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Association Graphs

Selim Mimaroglu

University of Massachusetts Boston

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This Presentation is Built On: A Graph-Based Approach for Discovering Various

Types of Association Rules (MAIN) by Show-Jane Yen and A.L.P. Chen at IEEE Transactions on Data and Knowledge Engineering 2001

Mining Multiple-Level Association Rules from Large Databases by Jiawei Han and Yongjian Fu at VLDB ’95

Mining Multilevel Association Rules from Transaction Databases section 6.3 ISBN 1558604898

Mining Generalized Association Rules by R. Srikant, and R. Agrawal at VLDB ’95

Introduction to Data Mining by Tan, Steinbach, Kumar ISBN:0321321367

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Organization Primitive Association Rules

Definition Association Graphs for finding large (frequent) item sets

Multiple-Level Association Rules Definition Why it’s important? Association Graphs for finding large item sets Interest Measure

Generalized Association Rules Definition Association Graphs for finding large item sets

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Association Rules Discovering patterns from a large database

(generally data warehouse) is computationally expensive

To find all the rules that satisfy minsupport and minconf

X → Y Transactions in the database contain the items

in X tend also contain the items in Y.

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Definitions Let I={i1, i2, …, id} be

set of all items in a market basked data

Let T={t1, t2, ..tN} be the set of all transactions

A collection of items is termed itemset

TID i1 i2 … id

t1 1 0 1

t2 1 0 0

… 0 0 1

tN 1 1 1

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DefinitionsLet X and Y be two disjoint itemsets

Support Count of X

Support of X → Y

Confidence of X → Y

( ) |{ | , } |i i iX t X t t T

( )( )

X Ys X Y

N

( )( )

( )

X Yc X Y

X

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Taxonomy (or Concept Hierarchy) This is created by the domain expert, e.g. store

manager. There may be more than one taxonomy for the same

items is-a relation “Folgers Coffee Classic Roast Singles - 19 ct”

(barcode 7890045) is-a “Regular Ground Coffee” “Regular Ground Coffee” is-a “Coffee” “Coffee” is-a “Beverage” “Beverage” is-a “General Grocery Item”

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Taxonomy Example

……………………………………………………

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Primitive Association Rules Deals with the lowest level items of the

taxonomy (most widely used) Very well studied

Algorithms using Apriori principle FP-Growth Algorithms

Steps Large (frequent) itemset generation Rule Generation

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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

Frequent Itemset Generation (Apriori)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

ABCDEPruned supersets

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Motivation Many database scans for

checking if the candidate itemsets are qualified (i.e. ≥ minsupport )

Reducing the database scans FP-Growth: 2 scans Association Graphs: 1 scan ?

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Mining Primitive Association Rules

Create Bit Vectors

TID A B C D E

100 1 0 1 1 0

200 0 1 1 0 1

300 1 1 1 0 1

400 0 1 0 0 1

Number the Attributes

A:1 D:4

B:2 E:5

C:3

BV1

1

0

1

0

BV2

0

1

1

1

BV3

1

1

1

0

BV4

1

0

0

0

BV5

0

1

1

1

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Bit Vectors (Column Store) This is a column store (as

opposed to a row store) Read Efficient Row stores are write

efficient For an item or itemset a 64

bit processor can count the support count of 64 rows in one instruction only

Logical AND, OR

minsupport = 50% (2 transactions)

Is item 1 (column A) frequent ? Yes

BV1

1

0

1

0

Is the itemset {1, 3} frequent?Yes

BV1

1

0

1

0

BV3

1

1

1

0

{1,3}

1

0

1

0

=

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Association Graph ConstructionProperty 1: The support for the itemset {i1, i2, …, ik} is

the number of 1s in BVi1 BVi∧ 2 … ∧ BVik, where the notation “∧” is a logical AND operator

ACG: Association Graph Construction: For every two large items i and j ( i < j), if the number of 1s in BVi ∧ BVj achieves the user-specified minimum support, a directed edge from item i to item j is created (The Association Graph is a Directed Acyclic Graph. Proven in Appendix A). Also, itemset (i, j) is a large 2 - itemset

