10 Assrule Eng

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Association rules Introduction

Association rules

A Binary database is a database where each row is composed ofbinary attributes

A B C D . . .T1 1 0 0 1 . . .T2 0 1 1 1 . . .T3 1 0 1 0 . . .T4 0 0 1 0 . . .

From this database frequent patterns of occurrence can be discovered

Javier Bejar cbea (LSI - Master AI) Association rules TAVA - Term 2010/2011 1 / 39


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Association rules Introduction

Association rules

This kind of databases appear for example on transactional data(market basket analysis)

An association rule represents coocurence of events in the database

TID Items1 Bread, Chips, Beer, Yogourt, Eggs2 Flour, Beer, Eggs, Butter,3 Bread, Ham, Eggs, Butter, Milk4 Flour, Eggs, Butter, Chocolate5 Beer, Chips, Bacon, Eggs

{Flour} {Eggs}{Beer} {Chips}{Bacon} {Eggs}



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Association rules Definitions

Definitions (I)

We define R as the set of attributes of the database

We define X as a subset of attributes from R.

We say that X is a pattern from DB if there are any row where all

the attributes of X are 1.We define the support (frequency) of a pattern X as the function:

fr(X) =|M(X, r)|

|r|

Where |M(X, r)| is the number of times that X appears in the DBand |r| is the size of the BD.



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Definitions (II)

Given a minimum support (min sup [0, 1]), we say that the patternX

is frequent if: fr(X) min sup

F(r, min sup) is the set of patterns that are frequent in :

F(r, min sup) = {X R/fr(X) min sup}


A i i l D fi i i


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Definitions (III)

Given a BD with R attributes and X and Y attribute subsets, withX Y = , an association rule is the expression:

X Y

conf(X Y) is the confidence of an association rule, computed as:

conf(X Y) =fr(X Y)

fr(X)

We consider a minimum value of confidence (min conf [0, 1])


A i i l ith Fi di i ti l


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Apriori algorithm Finding association rules

Finding association rules

Given a minimum confidence and a minimum support, a associationrule (X Y) exists in a DB if:

(fr(X Y) min sup) (conf(X Y) min conf )

Trivial approach:

It is enough to find all frequent subsets from RWe have to explore X Y ((X R) (Y X)) the association rule

(X Y) Y

There are 2|R| candidates




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Properties of frequent sets

Given X and Y with Y X:

If fr(Y) fr(X), if X is frequent, Y also is frequentIf any subset Y from X it is not frequent then X is not frequent

The feasible approach is to find the frequent sets starting from size 1and increasing its size




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Finding frequent sets

Fl(r, min sup) is the set of frequent sets from R of size l.

Given a set of patterns of size L, the only frequent set candidates oflength L + 1 will be those which all subsets are in the frequent sets of

length L.

C(Fl+1(r)) = {X R/|X| = l+1 Y X |Y| = l Y Fl(r)}

The computing of association rules can be done iteratively starting

with the smallest frequent subsets until the complete subset isanalyzed


Apriori algorithm Example: Properties


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A DCBL=1

L=2

L=3

L=4 ABCD

ABC ABD ACD BCD

AB AC AD BC BD CD




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A DC

L=1

L=2

L=3

L=4 ABCD

ABC ABD ACD BCD

AB AC AD BC BD CD




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p g p p

A DC

L=1

L=2

L=3

L=4 ABCD

ABC ABD ACD BCD

AC AD CD




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p g p p

A DC

L=1

L=2

L=3

L=4 ABCD

ACD

AC AD CD




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g

A DC

L=1

L=2

L=3

L=4

ACD

AC AD CD


Apriori algorithm Algorithm


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Association Rules Algorithm

Algorithm: Apriori (R,min sup,min conf)C,CCan,CTemp:set of frequent subsetsRAs:Set of association rules, L:integerL=1CCan=Frequent sets 1(R,fr min)RAs=while CCan = do

L=L+1CTemp=Candidate sets(L,R,CCan) = C(Fl+1(r))

