Transcript P(C i ) - Binus Repository

Matakuliah : M0614 / Data Mining & OLAP
Tahun
: Feb - 2010
Classification and Prediction
Pertemuan 08
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat menggunakan teknik analisis
classification by decision tree induction, Bayesian
classification, classification by back propagation, dan
lazy learners pada data mining. (C3)
3
Bina Nusantara
Acknowledgments
These slides have been adapted from Han, J.,
Kamber, M., & Pei, Y. Data Mining: Concepts
and Technique and Tan, P.-N., Steinbach, M.,
& Kumar, V. Introduction to Data Mining.
Bina Nusantara
Outline Materi
• Bayesian classification
5
Bina Nusantara
Bayesian Classification: Why?
• A statistical classifier: performs probabilistic prediction, i.e., predicts
class membership probabilities
• Foundation: Based on Bayes’ Theorem.
• Performance: A simple Bayesian classifier, naïve Bayesian classifier,
has comparable performance with decision tree and selected neural
network classifiers
• Incremental: Each training example can incrementally
increase/decrease the probability that a hypothesis is correct — prior
knowledge can be combined with observed data
• Standard: Even when Bayesian methods are computationally
intractable, they can provide a standard of optimal decision making
against which other methods can be measured
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Data Mining: Concepts and Techniques
6
Bayesian Theorem: Basics
•
•
•
•
•
•
Let X be a data sample (“evidence”): class label is unknown
Let H be a hypothesis that X belongs to class C
Classification is to determine P(H|X), (posteriori probability), the probability
that the hypothesis holds given the observed data sample X
P(H) (prior probability), the initial probability
– E.g., X will buy computer, regardless of age, income, …
P(X): probability that sample data is observed
P(X|H) (likelyhood), the probability of observing the sample X, given that
the hypothesis holds
– E.g., Given that X will buy computer, the prob. that X is 31..40, medium
income
April 1, 2016
Data Mining: Concepts and Techniques
7
Bayesian Theorem
•
Given training data X, posteriori probability of a hypothesis H, P(H|X),
follows the Bayes theorem
P(H | X)  P(X | H )P(H )
P(X)
•
Informally, this can be written as
posteriori = likelihood x prior/evidence
•
Predicts X belongs to C2 iff the probability P(Ci|X) is the highest among
all the P(Ck|X) for all the k classes
•
Practical difficulty: require initial knowledge of many probabilities,
significant computational cost
April 1, 2016
Data Mining: Concepts and Techniques
8
Example of Bayes Theorem
• 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
•
If a patient has stiff neck, what’s the probability he/she has meningitis?
P( S | M ) P( M ) 0.5 1 / 50000
P( M | S ) 

 0.0002
P( S )
1 / 20
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?
Bayesian Classifiers
• Approach:
– compute the posterior probability P(C | A1, A2, …, An) for all values of C
using the Bayes theorem
P ( A A  A | C ) P (C )
P (C | A A  A ) 
P( A A  A )
1
1
2
2
n
n
1
2
– 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 )?
n
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)
– Can estimate P(Ai| Cj) for all Ai and Cj.
– New point is classified to Cj if P(Cj)  P(Ai| Cj) is maximal.
al
a l Estimate
How rto
u sProbabilities from Data?
c
c
i
i
o
r
ca
Tid
10
Refund
t
o
g
e
ca
t
o
g
e
co
n
u
it n
s
s
a
cl
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:
P(Status=Married|No) = 4/7
P(Refund=Yes|Yes)=0
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)
a
ric
l
a
ric
l
u
o
u
s
Howegoto Estimate
Probabilities
from
Data?
o
n
s
i
g
t
s
e
t
a
c
Tid
Refund
t
a
c
Marital
Status
n
o
c
Taxable
Income
a
cl
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
• 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
1
P( Income  120 | No) 
e
2 (54.54)
10

( 120110) 2
2 ( 2975)
 0.0072
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

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
Naïve Bayes Classifier
• If one of the conditional probability is zero, then the entire
expression becomes zero
• Probability estimation:
N ic
c: number of classes
Original : P( Ai | C ) 
Nc
N ic  1
Laplace : P( Ai | C ) 
Nc  c
N ic  mp
m - estimate : P( Ai | C ) 
Nc  m
p: prior probability
m: parameter
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
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
?
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
Example Naïve Bayesian Classifier: Training Dataset
Class:
C1:buys_computer = ‘yes’
C2:buys_computer = ‘no’
Data sample
X = (age <=30,
Income = medium,
Student = yes
Credit_rating = Fair)
April 1, 2016
age
<=30
<=30
31…40
>40
>40
>40
31…40
<=30
<=30
>40
<=30
31…40
31…40
>40
income studentcredit_rating
buys_compu
high
no fair
no
high
no excellent
no
high
no fair
yes
medium
no fair
yes
low
yes fair
yes
low
yes excellent
no
low
yes excellent
yes
medium
no fair
no
low
yes fair
yes
medium yes fair
yes
medium yes excellent
yes
medium
no excellent
yes
high
yes fair
yes
medium
no excellent
no
Data Mining: Concepts and Techniques
19
Example Naïve Bayesian Classifier: Training Dataset
•
•
•
P(Ci):
P(buys_computer = “yes”) = 9/14 = 0.643
P(buys_computer = “no”) = 5/14= 0.357
Compute P(X|Ci) for each class
P(age = “<=30” | buys_computer = “yes”) = 2/9 = 0.222
P(age = “<= 30” | buys_computer = “no”) = 3/5 = 0.6
P(income = “medium” | buys_computer = “yes”) = 4/9 = 0.444
P(income = “medium” | buys_computer = “no”) = 2/5 = 0.4
P(student = “yes” | buys_computer = “yes) = 6/9 = 0.667
P(student = “yes” | buys_computer = “no”) = 1/5 = 0.2
P(credit_rating = “fair” | buys_computer = “yes”) = 6/9 = 0.667
P(credit_rating = “fair” | buys_computer = “no”) = 2/5 = 0.4
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(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
Therefore, X belongs to class (“buys_computer = yes”)
April 1, 2016
Data Mining: Concepts and Techniques
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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)
Naïve Bayesian Classifier: Comments
•
•
•
Advantages
– Easy to implement
– Good results obtained in most of the cases
Disadvantages
– Assumption: class conditional independence, therefore loss of accuracy
– 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
April 1, 2016
Data Mining: Concepts and Techniques
22
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Classification and Prediction (cont.)
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