Extracting knowledge from the universe and identifying the association between attributes values is a challenging task today. Granular computing proposed by Pawlak [1], Louis [2] and Zadeh [3] is a tool to identify the associations between the attributes values that are indiscernible. However, in much information system the attribute values are almost indiscernible rather than discernible. Therefore, it is essential to identify the associations between the attribute values where the attribute values are almost indiscernible. In this paper, we use rough set on fuzzy approximation spaces and ordering values to deal with almost indiscernibility. Finally, we use granular computing to find the associations between the attribute values.

There is a huge repository of the data available across various domains. At the present age of internet, data can be easily collected and accumulated. It is very hard to extract useful information from the voluminous data available in the universe. So, it has become one of the most popular areas of recent research to find knowledge about the universe. In order to transform the processed data intelligently and automatically into useful information and knowledge, there is need of new techniques and tools. Development of these techniques and tools are studied under different domains like knowledge discovery in database, computational intelligence and knowledge engineering etc. Various traditional tools are developed by researchers to mine knowledge from the accumulated voluminous data, but most of them are crisp and deterministic in nature. However, if we see the real time dataset, it is inconsistent and ambiguous. So, there is need of classification among the objects of the universe into similarity classes. The basic building block of knowledge about the universe is called granule. Creating granules from the objects of the universe by classification is called knowledge granulation and processing of these granules in order to find the knowledge about of the universe is termed as granular computing. It is observed that classification of the universe can be done on the basis of indiscernibility relation among the objects. Granular computing as proposed by Pawlak[1],Louie[2] and Zadeh[3] is a tool to identify the associations between the attribute values, that are indiscernible. However, if we consider the real life situation, the attribute values are almost indiscernible. In order to model the real life situation, the fundamental concept of classical sets has been extended in various directions. One of the approach in the direction was introduced by Fayyad et al. (1996), who developed and illustrated the knowledge discovery in database(KDD) and identifies some useful and understandable pattern in data, but if we take into account the factors affecting KDD, its complexity increases. Amit Singhal(2001) and Donaovan(2003) also provided some classification to classify dataset that are crisp in nature. Zadeh’s (1965) introduces fuzzy set theory concepts which were applied to knowledge discovery database, the concept was further extended to L fuzzy set by Goguen (1967); intuitionistic fuzzy set by Atanasov(1986);twofold fuzzy set by Dubosis et al.(1987) to name a few. But all of these methods lack uniqueness in choosing the membership and non membership function. Rough set introduced by Pawlak(1991) is a tool that depends upon the notion of equivalence relation defined over a universe. This concept is further extended to rough set on fuzzy approximation space which depends upon fuzzy proximity relation as discussed by D.P. Acharjya & Tripathy (2008). Rough set on fuzzy approximation space is an intelligent tool that finds out the significance of attributes in the given data set using the member function.

Here, in this paper we use rough set on fuzzy approximation space, deals with data that are almost indiscernible and use granular computing approach to find out association between the attribute values that are almost indiscernible rather than being indiscernible.

2. ROUGH SET

The classical set i.e. crisp set has been studied and extended in many directions to model real life situations. The notion of fuzzy set studied by Zadeh (1965), its generalizations and the notion of rough sets was studied by Pawlak and Skowron (2007) were the major research in this field. The rough set philosophy is based on the concept that there is some information associated with each object of the universe. So there is need to classify objects of the universe based on the indiscernibility relation among them, as there are various objects in the universe that are similar to each other. Rough set is a mathematical tool that is used to classify the objects of the universe based on the indiscernibility relation among the objects of the universe. The basic idea of rough set is based upon the approximation of sets by pair of sets known as lower approximation and upper approximation with respect to some imprecise information. In this section, we will study the basic concepts, definitions and different notations that will be used in the rest of the paper.

