Monotone set-valued measures: Choquet integral, $ f $-divergence and Radon-Nikodym derivatives

Zengtai Gong; Chengcheng Shen; Zengtai Gong; Chengcheng Shen

doi:10.3934/math.2022609

AIMS Mathematics

2022, Volume 7, Issue 6: 10892-10916. doi: 10.3934/math.2022609

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Research article

Monotone set-valued measures: Choquet integral, $ f $-divergence and Radon-Nikodym derivatives

Zengtai Gong ^{1
,
,},
Chengcheng Shen ^1,2

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2.
Department of Advance Mathematics Teaching, Lanzhou Technology and Business College, Lanzhou 730101, China

Received: 16 November 2021 Revised: 07 March 2022 Accepted: 22 March 2022 Published: 01 April 2022
MSC : 26E50, 28E10

Divergence as a degree of the difference between two data is widely used in the classification problems. In this paper, $ f $-divergence, Hellinger divergence and variation divergence of the monotone set-valued measures are defined and discussed. It proves that Hellinger divergence and variation divergence satisfy the triangle inequality and symmetry by means of the set operations and partial ordering relations. Meanwhile, the necessary and sufficient conditions of Radon-Nikodym derivatives of the monotone set-valued measures are investigated. Next, we define the conjugate measure of the monotone set-valued measure and use it to define and discuss a new version $ f $-divergence, and we prove that the new version $ f $-divergence is nonnegative. In addition, we define the generalized $ f $-divergence by using the generalized Radon-Nikodym derivatives of two monotone set-valued measures and examples are given. Finally, some examples are given to illustrate the rationality of the definitions and the operability of the applications of the results.
- $ f $-divergence,
- Hellinger divergence,
- variation divergence,
- Radon-Nikodym derivatives,
- Choquet integral
Citation: Zengtai Gong, Chengcheng Shen. Monotone set-valued measures: Choquet integral, $ f $-divergence and Radon-Nikodym derivatives[J]. AIMS Mathematics, 2022, 7(6): 10892-10916. doi: 10.3934/math.2022609

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Abstract

Divergence as a degree of the difference between two data is widely used in the classification problems. In this paper, $ f $-divergence, Hellinger divergence and variation divergence of the monotone set-valued measures are defined and discussed. It proves that Hellinger divergence and variation divergence satisfy the triangle inequality and symmetry by means of the set operations and partial ordering relations. Meanwhile, the necessary and sufficient conditions of Radon-Nikodym derivatives of the monotone set-valued measures are investigated. Next, we define the conjugate measure of the monotone set-valued measure and use it to define and discuss a new version $ f $-divergence, and we prove that the new version $ f $-divergence is nonnegative. In addition, we define the generalized $ f $-divergence by using the generalized Radon-Nikodym derivatives of two monotone set-valued measures and examples are given. Finally, some examples are given to illustrate the rationality of the definitions and the operability of the applications of the results.

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