Relationships between fuzzy probabilistic approximation spaces and their entropy measurement

Abstract

A fuzzy probability approximation space (FPA-space) is a approximation space (A-space) where three types of uncertainty (probability, fuzziness and roughness) are combined, which is obtained by putting probability distribution into a fuzzy approximation space (FA-space). This paper studies relationships between FPA-spaces and and their entropy measurement. Two types of fuzzy relation matrices are first defined by introducing the probability into a given fuzzy relation matrix in two ways, and on this basis, they are extended to two FA-spaces. Then, equality, dependence and independence between FPA-spaces are studied. Next, the distance between FPA-spaces is discussed. Moreover, the uncertainty for an FPA-space is measured by means of information entropy. Finally, the proposed information entropy is applied in the selection of classifier systems. Since fuzzy set theory, probability theory and rough set theory are aggregated together in an FPA-space, the obtained results of this paper may be helpful for dealing with practice problems with a sort of uncertainty.

Keywords:

• FPA-space
• fuzzy relation
• dependence
• distance
• uncertainty
• entropy
• classifier system

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China [11971420] and Natural Science Foundation of Guangxi [AD19245102, 2018GXNSFDA294003, 2018GXNSFDA281028, 2018GXNSFAA294134].

Notes on contributors

Zhaowen Li

Zhaowen Li received the MSc degree in Mathematics from Guangxi University, Nanning, China, in 1988 and the PhD degree in Mathematics from Hunan University, Changsha, China, in 2008. He is currently a professor in School of Mathematics and Statistics, Yulin Normal University. His research interests include granular computing, rough set theory, data mining, fuzzy set theory and information systems.

Damei Luo

Damei Luo is a master student at school of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, China. Her main research interests include granular computing, rough set theory and information systems.

Gangqiang Zhang

Gangqiang Zhang received the MSc degree in Software Engineering from Beihang University, Beijing, China, in 2006. He is currently an associate professor in School of Artificial Intelligence, Guangxi University for Nationalities. His main research interests include rough set theory, fuzzy set theory and information systems.

Liangdong Qu

Liangdong Qu received the MSc degree in Mathematics from Guangxi University for Nationalities, Nanning, China, in 2009. He is currently an associate professor in School of Artificial Intelligence, Guangxi University for Nationalities. His main research interests include rough set theory and information systems.

Ningxin Xie

Ningxin Xie received the MSc degree in Computer from Guangxi University, Nanning, China, in 2001. He is currently a professor in School of Artificial Intelligence, Guangxi University for Nationalities. His main research interests include rough set theory, fuzzy set theory and information systems.