Labeled Structures And Rough Approximations On Multi-scale Ordered Information Systems

Rough set theory, initiated by Pawlak in the early1980s, is a formal data analysistool for modeling and processing incomplete information. This theory is an extension ofclassical set theory, and has been demonstrated to have very successful applications inthe fields of artificial intelligence such as expert systems, machine learning, patternrecognition, decision analysis, process control, and knowledge discovery in databases.The basic structure of Pawlak’s rough set theory is an approximation spaceconsisting of a universe of discourse and an equivalence relation imposed on it. Based onthe approximation space, the notions of lower and upper approximations can beconstructed. Most applications based on rough set theory can fall into theobject-attribute-value representation model, called information systems. Using the lowerand upper approximations of decision classes with respect to the approximation spacedetermined by a conditional attribute set, knowledge hidden in decision informationsystems may be unraveled and expressed in the form of decision rules. It is well knownthat in a Pawlak information system, each object under each attribute can only take onone value. We call such system a single scale information system. However, in real-lifeapplication, objects are usually measured at different scales under the same attribute.Thus, in many real-life multi-scale information systems, an object can take on as manyvalues as there are scales under the same attribute. Therefore, granular computing, whichimitates human being’s thinking, is an approach for knowledge representation and datamining in rough set theory. Its basic computing unit is called granules, and its objective isto establish effective computation models for dealing with large scale complex data andinformation. The key to granular computing is to make use of granules in problemsolving.In real-world applications, there exist more complicated types of data. Withdifferent requirements, people have to rank objects according to their attribute valuesunder different levels of granularity. Thus, how to discover knowledge in hierarchicallyorganized ordered information is of particular importance in real-life data mining. In order to study knowledge acquisition in ordered information systems with multi-granularlabels, in this dissertation, we propose the notion of multi-granular labeled orderedstructures and discuss their induced rough set approximations. The concept of orderedlabeled structures is first introduced. A dominance relation on the universe of discoursefrom an ordered labeled structure is also defined. Dominated labeled blocks determinedby the dominance relation are then constructed. Ordered lower approximations andordered upper approximations as well as ordered labeled lower approximations andordered labeled upper approximations of sets based on dominance relations are furtherproposed. Finally, the notion of multi-scale ordered information systems is introduced,lower and upper approximations in multi-scale ordered information systems are defined,and their properties are examined.