C=Frequent sets(CTemp,min sup)RAs=RAs Confidence rules(C,min conf)CCan=C

end


Apriori algorithm Algorithm


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Effect of Support

The specific value used as min sup has effect on the computationalcost of the search

If it is too high, only a few patterns will appear (we could miss

interesting rare occurrences)If it is too low the computational cost will be too high (too manyassociation will appear)

Unfortunately the threshold value can no be known beforehand

Sometimes multiple minimum supports are needed (different fordifferent groups of items)


Apriori algorithm Measures of interestingness


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Measures for association rules interestingness

Sometimes the confidence of a rule does not represent well itsinterestingness

Y YX 15 5 20

X 75 5 8090 10 100

conf(X Y) = 0,75 but conf(X Y) = 0,9357


Apriori algorithm Measures of interestingness


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Measures for association rules interestingness

Other measures can be computed using probabilistic independence

Lift(X Y) =P(Y|X)

P(Y)

Interest(X Y) =P(X, Y)

P(X) P(Y)PS(X Y) = P(X, Y) P(X) P(Y)

There are plenty of measures in the literature, some good for certain

application, but not for others


Apriori algorithm Example


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Example (1)

A B C D E F

1 1 1 0 0 1 02 1 0 1 0 1 13 0 1 1 1 0 14 1 1 1 0 0 05 0 1 0 0 1 16 1 0 1 1 0 07 1 1 0 0 1 0

8 1 1 0 0 0 09 1 0 1 1 0 1

10 1 1 1 0 1 111 1 1 0 0 1 012 1 1 0 1 1 013 1 1 1 1 1 114 0 1 1 0 0 015 1 1 0 1 0 016 1 1 0 0 1 017 0 1 0 0 1 018 0 1 1 0 1 019 1 0 0 0 1 020 1 1 0 0 1 0

fr([A])=15/20

fr([B])=16/20fr([C])= 9/20fr([D])= 6/20fr([E])=13/20fr([F])= 6/20




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Example (2)

Supposing a minimum support of 25 %. All columns have larger frequency,thus:

F1= {[A],[B],[C],[D],[E],[F]}From this we can compute the candidates for frequent sets (all

combinations):

C(F1)= {[A,B],[A,C],[A,D],[A,E],[A,F],[B,C],[B,D],[B,E],[B,F],[C,D],[C,E],[C,F],

[D,E],[D,F],[E,F]}




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AB AC AD AE AF BC BD BE BF CD CE CF DE DF EF

FR=25%

A B C D E F




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Example (3)

fr([A,B])= 11/20 fr([B,C])= 6/20 fr([C,D])= 4/20 fr([D,E])= 2/20fr([A,C])= 6/20 fr([B,D])= 4/20 fr([C,E])= 4/20 fr([D,F])= 2/20fr([A,D])= 4/20 fr([B,E])= 11/20 fr([C,F])= 5/20 fr([E,F])= 4/20fr([A,E])= 10/20 fr([B,F])= 4/20fr([A,F])= 4/20

Of the candidates, the ones that have a support greater than 25 % are:

F2={[A,B],[A,C],[A,E],[B,C],[B,E],[C,F]}




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A B C D E F

AF BD BF CD CE DE DF EFAB AC AE BC BE CFAD

FR=25%

ABC ABE




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Example (4)

The confidence of the association rules are:

conf(A B)= 11/15 conf(A E)= 10/15 conf(B E)= 11/16conf(B A)= 11/16 conf(E A)= 10/13 conf(E B)= 11/13conf(A C)= 6/15 conf(B C)= 6/16 conf(C F)= 5/9conf(C A)= 6/9 conf(C B)= 6/9 conf(F C)= 5/13

If we choose as confidence value 2/3 the rules that are significant:

{A B, B A, C A, A E, E A, C B, B E, E B}




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Example (5)

From the set F2, we can compute the set of candidates for F3

C(F2)={[A,B,C],[A,B,E]}

fr([A,B,C])= 3/20 fr([A,B,E])= 8/20

Of the candidates, the ones that have support greater than 25 % are:

F3={[A,B,E]}




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A B C D E F

AF BD BF CD CE DE DF EFAD

FR=25%

AB AC AE BC BE CF

ABEABC




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Example (6)

The confidence of their corresponding association rules are:

conf(A B E)= 8/11conf(A E B)= 8/10conf(B E A)= 8/11

If we choose as confidence value 2/3 the rules that are significant:

{A B E, A E B, B E A}




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Example (7)

Now we choose a minimum support of 50 %. The columns that have largerfrequency are only A, B and E. Therefore:

F1= {[A],[B],[E]}

From this we can compute the support set candidates:

C(F1)={[A,B],[A,E],[B,E]}



( )


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Example (8)

All have a support greater than 50 %, therefore

F2={[A,B],[A,E],[B,E]}

And the candidates are:

C(F2)={[A,B,E]}

That also have a support greater than 50 %, thus:

F3={[A,B,E]}


FP-Grow Introduction

G


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FP-Grow

Han, Pei, Yin, Mao Mining Frequent Patterns without CandidateGeneration: A Frequent-Pattern Tree Approach (2004)

The problem of the apriori approach is that the number of candidates

to explore in order to find long patterns can be very largeThis approach tries to obtain patterns from transaction databaseswithout candidate generation

It is based on a specialized data structure (FP-Tree) that summarizesthe frequency of the patterns in the DB

The patterns are explored incrementally adding prefixes withoutcandidate generation


FP-Grow FP-tree

FP T


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FP-Tree

The goal of this structure is to avoid to query the DB to compute thefrequency of patterns

It assumes that there is an order among the elements of a transactionThis order allows to obtain common prefixes from the transactions

The transactions with common prefixes are merged in the structuremaintaining the frequency of each subprefix


FP-Grow FP-tree

FP T Al i h


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FP-Tree - Algorithm

1 Compute the frequency of each individual item in the DB

2 Create the tree root (empty prefix)3 For each transaction from the DB

1 Pick the more frequent items2 Order the items by their original frequency3 Iterate for each item, inserting it in the tree

If the node has a descendant equal to the actual item increase its

frequency

Otherwise, a new node is created


FP-Grow FP-tree

FP T E l


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FP-Tree - Example

BD Transactions

1 a, e, f 2 a, c, b, f, g3 b, a, e, d4 d, e, a, b, c5 c, d, f, b6 c, f, a, d

Item frequency

a 5b 4

c 4d 4e 3f 4g 1

BD Transactions (fr=4)

1 a, f 2 a, c, b, f 3 a, b, d4 a, b, c, d5 b, c, d6 a, c, d, f


FP-Grow FP-tree

FP T E l


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FP-Tree - Example

a

b

c

d

f4

4

4

4

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1

1

Trans(a,f)


FP-Grow FP-tree

FP T E l


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FP-Tree - Example

a

b

c

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f4

4

4

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f1

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FP-Grow FP-tree

FP Tree Example


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FP-Tree - Example

a

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f4

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Trans(a,b,d)

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FP-Grow FP-tree

FP Tree Example


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FP-Tree - Example

a

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Trans(a,b,cd)

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FP-Grow FP-tree

FP Tree Example


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FP-Tree - Example

a

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FP-Grow FP-tree

FP Tree Example


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FP-Tree - Example

a

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Trans(a,c,d,f

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FP-Grow Pattern extraction

Pattern extraction


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Pattern extraction

From this data structure we can obtain all frequent patterns using theproperties of a FP-tree:

For any given frequent item ai we can obtain all the frequent patterns

that contain the item following the links ofai

in the treeTo compute the frequent patterns with ai as suffix only the prefixsubpaths of the nodes ai have to be used and their frequencycorresponds to the frequency of node ai

The frequent patterns can be obtained from a recursive traversal of

the paths in the FP-tree and the combination of subpatterns


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10 Assrule Eng