Data Mining Association Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 6 Introduction to Data Mining

by Tan, Steinbach, Kumar

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

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

Example of Association Rules

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

1 2 3 4 5

Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke

Implication means co-occurrence, not causality!

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Definition: Frequent Itemset

!

Itemset
A collection of one or more items
!

Example: {Milk, Bread, Diaper}

TID Items

k-itemset
!

An itemset that contains k items

1 2 3 4 5

Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke

Support count (")

Frequency of occurrence of an itemset E.g. "({Milk, Bread,Diaper}) = 2

Support
Fraction of transactions that contain an itemset E.g. s({Milk, Bread, Diaper}) = 2/5

Frequent Itemset
An itemset whose support is greater than or equal to a minsup threshold

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Definition: Association Rule

!

Association Rule
An implication expression of the form X ! Y, where X and Y are itemsets Example: {Milk, Diaper} ! {Beer}

TID

Items

1 2 3 4 5

Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke

Rule Evaluation Metrics

Support (s)
!

Fraction of transactions that contain both X and Y Measures how often items in Y appear in transactions that contain X

Example:

{Milk, Diaper } ! Beer

s=

Confidence (c)
!

! (Milk, Diaper, Beer)

|T|

2 = 0.4 5

! (Milk, Diaper, Beer) 2 c= = = 0.67 ! (Milk, Diaper ) 3

Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

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!
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Mining Association Rules

TID Items

Example of Rules:
{Milk,Diaper} ! {Beer} (s=0.4, c=0.67) {Milk,Beer} ! {Diaper} (s=0.4, c=1.0) {Diaper,Beer} ! {Milk} (s=0.4, c=0.67) {Beer} ! {Milk,Diaper} (s=0.4, c=0.67) {Diaper} ! {Milk,Beer} (s=0.4, c=0.5) {Milk} ! {Diaper,Beer} (s=0.4, c=0.5)

1 2 3 4 5

Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke

Observations:
All the above rules are binary partitions of the same itemset: {Milk, Diaper, Beer} Rules originating from the same itemset have identical support but can have different confidence Thus, we may decouple the support and confidence requirements
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Mining Association Rules

!

Two-step approach:
1. Frequent Itemset Generation

Generate all itemsets whose support $ minsup

2. Rule Generation

Generate high confidence rules from each frequent itemset, where each rule is a binary partitioning of a frequent itemset

Frequent itemset generation is still computationally expensive

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Frequent Itemset Generation

null

AB

AC

AD

AE

BC

BD

BE

CD

CE

DE

ABC

ABD

ABE

ACD

ACE

ADE

BCD

BCE

BDE

CDE

ABCD

ABCE

ABDE

ACDE

BCDE

ABCDE

Given d items, there are 2d possible candidate itemsets

4/18/2004 #

Tan,Steinbach, Kumar

Introduction to Data Mining

Frequent Itemset Generation

! Brute-force

approach:

Each itemset in the lattice is a candidate frequent itemset Count the support of each candidate by scanning the database
Transactions
TID 1 2 3 4 5 Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke

List of Candidates

Match each transaction against every candidate Complexity ~ O(NMw) => Expensive since M = 2d !!!
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Computational Complexity
! Given

d unique items:

Total number of itemsets = 2d Total number of possible association rules:

'- d * - d ! k *$ R = / %+ ( . / + (" &, k ) , j )# = 3 ! 2 +1

d !1 k =1 d !k j =1 d d +1

If d=6, R = 602 rules

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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
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

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:

$X , Y : ( X # Y ) " s( X ) ! s(Y )
Support of an itemset never exceeds the support of its subsets This is known as the anti-monotone property of support
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Illustrating Apriori Principle

null

AB

AC

AD

AE

BC

BD

BE

CD

CE

DE

Found to be Infrequent

ABC

ABD

ABE

ACD

ACE

ADE

BCD

BCE

BDE

CDE

ABCD

ABCE

ABDE

ACDE

BCDE

Pruned supersets
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ABCDE

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Illustrating Apriori Principle

