Stronger association rules for positive attributes

Gábor Czédli

By a context we mean a binary table with crosses at some entries, i.e. a relation between two sets. The elements of these sets are called objects (= row labels) and attributes (= column labels). Each context determines a pair of Galois closure operators. This gives rise to formal concept analysis, cf. Ganter and Wille [6], and also to studying strong association rules in data mining, cf. Agrawal, Imielinski and Swami [1]; the term “association rule” being kept for the fuzzy version. There are cases where the Galois closure is too large or, in other words, even the strong association rules challenge decision making with too many choices. In [3], some stronger association rules (i.e., a smaller pair of closure operators) have been introduced. Their mathematical features and possible further applicability have been studied in [4] and [5]. While [3] makes it clear that the new operator is useful in (pure) algebra, [4] and [5] point out that we expect its use in applied fields only when all the attributes are advantageous or good or useful, shortly, if the attributes are positive. The goal of this paper is to introduce a more general pair of closure operators, smaller than the Galois one, such that the corresponding stronger association rules take into account that not all the attributes are positive. The main result confirms that our definition gives indeed closure operators. A strong emphasis is put on detailing how and why the new stronger association rules promise future applications although no concrete database has been analyzed from this aspect yet.