Measuring roughness of generalized rough sets induced by a covering

doi:10.1016/j.fss.2007.03.018

Fuzzy Sets and Systems

Volume 158, Issue 22, 16 November 2007, Pages 2443-2455

https://doi.org/10.1016/j.fss.2007.03.018 Get rights and content

Abstract

In this paper, we propose new lower and upper approximations and obtain some important properties in generalized rough set induced by a covering. Especially, these properties are compared with ones of Pawlak's rough sets and Bonikowski's covering generalized rough sets, respectively. Moreover, we define a measure of roughness based on generalized rough sets with the new approximations and discuss some significant properties of the measure.

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The knowledge trajectory and thematic evolution of the rough sets research: A main path and scientific mapping analysis
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Citation Excerpt :
Covering-based rough sets [27] (CbRSs), as one of the essential extension model of classical rough sets [10], were presented to deal with covering data. From a theoretical perspective, nearly 40 pairs of covering approximation operators [2,32] have been established, corresponding axiomatic systems of them [26,28] have been established, and several types of covering reduction [23] have been defined. In application, they have been used to decision rule synthesis [3,19], attribute reduction [14,24,33], and any other fields [7,20,34].
As two important tools of information descriptions, minimal description (MinD) and maximal description (MaxD) are used to eliminate redundant information when describing an object in covering-based rough sets (CbRSs). Hence, there are many key issues (such as reductions, neighborhoods and approximations) have to do with them in CbRSs. However, it is complicated and time-consuming to use set approaches to compute them in a large cardinal covering approximation space. So the matrix approaches have been used to calculate them by which computers can implement the corresponding calculations easily. Based on the previous research work and inspired by the demand of knowledge discovery under large scale covering information systems, we propose two new matrix methods to calculate MinD and MaxD in CbRSs in this paper. Firstly, a note on the existing methods (matrix approach-1 and approach-2) is presented. Secondly, a new matrix, called grained matrix, is presented. After combining grained matrices and the mean of “sum”, the new matrix approach-1 is presented to calculate MinD and MaxD. Thirdly, three new matrix operations and the notion of complementary matrix are given which can simplify the calculation of computers. Based on new operations, we present the new matrix approach-2 to compute MinD and MaxD. Finally, the computational efficiency of the presented approaches are shown and compared with other methods by experiments on several UCI data sets.

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^☆: This work is supported by the National 973 Program of China (2002CB31200).

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Measuring roughness of generalized rough sets induced by a covering☆

Abstract

Roughness of a fuzzy set

Inform. Sci.

On rough sets and fuzzy rough sets

Bull. Polish Acad. Sci. Math.

On rough fuzzy sets

Bull. Polish Acad. Sci. Math.

Algebraic structures of rough sets

Extensions and intentions in the rough set theory

Inform. Sci.

A calculus of rough sets of the first order

Bull. Polish Acad. Sci.

Abstract approximation spaces for rough theories

Fuzziness in rough sets

Fuzzy Sets and Systems

Rough fuzzy sets and fuzzy rough sets

Internat. J. Gen. Systems

Rough set approach to incomplete information systems

Inform. Sci.