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In: Proceedings of 20th very large databases (VLDB) conference, pp 487–499 Agrawal R, Imielinski T, Swami A (1993) Mining association rules between sets of items in large databases. In: Proceedings of ACM SIGMOD conference management of data, pp 207–216 Feller W (1968) An introduction to probability theory and its applications, vol 1, 3rd edn. fi/data Gregg JR (1998) Ones and zeros: understanding Boolean algebra, digital circuits, and the logic of sets. Wiley-IEEE Press, New York Han J, Pei J, Yiwen Y (2000) Mining frequent patterns without candidate generation.
We calculate supp〈Y, X, D〉 in terms of supports of relevant frequent itemsets, for Y ⊆ X. Let X = Y ∪ Z, where Z = {a1, a2, …, ap}. The following theorem is useful for synthesizing conditional supports using relevant frequent itemsets in D. 8 Let X, ÉY and Z are itemsets in database D such that X ¼ Y [ Z, where Z ¼ a1 ; a2 ; . ; ap . Then, 18 2 Synthesizing Conditional Patterns in a Database supphY; X; Di ¼ suppðY; DÞ À p X suppðY \ fai g; DÞ þ i¼1 p X À p X suppðY \ fai ; aj g; DÞ i\j; i;j¼1 suppðY \ fai ; aj ; ak g; DÞ þ Á Á Á þ ðÀ1Þp ð2:4Þ i\j\k; i;j;k¼1  suppðY \ fa1 ; a2 ; .
Proof The proof is based on induction on m. Now, P(Y, D) = P(Y ∧ a1, D) + P (Y ∧ ¬a1, D). The result is true for m = 1. Let the result is true for m ≤ k. We shall show that the result is true for m = k + 1. For m = k, we have P(Y, D) = P (Y ∧ a1 ∧ a2 ∧ ··· ∧ ak, D) + P(Y ∧ ¬a1 ∧ a2 ∧ ··· ∧ ak, D) + P (Y ∧ a1 ∧ ¬a2 ∧ ··· ∧ ak, D) + P(Y ∧ ¬a1 ∧ ¬a2 ∧ ··· ∧ ak, D) + ··· + P (Y ∧ ¬a1 ∧ ¬a2 ∧ ··· ∧ ¬ak−1 ∧ ak, D) + P(Y ∧ ¬a1 ∧ ¬a2 ∧ ··· ∧ ¬ak−1 ∧ ¬ak, D) [by induction hypothesis]. After incorporating ak+1, we get P(Y, D) = P (Y ∧ a1 ∧ a2 ∧ ··· ∧ ak ∧ ak+1, D) + P(Y ∧ a1 ∧ a2 ∧ ··· ∧ ak ∧ ¬ak+1, D) + P (Y ∧ ¬a1 ∧ a2 ∧ ··· ∧ ak ∧ ak+1, D) + P(Y ∧ ¬a1 ∧ a2 ∧ ··· ∧ ak ∧ ¬ak+1, D) + ··· + P (Y ∧ ¬a1 ∧ ¬a2 ∧ ··· ∧ ¬ak−1 ∧ ¬ak ∧ ak+1, D) + P(Y ∧ ¬a1 ∧ ¬a2 ∧ ··· ∧ ¬ak h −1 ∧ ¬ak ∧ ¬ak+1, D).