A characteristic rule is a rule of the form {p} ?? {q1, q2,...,qn}, where the rule antecedent contains only a single item. An itemset of size k can produce up to k characteristic rules. Let ? be the minimum confidence of all characteristic rules generated from a given itemset: Is ? monotone, anti-monotone, or non-monotone?

For each of the following measures, determine whether it is monotone, anti-


monotone, or non-monotone (i.e., neither monotone nor anti-monotone).

? is an anti-monotone measure because


?({A1, A2, ··· , Ak}) ? ?({A1, A2, ··· , Ak, Ak+1}) (6.2)


For example, we can compare the values of ? for {A, B} and {A, B, C}.








Since s(A, B, C) ? s(A, B) and max(s(A), s(B), s(C)) ? max(s(A), s(B)),


therefore ?({A, B}) ? ?({A, B, C}).