Item Bread Coke Milk Beer Diaper Eggs Count 4 2 4 3 4 1

Items (1-itemsets)
Itemset {Bread,Milk} {Bread,Beer} {Bread,Diaper} {Milk,Beer} {Milk,Diaper} {Beer,Diaper} Count 3 2 3 2 3 3

Pairs (2-itemsets) (No need to generate candidates involving Coke or Eggs)

Minimum Support = 3
If every subset is considered, 6C + 6C + 6C = 41 1 2 3 With support-based pruning, 6 + 6 + 1 = 13

Triplets (3-itemsets)
Itemset {Bread,Milk,Diaper} Count 3

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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

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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
Transactions
TID 1 2 3 4 5 Items Bread, Milk Bread, Diaper, Beer, Eggs Milk, Diaper, Beer, Coke Bread, Milk, Diaper, Beer Bread, Milk, Diaper, Coke
Buckets
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Hash Structure

Generate Hash Tree

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)

Hash function 3,6,9 1,4,7 2,5,8

145 124 457 125 458

234 567 345 136 159

356 357 689

367 368

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Association Rule Discovery: Hash tree

Hash Function

Candidate Hash Tree

1,4,7 2,5,8

3,6,9

234 567 145 Hash on 1, 4 or 7 124 457 125 458

Introduction to Data Mining 4/18/2004 #

136 345 159 356 357 689 367 368

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Association Rule Discovery: Hash tree

Hash Function

Candidate Hash Tree

1,4,7 2,5,8

3,6,9

234 567 145 Hash on 2, 5 or 8 124 457 125 458

Introduction to Data Mining 4/18/2004 #

136 345 159 356 357 689 367 368

Tan,Steinbach, Kumar

Association Rule Discovery: Hash tree

Hash Function

Candidate Hash Tree

1,4,7 2,5,8

3,6,9

234 567 145 Hash on 3, 6 or 9 124 457 125 458

Introduction to Data Mining 4/18/2004 #

136 345 159 356 357 689 367 368

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Subset Operation
Given a transaction t, what are the possible subsets of size 3?

Transaction, t
1 2 3 5 6

Level 1
1 2 3 5 6 2 3 5 6 3 5 6

Level 2
12 3 5 6 13 5 6 15 6 23 5 6 25 6 35 6

123 125 126

135 136

156

235 236

256

356

Level 3
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Subsets of 3 items
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Subset Operation Using Hash Tree

1 2 3 5 6 transaction 1+ 2356 2+ 356 3+ 56
234 567 145 136 345 124 457 125 458 159 356 357 689 367 368
Hash Function

1,4,7 2,5,8

3,6,9

Tan,Steinbach, Kumar

Introduction to Data Mining

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Subset Operation Using Hash Tree

1 2 3 5 6 transaction 1+ 2356 12+ 356 13+ 56 15+ 6
145 136 345 124 457
Tan,Steinbach, Kumar

Hash Function

2+ 356 3+ 56
234 567 356 357 689 367 368

1,4,7 2,5,8

3,6,9

125 458

159

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4/18/2004

Subset Operation Using Hash Tree

1 2 3 5 6 transaction 1+ 2356 12+ 356 13+ 56 15+ 6
145 136 345 124 457
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Hash Function

2+ 356 3+ 56
234 567 356 357 689 367 368

1,4,7 2,5,8

3,6,9

125 458

159

Match transaction against 11 out of 15 candidates

Introduction to Data Mining 4/18/2004 #

Factors Affecting Complexity

!

Choice of minimum support threshold

lowering support threshold results in more frequent itemsets this may increase number of candidates and max length of frequent itemsets

Dimensionality (number of items) of the data set

more space is needed to store support count of each item if number of frequent items also increases, both computation and I/O costs may also increase

Size of database
since Apriori makes multiple passes, run time of algorithm may increase with number of transactions

Average transaction width

transaction width increases with denser data sets This may increase max length of frequent itemsets and traversals of hash tree (number of subsets in a transaction increases with its width)

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Compact Representation of Frequent Itemsets

!