Let U ?‚?be a set of objects called the universe, and R be an equivalence relation over U. Then U/R we denote the family of equivalence classes of R (or classification of U) referred to as categories or concepts of R and [x]R denotes a category in R containing an element xU . By a knowledge base, we understand a relational system K (U,R) , where U is as above and R is a family of equivalence relations over U. For any subset P R and P ?‚?, IND(P) is the intersection of all equivalence relations in P. IND(P) is called the indiscernibility relation over P. The equivalence classes of IND(P) are called P-basic knowledge about U in K. For any QR , every equivalence class of Q is called Q-elementary concepts of knowledge R.

The family of all P-basic categories P ?‚?, P R will be called the family of basic categories in knowledge base K (U,R) . By IND(K) we denote the family of all equivalence relations defined in K; equivalently,

IND(K) {IND(P) : P R,P ?‚?}

For any X U and an equivalence relation RIND(K) , we associate two subsets X and RX called the R-upper and R-lower approximations of X respectively, which are given by:

RX U{Y U /R :Y X} and

X U{Y U/ R :Y I X ?‚?}

The R-boundary of X is denoted by BNR(X) and is given as BNR(X ) X RX . We say X is rough with respect to R if and only if X ?‚?RX or equivalently BNR(X) ?‚?. If BNR(X)or X RX then the target set X is a crisp set. It indicates that a rough set is more generalized then crisp set.

3. ROUGH SETS ON FUZZY APPROXIMATION SPACE

Let U be a universe. The elements of the universe may have crisp or fuzzy relations among them, based on the nature of the dataset. In this section, we present the definitions, notations and results on fuzzy approximation space and rough set on fuzzy approximation space. We will refer to these concepts in later section of the paper and it will be the base of our further discussion.

Definition 3.1: Let U be a universe. We define a fuzzy relation on U as a fuzzy relation on U as a fuzzy subset of (U ?- U).

Definition 3.2: A fuzzy relation R on U is said to be a fuzzy proximity relation if

µR (x, x) = 1 for all x U and (3.1)

µR (x, y) =µR (y, x) for x, y U (3.2)

Definition 3.3: Let R is a fuzzy proximity relation on U. Then for a given α  [0, 1], we say that two elements x and y are α-similar with respect to R if (x, y)  Ra and we write xRay.

Definition 3.4: Let R is a fuzzy proximity relation on U. Then for a given α  [0, 1], we say that two elements x and y are α-identical with respect to R if either x is α-similar to y or x is transitively α-similar to y with respect to R, i.e., there exists a sequence of elements u1,u2,u3,……..un in U such that xRau1, u1Rau2, u2Rau3,……….. unRay.

If x and y are α-identical with respect to fuzzy proximity relation R, then we write xR(α)y, where the relation R(α) for each fixed α  [0,1] is an equivalence relation on U.

Definition 3.5: The pair (U, R) is called a fuzzy approximation space. For any α  [0, 1], we denote by R*α,the set of all equivalence classes of R(α). Also, we call (U, R(α)), the generated approximation space associated with R and α.

Definition 3.6: Let (U, R) be a fuzzy approximation space and let X  U. Then the rough set of X in (U, R (α)) is denoted by (Xα, α) where Xα is the α-lower approximation of X whereas α is the α-upper approximation of X. We define Xα and α as

Xα ={Y: Y R*α and Y  X} and (3.3)

α= {Y: Y R*α and Y ∩ X ≠} (3.4)

Definition 3.7: Let X U. Then X is said to be α-discernible if and only if Xα=α and X is said to be α-rough if and only if Xα≠α.

Many properties of α-lower and α-upper approximations have been studied by De [4].

4. ORDERED INFORMATION SYSTEM

Let I= (U, A, {Va: aA}, {fa: a A}) be an information system, where U is finite non-empty set of objects called the universe and A is a non empty finite set of attributes. For every aA, Va is the set of values that attribute may take and fa: U??’ Va is an information function. In practical applications object can be cases, companies, institutions, processes and observations. Attributes can be interpreted as features, variables, and characteristics. A special case of information systems called information table or attribute value table where the columns are labelled by attributes and rows are by objects. For example: The information table assigns a value a(x) from Va to each attribute and object in the universe U. With any PA there is an associated equivalence relation such that

IND(P)={(x,y) U2 |aP,a(x) = a(y)}

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