1. Basic Association Analysis (IDM ch. 6) 1. Review 2. Maximal and Closed Itemsets 3. Rule...

1. Basic Association Analysis (IDM ch. 6)1. Review2. Maximal and Closed Itemsets3. Rule Generation4. Kuis

2. Support Vector Machines / SVM (IDM ch. 5.5 and DMBAR 14.4)3. Practical:

1. Association Rules Mining (DMBAR ch. 16) 1. online radio 2. predicting income

2. SVM1. “Support vector machines in R,” by Karatzoglou et al. (2006)

Pertemuan 9: 27 Oktober 2015

Association Rule Mining• Given a set of transactions, find rules that will predict the

occurrence of an item based on the occurrences of other items in the transaction

Market-Basket transactions

TID Items

1 Bread, Milk

2 Bread, Diaper, Beer, Eggs

3 Milk, Diaper, Beer, Coke

4 Bread, Milk, Diaper, Beer

5 Bread, Milk, Diaper, Coke

Example of Association Rules

{Diaper} {Beer},{Milk, Bread} {Eggs,Coke},{Beer, Bread} {Milk},

Implication means co-occurrence, not causality!

Definition: Association RuleTID Items

1 Bread, Milk

2 Bread, Diaper, Beer, Eggs

3 Milk, Diaper, Beer, Coke

4 Bread, Milk, Diaper, Beer

5 Bread, Milk, Diaper, Coke

Example:Beer}Diaper,Milk{

4.052

|T|)BeerDiaper,,Milk(

s

67.032

)Diaper,Milk()BeerDiaper,Milk,(

c

Association Rule– An implication expression of the form X

Y, where X and Y are itemsets– Example:

{Milk, Diaper} {Beer}

Rule Evaluation Metrics– Support (s)

Fraction of transactions that contain both X and Y

– Confidence (c) Measures how often items in Y

appear in transactions thatcontain X

Frequent Itemset Generation Strategies

• Reduce the number of candidates (M)– Complete search: M=2d

– Use pruning techniques to reduce M• Reduce the number of transactions (N)

– Reduce size of N as the size of itemset increases– Used by DHP and vertical-based mining algorithms

• Reduce the number of comparisons (NM)– Use efficient data structures to store the candidates or

transactions– No need to match every candidate against every

transaction

Reducing Number of Candidates• Apriori principle:

– If an itemset is frequent, then all of its subsets must also be frequent

• Apriori principle holds due to the following property of the support measure:

– Support of an itemset never exceeds the support of its subsets

– This is known as the anti-monotone property of support

)()()(:, YsXsYXYX

Found to be Infrequent

null

AB AC AD AE BC BD BE CD CE DE

A B C D E

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

ABCD ABCE ABDE ACDE BCDE

ABCDE

Illustrating Apriori Principlenull


A B C D E



ABCDE

Pruned supersets

Illustrating Apriori PrincipleItem CountBread 4Coke 2Milk 4Beer 3Diaper 4Eggs 1

Itemset Count{Bread,Milk} 3{Bread,Beer} 2{Bread,Diaper} 3{Milk,Beer} 2{Milk,Diaper} 3{Beer,Diaper} 3

Itemset Count {Bread,Milk,Diaper} 3

Items (1-itemsets)

Pairs (2-itemsets)

(No need to generatecandidates involving Cokeor Eggs)

Triplets (3-itemsets)Minimum Support = 3

If every subset is considered, 6C1 + 6C2 + 6C3 = 41

With support-based pruning,6 + 6 + 1 = 13

Apriori Algorithm

• Method: – Let k=1– Generate frequent itemsets of length 1– Repeat until no new frequent itemsets are identified

• Generate length (k+1) candidate itemsets from length k frequent itemsets

• Prune candidate itemsets containing subsets of length k that are infrequent

• Count the support of each candidate by scanning the DB• Eliminate candidates that are infrequent, leaving only those

that are frequent

Reducing Number of Comparisons• Candidate counting:

– Scan the database of transactions to determine the support of each candidate itemset

– To reduce the number of comparisons, store the candidates in a hash structure

• Instead of matching each transaction against every candidate, match it against candidates contained in the hashed buckets

TID Items 1 Bread, Milk 2 Bread, Diaper, Beer, Eggs 3 Milk, Diaper, Beer, Coke 4 Bread, Milk, Diaper, Beer 5 Bread, Milk, Diaper, Coke

