One of the main concepts in grey system theory is how systems should be controlled under incomplete or lack of information situation. Grey number denoting an uncertain value is described in real interval from this concept. In this paper, we introduce the real formal concept analysis based on grey-rough set theory by using grey numbers, instead of binary values. We propose, to extend the notion of Galois connection in a real binary relation as well as the notions of formal concept and Galois lattice. The relationships between the new notions and old ones are discussed. Finally, we present a grey-rough set approach to Galois lattices reduction.
Grey system theory [1], [2], [3] and [4], proposed by Deng, covers grey classification [5], [6] and [7], grey control, grey decision-making [8], grey prediction [9], [10], [11], [12] and [13], grey structural modelling [14] and [15], grey relational analysis as well as grey-rough sets [8], [16], [17] and [18], etc. It deals with the uncertainty over how systems with incomplete or lack of information should be controlled. One of the important concepts is a grey number. It is a number whose exact value is unknown but range is known. One of the practical applications of grey numbers is in error analysis [19] using a form x = x best + δ x, where x is the measured value, x best is the best estimate and δ x is the error in the measurement of x . This form is equal to View the MathML source⊗x←→[xbest-δx,xbest+δx],⊗˜x=xbest in grey system theory: ⊗x is the interval where the best estimate exists and View the MathML source⊗˜x is one of the exact values that seems to be the best estimate. In real applications, a solution of equations or an optimized parameter is View the MathML source⊗˜x and their condition given in advance is ⊗x. Thus grey system theory deals with uncertainty unlike those of fuzzy set theory or rough set theory. The grey lattice operation [5] and [7] is one of the operations for grey numbers that modifies a range of given intervals of grey numbers. It is more suitable to handling information tables containing interval data. With the motivation, Yamaguchi et al. [20] proposed a new rough set model named grey-rough set, which is a new collaboration of rough set theory and grey system theory. A grey-rough approximation is based on the grey lattice relation instead of an equivalence class and an indiscernibility relation in Pawlak’s model. Compared with the classical rough set, the proposal extends a treatable value into interval data. It provides a maximum solution and minimum solution both in upper and lower approximations. It give us a new mathematical background to develop a data set containing interval data.
Wille, in his Formal Concept Analysis [21], proposes a theory that allows us to formalize the three basic ideas of the conceptual knowledge, the objects, the attributes and the concepts. These ideas are linked through three basic relations: one object has an attribute, one object belongs to a concept and one concept is a subconcept of another one. With this analysis, a model to represent the concepts and to set up hierarchies among them is defined by Wille. We notice that most existing work focuses only on binary data. In order to generalize this work, the Galois lattice formalism was extended to symbolic data by [22] and further developed by [23], [24] and [25]. Nevertheless, the general formalism of Galois lattice was addressed by [26] and [27]. The rationale for this generalization is that nowadays, either descriptions of data are complex, or the size of datasets is drastically growing up so that if, for example, we want to deal with classes of data, we need descriptions that are much more complex than 0 or 1.
Polaillon and Diday [28] proposed an extension of two classical algorithms (Ganter and Chein) and an incremental one (Godin et al.) to multivariate, interval and histogram data with missing values. In [29], Baklouti et al. proposed a fast Galois lattice-building algorithm based on dichotomic search and working for objects having general description. Although the algorithm can deal with general data, they have to redefine the set, the order relation, the operation of infimum and the largest element in different processes. Jaoua and Elloumi [30] introduced the notion of a real set as an extension of a crisp and a fuzzy set by using sequences of intervals as membership degrees, instead of a single value in [0,1]. They also proposed, to extend the notion of Galois connection in a real binary relation as well as the notions of rectangular relation, formal concept and Galois lattice. They had applied the strict Galois connection to build a real classifier system. But, their Galios lattices are different from classical concept lattices. Its intent and extent of bottom nodes are both empty. The object and attribute universe are intent, extent of top node, respectively.
In this paper, a new formal concept analysis approach for interval data based on grey-rough set theory is proposed. Since it is suitable for interval data reduction of attributes, we also propose, to use grey-rough set approach reducing real Galois lattices.
In this paper, we have proposed an extension of the notion of Galois connection in a real binary relation as well as the notions of formal concept and Galois lattice. We have also proposed to organize the real concepts in a Galois lattice according to a generalize consideration. The generalize consideration allows us to deal with binary values and discrete values. We have cited that the representation of the formal concept, can be more convenient by considering real intervals, since different possible values for a information system can be considered. In fact, to fuzzy problem the fuzzification step can be avoided, by considering only different imprecision levels, related to the different properties in a data table.
