TAILIEUCHUNG - Data Mining and Knowledge Discovery Handbook, 2 Edition part 48

Data Mining and Knowledge Discovery Handbook, 2 Edition part 48. Knowledge Discovery demonstrates intelligent computing at its best, and is the most desirable and interesting end-product of Information Technology. To be able to discover and to extract knowledge from data is a task that many researchers and practitioners are endeavoring to accomplish. There is a lot of hidden knowledge waiting to be discovered – this is the challenge created by today’s abundance of data. Data Mining and Knowledge Discovery Handbook, 2nd Edition organizes the most current concepts, theories, standards, methodologies, trends, challenges and applications of data mining (DM) and knowledge discovery. | 450 Tsau Young T. Y. Lin and Churn-Jung Liau Proposition 2 An equivalence relation onU a partition on U In RST the pair U A is called an approximation space and its topological properties are studied. Binary Relation Granulation - Topological Partitions In Lin 1998b we observe that there is a derived partition for each BNS that is the map B V 2U p B p induces a partition on V the equivalence class C p B-1 B p is the center of B p . In the case V U the B p is the neighborhood of C p and C p consists of all the points that have the same neighborhood. So B p B C p . We observe that C p is a partition. Since each B p is a neighborhood of the set C p . The quotient set is a BNS Lin 1989a . We will call the collection of C p topological partition with the understanding that there is a neighborhood B p for each equivalence class C p . The neighborhoods capture the interaction among equivalence classes Lin 2000 . Fuzzy Binary Granulations Fuzzy Binary Relations In Lin 1996 we have discussed various fuzzy sets. In this chapter a fuzzy set is uniquely defined by its membership function. So a fuzzy set is a w-sofset if we use the language of the cited paper. A fuzzy binary relation is a fuzzification of a binary relation. Let I be the unit interval 0 1 . Let FBR be a fuzzy binary relation that is there is a membership function FBR V x U I p u r. For each p e V there is a fuzzy set whose membership function FMp U I is defined by FMp u FBR p u we call FMp a fuzzy binary neighborhood set. Again we can view the idea geometrically. We assume a fuzzy binary neighborhood system FBNS is imposed for V on U. For each object p e V we associate a fuzzy subset denoted by FB p C U. In other words we have a map FB V FZ U p FB p where FZ U means all fuzzy subsets on U. FB p is called a fuzzy binary neighborhood and FB a fuzzy binary granulation FBG and the collection FB p p e V a fuzzy binary neighborhood system FBNS . It is clear that given a map FB there is a binary relation

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