N

Transactions Hash Structure

k

Buckets

Generate Hash Tree

2 3 45 6 7

1 4 51 3 6

1 2 44 5 7 1 2 5

4 5 81 5 9

3 4 5 3 5 63 5 76 8 9

3 6 73 6 8

1,4,7

2,5,8

3,6,9Hash function

Suppose you have 15 candidate itemsets of length 3:

{1 4 5}, {1 2 4}, {4 5 7}, {1 2 5}, {4 5 8}, {1 5 9}, {1 3 6}, {2 3 4}, {5 6 7}, {3 4 5}, {3 5 6}, {3 5 7}, {6 8 9}, {3 6 7}, {3 6 8}

You need:

• Hash function

• Max leaf size: max number of itemsets stored in a leaf node (if number of candidate itemsets exceeds max leaf size, split the node)

Association Rule Mining Task

• Given a set of transactions T, the goal of association rule mining is to find all rules having

• support ≥ minsup threshold• confidence ≥ minconf threshold

• Brute-force approach:• List all possible association rules• Compute the support and confidence for each rule• Prune rules that fail the minsup and minconf

thresholds Computationally prohibitive!

Maximal Frequent Itemset

null


A B C D E



ABCDE

BorderInfrequent Itemsets

Maximal Itemsets

An itemset is maximal frequent if none of its immediate supersets is frequent

Closed Itemset

TID Items1 {A,B}2 {B,C,D}3 {A,B,C,D}4 {A,B,D}5 {A,B,C,D}

• An itemset is closed if none of its immediate supersets has the same support as the itemset

Itemset Support{A} 4{B} 5{C} 3{D} 4

{A,B} 4{A,C} 2{A,D} 3{B,C} 3{B,D} 4{C,D} 3

Itemset Support{A,B,C} 2{A,B,D} 3{A,C,D} 2{B,C,D} 3

{A,B,C,D} 2

Maximal vs Closed Itemsets

TID Items

1 ABC

2 ABCD

3 BCE

4 ACDE

5 DE

null


A B C D E



ABCDE

124 123 1234 245 345

12 124 24 4 123 2 3 24 34 45

12 2 24 4 4 2 3 4

2 4

Transaction Ids

Not supported by any transactions

Maximal vs Closed Frequent Itemsets

null


A B C D E



ABCDE

124 123 1234 245 345

12 124 24 4 123 2 3 24 34 45

12 2 24 4 4 2 3 4

2 4

Minimum support = 2

# Closed = 9

# Maximal = 4

Closed and maximal

Closed but not maximal

Maximal vs Closed Itemsets

FrequentItemsets

ClosedFrequentItemsets

MaximalFrequentItemsets

Rule Generation

• Given a frequent itemset L, find all non-empty subsets f L such that f L – f satisfies the minimum confidence requirement• If {A,B,C,D} is a frequent itemset, candidate rules:

ABC D, ABD C, ACD B, BCD A, A BCD, B ACD, C ABD, D ABC,AB CD, AC BD, AD BC, BC AD, BD AC, CD AB,

• If |L| = k, then there are 2k – 2 candidate association rules (ignoring L and L)

Rule Generation• How to efficiently generate rules from

frequent itemsets?– In general, confidence does not have an anti-

monotone propertyc(ABC D) can be larger or smaller than c(AB D)

– But confidence of rules generated from the same itemset has an anti-monotone property

– e.g., L = {A,B,C,D}:c(ABC D) c(AB CD) c(A BCD)

• Confidence is anti-monotone w.r.t. number of items on the RHS of the rule

Rule Generation for Apriori Algorithm

ABCD=>{ }

BCD=>A ACD=>B ABD=>C ABC=>D

BC=>ADBD=>ACCD=>AB AD=>BC AC=>BD AB=>CD

D=>ABC C=>ABD B=>ACD A=>BCD

Lattice of rules

ABCD=>{ }

BCD=>A ACD=>B ABD=>C ABC=>D

BC=>ADBD=>ACCD=>AB AD=>BC AC=>BD AB=>CD

D=>ABC C=>ABD B=>ACD A=>BCD

Pruned Rules

Low Confidence Rule

Rule Generation for Apriori Algorithm

• Candidate rule is generated by merging two rules that share the same prefixin the rule consequent

