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1

Rough Fuzzy Concept Analysis

Bartl E., Konecny J.

Fundamenta Informaticae

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2017

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Vol. 156, nr 2

141--168

EN

We provide a new approach to fusion of Fuzzy Formal Concept Analysis and Rough Set Theory. As a starting point we take into account a couple of fuzzy relations, one of them represents the lower approximation, while the other one the upper approximation of a given data table. By defining appropriate concept-forming operators we transfer the roughness of the input data table to the roughness of corresponding formal fuzzy concepts in the sense that a formal fuzzy concept is considered as a collection of objects accompanied with two fuzzy sets of attributes— those which are shared by all the objects and those which at least one object has. In the paper we study the properties of such formal concepts and show their relationship with concepts formed by well-known isotone and antitone operators.

2

A comprehensive survey on formal concept analysis, its research trends and applications

Singh P. K., Aswani Kumar C., Gani A.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 2

495--516

EN

In recent years, FCA has received significant attention from research communities of various fields. Further, the theory of FCA is being extended into different frontiers and augmented with other knowledge representation frameworks. In this backdrop, this paper aims to provide an understanding of the necessary mathematical background for each extension of FCA like FCA with granular computing, a fuzzy setting, interval-valued, possibility theory, triadic, factor concepts and handling incomplete data. Subsequently, the paper illustrates emerging trends for each extension with applications. To this end, we summarize more than 350 recent (published after 2011) research papers indexed in Google Scholar, IEEE Xplore, ScienceDirect, Scopus, SpringerLink, and a few authoritative fundamental papers.

3

Research on Interval Concept Lattice and its Construction Algorithm

Liu B., Zhang C., Yang Y.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 1b

1--6

EN

In classic concept lattice and rough concept lattice, the concept extents have all the attributes or only one attribute sometimes. So the support and confidence degree of the extracted association rules would be reduced greatly. To solve this problem, authors have put forward a new concept lattice structure: interval concept lattice Lαβ (Mα, Mβ,Y) based on the parameter interval [ α,β ] (0 ≤ α ≤ β ≤ 1). The concept extent is an object sets which meet the properties in the intent in the interval [ α,β ] 0 ≤ α ≤ β ≤ 1. It has been proved that interval concept lattice degenerate into classic concept lattice when ( α = β = 1), and when ( α > 0, β = 1), interval concept lattice degenerate into rough concept lattice. Then some unique properties of interval concept lattice have been proved. The construction algorithm of interval concept lattice was designed. Finally, the necessity and practicability were verified through a case study.

PL

W klasycznej i przybliżonej kracie pojęć ich obszar obejmuje każdy lub czasami tylko jeden atrybut. A więc podstawa i stopień poufności wydobywanych relacji mogą zostać poważnie zredukowane. Aby rozwiązać ten problem autorzy proponują nową strukturę kraty pojęć: przedziałową kratę pojęć Lαβ (Mα, Mβ,Y) zdefiniowaną w przedziale [ α,β ] (0 ≤ α ≤ β ≤ 1). Udowodniono, że przedziałowa krata pojęć przekształca się w klasyczną jeśli ( α = β = 1) i w przybliżoną gdy ( α > 0, β = 1). Zbadano unikalne własności kraty przedziałowej i zaprojektowano algorytm jej budowy. W końcu zweryfikowano , w przypadku studialnym, potrzebę jej wprowadzenia i możliwość wykonania.

4

Vector-based Attribute Reduction Method for Formal Contexts

Shao M., Liu M., Guo L.

Fundamenta Informaticae

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2013

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Vol. 126, nr 4

397--414

EN

Attribute reduction is one basic issue in knowledge discovery of information systems. In this paper, based on the object oriented concept lattice and classical concept lattice, the approach of attribute reduction for formal contexts is investigated. We consider attribute reduction and attribute characteristics from the perspective of linear dependence of vectors. We first introduce the notion of context matrix and the operations of corresponding column vectors, then present some judgment theorems of attribute reduction for formal contexts. Furthermore, we propose a new method to reducing formal context and show corresponding reduction algorithms. Compared with previous reduction approaches which employ discernibility matrix and discernibility function to determine all reducts, the proposed approach is more simpler and easier to implement.

5

Decomposition of Relations and Concept Lattices

Berghammer R., Winter M.