Some itemsets are redundant because they have identical support as their supersets

TID A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

'10 $ " ! Number of frequent itemsets = 3 ( ! % &k#

10 k =1

Need a compact representation

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Maximal Frequent Itemset

An itemset is maximal frequent if none of its immediate supersets is frequent null
Maximal Itemsets
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

ABCD

ABCE

ABDE

ACDE

BCDE

Infrequent Itemsets
Tan,Steinbach, Kumar

ABCD E

Border
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Closed Itemset
!

An itemset is closed if none of its immediate supersets has the same support as the itemset
Itemset {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} Support 4 5 3 4 4 2 3 3 4 3

TID 1 2 3 4 5

Items {A,B} {B,C,D} {A,B,C,D} {A,B,D} {A,B,C,D}

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

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Maximal vs Closed Itemsets

TID 1 2 3 4 5 Items ABC ABCD BCE ACDE DE
12
AB null

Transaction Ids
245
D E

124
A

123
B

1234
C

345

124
AC

24
AD

4
AE

123
BC

2
BD

BE

24

CD

34

CE

45

DE

12
ABC

2
ABD ABE

24
ACD

4
ACE

4
ADE

2
BCD

BCE

BDE

CDE

4
ABCD ABCE ABDE ACDE BCDE

Not supported by any transactions

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ABCDE

4/18/2004

Maximal vs Closed Frequent Itemsets

Minimum support = 2
124
A null

Closed but not maximal

245
D E

123
B

1234
C

345

Closed and maximal

34 45

12
AB

124
AC

24
AD

4
AE

123
BC

2
BD

BE

24

CD

CE

DE

12
ABC

2
ABD ABE

24
ACD

4
ACE

4
ADE

2
BCD

BCE

BDE

CDE

4
ABCD ABCE ABDE ACDE BCDE

# Closed = 9 # Maximal = 4

ABCDE

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Maximal vs Closed Itemsets

Frequent Itemsets Closed Frequent Itemsets Maximal Frequent Itemsets

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Alternative Methods for Frequent Itemset Generation

! Traversal

of Itemset Lattice
Frequent itemset border

General-to-specific vs Specific-to-general
Frequent itemset border null null null

.. ..
{a1,a2,...,an}

.. ..
{a1,a2,...,an} Frequent itemset border

.. ..
{a1,a2,...,an}

(a) General-to-specific
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(b) Specific-to-general

(c) Bidirectional
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Introduction to Data Mining

Alternative Methods for Frequent Itemset Generation

! Traversal

of Itemset Lattice
null null

Equivalent Classes

AB

AC

AD

BC

BD

CD

AB

AC

BC

AD

BD

CD

ABC

ABD

ACD

BCD

ABC

ABD

ACD

BCD

ABCD

ABCD

(a) Prefix tree

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(b) Suffix tree

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Alternative Methods for Frequent Itemset Generation

! Traversal

of Itemset Lattice

Breadth-first vs Depth-first

(a) Breadth first

(b) Depth first

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Alternative Methods for Frequent Itemset Generation

! Representation

of Database

horizontal vs vertical data layout

Horizontal Data Layout
TID 1 2 3 4 5 6 7 8 9 10 Items A,B,E B,C,D C,E A,C,D A,B,C,D A,E A,B A,B,C A,C,D B

Vertical Data Layout

A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6

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FP-growth Algorithm
! Use

a compressed representation of the database using an FP-tree an FP-tree has been constructed, it uses a recursive divide-and-conquer approach to mine the frequent itemsets

! Once

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FP-tree construction
After reading TID=1: null A:1 B:1 After reading TID=2: A:1 B:1

TID 1 2 3 4 5 6 7 8 9 10

Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}

null B:1 C:1 D:1

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FP-Tree Construction
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}

Transaction Database

null A:7 B:3 C:3 D:1 D:1 D:1 E:1 Pointers are used to assist frequent itemset generation E:1 E:1

