Mining Triadic Association Rules from Ternary Relations

Abstract

Ternary and more generally n-ary relations are commonly found in real-life applications and data collections. In this paper, we define new notions and propose procedures to mine closed tri-sets (triadic concepts) and triadic association rules within the framework of triadic concept analysis. The input data is represented as a formal triadic context of the form $\mathbb{K}:=(K_1, K_2, K_3, Y)$, where K ₁, K ₂ and K ₃ are object, attribute and condition sets respectively, and Y is a ternary relation between the three sets. While dyadic association rules represent links between two groups of attributes (itemsets), triadic association rules can take at least three distinct forms. One of them is the following: A ${\underrightarrow{c}}$ D, where A and D are subsets of K ₂, and C is a subset of K ₃. It states that A implies D under the conditions in C. In particular, the implication holds for any subset in C.

The benefits of triadic association rules of this kind lie in the fact that they represent patterns in a more compact and meaningful way than association rules that can be extracted for example from the formal (dyadic) context $\mathbb{K}^{(1)}:= (K_1, K_2 \times K_3, Y^{(1)}) \text{ with } (a_i, (a_j, a_k)) \in Y^{(1)} : \iff (a_i, a_j, a_k) \in Y.$