Fundamenta Informaticae

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2013

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Vol. 126, nr 1

37--82

EN

We introduce the decomposition of an arbitrary relation into a sequential composition of three relations, viz. of a mapping with a partial order and then the transpose of a mapping. After presenting some basic properties, we investigate the specific classes of junkfree, irreducible and minimal decompositions and show that for all relations a minimal decomposition exists. We also study decompositions with regard to DedekindMacNeille completions and concept lattices. These constructions are closely related to decompositions of relations. In our setting the fundamental theorem of concept lattices states that concept lattices are minimal-complete decompositions and all such decompositions are isomorphic. As a further main result we prove that the cutDedekindMacNeille completion of the order that belongs to the minimal decomposition of a relation is isomorphic to the concept lattice of that relation. Instead of considering binary relations on sets, we will work point-free within the general framework of allegories. This complement-free approach implies that the results of the paper can be applied to all models of these algebraic structures, including, for instance, lattice-valued fuzzy relations.

6

Attribute Reduction in Formal Contexts: A Covering Rough Set Approach

Li T-J., Wu W. Z.

Fundamenta Informaticae

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2011

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Vol. 111, nr 1

15-32

EN

This paper proposes an approach to attribute reduction in formal contexts via a covering rough set theory. The notions of reducible attributes and irreducible attributes of a formal context are first introduced and their properties are examined. Judgment theorems for determining all attribute reducts in the formal context are then obtained. According to the attribute reducts, all attributes of the formal context are further classified into three types and the characteristic of each type is characterized by the properties of irreducible classes of the formal context. Finally, by using the discernibility attribute sets, a method of distinguishing the reducible attributes and the irreducible attributes in formal contexts is presented.

7

Normalized-scale Relations and Their Concept Lattices in Relational Databases

Lei Y., Sui Y., Cao C.

Fundamenta Informaticae

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2009

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Vol. 93, nr 4

393-409

EN

Formal Concept Analysis (FCA) is a valid tool for data mining and knowledge discovery, which identifies conceptual structures from (formal) contexts. As many practical applications involve non-binary data, non-binary attributes are introduced via a many-valued context in FCA. In FCA, conceptual scaling provides a complete framework for transforming any many-valued context into a context, in which each non-binary attribute is given a scale, and the scale is a context. Each relation in relational databases is a many-valued context of FCA. In this paper, we provide an approach toward normalizing scales, i.e., each scale can be represented by a nominal scale and/or a set of statements. One advantage of normalizing scales is to avoid generating huge (binary) derived relations. By the normalization, the concept lattice of a derived relation is reduced to a combination of the concept lattice of a derived nominal relation and a set of statements. Hence, without transforming a relation into a derived relation, one can not only determine concepts of the derived relation from concepts of given scales, but also determine concepts of the derived relation from concepts of a derived nominal relation and a set of statements. The connection between the concept lattice of a derived nominal relation and the concept lattice of a derived relation is also considered.

8

Concept Lattices of Subcontexts of a Context

Wang X., Zhang W.

Fundamenta Informaticae

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2009

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Vol. 90, nr 1-2

157-169

EN

As an effective tool for data analysis and knowledge processing, the theory of concept lattices has been studied extensively and applied to various fields. In order to discover useful knowledge, one often ignores some attributes according to a particular purpose and merely considers the subcontexts of a rather complex context. In this paper, we make a deep investigation on the theory of concept lattices of subcontexts. An approach to construct the concept lattice of a context is first presented by means of the concept lattices of its subcontexts. Then the concept lattices induced by all subcontexts of the context are considered as a set, and an order relation is introduced into the set. It is proved that the set together with the order relation is a complete lattice. Finally, the top element and the bottom element of the complete lattice are also obtained.

9

Concept lattices and similarity in non-commutative fuzzy logic

Georgescu G., Popescu A.

Fundamenta Informaticae

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2002

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Vol. 53, nr 1

23-54

EN

A classical (crisp) concept is given by its extent (a set of objects) and its intent (a set of properties). In commutative fuzzy logic, the generalization comes naturally, considering fuzzy sets of objects and properties. In both cases (the first being actually a particular case of the second), the situation is perfectly symmetrical: a concept is given by a pair (A,B), where A is the largest set of objects sharing the attributes from B and B is the largest set of attributes shared by the objects from A (with the necessary nuance when fuzziness is concerned). Because of this symmetry, working with objects is the same as working with properties, so there is no need to make any choice. In this paper, we define concepts in a "non-commutative fuzzy world", where conjunction of sentences is not necessarily commutative, which leads to the following non-symmetrical situation: a concept has one extent (because, at the end of the day, concepts are meant to embrace, using certain descriptions, diverse sets of objects), but two intents, given by the two residua (implications) of the non-commutative conjunction.