B:5 C:3 D:1

C:1

D:1

Header table Item Pointer A B C D E

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FP-growth

null A:7 B:5 C:3 D:1 D:1 B:1 C:1 D:1

C:1 D:1

D:1

Conditional Pattern base for D: P = {(A:1,B:1,C:1), (A:1,B:1), (A:1,C:1), (A:1), (B:1,C:1)} Recursively apply FPgrowth on P Frequent Itemsets found (with sup > 1): AD, BD, CD, ACD, BCD

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Tree Projection
Set enumeration tree:
Possible Extension: E(A) = {B,C,D,E}
AB AC A B null

AD

AE

BC

BD

BE

CD

CE

DE

ABC

ABD

ABE

ACD

ACE

ADE

BCD

BCE

BDE

CDE

Possible Extension: E(ABC) = {D,E}

ABCD ABCE ABDE ACDE BCDE

ABCDE

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Tree Projection
! Items

are listed in lexicographic order ! Each node P stores the following information:
Itemset for node P List of possible lexicographic extensions of P: E(P) Pointer to projected database of its ancestor node Bitvector containing information about which transactions in the projected database contain the itemset

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Projected Database
Original Database: Projected Database for node A:

TID 1 2 3 4 5 6 7 8 9 10

Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}

TID 1 2 3 4 5 6 7 8 9 10

Items {B} {} {C,D,E} {D,E} {B,C} {B,C,D} {} {B,C} {B,D} {}

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For each transaction T, projected transaction at node A is T % E(A)

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ECLAT
!

For each item, store a list of transaction ids (tids)

Horizontal Data Layout
TID 1 2 3 4 5 6 7 8 9 10 Items A,B,E B,C,D C,E A,C,D A,B,C,D A,E A,B A,B,C A,C,D B

Vertical Data Layout

A 1 4 5 6 7 8 9 B 1 2 5 7 8 10 C 2 3 4 8 9 D 2 4 5 9 E 1 3 6

TID-list
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ECLAT
!

Determine support of any k-itemset by intersecting tid-lists of two of its (k-1) subsets.
A 1 4 5 6 7 8 9
B 1 2 5 7 8 10

&

AB 1 5 7 8

! !

3 traversal approaches:
top-down, bottom-up and hybrid

Advantage: very fast support counting ! Disadvantage: intermediate tid-lists may become too large for memory
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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, A !BCD, AB !CD, BD !AC, ABD !C, B !ACD, AC ! BD, CD !AB, ACD !B, C !ABD, AD ! BC, BCD !A, D !ABC BC !AD,

! If

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

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Rule Generation
! How

to efficiently generate rules from frequent itemsets?

In general, confidence does not have an antimonotone property
c(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

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Rule Generation for Apriori Algorithm

Lattice of rules
Low Confidence Rule
ABCD=>{ }

BCD=>A

ACD=>B

ABD=>C

ABC=>D

CD=>AB

BD=>AC

BC=>AD

AD=>BC

AC=>BD

AB=>CD

D=>ABC

C=>ABD

B=>ACD

A=>BCD

Pruned Rules
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Rule Generation for Apriori Algorithm

! Candidate

rule is generated by merging two rules that share the same prefix in the rule consequent
CD=>AB BD=>AC

! join(CD=>AB,BD=>AC)

would produce the candidate rule D => ABC

! Prune

rule D=>ABC if its subset AD=>BC does not have high confidence
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D=>ABC

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Effect of Support Distribution

! Many

real data sets have skewed support distribution

Support distribution of a retail data set

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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

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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% is infrequent but {Milk,Coke,Broccoli} is frequent
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! {Milk,Coke}

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Multiple Minimum Support

Item MS(I) Sup(I)

X
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
4/18/2004 #

AB AC A AD B C BD D BE CD AE BC

0.10% 0.25%

0.20% 0.26%

0.30% 0.29%

0.50% 0.05%

3%

4.20%

CE DE

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Multiple Minimum Support X

Item MS(I) Sup(I)

AB AC A

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
4/18/2004 #

0.10% 0.25% B C

AD AE BC

0.20% 0.26%

0.30% 0.29% D

BD BE CD E CE DE

0.50% 0.05%

3%

4.20%

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Multiple Minimum Support (Liu 1999) X

! 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
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Multiple Minimum Support (Liu 1999) X

! Modifications
! A

to Apriori:

In traditional Apriori,
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.
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

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 the original formulation of association rules, support & confidence are the only measures used
Introduction to Data Mining 4/18/2004 #

! In

Tan,Steinbach, Kumar

Application of Interestingness Measure

Interestingness Measures

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Computing Interestingness Measure

!

