=“yes”)=0.028
P(X|buys_computer=“no”) * P(buys_computer=“no”)=0.007
=> X belongs to class “buys_computer=yes”
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
68
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

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

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
4/18/2004
69
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
Class
?
Introduction to Data Mining
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
P(A|M)P(M) > P(A|N)P(N)
=> Mammals
4/18/2004
70
How to Estimate Probabilities from Data?

For continuous attributes:
– Discretize the range into bins
one ordinal attribute per bin
 violates independence assumption

– 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
71
l
s
al
uProbabilities
ca
c
How tooriEstimate
from
Data?
i
o
r
u
o
c
Tid
e
at
Refund
g
c
e
at
Marital
Status
g
c
t
n
o
Taxable
Income
in
as
l
c
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
s

Normal distribution:
1
P( A | c ) 
e
2
i
j

( Ai   ij ) 2
2  ij2
2
ij
– One for each (Ai,ci) pair

For (Income, Class=No):
– If Class=No

sample mean = 110

sample variance = 2975
10
1
P( Income  120 | No) 
e
2 (54.54)
© Tan,Steinbach, Kumar
Introduction to Data Mining

( 120110 ) 2
2 ( 2975 )
 0.0072
4/18/2004
72
Naïve Bayes Classifier
If one of the conditional probability is zero, then the
entire expression becomes zero
 Probability estimation:

N ic
Original : P( Ai | C ) 
Nc
N ic  1
Laplace : P( Ai | C ) 
Nc  c
N ic  mp
m - estimate : P( Ai | C ) 
Nc  m
© Tan,Steinbach, Kumar
Introduction to Data Mining
c: number of classes
p: prior probability
m: parameter
4/18/2004
73
Naïve Bayes (Summary)

Robust to isolated noise points

Handle missing values by ignoring the instance during
probability estimate calculations

Robust to irrelevant attributes

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
74
Naïve Bayesian Classifier: Comments

Advantages :
– Easy to implement
– Good results obtained in most of the cases

Disadvantages
– Assumption: class conditional independence
– Practically, dependencies exist among variables
– E.g., hospitals: patients: Profile: age, family history etc
Symptoms: fever, cough etc., Disease: lung cancer, diabetes etc
– Dependencies among these cannot be modeled by Naïve Bayesian
Classifier

How to deal with these dependencies?
– Bayesian Belief Networks
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
75
Bayesian Networks

Bayesian belief network allows a subset of the variables
conditionally independent

A graphical model of causal relationships
– Represents dependency among the variables
– Gives a specification of joint probability distribution
Y
X
Z
P
© Tan,Steinbach, Kumar
2010/03/26
Nodes: random variables
Links: dependency
X,Y are the parents of Z, and Y is the
parent of P
No dependency between Z and P
Has no loops or cycles
Introduction to Data Mining
4/18/2004
76
76
Bayesian Belief Network: An Example
•One conditional probability table (CPT) for each variable
Family
History
Smoker
The CPT for the variable LungCancer:
Shows the conditional probability for each
possible combination of its parents
(FH, S)
LungCancer
PositiveXRay
Emphysema
Dyspnea
Bayesian Belief Networks
© Tan,Steinbach, Kumar
(FH, ~S) (~FH, S) (~FH, ~S)
LC
0.8
0.5
0.7
0.1
~LC
0.2
0.5
0.3
0.9
Derivation of the probability of a particular
combination of values of X, from CPT:
n
P( x1 ,..., xn )   P( xi | Parents(Y i))
i 1
Introduction to Data Mining
4/18/2004
77
Classification: A Mathematical Mapping
Classification:
– predicts categorical class labels
 E.g., Personal homepage classification
– xi = (x1, x2, x3, …), yi = +1 or –1

 x1
: # of a word “homepage”
x2 : # of a word “welcome”
 x3 : …

Mathematically
– x  X = n, y  Y = {+1, –1}
– We want a function f: X  Y
© Tan,Steinbach, Kumar
2010/03/26
Introduction to Data Mining
4/18/2004
78
78
Linear Classification


x
x
x
x
x

x
x
x
x
o
o
x
o
o
o
© Tan,Steinbach, Kumar
o
o
o
o
o

o
o
o
Introduction to Data Mining
Binary Classification problem
The data above the line
belongs to class ‘x’
The data below red line
belongs to class ‘o’
Examples:
– Perceptron
– SVM
– Probabilistic Classifiers
4/18/2004
79
Neural Networks

