Real formal concept analysis based on grey-rough set theory

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Abstract

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.

Introduction

Grey system theory [1], [2], [3], [4], proposed by Deng, covers grey classification [5], [6], [7], grey control, grey decision-making [8], grey prediction [9], [10], [11], [12], [13], grey structural modelling [14], [15], grey relational analysis as well as grey-rough sets [8], [16], [17], [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 = xbest + δx, where x is the measured value, xbest is the best estimate and δx is the error in the measurement of x. This form is equal to x[xbest-δx,xbest+δx],˜x=xbest in grey system theory: ⊗x is the interval where the best estimate exists and ˜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 ˜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], [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], [25]. Nevertheless, the general formalism of Galois lattice was addressed by [26], [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.

Section snippets

Grey lattice operation and grey-rough sets

Let G be the universal set, x be an element of G (x  G), Rbe the set of real numbers and X  R be the set of value range that x may hold.

Definition 2.1

[20]

Let Gr be a grey set of G defined by two mappings of the upper membership function μ¯Gr(x) and the lower membership function μGr(x) as follows:μ¯Gr(x):G[0,1];μ̲Gr(x):G[0,1]where μ̲Gr(x)μ¯Gr(x) and x  G. When μ̲Gr(x)=μ¯Gr(x), the grey set becomes a fuzzy set, which means that grey system theory deals with fuzzy situation more flexibly.

Definition 2.2

[20]

When two values x̲,x¯(x̲=inf

Grey information system and formal concept analysis

Definition 3.1

Let IS = (G,M,V,f) be a grey information system, M = {a1,a2,  , an}, S = {f(s,a1), f(s,a2),  , f(s,an)} be a set of n values on attributes of M denoting an objective and all such S on M be SM, S  SM. If x, y be an object holding n values on M as Sx = {f(x,a1),f(x,a2) ,… , f(x,an)}, Sy = {f(y,a1),f(y,a2),  , f(y,an)}, respectively, then Sx  Sy = {f(x,a1)  f(y,a1),f(x,a2)  f(y,a2),  , f(x,an)  f(y,an)}, Sx  Sy = {f(x,a1)  f(y,a1),f(x,a2)  f(y,a2),  , f(x,an)  f(y,an)}, Sx  Sy = {f(x,a1)  f(y,a1),f(x,a2)  f(y,a2

A definition and theorem of attribute reduction based on significance attributes

Definition 4.1

Let IS = (G,M,V,f) is a grey information system, G = {x1, x2,  , xm}, M = (a1, a2,  , an). Constructing the binary discernibility matrix of IS: the each column of matrix corresponding a condition attribute, there are n column in total, each row corresponding to an object (xp,xq) in universe. The matrix possesses at most m(m  1)/2 rows. Let the matrix have a element w((p,q),i) in its row corresponding to an object (xp,xq), in its column corresponding condition attribute is ai, thenw((p,q),i)=1f(xp,ai)f(xq

Proposed approach based on real Galois lattice

Supposing a grey table of supplier’s selection is IS = (G,M,V,f), where G = {S1, S2,  , Sm} is a sets of m suppliers alternatives called the universe. M = {a1, a2,  , an} is a set of n attributes of suppliers. f : G × M  V is grey description function. T = (G,A  D,f) is called a grey decision table. D is decision attribute. The element of A is called conditions, which include quantitative or qualitative attributes.

The attribute ratings ⊗v are described by linguistic estimations and they can be expressed in grey

Conclusion

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

Acknowledgements

The authors thank the anonymous referees for their constructive comments. This work was supported by a grant from the National Natural Science Foundation of China (No. 60575035).

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    This work is partly supported by the National Natural Science Foundation of China with Grant No. 60575035.

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