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We deal with logical calculi of associational rules. Formulae of these calculi correspond to association rules. Formula – association rules concern data matrices. Association rule can be true or false in a data matrix M.
We deal with association rules that are generalization of rules mined by the procedure 4ft-Miner. We do not deal here with conditional associational rules. The association rule have the form of φ ≈ ψ where φ and ψ are the Boolean attributes.
The meaning of the association rule φ ≈ ψ is that the Boolean attributes φ and ψ are associated in the way given by the symbol ≈. The symbol ≈ is called 4ft-quantifier.
There are predicate calculi of association rules. Examples of predicate association rules are
P_{1} ∧ P_{2} ⇒_{0.9, 100} P_{3} and P_{1} ∨ P_{2} ⇔_{0.95, 50} P_{3} ∧ ¬P_{4}.
Here ⇒_{0.9, 100} is the 4ft-quantifier of founded implication and ⇔_{0.95, 50} is the 4ft-quantifier of double founded implication.
Predicate calculi of association rules correspond to four-fold table predicate calculi [Ra 98A] and they are special case of monadic observational predicate calculi [Ha 78].
The calculi of association rules concern rules φ ≈ ψ where φ and ψ can be any correct combination of some basic Boolean attributes and Boolean connectives ∧, ∨, ¬.
Examples of basic Boolean attributes and theirs values in data matrix M are in Fig. 1.
Rows of M | Attributes of M | Examples of basic Boolean attributes | |||||
A_{1} | A_{2} | … | A_{K} | A_{1}(1,2) | A_{1}(3) | A_{1}(4) | |
o_{1} | 1 | 6 | … | B | 1 | 0 | 0 |
o_{2} | 2 | 4 | … | C | 1 | 0 | 0 |
o_{3} | 4 | 7 | … | G | 0 | 0 | 1 |
… | … | … | … | … | … | … | … |
o_{n - 1} | 4 | 9 | … | F | 0 | 0 | 1 |
o_{n} | 3 | 8 | … | H | 0 | 1 | 0 |
Examples of association rules are
A_{1}(1,2) ∧ A_{K}(B,G,H) ⇒_{0.9, 100} A_{2}(4,7,9) and A_{3}(2,3,9) ∨ A_{4}(1,2,9) ⇔_{0.95, 50} ¬A_{5}(3,6,8).
Calculi of association rules are defined and studied e.g. in [Ra 98C], see also [Ra 01B]. These calculi are special case of monadic predicate calculi defined [Ha 78].
The value of association rule φ ≈ ψ in data matrix M can be true or false. The true value of φ ≈ ψ in M depends on four-fold contingency table 4ft(φ, ψ, M) of φ and ψ in M, see Tab. 1.
M | ψ | ¬ψ |
---|---|---|
φ | a | b |
¬φ | c | d |
Here a is the number of objects satisfying both φ and ψ, b is the number of objects satisfying φ and not satisfying ψ, c is the number of objects not satisfying φ and satisfying ψ, and d is the number of objects satisfying neither φ nor ψ.
There is a {0,1} function F_{≈} associated to each 4ft-quantifier ≈. The function F_{≈} is defined for all quadruples 〈a,b,c,d〉 of integer non-negative numbers such that a + b + c + d > 0.
The association rule φ ≈ ψ is true in data matrix M if it is
F_{≈}(a,b,c,d) = 1 where 〈a,b,c,d〉 = 4ft(φ, ψ, M).
We usually write only ≈ (a,b,c,d) instead of F_{≈}(a,b,c,d).
Examples of 4ft quantifiers and their associated functions are 4ft quantifiers implemented in the procedure 4ft-Miner.
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