Given a rule X ! Y, information needed to compute rule interestingness can be obtained from a contingency table
Contingency table for X
Y X X f11 f01 f+1 Y f10 f00 f+0 f1+ fo+ |T|

!Y f11: support of X and Y f10: support of X and Y f01: support of X and Y f00: support of X and Y Used to define various measures
" support,

confidence, lift, Gini, J-measure, etc.

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Tan,Steinbach, Kumar

Introduction to Data Mining

Drawback of Confidence

Coffee Tea Tea 15 75 90

Coffee 5 5 10 20 80 100

Association Rule: Tea ! Coffee

Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9 # Although confidence is high, rule is misleading # P(Coffee|Tea) = 0.9375
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

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(S&B) = 420/1000 = 0.42 P(S) ) P(B) = 0.6 ) 0.7 = 0.42 P(S&B) = P(S) ) P(B) => Statistical independence P(S&B) > P(S) ) P(B) => Positively correlated P(S&B) < P(S) ) P(B) => Negatively correlated
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Statistical-based Measures
! Measures

that take into account statistical dependence

P (Y | X ) Lift = P (Y ) P( X , Y ) Interest = P ( X ) P(Y ) PS = P ( X , Y ) ! P ( X ) P(Y ) P ( X , Y ) ! P ( X ) P(Y ) " ! coefficien t = P( X )[1 ! P( X )]P (Y )[1 ! P (Y )]
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Example: Lift/Interest

Coffee Tea Tea 15 75 90

Coffee 5 5 10 20 80 100

Association Rule: Tea ! Coffee

Confidence= P(Coffee|Tea) = 0.75 but P(Coffee) = 0.9 # Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Drawback of Lift & Interest

Y X X 10 0 10

Y 0 90 90 10 90 100 X X

Y 90 0 90

Y 0 10 10 90 10 100

0.1 Lift = = 10 (0.1)(0.1)

0.9 Lift = = 1.11 (0.9)(0.9)

Statistical independence: If P(X,Y)=P(X)P(Y) => Lift = 1

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

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 Aprioristyle support based pruning? How does it affect these measures?

Properties of A Good Measure

! Piatetsky-Shapiro:

3 properties a good measure M must satisfy:

M(A,B) = 0 if A and B are statistically independent M(A,B) increase monotonically with P(A,B) when P(A) and P(B) remain unchanged M(A,B) decreases monotonically with P(A) [or P(B)] when P(A,B) and P(B) [or P(A)] remain unchanged

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Comparing Different Measures

10 examples of contingency tables:
Example
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10

f11
8123 8330 9481 3954 2886 1500 4000 4000 1720 61

f10

f01

f00

Rankings of contingency tables using various measures:

83 424 1370 2 622 1046 94 127 298 3080 5 2961 1363 1320 4431 2000 500 6000 2000 1000 3000 2000 2000 2000 7121 5 1154 2483 4 7452

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Property under Variable Permutation

B p r
B q s

A A

B B

A p q

A r s

Does M(A,B) = M(B,A)? Symmetric measures:

" support,

lift, collective strength, cosine, Jaccard, etc

Asymmetric measures:
" confidence,
Tan,Steinbach, Kumar

conviction, Laplace, J-measure, etc

Introduction to Data Mining 4/18/2004 #

Property under Row/Column Scaling

Grade-Gender Example (Mosteller, 1968):
Male High Low 2 1 3 Female 3 4 7 5 5 10 High Low Male 4 2 6 Female 30 40 70 34 42 76

Mosteller: Underlying association should be independent of the relative number of male and female students in the samples
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

2x

10x

Property under Inversion Operation

A
Transaction 1

B 0 0 0 0 1 0 0 0 0 0 (a)

C 0 1 1 1 1 1 1 1 1 0

D 1 1 1 1 0 1 1 1 1 1 (b)

E 0 1 1 1 1 1 1 1 1 0 (c)
4/18/2004

F 0 0 0 0 1 0 0 0 0 0

. . . . .

