Co-occurrence, interest, and fuzzy events

The objective of data mining is to discover new and interesting relationships among the data in large databases. The relationships are expressed in terms of predicates that describe properties of the data. The two most common types of data relationships, co-occurrence and deviation from expectation, are assessed based on the number or distribution of the tuples in the database that are examples of the predicates. Fuzzy predicates were introduced into the partitioning of attribute domains to produce smooth transitions between classes and to facilitate the modeling with linguistic terms. When predicates are fuzzy, a tuple may partially satisfy a predicate and the notion of being an example is also fuzzy. For mining fuzzy relationships, the standard measures of validity for crisp predicates have been extended to fuzzy predicates based on the cardinality of fuzzy sets or on the degree to which the tuples satisfy a fuzzy implication. In this paper we use the notions of fuzzy events and fuzzy partitions to better understand the variations of the validity measures for relationships between fuzzy predicates.