IJPAM: Volume 13, No. 2 (2004)

COMPLETION OF THE SHAPLEY ENTROPY
OF FUZZY MEASURES

Guo-Jun Wang$^1$, Ning He$^2$
$^1$Institute of Mathematics
Shaanxi Normal University
Xi'an 710062, P.R. CHINA
e-mail: gjwang@snnu.edu.cn
$^2$Shaanxi University of Technology
Hanzhong 723001, P.R. CHINA
e-mail: he_ning0916@sina.com.cn


Abstract.A fuzzy measure with complete certainty has the minimal Shapley entropy and vice versa. This paper points out that a fuzzy measure with complete uncertainty has the maximal Shapley entropy but not vice versa. An example shows that fuzzy measures with maximal Shapley entropy may far from complete uncertainty. A complementary entropy of the Shapley entropy called the partitional entropy is proposed which behaves well if used together with the Shapley entropy. The sum of the two entropies is called the absolute entropy. It is proved that a fuzzy measure possesses complete certainty or complete uncertainty if and only if its absolute entropy attains the minimal value or the maximal value respectively. Moreover, extension of fuzzy measures is discussed, the regular extension of fuzzy measures is introduced which keeps certain basic properties of the original fuzzy measure unchanged.

Received: March 6, 2004

AMS Subject Classification: 28E10, 28D20

Key Words and Phrases: fuzzy measure, Shapley entropy, partitional entropy, regular extension

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 13
Issue: 2