Generalization of One-Sided Concept Lattices

We provide a generalization of one-sided (crisp-fuzzy) concept lattices, based on Galois connections. Our approach allows analysis of object-attribute models with different structures for truth values of attributes. Moreover, we prove that this method of creating one-sided concept lattices is the most general one, i.e., with respect to the set of admissible formal contexts, it produces all Galois connections between power sets and the products of complete lattices. Some possible applications of this approach are also included

Interpretation of Fuzzy Attribute Subsets in Generalized One-Sided Concept Lattices

In this paper we describe possible interpretation and reduction of fuzzy attributes in Generalized One-sided Concept Lattices (GOSCL). This type of concept lattices represent generalization of Formal Concept Analysis (FCA) suitable for analysis of datatables with different types of attributes. FCA as well as generalized one-sided concept lattices represent conceptual data miningmethods. With growing number of attributes the interpretation of fuzzy subsets may become unclear, hence another interpretation of this fuzzy attribute subsets can be valuable. The originality of the presented method is based on the usage of one-sided concept lattices derived from submodels of former object-attribute model by grouping attributes with the same truth value structure. This leads to new method for attribute reduction in GOSCL environment

Reduction and simple proof of characterization of fuzzy concept lattices

Presented is a reduction of fuzzy Galois connections and fuzzy concept lattices to (crisp) Galois connections and concept lattices: each fuzzy concept lattice can be viewed as a concept lattice (in a natural way). As a result, a simple proof of the characterization theorem for fuzzy concept lattices is obtained. The reduction enables us to apply the results worked out for concept lattices to fuzzy concept lattices

Identifying Non-Sublattice Equivalence Classes Induced by an Attribute Reduction in FCA

The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of the obtained quotient set and their classes. This paper studies the algebraic structure of the induced equivalence classes and characterizes those classes that are convex sublattices of the original concept lattice. Particular attention is given to the reductions removing FCA's unnecessary attributes. The obtained results will be useful to other complementary reduction techniques, such as the recently introduced procedure based on local congruences

Distributed Computation of Generalized One-Sided Concept Lattices on Sparse Data Tables

In this paper we present the study on the usage of distributed version of the algorithm for generalized one-sided concept lattices (GOSCL), which provides a special case for fuzzy version of data analysis approach called formal concept analysis (FCA). The methods of this type create the conceptual model of the input data based on the theory of concept lattices and were successfully applied in several domains. GOSCL is able to create one-sided concept lattices for data tables with different attribute types processed as fuzzy sets. One of the problems with the creation of FCA-based models is their computational complexity. In order to reduce the computation times, we have designed the distributed version of the algorithm for GOSCL. The algorithm is able to work well especially for data where the number of newly generated concepts is reduced, i.e., for sparse input data tables which are often used in domains like text-mining and information retrieval. Therefore, we present the experimental results on sparse data tables in order to show the applicability of the algorithm on the generated data and the selected text-mining datasets

Use of Concept Lattices for Data Tables with Different Types of Attributes

In this paper we describe the application of Formal Concept Analysis (FCA) for analysis of data tables with different types of attributes. FCA represents one of the conceptual data mining methods. The main limitation of FCA in classical case is the exclusive usage of binary attributes. More complex attributes then should be converted into binary tables. In our approach, called Generalized One-Sided Concept Lattices, we provide a method which deal with different types of attributes (e.g., ordinal, nominal, etc.) within one data table. Therefore, this method allows to create same FCA-based output in form of concept lattice with the precise many-valued attributes and the same interpretation of concept hierarchy as in the classical FCA, without the need for specific unified preprocessing of attribute values

Note on formal contexts of generalized one-sided concept lattices

Generalized one-sided concept lattices represent one of the conceptual data mining methods, suitable for an analysis of object-attribute models with the different types of attributes. It allows to create FCA-based output in form of concept lattice with the same interpretation of concept hierarchy as in the case of classical FCA. The main aim of this paper is to investigate relationship between formal contexts and generalized one-sided concept lattices. We show that each one uniquely determines the other one and we also derive the number of generalized one-sided concept lattices defined within the given framework of formal context. The order structure of all mappings involved in some Galois connections between a power set and a direct product of complete lattices is also dealt with. Keywords: Galois connection, generalized one-sided concept lattice, formal context

Galois Connection with Truth Stressers: Foundation for Formal Concept Analysis of Object-Attribute Data with Fuzzy Stressers

Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair ↑ , ↓ of mappings ↑ assigning subcollections of Y to subcollections of X, and ↓ assigning subcollections of X to subcollections of Y . By definition, Galois connections have to satisfy certain conditions. Galois connections can be interpreted in the following manner: For subcollections A and B of X and Y , respectively, A ↑ is the collection of all elements of Y which are in a certain relationship to all elements from A, and B ↓ is the collection of all elements of X which are in the relationship to all elements in B. From the very many examples of Galois connections in mathematics, let us recall the following. Let X be the set of all logical formulas of a given language, Y be the set of all structures (interpretations) of the same language. For A ⊆ X and B ⊆ Y , let A ↑ consist of all structures in which each formula from A is true, let B ↓ denote the set of all formulas which are true in each structure from B. Then, ↑ and ↓ is a Galois connection. As an example of applications of Galois connections, consider the following example which is the main source of inspiration for the present paper. Let X and Y denote a set of objects and attributes, respectively, Let I denote the relationship "to have" between objects and attributes. Then X, Y , and I can be seen as representing an object-attribute data table (for instance, organisms as objects, and their properties as attributes). If, for subcollections A of X and B of Y , A ↑ denotes the collection of all attributes shared by all objects from A, and B ↓ denotes the collection of all objects sharing all attributes from B, then ↑ and ↓ form a Galois connection. These connections form the core of socalled formal concept analysis (FCA) of object-attribute data, se