الفهرس 
Chapter 1 PreliminariesOnly 14 pages are availabe for public view
 |  | from | 154 |  |  |
| | | from | 154 | | |
Abstract
Methods of data classifications are considered as a major pre- processing step for pattern recognition, machine learning, and data mining.
We give two topological approaches to generalize multi- granular rough sets using families of binary relations. In the first approach, we define a family of topological spaces using families of relations to maximize the interiors and minimize the closures. In the second approach, we define minimal neighbor- hoods to classify multi-data of information systems and gen- erate a multi-granular knowledge base. Moreover, we present some important algorithms to reduce all topological reductions of the information system using topological bases. We round off by studying real life applications of this work using medical data.
The properties of three types of lower and upper approx- imations of a non-empty finite set for dominance relations are studied. These are based on the dominance class generated by the dominance relation.
the union of all equivalence classes contained in A, while the up- per approximation is the intersection of all equivalence classes which intersect A non-trivially. A rough set is a pair of two exact sets the lower approximation of A and the upper approximation of A.
Since, the equivalence relations are too restrictive for many real life applications, the classical rough set theory of Pawlak needed to be generalized. The generalization process has twofold. A first one is to replace the equivalence relation by tolerance re- lation [9, 50], similarity relation [51], characteristic relation [23, 35] and arbitrary binary relation [26]. Greco, Matarazzo, and Slowinski [19, 20, 21, 22] proposed an extension of rough set the- ory, called the dominance-based rough set approach (DRSA) to solve the ordering problem of objects. Recently, further stud- ies have been made in DRSA [12, 13, 38, 47]. In [2], Abu-Donia had achieved a new comparison between different kinds of ap- proximations by using a family of binary relations without any conditions. The second, is to replace the partition induced by the equivalence relation by a covering and use it to approximate any subset of the universe [56].
These frameworks are called granular computing, which are models providing solutions to problems in data mining, ma- chine learning, pattern recognition and cognitive science. But still there are problems that require more extensions. In 2006, Y.Qian introduced the multi-granular computing using rough set instead of a single granular. Multi-granular computing ap- proach is replacing the single relation used in a single granular by a set of relations on the same universe, see [36, 37, 55].
Many extensions of rough sets proofs to be suitable in dif- ferent applications. Many of these extensions appeared using fuzzy sets [10, 54]. Other approaches to fuzzy sets based on fuzzy covering and comparison of different types of rough sets were introduced by covering and are addressed in [14, 31]. The generalizations of fuzzy rough sets constructed by fuzzy cover- ing were studied in [11, 28]. New Different kinds of generalized rough sets based on neighborhoods with a medical application studied in [4, 3, 5, 6, 7, 8, 16, 24, 45, 46].
One of the important branches in mathematics is topology.
Topology is the best implementation of relationship between ob- jects or features so when we deal with complicated relation- ships topology becomes a very satisfactory tool. Pawlak has pointed out that topology is closely linked to rough set theory and on the full conviction that the topological structure of the rough sets is one of the key issues of rough set theory. This convenient relationship has prompted researchers to study this relationship, it’s properties and its applications in real life see [1, 17, 18, 39, 40, 41, 42, 43, 44]. In 2013, Y.Qian has investigated a new theory on multi-granulation rough sets from the topo- logical point of view, by inducing n-topological spaces on the universe set U from n-equivalence relations on U. He also has studied the multi-granulation topological rough space and its topological properties see [29].
In this thesis, we will discuss the fundamental goals of gen- eralized information systems in topology, as well as their tools
and limitations. We will demonstrate how new topological meth- ods for attribute reduction, significance testing, and data filter- ing can improve it. This thesis can be considered as a topologi- cal generation of Pawlak information system concepts about ap- proximations defined by general relations. The primary purpose of this thesis is to introduce new topological approaches for data classification and study rough set approximation properties de- fined by dominance relations.
This thesis is organized into four chapters. In what follows, the contents of each is briefly outlined.
Chapter 1 provides some basic definitions and theorems that will be used throughout this thesis.
Chapter 2 contains some basic concepts of Pawlak informa- tion systems and rough set theory to be used and generalized throughout this thesis.
In Chapter 3, we offer a convenient hybrid method using
topology and rough set theory to solve the problem of multi- source and variable. We also developed algorithms for the re- duction of attributes. Two topological approaches to general- ized multi-granulation are presented in two categories. The first method relied on minimization in the boundary region, whereas the second method relied on the concept of minimal neighbor- hoods. We apply our results to the problem of attribute reduc- tion in medical information systems. All the results in this
chapter have been accepted for publication in Italian Journal
of Pure and Applied Mathematics (IJPAM), under title Topolog- ical Approaches for Generalized Multi-granula-tion Rough Sets with Applications.
Chapter 4 studies three types of rough approximations defi- nitions. These approximations are built on dominance relations. There are two parts: The first part discusses the generalization of the previous approximations to a family of n-dominance re- lations. The second part studies the properties of these gener- alized approximations. This chapter’s results have been pub- lished in the International Journal of Fuzzy Logic and Intelli- gent Systems (Vol. 22, No. 2, June 2022, pp. 193-201), under title
Properties of Different Types of Rough ApproxiMations Deflned by a FaMily of DoMinance Relations.
Finally, the conclusion and future plane are introduced.
Appendix provides our python implementation that has been used for the calculations.