Formal Concept Analysis Of Data Research Articles

Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy

In this paper we propose a method for reducing the number of formal concepts in formal concept analysis of data with fuzzy attributes. We compute the weight of fuzzy formal concepts based on Shannon entropy. Further, the number of fuzzy formal concepts is reduced at chosen granulation of their computed weight. We show that the results obtained from the proposed method are in good agreement with Levenshtein distance method and interval–valued fuzzy formal concepts method but with less computational complexity.

Formal concept analysis and linguistic hedges

This paper presents an application of linguistic hedges to formal concept analysis of data with fuzzy attributes. Formal concept analysis aims at extraction of particular (bi-)clusters, called formal concepts, from data. The clusters link collections of objects (extents) and attributes (intents), and have a clear interpretation due to a simple verbal description of the concept-forming operators. We insert linguistic hedges such as ‘very’ or ‘extremely’ in the description of the operators. In this way, linguistic hedges become parameters for formal concept analysis that control the number of clusters extracted from data. Namely, as we show theoretically as well as experimentally, stronger hedges result in a smaller number of clusters. The new concept-forming operators form Galois-like connections. We study their properties and axiomatize them. Then, we show that a concept lattice with hedges, i.e. the set of all formal concepts of the new operators is indeed a complete lattice which is isomorphic to a particular ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the well-known main theorem of ordinary concept lattices. The isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. We demonstrate by experiments that when selecting various hedges from the strongest to weaker hedges, the reduction in size of the corresponding concept lattices is smooth. From a broader perspective, we argue that linguistic hedges represent mathematically and computationally a feasible way to parameterize methods for knowledge extraction from data that enable one to emphasize or to suppress extracted patterns while keeping their interpretation.

Isotone fuzzy Galois connections with hedges

We study isotone fuzzy Galois connections and concept lattices parameterized by particular unary operators. The operators represent linguistic hedges such as “very”, “rather”, “more or less”, etc. Isotone fuzzy Galois connections and concept lattices provide an alternative to their antitone counterparts which are the fundamental structures behind formal concept analysis of data with fuzzy attributes. We show that hedges enable us to control the number of formal concepts in the associated concept lattice. We also describe the structure of the concept lattice and provide a counterpoint to the main theorem of concept lattices.

Fast factorization by similarity in formal concept analysis of data with fuzzy attributes

We present a method of fast factorization in formal concept analysis (FCA) of data with fuzzy attributes. The output of FCA consists of a partially ordered collection of clusters extracted from a data table describing objects and their attributes. The collection is called a concept lattice. Factorization by similarity enables us to obtain, instead of a possibly large concept lattice, its factor lattice. The elements of the factor lattice are maximal blocks of clusters which are pairwise similar to degree exceeding a user-specified threshold. The factor lattice thus represents an approximate version of the original concept lattice. We describe a fuzzy closure operator the fixed points of which are just clusters which uniquely determine the blocks of clusters of the factor lattice. This enables us to compute the factor lattice directly from the data without the need to compute the whole concept lattice. We present theoretical solution and examples demonstrating the speed-up of our method.

A note on variable threshold concept lattices: Threshold-based operators are reducible to classical concept-forming operators

The present paper deals with formal concept analysis of data with fuzzy attributes. We clarify several points of a new approach of [S.Q. Fan, W.X. Zhang, Variable threshold concept lattice, Inf. Sci., accepted for publication] which is based on using thresholds in concept-forming operators. We show that the extent- and intent-forming operators from [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication] can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied e.g. in [R. Belohlavek, Fuzzy Galois connections, Math. Logic Quarterly 45 (4) (1999) 497–504; R. Belohlavek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277–298; S. Pollandt, Fuzzy Begriffe, Springer-Verlag, Berlin/Heidelberg, 1997]. As a consequence, main properties of the new operators from [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication], including the properties studied in [S.Q. Fan, W.X. Zhang, Inf. Sci., accepted for publication], can be obtained as consequences of the original operators from [R. Belohlavek, 1999; R. Belohlavek, 2004; S. Pollandt, 1997].