Accurate Topological Measures for Rough Sets

—Data granulation is considered a good tool of decision making in various types of real life applications. The basic ideas of data granulation have appeared in many fields, such as interval analysis, quantization, rough set theory, Dempster-Shafer theory of belief functions, divide and conquer, cluster analysis, machine learning, databases, information retrieval, and many others. Some new topological tools for data granulation using rough set approximations are initiated. Moreover, some topological measures of data granulation in topological information systems are defined. Topological generalizations using -open sets and their applications of information granulation are developed.


INTRODUCTION
Granulation of the universe involves the decomposition of the universe into parts.In other words, the grouping individual elements or objects into classes, based on offering information and knowledge [7,14,15,21,36,37,[42][43][44][45].Elements in a granule are pinched together by indiscernibility, similarity, proximity or functionality [43].The starting point of the theory of rough sets is the indiscernibility of objects or elements in a universe of concern [14,15, 17-20, 51,52, 21-22].
The original rough set theory was based on an equivalent relation on a finite universe U.For practical use, there have been some extensions on it.One extension is to replace the equivalent relation by an arbitrary binary relation; the other direction is to study rough set via topological method [8,14].In this work, we construct topology for a family covering rough sets.
In [40] addressed four operators on a knowledge base, which are sufficient for generating new knowledge structures.Also, they addressed an axiomatic definition of knowledge granulation in knowledge bases.

Topological notions like semi-open, pre-open,   open
sets are as basic to mathematicians of today as sets and functions were to those of last century [48][49][50][51][52].Then, we think the topological structure will be so important base for knowledge extraction and processing.
The topology induced by binary relations on the universes of information systems is used to generalize the basic rough set concepts.The suggested topological operations and structure open up the way for applying affluent more of topological facts and methods in the process of granular computing.In particular, the notion of topological membership function is introduced that integrates the concept of rough and fuzzy sets [17][18][19][20].
In this paper, we indicated some topological tools for data granulation by using new topological tools for rough set approximations.Moreover, we introduced using general binary relations a refinement data granulation instead of the classical equivalence relations.Section 1 gives a brief overview of data granulation structures in the universe using equivalence and general relations.Fundamentals of rough set theory under general binary relations are the main purpose of Section 2. Section 3 studies the topological data granulation properties of topological information systems.Explanation of topological data granulation in information systems appears in www.ijarai.thesai.orgSection 4. In Section 5 we are given some more accurate topological tools for data granulation using   open sets approach.The conclusions of our work are presented in Section 6.

II. ESSENTIALS OF ROUGH SET APPROXIMATIONS UNDER GENERAL BINARY RELATIONS
In rough set theory, it is usually assumed that the knowledge about objects is restricted by some indiscernibility relations.The Indiscernibility relation is an equivalence relation which is interpreted so that two objects are equivalent if we can't distinguish them using our information.This means that the objects of the given universe U indiscernible by R into three classes with respect to any subset XU  : Class 1: the objects which surely belong to X , Class 2: the objects which possibly belong to X , Class 3: the objects which surely not belong to X , The object in Class 1 form the lower approximation of X , and the objects of Class 1 and 3 form together its upper approximation.The boundary of X consists of objects in Class 3. Some subsets of U are identical to both of them approximations and they are called crisp or exact; otherwise, the set is called rough.

For any approximation space ( , ) A U R 
, where R is an equivalence relation, lower and upper approximations of a subset XU  , namely () RX and () RX are defined as follows: The lower and upper approximations have the following properties: For every , X Y U  from the approximation space ( , ) A U R  we have: The equality in all properties happens when ( ) ( ) R X R X X  .The proof of all these properties can be found in [17][18][19][20][21][22][23]51].
Furthermore, for a subset XU  , a rough membership function is defined as follows: , where X denotes the cardinality of the set X .The rough membership value () X x  may be interpreted as the conditional probability that an arbitrary element belongs to X given that the element belongs to [] R x .
Based on the lower and upper approximations, the universe U can be divided into three disjoint regions, the positive () POS X , the negative () NEG X and the boundary () BND X ,where: Considering general binary relations in [18,52] is an extension to the classical lower and upper approximations of any subset X of U .
{ : is the base generated by the general relation defined in [17,52].The general forms based on  are defined as follows: For data granulation by any binary relation, in [E.Lashein (2005) ] a rough membership function is defined as follows: () ()

III. ROUGH SETS OF EQUIVALENCE AND GENERAL BINARY RELATIONS
Indiscernibility as defined by equivalence relation represents a very restricted type of relationships between elements and universes.The procedure to granule the universe by general binary relations is introduced in [6].
A topological space [1,2] is a pair ( , ) X  consisting of a set X and a family  of subset of X satisfying the following conditions:  In later years a number of generalizations of open sets have been considered [21][22][23].We talk about some of these generalizations concepts in the following definitions.
Let U be a finite universe set and R is any binary relation defined onU , and () rR x be the set of all elements which are in relation to certain elements x in U from right for all xU  Let  be the general knowledge base (topological base) using all possible intersections of the members of () rR x .The component that will be equal to any union of some members of  must be misplaced.

Let ( , ) A U R 
be an approximation space where R is any binary relation defined on U .Then we can define two new approximations as follows: The topological lower and the topological upper approximations have the following properties: For every , X Y U  and every approximation space ( , ) A U R we have: Given that topological lower and topological upper approximations satisfy that: this enables us to divide the universe U into five disjoint regions (granules) as follows: (See Figure 1) The following theorems study the properties and relationships among the above regions namely boundary, positive and negative regions. () (1) and ( 2) is obvious, by definitions. (  be a topological information system and for any subset , X Y U  we have: ( () 2), ( 3) and (4) are obvious. ( { , , , , , , } U u u u u u u u  be the universe of 7 patients have data sheets shown in Table I with possible dengue symptoms.If some experts give us the general relation R defined among those patients as follows: According to the topological knowledge base we can easily see that: Let ( , ) UR  be a  -approximation space. -lower approximation and  -upper approximation of any non- empty subset X of U is defined as: We see that: The Universe U can be separated into divergent 24  We see that the degree of accuracy of the granule { , , , } b c d e using Pawlak's accuracy measure equal to 60% , using  -accuracy measure equal to 80% and using  - accuracy measure equal to 100% .Accordingly  - accuracy measure is more precise than Pawlak's accuracy and  -accuracy measures.

VI. CONCLUSIONS AND APPLICATION NOTES
In the near future is the completion of a new paper for the application of the granules concepts of this paper in medicine especially in the field of heart disease in collaboration with specialists in this field.We designed a JAVA application program novelty to generate granules division automatically once you select points covered by the heart scan and the medical relationship among them using topology defined on it.The program works under any operating system but needs to be a great RAM memory and strong processor to end the division of the millions of points to the granules in seconds.
 is closed under arbitrary union, (3)  is closed under finite intersection.The pair ( , ) X  is called a topological space.The elements of X are called points .The subsets of X belonging to  are called open sets.The complement of the open subsets are called closed sets.The family  of all opensubsets of X is also called a topology for X .
information system and for any subset XU  we have:

7 (.
Then we have the following granules of the universe: granules with respect to any XU  .According to Example 5.1 we can construct the following table (TableII) showing the degree of accuracy measure