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Mining Association Rules

Mohamed G. Elfeky

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Introduction

Data mining is the discovery of knowledge and useful information from the large amounts of data stored in databases.

Association Rules: describing association relationships among the attributes in the set of relevant data.

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RulesBody ==> Consequent [ Support , Confidence ]

Body: represents the examined data. Consequent: represents a discovered property for the examined data. Support: represents the percentage of the records satisfying the body or the consequent. Confidence: represents the percentage of the records satisfying both the body and the consequent to those satisfying only the body.

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Association Rules Examples

Basket DataTea ^ Milk ==> Sugar [0.3 , 0.9]

Relational Datax.diagnosis = Heart ^ x.sex = Male ==> x.age > 50 [0.4 , 0.7]

Object-Oriented Datas.hobbies = { sport , art } ==> s.age() = Young [0.5 , 0.8]

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Topics of DiscussionFormal Statement of the ProblemDifferent Algorithms AIS SETM Apriori AprioriTid AprioriHybrid

Performance Analysis

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Formal Statement of the Problem

I = { i1 , i2 , … , im } is a set of itemsD is a set of transactions TEach transaction T is a set of items (subset of I)TID is a unique identifier that is associated with each transactionThe problem is to generate all association rules that have support and confidence greater than the user-specified minimum support and minimum confidence

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Problem Decomposition

The problem can be decomposed into two subproblems:

1. Find all sets of items (itemsets) that have support (number of transactions) greater than the minimum support (large itemsets).

2. Use the large itemsets to generate the desired rules.

For each large itemset l, find all non-empty subsets, and for each subset a generate a rule a ==> (l-a) if its confidence is greater than the minimum confidence.

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General Algorithm1. In the first pass, the support of each

individual item is counted, and the large ones are determined

2. In each subsequent pass, the large itemsets determined in the previous pass is used to generate new itemsets called candidate itemsets.

3. The support of each candidate itemset is counted, and the large ones are determined.

4. This process continues until no new large itemsets are found.

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AIS AlgorithmCandidate itemsets are generated and counted on-the-fly as the database is scanned.

1. For each transaction, it is determined which of the large itemsets of the previous pass are contained in this transaction.

2. New candidate itemsets are generated by extending these large itemsets with other items in this transaction.The disadvantage is that this results in unnecessarily generating and counting too many candidate itemsets that turn out to be small.

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Example

TID Items

100 1 3 4

200 2 3 5

300 1 2 3 5

400 2 5

Database

Itemset

Support

{1} 2

{2} 3

{3} 3

{5} 3

L1

Itemset

Support

{1 3}* 2

{1 4} 1

{3 4} 1

{2 3}* 2

{2 5}* 3

{3 5}* 2

{1 2} 1

{1 5} 1

C2

Itemset

Support

{1 3 4} 1

{2 3 5}*

2

{1 3 5} 1

C3

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SETM AlgorithmCandidate itemsets are generated on-the-fly as the database is scanned, but counted at the end of the pass.

1.New candidate itemsets are generated the same way as in AIS algorithm, but the TID of the generating transaction is saved with the candidate itemset in a sequential structure.

2.At the end of the pass, the support count of candidate itemsets is determined by aggregating this sequential structureIt has the same disadvantage of the AIS algorithm.Another disadvantage is that for each candidate itemset, there are as many entries as its support value.

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Example

TID Items

100 1 3 4

200 2 3 5

300 1 2 3 5

400 2 5

Database

Itemset

Support

{1} 2

{2} 3

{3} 3

{5} 3

L1

Itemset

TID

{1 3} 100

{1 4} 100

{3 4} 100

{2 3} 200

{2 5} 200

{3 5} 200

{1 2} 300

{1 3} 300

{1 5} 300

{2 3} 300

{2 5} 300

{3 5} 300

{2 5} 400

C2

Itemset

TID

{1 3 4} 100

{2 3 5} 200

{1 3 5} 300

{2 3 5} 300

C3

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Apriori Algorithm

Candidate itemsets are generated using only the large itemsets of the previous pass without considering the transactions in the database.

1.The large itemset of the previous pass is joined with itself to generate all itemsets whose size is higher by 1.

2.Each generated itemset, that has a subset which is not large, is deleted. The remaining itemsets are the candidate ones.

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Example

TID Items

100 1 3 4

200 2 3 5

300 1 2 3 5

400 2 5

Database

Itemset

Support

{1} 2

{2} 3

{3} 3

{5} 3

L1

Itemset

Support

{1 2} 1

{1 3}* 2

{1 5} 1

{2 3}* 2

{2 5}* 3

{3 5}* 2

C2

Itemset

Support

{2 3 5}*

2

C3 {1 2 3}

{1 3 5}

{2 3 5}

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AprioriTid AlgorithmThe database is not used at all for counting the support of candidate itemsets after the first pass.

1.The candidate itemsets are generated the same way as in Apriori algorithm.

2.Another set C’ is generated of which each member has the TID of each transaction and the large itemsets present in this transaction. This set is used to count the support of each candidate itemset.The advantage is that the number of entries in C’ may be smaller than the number of transactions in the database, especially in the later passes.

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Example

TID Items

100 1 3 4

200 2 3 5

300 1 2 3 5

400 2 5

Database

Itemset

Support

{1} 2

{2} 3

{3} 3

{5} 3

L1

Itemset

Support

{1 2} 1

{1 3}* 2

{1 5} 1

{2 3}* 2

{2 5}* 3

{3 5}* 2

C2

Itemset

Support

{2 3 5}*

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C3

100 {1 3}

200 {2 3}, {2 5}, {3 5}

300 {1 2}, {1 3}, {1 5}, {2 3}, {2 5}, {3 5}

400 {2 5}

C’2

200 {2 3 5}

300 {2 3 5}

C’3

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Performance Analysis

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AprioriHybrid Algorithm

Performance Analysis shows that:1. Apriori does better than AprioriTid in the

earlier passes.2. AprioriTid does better than Apriori in the

later passes.Hence, a hybrid algorithm can be designed that uses Apriori in the initial passes and switches to AprioriTid when it expects that the set C’ will fit in memory.