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Example 1:

BV1

1

0

1

0

BV2

0

1

1

1

BV3

1

1

1

0

BV4

1

0

0

0

BV5

0

1

1

1

1

2 5

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Lemma 1: If an itemset is not a large itemset, then any itemset which contains this itemset cannot be a large itemset (Proof: Apriori principle)

Lemma 2: For a large itemset (i1, i2, …, ik), if there is no directed edge from ik to an item v, then itemset (i1, i2, …, ik, v) cannot be a large itemset

Proof of Lemma 2: If there is not an edge from ik to v then (ik, v) is not a large itemset. If (ik, v) is not a large itemset then none of the supersets of (ik, v) can be a large itemset (e.g. (i1, i2, …, ik, v) isn’t a large itemset )

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Finding All Large Itemsets by Using an Association Graph

Suppose (i1, i2, …, ik) is a large k-itemset. If there is no directed edge from ik to v then the itemset need not be extended into k+1-itemset because (i1, i2, …, ik, v) must not be a large itemset according to Lemma 2

If there is a directed edge from ik to u, then the itemset (i1, i2, …, ik) is extended to k+1-itemset (i1, i2, …, ik, u). The itemset (i1, i2, …, ik, u) is a large k+1 itemset if the number of 1s in BVi1 ∧BVi2 … ∧ BVik BVi∧ u achieves the minsupport

1

2 5

3

Fig 1: Association Graph

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Example 2: Finding all large itemsets by using the Association Graph of Example1

Large 2-itemsets: (1, 3) (2, 3) (2, 5) (3,5)

minsupport :

50% (2 rows)1

2 5

3

Fig 1: Association Graph

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Large 3-itemset Generation (1, 3, 5) (2, 3, 5)

1

2 5

3

BV1

1

0

1

0

BV3

1

1

1

0

BV5

0

1

1

1

(1, 3, 5)

0

0

1

0

=

BV3

1

1

1

0

BV5

0

1

1

1

=

(2, 3, 5)

0

1

1

0

BV2

0

1

1

1

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Is it really 1 (one) scan? Creating an Association

Graph may be computationally expensive

This really is not 1 scan only.

Bit Vectors make reading extremely efficient.

This is almost as good as it gets

1

2 5

3

Fig 2: An itemset lattice (at level 2)

Fig 1: An Association Graph

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Multilevel Association Rulesall

computer software printer

desktop laptop

IBM Dell

computeraccessory

Sony Toshiba

educationalfinancial

management

Microsoft … …

color b/w

HP … Sony …

wrist pad

Ergoway

mouse

Logitech

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Why Multilevel Association Rules? Support at the lower concept levels is low You can miss the association rules at higher

levels (bigger picture) Support:

Uniform support for all levels Reduced minimum support at lower levels Level by level independent Other approaches

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Mining Multiple level Association Rules Replace items/concepts

at level k with the concepts at level k-1

Apply an Association Rules Algorithm at the items at level k-1

computer

desktop laptop

IBM Dell Sony Toshiba

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Example 3 for Multilevel Associations For level 4 Generate Association

Graphs on this table

Upgrade the items at Table 1 to level 3 (by looking at the taxonomy tree)

TID Items Purchased

T1 IBM desktop computer,

Sony b/w printer

T2 Microsoft educational software,

Microsoft financial software

T3 Logitech mouse,

Microsoft financial software

T4 IBM desktop computer,

Microsoft financial software

T5 IBM desktop computer

Table 1: Market basket data (at lowest level taxonomy: level 4)

TID Items Purchased

T1 desktop computer,

b/w printer

T2 educational software,

financial software

T3 mouse,

financial software

T4 desktop computer,

financial software

T5 desktop computer

Table 2: Market basket data (at level 3)

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Redundant Multilevel Rulesdesktop computer → b/w printer [support=8%, confidence = 70%]

(R1)

IBM desktop computer → b/w printer [support=2%, confidence=72%] (R2)

A rule is interesting if: it doesn’t have a parent rule it can’t be deduced from its parent rule