Analogy to Biological Systems (Indeed a great example of a
good learning system)

Massive Parallelism allowing for computational efficiency

The first learning algorithm came in 1959 (Rosenblatt)
– If a target output value is provided for a single neuron
with fixed inputs, one can incrementally change weights
to learn to produce these outputs using the perceptron
learning rule
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
80
Neural Networks
Terminal Branches
of Axon
Dendrites
Axon
Terminal Branches
of Axon
Dendrites
x1
w1
x2
x3
w2
w3
Binary classifier functions
Threshold activation function
S
Axon
wn
xn
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
81
Activation Function
1
y hid (u ) 
1  e u
Step Threshold
Threshold Activation Function
Sigmoid
Sigmoid Neuron unit function
N
y  u   wn xn
n 1
Linear
Linear Activation functions
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
82
Neural Networks
The general artificial neuron model has five components,
shown in the following list
1.
A set of inputs, xi.
2.
A set of weights, wi.
3.
A bias, u.
4.
An activation function, f.
x0
w0
5.
Neuron output, y
x1
w1
xn
wn
A Neuron
Input
weight
vector x vector w
© Tan,Steinbach, Kumar
Introduction to Data Mining

weighted
sum
k
f
output y
Activation
function
4/18/2004
83
Artificial Neural Networks (ANN)
X1
X2
X3
Y
Input
1
1
1
1
0
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
1
0
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
84
Artificial Neural Networks (ANN)
X1
X2
X3
Y
1
1
1
1
0
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
1
0
0
1
1
1
0
0
1
0
Input
nodes
Black box
X1
X2
X3
Output
node
0.3
0.3
0.3
Y
S
t=0.4
 1, if 0.3x 1  0.3x 2  0.3x 3 - 0.4  0;
yˆ  
 1, if 0.3x 1  0.3x 2  0.3x 3 - 0.4  0.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
85
Artificial Neural Networks (ANN)


Model is an assembly of
inter-connected nodes and
weighted links
Input
nodes
Black box
X1
w1
w2
X2
Output node sums up each
of its input value according
to the weights of its links
Output
node
Y
S
w3
X3
t
Perceptron Model

Compare output node
against some threshold t
Y  I (  wi X i  t )
or
i
Y  sign(  wi X i  t )
i
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
86
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
87
Layered Networks
Inputs
x1
© Tan,Steinbach, Kumar
.
Hidden Neurons
Output Neurons
Outputs
yh1(u)
yo1(u)
y1
x2
yh2(u)
yo2(u)
y2
xn
yhn(u)
yom(u)
ym
Introduction to Data Mining
4/18/2004
88
Perceptron Learning Algorithm
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Let D={xi,yi|i=1,2,…,N} be the training instances
Initialize the weight vector with random values w(0)
repeat
for each instance (xi,yi) D
Compute ŷ
for each weight wj do
(k )
(k+1)
(k)
y
ˆ
y
wj
= wj + λ( i - i ) xij
error
end for
end for
until stopping condition is met
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
89
Algorithm for learning ANN

Initialize the weights (w0, w1, …, wk)

Adjust the weights in such a way that the output of
ANN is consistent with class labels of training
examples
2
N
– Objective function: SSE    yi  yˆ i 
i
– Find the weights wi’s that minimize the above
objective function

e.g., backpropagation algorithm
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
90
Error Back-Propagation (BP)
Training error term: e
Hidden Neurons
uhid,1
yhid(uhid,1)
N1
whid,1
x
yhid
uout,1
yhid(uhid,2)
whid,2
wout,2
N2
x
ytrain
wout,1
yout(Suout,n)
Output Neuron
whid,n
wout,n
yout
e(yout, ytrain)
e
1
e  ( y  yˆ ) 2
2
yhid(uhid,n)
Nn
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
91
Support Vector Machines

Find a linear hyperplane (decision boundary) that will separate the data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
92
Support Vector Machines
B1

One Possible Solution
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
93
Support Vector Machines
B2

Another possible solution
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
94
Support Vector Machines
B2

Other possible solutions
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
95
Support Vector Machines
B1
B2


Which one is better? B1 or B2?
How do you define better?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
96
Support Vector Machines
B1
B2
b21
b22
margin
b11
b12

