Dominance-Based Rough Set Approach to Granular Computing

Dominance-Based Rough Set Approach to Granular Computing

Salvatore Greco, Benedetto Matarazzo, Roman Slowinski
DOI: 10.4018/978-1-60566-324-1.ch019
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Dominance-based Rough Set Approach (DRSA) was introduced as a generalization of the rough set approach for reasoning about preferences. While data describing preferences are ordinal by the nature of decision problems they concern, the ordering of data is also important in many other problems of data analysis. Even when the ordering seems irrelevant, the presence or the absence of a property (possibly graded or fuzzy) has an ordinal interpretation. Since any granulation of information is based on analysis of properties, DRSA can be seen as a general framework for granular computing. After recalling basic concepts of DRSA, the article presents their extensions in the fuzzy context and in the probabilistic setting. This permits to define the rough approximation of a fuzzy set, which is the core of the subject. The article continues with presentation of DRSA for case-based reasoning, where granular computing based on DRSA has been successfully applied. Moreover, some basic formal properties of the whole approach are presented in terms of several algebras modeling the logic of DRSA. Finally, it is shown how the bipolar generalized approximation space, being an abstraction of the standard way to deal with roughness within DRSA, can be induced from one of the algebras modeling the logic of DRSA, the bipolar Brower-Zadeh lattice.
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Dominance-Based Rough Set Approach

This section presents the main concepts of the Dominance-based Rough Set Approach (DRSA) (for a more complete presentation see, for example, Greco et al. (1999, 2001a, 2004b,c, 2005a); Słowiński et al. (2005)).

Information about objects is represented in the form of an information table. The rows of the table are labeled by objects, whereas columns are labeled by attributes and entries of the table are attribute-values. Formally, an information table (system) is the 4-tuple 978-1-60566-324-1.ch019.m01, where 978-1-60566-324-1.ch019.m02 is a finite set of objects, 978-1-60566-324-1.ch019.m03 is a finite set of attributes, 978-1-60566-324-1.ch019.m04 and 978-1-60566-324-1.ch019.m05 is the value set of the attribute 978-1-60566-324-1.ch019.m06, and 978-1-60566-324-1.ch019.m07 is a total function such that 978-1-60566-324-1.ch019.m08 for every 978-1-60566-324-1.ch019.m09, 978-1-60566-324-1.ch019.m10, called an information function (Pawlak, 1991). The set 978-1-60566-324-1.ch019.m11 is, in general, divided into set 978-1-60566-324-1.ch019.m12 of condition attributes and set 978-1-60566-324-1.ch019.m13 of decision attributes.

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