1. CS 685: Special Topics in Data Mining Spring 2008 Jinze Liu
Classification
CS 685: Special Topics in Data Mining
Spring 2008
Jinze Liu

2. Bayesian Classification: Why?
Probabilistic learning: Calculate explicit probabilities for hypothesis, among the most practical approaches to certain types of learning problems
Incremental: Each training example can incrementally increase/decrease the probability that a hypothesis is correct. Prior knowledge can be combined with observed data.
Probabilistic prediction: Predict multiple hypotheses, weighted by their probabilities
Standard: Even when Bayesian methods are computationally intractable, they can provide a standard of optimal decision making against which other methods can be measured

3. Bayesian Theorem: Basics
Let X be a data sample whose class label is unknown
Let H be a hypothesis that X belongs to class C
For classification problems, determine P(H/X): the probability that the hypothesis holds given the observed data sample X
P(H): prior probability of hypothesis H (i.e. the initial probability before we observe any data, reflects the background knowledge)
P(X): probability that sample data is observed
P(X|H) : probability of observing the sample X, given that the hypothesis holds

4. Bayesian Theorem
Given training data X, posteriori probability of a hypothesis H, P(H|X) follows the Bayes theorem
Informally, this can be written as
posterior =likelihood x prior / evidence
MAP (maximum posteriori) hypothesis
Practical difficulty: require initial knowledge of many probabilities, significant computational cost

5. Naïve Bayes Classifier
A simplified assumption: attributes are conditionally independent:
The product of occurrence of say 2 elements x1 and x2, given the current class is C, is the product of the probabilities of each element taken separately, given the same class P([y1,y2],C) = P(y1,C) * P(y2,C)
No dependence relation between attributes
Greatly reduces the computation cost, only count the class distribution.
Once the probability P(X|Ci) is known, assign X to the class with maximum P(X|Ci)*P(Ci)

6. Training dataset Class: C1:buys_computer= ‘yes’ C2:buys_computer= ‘no’
Data sample
X =(age<=30,
Income=medium,
Student=yes
Credit_rating=
Fair)

7. Naïve Bayesian Classifier: Example
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”)= x x x =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
X belongs to class “buys_computer=yes”

8. 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

9. 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
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 Y Z P X

10. Bayesian Belief Network: An Example
Family
History
Smoker
(FH, S)
(FH, ~S)
(~FH, S)
(~FH, ~S) LC
0.8
0.5
0.7
0.1
LungCancer
Emphysema
~LC
0.2
0.5
0.3
0.9
The conditional probability table for the variable LungCancer:
Shows the conditional probability for each possible combination of its parents
PositiveXRay
Dyspnea
Bayesian Belief Networks

11. Learning Bayesian Networks
Several cases
Given both the network structure and all variables observable: learn only the CPTs
Network structure known, some hidden variables: method of gradient descent, analogous to neural network learning
Network structure unknown, all variables observable: search through the model space to reconstruct graph topology
Unknown structure, all hidden variables: no good algorithms known for this purpose
D. Heckerman, Bayesian networks for data mining

12. Association Rules Itemset X = {x1, …, xk}
Transaction-id
Items bought
100
f, a, c, d, g, I, m, p
200
a, b, c, f, l,m, o
300
b, f, h, j, o
400
b, c, k, s, p
500
a, f, c, e, l, p, m, n
Itemset X = {x1, …, xk}
Find all the rules X  Y with minimum support and confidence
support, s, is the probability that a transaction contains X  Y
confidence, c, is the conditional probability that a transaction having X also contains Y
Customer
buys diaper
buys both
buys beer
Let supmin = 50%, confmin = 50%
Association rules:
A  C (60%, 100%)
C  A (60%, 75%)

13. Classification based on Association
Classification rule mining versus Association rule mining
Aim
A small set of rules as classifier
All rules according to minsup and minconf
Syntax
X  y
X Y

14. Why & How to Integrate
Both classification rule mining and association rule mining are indispensable to practical applications.
The integration is done by focusing on a special subset of association rules whose right-hand-side are restricted to the classification class attribute.
CARs: class association rules

15. CBA: Three Steps Discretize continuous attributes, if any
Generate all class association rules (CARs)
Build a classifier based on the generated CARs.