Find hyperplane maximizes the margin => B1 is better than B2
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
97
Support Vector Machines

x  {x1 , x2 , x3 , x 4 ...}
B1
 
w x  b  0
 
w  x  b  1
 
w  x  b  1
b11
 
if w  x  b  1
1

f ( x)  
 
 1 if w  x  b  1
© Tan,Steinbach, Kumar
Introduction to Data Mining
b12
2
Margin   2
|| w ||
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Support Vector Machines

We want to maximize:
2
Margin   2
|| w ||
 2
– Which is equivalent to minimizing: L( w)  || w ||
2
– But subjected to the following constraints:
 
if w  x i  b  1
1

f ( xi )  
 
 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
99
Support Vector Machines

What if the problem is not linearly separable?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
100
Support Vector Machines

What if the problem is not linearly separable?
– Introduce slack variables
 2
 Need to minimize:
|| w ||
 N k
L ( w) 
 C   i 
2
 i 1 

Subject to:
 
if w  x i  b  1 - i
1

f ( xi )  
 
 1 if w  x i  b  1  i
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
101
Nonlinear Support Vector Machines

What if decision boundary is not linear?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
102
Nonlinear Support Vector Machines

Transform data into higher dimensional space
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
103
Ensemble Methods

Construct a set of classifiers from the training data

Predict class label of previously unseen records by
aggregating predictions made by multiple classifiers
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
104
General Idea
D
Step 1:
Create Multiple
Data Sets
Step 2:
Build Multiple
Classifiers
D1
D2
C1
C2
Step 3:
Combine
Classifiers
© Tan,Steinbach, Kumar
....
Original
Training data
Dt-1
Dt
Ct -1
Ct
C*
Introduction to Data Mining
4/18/2004
105
Why does it work?

Suppose there are 25 base classifiers
– Each classifier has error rate,  = 0.35
– Assume classifiers are independent
– Probability that the ensemble classifier makes a
wrong prediction:
 25  i
25i

(
1


)
 0.06



 i 
i 13 

25
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
106
Examples of Ensemble Methods

How to generate an ensemble of classifiers?

Method
– By manipulating the training set
Bagging
Boosting
– By manipulating the input features
– By manipulating the learning algorithm
– By manipulating the class label
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
107
Bagging

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

Build classifier on each bootstrap sample

Each sample has probability (1 – 1/n)n of being
selected
© Tan,Steinbach, Kumar
Introduction to Data Mining
10
9
2
7
4/18/2004
108
Boosting

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
109
Boosting
Records that are wrongly classified will have their
weights increased
 Records that are classified correctly will have their
weights decreased

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
110
Example: AdaBoost

Base classifiers: C1, C2, …, CT

Error rate:
1
i 
N

 w  C ( x )  y 
N
j 1
j
i
j
j
Importance of a classifier:
1  1  i 

 i  ln 
2  i 
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
111
Example: AdaBoost

Weight update:
 j

if C j ( xi )  yi
w exp
( j 1)
wi



Z j  exp j if C j ( xi )  yi
where Z j is the normalizat ion factor
( j)
i
If any intermediate rounds produce error rate higher
than 50%, the weights are reverted back to 1/n and
the resampling procedure is repeated
 Classification:
T

C * ( x )  arg max  j C j ( x )  y 
y
© Tan,Steinbach, Kumar
Introduction to Data Mining
j 1
4/18/2004
112
Illustrating AdaBoost
Initial weights for each data point
Original
Data
0.1
0.1
- - - - -
+++
Data points
for training
0.1
++
B1
Boosting
Round 1
0.0094
+++
© Tan,Steinbach, Kumar
0.0094
0.4623
- - - - - - -
Introduction to Data Mining
 = 1.9459
4/18/2004
113
Illustrating AdaBoost
B1
Boosting
Round 1
0.0094
0.0094
+++
0.4623
- - - - - - -
 = 1.9459
B2
Boosting
Round 2
0.0009
0.3037
- - -
- - - - -
0.0422
++
 = 2.9323
B3
0.0276
0.1819
0.0038
Boosting
Round 3
+++
++ ++ + ++
Overall
+++
- - - - -
© Tan,Steinbach, Kumar
Introduction to Data Mining
 = 3.8744
++
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114