covering based rough set approach to uncertainty management in databases

8/6/2019 Covering Based Rough Set Approach to Uncertainty Management in Databases

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Covering Based Rough Set Approach to Uncertainty Management in Databases

Abstract:

Relational Databases were extended by Beauboef and Petry to introduce rough relational databases,

fuzzy rough relational databases and intuitions tic rough relational databases. The introduction of

these concepts into the realm of databases, enhanced capabilities of databases by allowing for themanagement of uncertainty in them. Rough set, due to its versatility can be integrated into an

underlying database model, relational or object oriented and also used in the design and querying of

the databases. Covering based rough sets provide generality as well as better modelling power to

basic rough sets. Also, this new model unifies many other extensions of the basic rough set model, in

this article, we introduce the concept of covering based rough relational databases and define basic

operations on them. Besides with previous approaches, it is our objective to illustrate the usefulness

and versatility of covering based rough sets for uncertainty management in databases.

Keywords

1. Introduction:Recently, there has been a trend to develop for formal modelling; which take care of

handling uncertainty and impreciseness due to the fact that real life situations are very often

in crisp and deterministic and they cannot be described precisely; which makes the existing

precise and crisp models of little use. Rough sets, introduced by Pawlak(1982) has been

found to be an crisp sets, called the lower and upper approximations of the sets. The lower

approximation of a rough set comprises of those elements of the universe, which can be said

to belong to it definitely with the available knowledge. The upper approximation on the

other hand comprises of those elements which are possibly in the set with respect to the

available information. Because of rough sets ability to define uncertain things in terms of

certain, definable things, it is a natural mechanism for integrating real-world uncertainty in

computerized databases .Moreover, other uncertainty management techniques may be

combined with rough sets to offer even greater uncertainty management in databases.

The major concepts used in rough set theory are the use of indiscernibility relations to

partition domains into equivalence classes and the concept of lower and upper

approximation regions to allow the distinction between certain and possible (or partial)

inclusion of elements in rough set. Indiscernibility is the ability to distinguish between two or

more elements. Indiscernibility can arise from lack of precision in measurement, limitation

of computational representation or the granularity or resolution of sampling or

observations.


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In the past few years the basic concept rough set has been extended in many different

directions to enhance their modelling power and make them more applicable in real life

situations. One of the recent generalizations is the notion of covering based rough sets,

introduced by Bonikowski et al. A cover is a generalization of the notion of partition. The

covering based rough sets are models with promising potential for applications in data

mining. Several properties of the different types of covering based rough sets have beenderived by different researchers (Zhu and Wang 2003, 2006, Zhu 2007a, b, c).

2. Definitions And Notations:Let U be a universe we shall introduce some of the definitions and notations, which shall be

referred in the rest of the article.

2.1Basic Rough Sets:Let U be a universe of discourse and R be an equivalence relation over U. By U/R we

denote family of all equivalence class of R, referred to as categories or concepts of R and

the equivalence class of an element x U is denoted by [. By a knowledge, weunderstand a relational system K= (U,R), when u is as above and R is family of

equivalence relations over U.

For U any subset P () , the intersection of all equivalence relations in P is denotedby IND (P) and is called the indiscernibility relation over P. Given any X U and R

IND(K), we associate two subsets, R X =U{Y U/R : Y X} and R X ={Y U/R : Y

} , called (X) = R X - R X. The R X elements of are those elements of U which can

certainly be classified as elements of X and elements of R X are those elements of u

which can be possibly classified as elements of X, employing knowledge of R. We say

that X is rough with respect to R if and only ifR X R X, equivalently . X is

said to be R-definable if and only if R X =R X , or .

2.2Covering Based Rough Sets:Unlike the basic, there are four different methods to define the upper approximation for

covering based rough sets but there is only one definition of the lower approximation.

So, this leads to four different types of covering based rough sets introduced in the

literature. First introduce some definition which to be used in a sequel.

Definition 2.2.1: Let U be a universe of discourse and C be a family of subsets of U.C is

called a cover of U if no subset is C is empty and =U .We call (U,C) the coveringapproximation space and the covering C is called the family of approximation sets.


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Definition 2.2.2: Let (U,C) be an approximation space and x be any element of U. Then

the family Md (x) = { is called theminimal description of the object x.

Definition 2.2.3: For any X U, the family of sets (X) = { } is called the

family of sets bottom approximating the set X.

Definition 2.2.4: The set= (X) is called the lower approximation of set X.

Definition 2.2.5: The set= X\ is called the boundary of the set X.

Definition 2.2.6: The family of the sets Bn(X) = is called the family ofsets approximating the boundary of the set X

Definition 2.2.7: The family of sets (X)=(X) is called the family of sets topapproximating the set X.

Definition 2.2.8: The set is called the upper approximation of the set X.

Definition 2.2.9: A set X is called exact when (X) = (X).X is said to be a coveringbased rough set of the first type,otherwise.

The lower approximation is same for same for all types of covering based rough sets.

2.3Other Types of Covering Based Rough Sets:In this section we introduce the other types of covering based rough sets.

Definition 2.3.1: Let be a covering approximation space. For a set X

is called the second type of covering upper approximationof X.If =

, X is said to be exact. Otherwise, X is called a covering based rough set of

the second type.

Definition 2.3.2: Let be a covering approximation space. For a set X

is called the third type of covering upper approximation of X.If X. If =

, X is said to be exact. Otherwise, X is called a covering based rough set of

the third type.

Definition 2.3.3: Let be a covering approximation space. For a set X is called the fourth type of covering upper

approximation of X. If = , X is said to be exact. Otherwise, X is called a covering

based rough set of the fourth type.


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covering based rough set approach to uncertainty management in databases

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