Transaction N

1 0 0 0 0 0 0 0 0 1

Tan,Steinbach, Kumar

Introduction to Data Mining

Example: *-Coefficient
! *-coefficient
Y X X 60 10 70

is analogous to correlation coefficient for continuous variables

Y 10 20 30 70 30 100 X X Y 20 10 30 Y 10 60 70 30 70 100

0.6 " 0.7 ! 0.7 #= 0 . 7 ! 0 .3 ! 0 .7 ! 0 .3 = 0.5238

Tan,Steinbach, Kumar

0 . 2 " 0 .3 ! 0 .3 #= 0 . 7 ! 0 .3 ! 0 .7 ! 0 .3 = 0.5238
4/18/2004 #

* Coefficient is the same for both tables

Introduction to Data Mining

Property under Null Addition

B p r
B q s

A A

A A

B p r

B q s+k

Invariant measures:
" support,

cosine, Jaccard, etc

Non-invariant measures:
" correlation,

Gini, mutual information, odds ratio, etc

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Different Measures have Different Properties

Sym bol Measure Correlation Lambda Odds ratio Yule's Q Yule's Y Cohen's Mutual Information J-Measure Gini Index Support Confidence Laplace Conviction Interest IS (cosine) Piatetsky-Shapiro's Certainty factor Added value Collective strength Jaccard Klosgen's
, * * +

Range -1 ! 0 ! 1 0!1 0!1!$ -1 ! 0 ! 1 -1 ! 0 ! 1 -1 ! 0 ! 1 0!1 0!1 0!1 0!1 0!1 0!1 0.5 ! 1 ! $ 0!1!$ 0 .. 1 -0.25 ! 0 ! 0.25 -1 ! 0 ! 1 0.5 ! 1 ! 1 0!1!$ 0 .. 1

P1 Yes Yes Yes* Yes Yes Yes Yes Yes Yes No No No No Yes* No Yes Yes Yes No No

P2 Yes No Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

P3 Yes No Yes Yes Yes Yes Yes No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes

O1 Yes Yes Yes Yes Yes Yes Yes No No Yes Yes Yes Yes** Yes Yes Yes No No Yes Yes No

O2 No No Yes Yes Yes No No No No No No No No No No No No No No No No

O3 Yes No* Yes* Yes Yes No No* No No* No No No No No No Yes No No Yes* No No

O3' Yes Yes Yes Yes Yes Yes Yes No Yes No No No Yes No No Yes Yes No Yes No No
#

O4 No No No No No No No No No No Yes No No No Yes No No No No Yes No

! " # Q Y % M J G s c L V I IS PS F AV S & K

Tan,Steinbach, Kumar

), 2 1 ) 2 - 1 '* 2 - 3 ' 0 Yes ' '* 3 (+ Introduction to Data Mining 3( 3 3

4/18/2004

Support-based Pruning
! Most

of the association rule mining algorithms use support measure to prune rules and itemsets effect of support pruning on correlation of itemsets
Generate 10000 random contingency tables Compute support and pairwise correlation for each table Apply support-based pruning and examine the tables that are removed

! Study

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Effect of Support-based Pruning

All Itempairs
1000 900 800 700 600 500 400 300 200 100 0
-1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 1 2 3 4 5 6 7 8 0. 0. 0. 0. 0. 0. 0. 0. 0. 9 1

Correlation
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #

Effect of Support-based Pruning

Support < 0.01
300 250 200 150 100 50 0
-1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Support < 0.03

300 250 200 150 100 50 0
-1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 1 2 3 4

8 0.

0.

0.

0.

0.

0.

0.

0.