• Join(CD=>AB,BD=>AC)would produce the candidaterule D => ABC

• Prune rule D=>ABC if itssubset AD=>BC does not havehigh confidence

BD=>ACCD=>AB

D=>ABC

Effect of Support Distribution

• Many real data sets have skewed support distribution

Support distribution of a retail data set

Effect of Support Distribution

• How to set the appropriate minsup threshold?– If minsup is set too high, we could miss

itemsets involving interesting rare items (e.g., expensive products)

– If minsup is set too low, it is computationally expensive and the number of itemsets is very large

• Using a single minimum support threshold may not be effective

Multiple Minimum Support• How to apply multiple minimum supports?

• MS(i): minimum support for item i • e.g.:

• MS(Milk)=5%, • MS(Coke) = 3%, • MS(Broccoli)=0.1%, • MS(Salmon)=0.5%

• MS({Milk, Broccoli}) = min (MS(Milk), MS(Broccoli))= 0.1%

• Challenge: Support is no longer anti-monotone• Suppose: Support(Milk, Coke) = 1.5% and

Support(Milk, Coke, Broccoli) = 0.5%• {Milk,Coke} is infrequent but {Milk,Coke,Broccoli} is frequent

Multiple Minimum Support

A

Item MS(I) Sup(I)

A 0.10% 0.25%

B 0.20% 0.26%

C 0.30% 0.29%

D 0.50% 0.05%

E 3% 4.20%

B

C

D

E

AB

AC

AD

AE

BC

BD

BE

CD

CE

DE

ABC

ABD

ABE

ACD

ACE

ADE

BCD

BCE

BDE

CDE

Multiple Minimum Support

A

B

C

D

E

AB

AC

AD

AE

BC

BD

BE

CD

CE

DE

ABC

ABD

ABE

ACD

ACE

ADE

BCD

BCE

BDE

CDE

Item MS(I) Sup(I)

A 0.10% 0.25%

B 0.20% 0.26%

C 0.30% 0.29%

D 0.50% 0.05%

E 3% 4.20%

Multiple Minimum Support (Liu 1999)• Order the items according to their minimum support

(in ascending order)– e.g.: MS(Milk)=5% MS(Coke) = 3%,

MS(Broccoli)=0.1% MS(Salmon)=0.5%– Ordering: Broccoli, Salmon, Coke, Milk

• Need to modify Apriori such that:– L1 : set of frequent items

– F1 : set of items whose support is MS(1)where MS(1) is mini( MS(i) )

– C2 : candidate itemsets of size 2 is generated from F1

instead of L1

Multiple Minimum Support (Liu 1999)• Modifications to Apriori:

• In traditional Apriori, • A candidate (k+1)-itemset is generated by merging two

frequent itemsets of size k• The candidate is pruned if it contains any infrequent

subsets of size k

• Pruning step has to be modified:• Prune only if subset contains the first item• e.g.: Candidate={Broccoli, Coke, Milk} (ordered

according to minimum support)• {Broccoli, Coke} and {Broccoli, Milk} are frequent but

{Coke, Milk} is infrequent• Candidate is not pruned because {Coke,Milk} does not contain

the first item, i.e., Broccoli.

Pattern Evaluation• Association rule algorithms tend to produce

too many rules – many of them are uninteresting or redundant– Redundant if {A,B,C} {D} and {A,B} {D}

have same support & confidence• Interestingness measures can be used to

prune/rank the derived patterns• In the original formulation of association

rules, support & confidence are the only measures used

Application of Interestingness Measure

Feature

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

Pro

duct

FeatureFeatureFeatureFeatureFeatureFeatureFeatureFeatureFeature

Selection

Preprocessing

Mining

Postprocessing

Data

SelectedData

PreprocessedData

Patterns

KnowledgeInterestingness Measures

Computing Interestingness Measure

• Given a rule X Y, information needed to compute rule interestingness can be obtained from a contingency table

Y Y

X f11 f10 f1+

X f01 f00 fo+

f+1 f+0 |T|

Contingency table for X Yf11: support of X and Yf10: support of X and Yf01: support of X and Yf00: support of X and Y

Used to define various measures support, confidence, lift, Gini, J-measure, etc.

