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A Theoretical Framework for Association Mining

based on the Boolean Retrieval Model on the

Boolean Retrieval Model

- Peter Bollmann-Sdorra

Contents

- Introduction
- Background
- Boolean Association Mining
- Expressing item-sets as queries
- Conclusions
- Future Work

Introduction

- Researchers focus on discovering rules in the

form of implications between itemsets which have

adequate supports. - Having frequent itemsets as both antecedent and

precedent parts of rules represent only the

simplest form of predicates. - This simplicity is due in part to the lack of a

theoretical framework that includes more

expressive predicates.

Motivation

- In Information retrieval systems, a strong

theoretical background gives the user the power

to ask more sophisticated and pertinent

questions. - Information retrieval and association mining are

two complementary processes on the same data

records or transactions. - In information retrieval, given a query, we need

to find the subset of records that matches the

query. - In contrast, in data mining, we need to find the

queries (rules) having adequate number of records

that support them.

Proposed Solution

- we introduce the theory of association mining

that is based on a model of retrieval known as

the Boolean Retrieval Model, where - a Boolean query that uses only the AND operator

is analogous to an itemset, - a general Boolean query (AND, OR or NOT) has

interpretation as a generalized itemset, - notions of support of itemsets and confidence of

rules can be dealt with uniformly, and - an event algebra can be defined, involving all

possible transaction subsets, to formally obtain

a probability space.

Background

- Deriving association rules from data
- Given a set of items Ii1,i2, . . . , in,

and a set of transactions T t1, t2, . . .,

tm, each transaction ti? T , such that ti ? I, - an association rule is defined as X ? Y, where

X ? I, Y ? I, and X ? Y ?, describes the

existence of a relationship between the two

itemsets X and Y.

Measure for Significance

- The percentage of transactions in the database

that contain both X and Y.

Measure for Importance

- The percentage of transactions that contain Y

among those transactions containing X.

Measure for Importance

- Represents a test of statistical independence.

Boolean Association Mining

- Given a set of items I i1, i2, , in, a

transaction t is defined as a subset of items

such that t?2I, where 2I ?, i1, i2, ,

in, i1, i2, , i1, i2, , in. - Let T ? 2I be a given set of transactions t1,

t2, , tm. Every transaction t?T has an assigned

weight w(t).

Possible Weights

- weights ws are normalized to
- and

Example

- Let I beer, milk, bread be the set of all

items, where price(beer) 5, price(milk) 3,

and price(bread) 2. The set of transactions T

is - f(t) is the frequency of transaction t

Case 1 W(t) 1,

Case 2 W(t) f(t),

Case 3 W(t) t g(t),

Let g(t)f(t),

Case 4 W(t) v(t) g(t),

Let g(t)f(t) and v(t)Price(t)

Expressing item-sets as queries (logical

expressions)

- Definition 1 For a given set of items I, the set

Q of all possible queries associated with

item-sets created from I is defined as follows. - i ? I ? i ? Q,
- q, q ? Q ? q ? q? Q
- These are all.

- Definition 2 For any query q ? Q, the response

set of q, RS(q), is defined as follows - For all atomic i ? Q, RS(i) t?T i?t
- RS (q ? q) RS(q) ? RS(q)

- Definition 3 Let q (i1?i2??ik) and Aq denote

the item-set associated with q that is, Aq

i1, i2, , ik, the support of Aq is defined as - where q (i1? i2? ? ik).

- Lemma 1
- The support set of Aq SS(Aq), equals to RS(q).
- Lemma 2
- For queries q, q1, q2 and q3, the following

axioms hold - RS(q ? q) RS(q)
- RS((q1 ? q2) ? q3) RS(q1 ? (q2 ? q3))
- RS(q1 ? q2) RS(q2 ? q1)

Example

- RS((x1 ? x2) ? (x3 ? x2)) RS(x1 ? x2 ? x3)

- Definition 4
- For a given set of items I, the set Q of all

possible queries is defined as follows. - i ? I ? i ? Q,
- q, q ? Q ? q ? q? Q
- q, q ? Q ? q ? q ? Q
- q ? Q ? ?q ? Q

- Definition 5
- For any query q ? Q, the response set of

transactions, R (q) is defined as - For all i ? Q, RS (i) t?T i?t
- RS (q ? q) RS (q) ? RS (q)
- RS (q ? q) RS (q) ? RS (q)
- RS (?q) T - RS (q)

Theorem

- If q is a transformation of q that is obtained

by applying the rules of Boolean algebra, then - RS(q) RS(q)
- Each q ?Q can be considered as a generalized

itemset. The itemsets investigated in earlier

works only consider q ?Q.

- Lemma 3
- RS(q) q ?Q2T
- Theorem
- (T, 2T, P) is a probability space.

Rules and Their Response Strengths

- Definition 6 The confidence of a rule
- Aq ? Aq is defined as
- Definition 7 The interest of a rule Aq ? Aq is

defined as - Definition 8 The support of a rule Aq ? Aq is

defined as

- Lemma 4 For a rule Aq ? Aq,
- Lemma 5 For a rule Aq ? Aq,

Conclusions

- The theory of association mining that is based on

a model of retrieval known as the Boolean

Retrieval Model has been introduced. - The framework we develop derives from the

observation that information retrieval and

association mining are two complementary

processes on the same data records or

transactions. - Based on the theory of Boolean retrieval, we

generalize the itemset structure by using all

Boolean operators.

Conclusions (cont.)

- By introducing the notion of support of

generalized itemsets, a uniform measure for both

itemsets and rules (generalized itemsets) has

been developed. - Support of a generalized itemset is extended to

allow transactions to be weighted so that they

can contribute to support unequally.

Future Work

- In order to only generate understandable

queries, new restrictions or measures, such as,

compactness and simplicity, should be introduced.

- (These restrictions or measures could eliminate

a large number of frequent generalized itemsets,

many of which could have complex structures.)

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