16. Our Objectives
To generate the complete set of CARs that satisfy the user-specified minimum support (minsup) and minimum confidence (minconf) constraints.
To build a classifier from the CARs.

17. Rule Generator: Basic Concepts
Ruleitem
<condset, y> :condset is a set of items, y is a class label
Each ruleitem represents a rule: condset->y
condsupCount
The number of cases in D that contain condset
rulesupCount
The number of cases in D that contain the condset and are labeled with class y
Support=(rulesupCount/|D|)*100%
Confidence=(rulesupCount/condsupCount)*100%

18. RG: Basic Concepts (Cont.)
Frequent ruleitems
A ruleitem is frequent if its support is above minsup
Accurate rule
A rule is accurate if its confidence is above minconf
Possible rule
For all ruleitems that have the same condset, the ruleitem with the highest confidence is the possible rule of this set of ruleitems.
The set of class association rules (CARs) consists of all the possible rules (PRs) that are both frequent and accurate.

19. RG: An Example A ruleitem:<{(A,1),(B,1)},(class,1)> assume that
the support count of the condset (condsupCount) is 3,
the support of this ruleitem (rulesupCount) is 2, and
|D|=10
then (A,1),(B,1) -> (class,1)
supt=20% (rulesupCount/|D|)*100%
confd=66.7% (rulesupCount/condsupCount)*100%

20. RG: The Algorithm 1 F 1 = {large 1-ruleitems};
2 CAR 1 = genRules (F 1 );
3 prCAR 1 = pruneRules (CAR 1 ); //count the item and class occurrences to
determine the frequent 1-ruleitems and prune it
4 for (k = 2; F k-1 Ø; k++) do
C k = candidateGen (F k-1 ); //generate the candidate ruleitems Ck
using the frequent ruleitems Fk-1
6 for each data case d D do //scan the database
C d = ruleSubset (C k , d); //find all the ruleitems in Ck whose condsets
are supported by d
for each candidate c C d do
c.condsupCount++;
if d.class = c.class then
c.rulesupCount++; //update various support counts of the candidates in Ck
end
end

21. RG: The Algorithm(cont.)
F k = {c C k | c.rulesupCount minsup};
//select those new frequent ruleitems to form Fk
CAR k = genRules(F k ); //select the ruleitems both accurate and frequent
prCAR k = pruneRules(CAR k );
16 end
17 CARs =  k CAR k ;
18 prCARs =  k prCAR k ;

22. SVM – Support Vector Machines
Support Vectors
Large Margin
Small Margin

23. SVM – Cont. Linear Support Vector Machine
Given a set of points with label
The SVM finds a hyperplane defined by the pair (w,b)
(where w is the normal to the plane and b is the distance from the origin)
s.t.
x – feature vector, b- bias, y- class label, 2/||w|| - margin

24. SVM – Cont.

25. SVM – Cont. What if the data is not linearly separable?
Project the data to high dimensional space where it is linearly separable and then we can use linear SVM – (Using Kernels)
(1,0)
(0,0)
(0,1) + - -1 +1 + -

26. Non-Linear SVM Classification using SVM (w,b)
In non linear case we can see this as
Kernel – Can be thought of as doing dot product
in some high dimensional space

27. Example of Non-linear SVM

28. Results

29. SVM Related Links http://svm.dcs.rhbnc.ac.uk/
C. J. C. Burges. A Tutorial on Support Vector Machines for Pattern Recognition. Knowledge Discovery and Data Mining, 2(2), 1998.
SVMlight – Software (in C)
BOOK: An Introduction to Support Vector Machines N. Cristianini and J. Shawe-Taylor Cambridge University Press