Drawback of Confidence

Coffee Coffee

Tea 15 5 20

Tea 75 5 80

90 10 100

Association Rule: Tea CoffeeConfidence= P(Coffee|Tea) = 0.75

but P(Coffee) = 0.9

Þ Although confidence is high, rule is misleading

Þ P(Coffee|Tea) = 0.9375

Statistical Independence• Population of 1000 students

• 600 students know how to swim (S)• 700 students know how to bike (B)• 420 students know how to swim and bike (S,B)

• P(SB) = 420/1000 = 0.42• P(S) P(B) = 0.6 0.7 = 0.42

• P(SB) = P(S) P(B) => Statistical independence• P(SB) > P(S) P(B) => Positively correlated• P(SB) < P(S) P(B) => Negatively correlated

Statistical-based Measures• Measures that take into account statistical

dependence

)](1)[()](1)[(

)()(),(

)()(),(

)()(

),(

)(

)(

)(

)|(

YPYPXPXP

YPXPYXPtcoefficien

YPXPYXPPS

YPXP

YXPInterest

Ys

YXc

YP

XYPLift

Example: Lift/Interest

Coffee Coffee

Tea 15 5 20

Tea 75 5 80

90 10 100

Association Rule: Tea CoffeeConfidence= P(Coffee|Tea) = 0.75

but P(Coffee) = 0.9

Þ Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)

Drawback of Lift & Interest

Y Y

X 10 0 10

X 0 90 90

10 90 100

Y Y

X 90 0 90

X 0 10 10

90 10 100

10)1.0)(1.0(

1.0 Lift 11.1)9.0)(9.0(

9.0 Lift

Statistical independence:

If P(X,Y)=P(X)P(Y) => Lift = 1

There are lots of measures proposed in the literature

Some measures are good for certain applications, but not for others

What criteria should we use to determine whether a measure is good or bad?

What about Apriori-style support based pruning? How does it affect these measures?

Quiz: Mining Association Rules

1. Berapakah nilai support untuk itemset {B,D,E}?2. Berapakah nilai support dan confidence untuk aturan BD E?3. Gunakan algoritma Apriori untuk membentuk itemset dengan nilai minimal

frekuensi 3. Perhatikan bahwa dalam algoritma Apriori sebuah transaksi akan di-scan beberapa kali, dengan iterasi scan ke-i membangkitkan himpunan frekuensi i-itemset. Berikan semua himpunan itemset yang terbentuk dalam setiap iterasi.

4. Berikan aturan-aturan yang memiliki nilai support minimal 3 dan confidence ¾, berdasarkan itemset yang terbentuk pada nomor 3.

Support Vector Machines

• Find a linear hyperplane (decision boundary) that will separate the data


• One Possible Solution

B1


• Another possible solution

B2


• Other possible solutions

B2


• Which one is better? B1 or B2?• How do you define better?

B1

B2


• Find hyperplane maximizes the margin => B1 is better than B2

B1

B2

b11

b12

b21

b22

margin

Support Vector MachinesB1

b11

b12

0 bxw

1 bxw 1 bxw

1bxw if1

1bxw if1)(

xf 2||||

2 Margin

w


• We want to maximize:

– Which is equivalent to minimizing:

– But subjected to the following constraints:

• This is a constrained optimization problem– Numerical approaches to solve it (e.g., quadratic programming)

2||||

2 Margin

w

1bxw if1

1bxw if1)(

i

i

ixf

2

||||)(

2wwL


• What if the problem is not linearly separable?


• What if the problem is not linearly separable?– Introduce slack variables

• Need to minimize:

• Subject to:

ii

ii

1bxw if1

-1bxw if1)(

ixf

N

i

kiC

wwL

1

2

2

||||)(

Nonlinear Support Vector Machines

• What if decision boundary is not linear?

Nonlinear Support Vector Machines

• Transform data into higher dimensional space

Practical Work

• Association Rules Mining (DMBAR ch. 16) – online radio – predicting income

• SVM (package: kernlab)– “Support vector machines in R,” by Karatzoglou et

al. (2006), section 4 ksvm in kernlab– “Introduction to SVM in R” (Jean-Philippe Vert):

questions 1-10

1. Basic Association Analysis (IDM ch. 6) 1. Review 2. Maximal and Closed Itemsets 3. Rule...

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