Representing Numeric Values in Concept Lattices

Formal Concept Analysis is based on the occurrence of symbolic attributes in individual objects, or observations. But, when the attribute is numeric, treatment has been awkward. In this paper, we show how one can derive logical implications in which the atoms can be not only boolean symbolic attributes, but also ordinal inequalities, such as x ≤ 9. This extension to ordinal values is new. It employs the fact that orderings are antimatroid closure spaces. 1 Extending Formal Concept Analysis Formal Concept Analysis (FCA), which was initially developed by Rudolf Wille and Bernhard Ganter [3], provides a superb way of describing “concepts”, that is closed sets of attributes, or properties, within a context of occurrences, or objects. One can regard the concept as a closed set of objects with common attributes. Frequently clusters of these concepts, together with their structure, stand out with vivid clarity. However, two unresolved problems are often encountered. First, when concept lattices become large, it is hard to discern or describe significant clusters of related concepts. Gregor Snelting used formal concept analysis to analyze legacy code [6, 14]. Snelting’s goal was to reconstruct the overall system structure by determining which variables (attributes or columns) were accessed by which modules (objects or rows). It was hoped that the concept structure would become visually apparent. Unfortunately, the resulting concept lattice shown on page 356 of [6] is little more than a black blob. Visual interpretation of closure concepts does not seem to scale well. Second, when the attribute values are numeric, as is often the case with technical data, formal concept analysis becomes difficult. It is easy to comprehend the set of all objects whose color is “red”; But what precisely constitutes the set of objects whose weight is “9.5”, or “near 9.5”, or “less than 9.5” or otherwise similar to “9.5”. Fuzzy set theory [5, 18] might be appropriate here, but we prefer a deterministic approach. To cope with the first issue, our approach has been to work with the logical implications that can be deduced from adjacent closed concepts in the lattice, rather than to seek clusters of such concepts. So long as all the attributes are boolean this seems to work rather well. We have applied formal concept analysis to relations (contexts) of sizes 8, 124 × 85 and 1, 272 × 144. The resulting closed set lattices were large, 104,104 and 1,804 nodes respectively, but nevertheless we were able to extract useful logical implications. In Section 3 we briefly review some of this older work. In Section 4 we will develop our new approach to representing numeric predicates. Very briefly, it will consist of viewing a concept, not as being a closed set of attributes associated with a set of objects, but rather regarding it as a closed set of “closed sets” associated with the objects. 1 Siff and Reps [13] published shortly after. A crucial contribution of FCA has been to establish the role and importance of closure operators and closed sets in mathematical thinking. In Section 2 we review those closure concepts that we will need later. 2 Closure Operators and Closure Spaces The notion of “closure” plays a major role in our representation of the real world. In particular we will be concerned with closed sets of objects, closed sets of predicates and closed sets of numbers. 2.1 Closure Concepts By a “closure system” over a “universe” U, we mean a collection C of sets X,Y, . . . Z ⊆ U, including U, satisfying the property that if X,Y ∈ C then X ∩ Y ∈ C. The sets of C are said to be the closed sets of U. As an alternative to this “intersection” characterization, one can define a closure operator φ : 2U → 2U satisfying the following 3 axioms for all X,Y, Z: X ⊆ X.φ, X ⊆ Y implies X.φ ⊆ Y.φ X.φ.φ = X.φ. (For technical reasons we prefer to use suffix operator notation, so readX.φ as “X closure”.) Readily, a set X is closed in C, if X.φ = X . The equivalence of these two alternative definitions is well known [8], and we will use both in the following sections. Closure systems can satisfy many other axioms, and those that do give rise to different varieties of mathematical systems. If (X ∪ Y ).φ = X.φ ∪ Y.φ we say φ is a topological closure. If the system satisfies the “exchange axiom”, that is if p, q 6∈ X.φ but q ∈ (X ∪ {p}).φ then necessarily p ∈ (X ∪ {q}).φ, the system can be viewed as a kind of linear algebra, or more generally a “matroid”. The Galois closure we will be using in this section satisfies neither of these additional axioms. But later in Section 4, we will be using “antimatroid” closure operators, that is those which satisfy the “anti-exchange axiom” if p, q 6∈ X.φ and q ∈ (X ∪ {p}).φ then p 6∈ (X ∪ {q}).φ . 2.2 Galois Closure and Concept Lattices The approach we will follow is similar to that was first developed by Rudolf Wille and is best presented in [3]. Formal concept analysis begins with a relation R between two sets, say a set O of objects and a set P of object predicates, or attributes. Using standard relational terminology, each object oj ∈ O can be regarded as a row in R and each predicate pk ∈ P is a column. Each attribute pk is a binary, logical property, i.e. pk(o) is either true or false, because the object exhibits property pk or it doesn’t. A concept Cn is a pair of closed subsets Cn = (On, Pn) where On ⊆ O, Pn ⊆ P with the property T that for every oi ∈ On, every pk ∈ Pn, pk(oi) is true. Each concept is assumed to be maximal, that is for the set On there is no larger subset P ′ n ⊃ Pn satisfying property T , and for Pn there is no larger subset O ′ n ⊃ On satisfying T . The collection C of all concepts Cn, so defined, forms a closure system; that is, the intersection of any two concepts in C is a concept. Consequently, the collection C of concepts forms a lattice when partially ordered by containment with respect to the predicate sets Pn. 2 If we start with the relation 2 Ganter and Wille prefer to order with respect to object set containment yielding the dual lattice, c.f. page 20 [3].