Correlation

Correlation

Support < 0.05

300 250

Support-based pruning eliminates mostly negatively correlated itemsets

Tan,Steinbach, Kumar

200 150 100 50 0

-1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Correlation

Introduction to Data Mining

4/18/2004

0.

Effect of Support-based Pruning

! Investigate

how support-based pruning affects other measures

! Steps:

Generate 10000 contingency tables Rank each table according to the different measures Compute the pair-wise correlation between the measures

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Effect of Support-based Pruning

"

Without Support Pruning (All Pairs)

All Pairs (40.14%)
1 0.9 0.8 0.7 0.6

Conviction Odds ratio Col Strength Correlation Interest PS CF

Reliability Kappa Klosgen Yule Q Confidence Laplace IS Support Jaccard Lambda Gini J-measure Mutual Info 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Jaccard

Yule Y

0.5 0.4 0.3 0.2 0.1 0 -1

-0.8

-0.6

-0.4

-0.2

0 0.2 Correlation

0.4

0.6

0.8

" Red

cells indicate correlation between the pair of measures > 0.85 pairs have correlation > 0.85
Introduction to Data Mining

Scatter Plot between Correlation & Jaccard Measure

" 40.14%

Tan,Steinbach, Kumar

4/18/2004

Effect of Support-based Pruning

"

0.5% + support + 50%

0.005 <= support <= 0.500 (61.45%)
1 0.9 0.8 0.7 0.6

Interest Conviction Odds ratio Col Strength Laplace Confidence Correlation

Reliability PS Yule Q CF Yule Y Kappa IS Jaccard Support Lambda Gini J-measure Mutual Info 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Jaccard

Klosgen

0.5 0.4 0.3 0.2 0.1 0 -1

-0.8

-0.6

-0.4

-0.2

0 0.2 Correlation

0.4

0.6

0.8

Scatter Plot between Correlation & Jaccard Measure:

" 61.45%

pairs have correlation > 0.85

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Effect of Support-based Pruning

"

0.5% + support + 30%

0.005 <= support <= 0.300 (76.42%)
1 0.9 0.8 0.7 0.6

Support Interest Reliability Conviction Yule Q Odds ratio Confidence CF Yule Y Kappa Correlation Col Strength IS Jaccard Laplace PS Klosgen Lambda Mutual Info Gini J-measure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Jaccard

0.5 0.4 0.3 0.2 0.1 0 -0.4

-0.2

0.2 0.4 Correlation

0.6

0.8

Scatter Plot between Correlation & Jaccard Measure

" 76.42%

pairs have correlation > 0.85

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Subjective Interestingness Measure

! Objective

measure:

Rank patterns based on statistics computed from data e.g., 21 measures of association (support, confidence, Laplace, Gini, mutual information, Jaccard, etc).
! Subjective
! A ! A

measure:

Rank patterns according to user s interpretation

pattern is subjectively interesting if it contradicts the expectation of a user (Silberschatz & Tuzhilin) pattern is subjectively interesting if it is actionable (Silberschatz & Tuzhilin)

Tan,Steinbach, Kumar

Introduction to Data Mining

4/18/2004

Interestingness via Unexpectedness

!

Need to model expectation of users (domain knowledge)

+ -

Pattern expected to be frequent Pattern expected to be infrequent Pattern found to be frequent Pattern found to be infrequent

+ - +

Expected Patterns Unexpected Patterns

Need to combine expectation of users with evidence from data (i.e., extracted patterns)
Introduction to Data Mining 4/18/2004 #

Tan,Steinbach, Kumar

Interestingness via Unexpectedness

! Web

Data (Cooley et al 2001)

Domain knowledge in the form of site structure Given an itemset F = {X1, X2, ", Xk} (Xi : Web pages)
! L:

number of links connecting the pages = L / (k ) k-1) = 1 (if graph is connected), 0 (disconnected graph)

! lfactor ! cfactor

Structure evidence = cfactor ) lfactor

P( X ! X ! ... ! X ) Usage evidence = P( X ! X ! ... ! X )

1 2 k 1 2 k

Use Dempster-Shafer theory to combine domain knowledge and evidence from data
Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 #