15 research outputs found
Rough sets based on Galois connections
Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of approximation operators are based on an indiscernibility relation, which is an equivalence one. Lately. different papers have motivated the possibility of considering arbitrary relations. Nevertheless, when those are taken into account, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution to the arising problems by presenting an alternative definition of approximation operators based on the closure and interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks
Impact of local congruences in variable selection from datasets
Formal concept analysis (FCA) is a useful mathematical tool for obtaining
information from relational datasets. One of the most interesting research
goals in FCA is the selection of the most representative variables of the
dataset, which is called attribute reduction. Recently, the attribute reduction
mechanism has been complemented with the use of local congruences
in order to obtain robust clusters of concepts, which form convex sublattices
of the original concept lattice. Since the application of such local congruences
modifies the quotient set associated with the attribute reduction, it
is fundamental to know how the original context (attributes, objects and
relationship) has been modified in order to understand the impact of the
application of the local congruence in the attribute reduction.Partially supported by the the 2014-2020 ERDF Operational Programme in collaboration
with the State Research Agency (AEI) in project TIN2016-76653-P and PID2019-
108991GB-I00, and with the Department of Economy, Knowledge, Business and University
of the Regional Government of Andalusia in project FEDER-UCA18-108612, and
by the European Cooperation in Science & Technology (COST) Action CA17124
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
Characterizing reducts in multi-adjoint concept lattices
The construction of reducts, that is, minimal sets of attributes containing the main in- formation of a database, is a fundamental task in different frameworks, such as in For- mal Concept Analysis (FCA) and Rough Set Theory (RST). This paper will be focused on a general fuzzy extension of FCA, called multi-adjoint concept lattice, and we present a study about the attributes generating meet-irreducible elements and on the reducts in this framework. From this study, we introduce interesting results on the cardinality of reducts and the consequences in the classical case.Partially supported by the Spanish Economy and Competitiveness Ministry (MINECO) project TIN2016-76653-
Algebraic structure and characterization of adjoint triples
Implications pairs, adjoint pairs and adjoint triples provide general residuated structures considered in different mathematical theories. In this paper, we carry out a deep study on the operators involved in these structures, showing how they are characterized by means of the irreducible elements of a complete lattice. Moreover, the structure of each class of these operators will be analyzed. As a consequence, the use of these operators in real problems will be more tractable, fostering their consideration as basic and useful operators for providing, for instance, preferences among attributes and objects in a given database.Partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects TIN2016-76653-P and PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124
Implication operators generating pairs of weak negations and their algebraic structure
Negations operators have been developed and applied in many fields such as image processing, decision making, mathematical morphology, fuzzy logic, etc. One of the most effective non-monotonic operators are weak negations. This paper studies the algebraic structure and the characterization of the adjoint triples and Galois implication pairs which provides a fixed pair of weak negations. The obtained results allow the user to select the best conjunctor and implications associated with the most suitable negation to be used in the computations of the problem to be solved.Partially supported by the State Research Agency (AEI) and the European Regional Development Fund (ERDF) project TIN2016-76653-P, European Cooperation in Science & Technology (COST) Action CA17124
Characterizing One-Sided Formal Concept Analysis by Multi-Adjoint Concept Lattices
Managing and extracting information from databases is one of the main goals in several
fields, as in Formal Concept Analysis (FCA). One-sided concept lattices and multi-adjoint concept
lattices are two frameworks in FCA that have been developed in parallel. This paper shows that
one-sided concept lattices are particular cases of multi-adjoint concept lattices. As a first consequence
of this characterization, a new attribute reduction mechanism has been introduced in the one-side
framework.This research was partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in Project PID2019-108991GB-I00 and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in Project FEDER-UCA18-108612 and by the European Cooperation in Science & Technology (COST) Action CA17124
Attribute Classification and Reduct Computation in Multi-Adjoint Concept Lattices
The problem of reducing information in databases is an important topic in formal concept analysis, which has been studied in several articles. In this article, we consider the fuzzy en- vironment of the multi-adjoint concept lattices, since it is a general fuzzy framework that allows us to easily establish degrees of pref- erence on the elements of the considered database. We introduce algorithms to discover the information contained in the relational system. By means of these algorithms, we classify the attributes of a multi-adjoint context, and build a minimal subset of attributes preserving the information of the original knowledge system.The work of L. Antoni was supported in part by the Slovak Research and Development Agency under Contract APVV-15-0091. The work of M. E. Cornejo, J. Medina, and E. Ramírez-Poussa was supported in part by the Spanish Economy and Competitiveness Ministry (MINECO) under Project TIN2016-76653-P, in part by the Department of Economy, Knowl- edge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and in part by the European Cooperation in Science & Technology (COST) Action CA17124
Rough set decision algorithms for modeling with uncertainty
The use of decision rules allows to extract information and to infer conclusions from relational databases in a reliable way, thanks to some indicators like support and certainty. Moreover, decision algorithms collect a group of decision rules that satisfies desirable properties to describe the relational system. However, when a decision table is considered within a fuzzy environment, it is necessary to extend all notions related to decision algorithms to this framework. This paper presents a generalization of these notions, highlighting the new definitions of indicators of relevance to describe decision rules and decision algorithm
Value reducts and bireducts: A comparative study
In Rough Set Theory, the notion of bireduct allows to simultaneously reduce the sets of objects and attributes contained in a dataset. In addition, value reducts are used to remove some unnecessary values of certain attributes for a specific object. Therefore, the combination of both notions provides a higher reduction of unnecessary data. This paper is focused on the study of bireducts and value reducts of information and decision tables. We present theoretical results capturing different aspects about the relationship between bireducts and reducts, offering new insights at a conceptual level. We also analyze the relationship between bireducts and value reducts. The studied connections among these notions provide important profits for the efficient information analysis, as well as for the detection of unnecessary or redundant information