Generating Positive and Negative Exact Rules Using Formal Concept Analysis: Problems and Solutions

Abstract

The objective of this article is to investigate the problem of generating both positive and negative exact association rules when a formal context K of (positive) attributes is provided. A straightforward solution to this problem consists of conducting an apposition of the initial context K with its complementary context $\Tilde{K}$, construct the concept lattice $\mathfrak{B}(K|\Tilde{K})$ of apposed contexts and then extract rules. A more challenging problem consists of exploiting rules generated from each one of the contexts K and $\Tilde{K}$ to get the whole set of rules for the context $K|\Tilde{K}$.

In this paper, we analyze a set of identified situations based on distinct types of input, and come out with a set of properties. Obviously, the global set of (positive and negative) rules is a superset of purely positive rules (i.e., rules with positive attributes only) and purely negative ones since it generally contains mixed rules (i.e., rules in which at least a positive attribute and a negative attribute coexist).

The paper presents also a set of inference rules to generate a subset of all mixed rules from positive, negative and mixed ones. Finally, two key conclusions can be drawn from our analysis: (i) the generic basis containing negative rules, $\Sigma_{\Tilde{K}}$, cannot be completely and directly inferred from the set Σ _K of positive rules or from the concept lattice $\mathfrak{B}(K)$, and (ii) the whole set of mixed rules may not be completely generated from Σ _K alone, $\Sigma_K \cup \Sigma_{\Tilde{K}}$ alone, or $\mathfrak{B}(K)$ alone.