Suppose that one quarter of all “desktop computers” are “IBM desktop computers”, so we expect following Confidence of R2 shall be around 70% Support of R2 shall be around 8% * ¼ = 2%

R2 is redundant (uninteresting) because it does not convey any additional information

All redundant rules shall be pruned

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Mining Generalized Association Rules

all

computer software printer

desktop laptop

IBM Dell

computeraccessory

Sony Toshiba

educationalfinancial

management

Microsoft … …

color b/w

HP … Sony …

wrist pad

Ergoway

mouse

Logitech

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Rule Generation To generate generalized association patterns one can

add all ancestors of each item and then apply the basic algorithm

{IBM desktop computer, Sony b/w printer} {IBM desktop computer, desktop computer,

computer, Sony b/w printer, b/w printer, printer} This way works, but inefficient

If {IBM desktop computer} is a large itemset then {IBM desktop computer, desktop computer}, {IBM

desktop computer, computer} and {desktop computer, computer} are large itemsets but redundant

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Definition A generalized association rule is an

implication of the form X→Y, where X⊂I, Y ⊂ I, X∩Y=∅, and no item in Y is an ancestor of any item in X.

X →ancestor(X) is trivially true with 100% confidence and hence redundant

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Support for an item and its ancestorLemma 3: The support for an itemset X that contains an item x i and

its ancestor yi will be the same as the support for the itemset X- yi

Proof of Lemma 3: X={x1, ..., xi, yi,…, xn}The support for the itemset X is:

BV x1 ∧ … ∧ BV xi ∧ BV yi …∧ BV xn Note that:

BV xi = BV xi ∧ BV yi

The support of X becomes as the support of X- yi

BV x1 ∧ … ∧ BV xi ∧ …∧ BV xn

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Post Order Numbering Number the items

using the Post Order Numbering method (PON)

Lemma 4: For every two items i and j (i<j), item v is an ancestor of item i, but not an ancestor of item j , then v<j An example of concept hierarchy

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Support for an ancestor Lemma 5: Suppose items i1, i2, …, and im are all

specific descendants of the generalized item in. The bit vector BV in associated with item in is BV i1∨BV i2 ∨ …… ∨BV im and the number of 1s in this bit vector is the support for item in where ∨ stands for logical OR operation.

Lemma 6: If an itemset X is large itemset, then any itemset generated by replacing an item in itemset X with its ancestor is also a large itemset

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Creating the Generalized Association Graph Lemma 7: if (the number of 1s in BVi∧BVj)≥ minsupport, then

for each ancestor u of item i and for each ancestor v of item j, (the number of 1s in BVu∧BVj)≥minsupport and (the number of 1s in BVi∧BVv)≥minsupport

From Lemma 7, if an edge from item i to item j is created, the edges from item i to the ancestors of item j, which are not ancestors of item i, are also created

From Lemma 4, the ancestors of item i, which are not ancestors of item j, are all less than j. Hence if an edge from item i to item j is created, the edges from the ancestors of item i, which are not ancestors of j, to item j are also created (A. Graph Construction Pr.)

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Example of Generalized Association Rule with minsupport 40% (2 transactions)

The generalized Association Graph

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FinallyTheorem 1: Any itemset generated by traversing the Generalized Association

Graph (GAG) will not contain both an item and its ancestor

Proof of Theorem 1: Basis of Induction: In GAG there will be no edge between an item and its

ancestor. Therefore all 2-itemsets are free of ancestors Inductive Hypothesis: We assume that any large k-itemset (i1, i2, …, ik) does

not contain both an item and its ancestor Inductive Step: Large k-itemset (i1, i2, …, ik) is extended to k+1-itemset (i1,

i2, …, ik,, w). Suppose v1, v2, …, vk-1 are ancestors of i1, i2, …, ik-1 respectively, but none are ancestors of item ik. Because items are numbered by PON method, ik> vj (1≤ j ≤ k-1). Hence, there are no edges from item ik to the ancestors of items i1, i2, …, ik. So, item w cannot be an ancestor of item i1, i2, …, ik