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A Fractal-based Complex Belief Entropy for Uncertainty Measure in Complex Evidence Theory

Keming Wu, Fuyuan Xiao, Yi Zhang

TL;DR

The paper addresses how to quantify uncertainty within Complex Evidence Theory (CET) by linking Complex Basic Belief Assignments (CBBA) to probability through a fractal-inspired Complex Pignistic Belief Transformation (CPBT). It introduces Fractal-Based Complex Belief (FCB) entropy, derives a CPBT-driven mass redistribution M_F, and defines Com_F to compute an entropy over the CPBT-generated masses. The authors establish properties including probabilistic consistency, monotonicity, additivity, and subadditivity, and provide a maximum-entropy model for FCB entropy. Through numerical examples, pattern classification (Breast Cancer and Glass datasets), and information fusion scenarios, FCBE demonstrates superior sensitivity to focal-element intersections and richer information content than existing complex entropies. These results suggest FCB entropy as a robust, CET-consistent uncertainty measure with practical benefits for complex decision-making tasks.

Abstract

Complex Evidence Theory (CET), an extension of the traditional D-S evidence theory, has garnered academic interest for its capacity to articulate uncertainty through Complex Basic Belief Assignment (CBBA) and to perform uncertainty reasoning using complex combination rules. Nonetheless, quantifying uncertainty within CET remains a subject of ongoing research. To enhance decision-making, a method for Complex Pignistic Belief Transformation (CPBT) has been introduced, which allocates CBBAs of multi-element focal elements to subsets. CPBT's core lies in the fractal-inspired redistribution of the complex mass function. This paper presents an experimental simulation and analysis of CPBT's generation process along the temporal dimension, rooted in fractal theory. Subsequently, a novel Fractal-Based Complex Belief (FCB) entropy is proposed to gauge the uncertainty of CBBA. The properties of FCB entropy are examined, and its efficacy is demonstrated through various numerical examples and practical application.

A Fractal-based Complex Belief Entropy for Uncertainty Measure in Complex Evidence Theory

TL;DR

The paper addresses how to quantify uncertainty within Complex Evidence Theory (CET) by linking Complex Basic Belief Assignments (CBBA) to probability through a fractal-inspired Complex Pignistic Belief Transformation (CPBT). It introduces Fractal-Based Complex Belief (FCB) entropy, derives a CPBT-driven mass redistribution M_F, and defines Com_F to compute an entropy over the CPBT-generated masses. The authors establish properties including probabilistic consistency, monotonicity, additivity, and subadditivity, and provide a maximum-entropy model for FCB entropy. Through numerical examples, pattern classification (Breast Cancer and Glass datasets), and information fusion scenarios, FCBE demonstrates superior sensitivity to focal-element intersections and richer information content than existing complex entropies. These results suggest FCB entropy as a robust, CET-consistent uncertainty measure with practical benefits for complex decision-making tasks.

Abstract

Complex Evidence Theory (CET), an extension of the traditional D-S evidence theory, has garnered academic interest for its capacity to articulate uncertainty through Complex Basic Belief Assignment (CBBA) and to perform uncertainty reasoning using complex combination rules. Nonetheless, quantifying uncertainty within CET remains a subject of ongoing research. To enhance decision-making, a method for Complex Pignistic Belief Transformation (CPBT) has been introduced, which allocates CBBAs of multi-element focal elements to subsets. CPBT's core lies in the fractal-inspired redistribution of the complex mass function. This paper presents an experimental simulation and analysis of CPBT's generation process along the temporal dimension, rooted in fractal theory. Subsequently, a novel Fractal-Based Complex Belief (FCB) entropy is proposed to gauge the uncertainty of CBBA. The properties of FCB entropy are examined, and its efficacy is demonstrated through various numerical examples and practical application.
Paper Structure (16 sections, 71 equations, 8 figures, 5 tables)

This paper contains 16 sections, 71 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dempster-Shafer evidence theory
  4. Complex evidence theory
  5. Entropy
  6. Fractal-based complex belief entropy
  7. The importance of Deng entropy and CET in modeling uncertainty
  8. Theoretical Analysis of CPBT from a Fractal Perspective
  9. Fractal-based complex belief entropy
  10. Maximum fractal-based complex belief entropy
  11. The properties of the proposed FCB entropy
  12. APPLICATION
  13. Discuss from numerical examples
  14. Application in pattern classification
  15. Application in information fusion
  16. ...and 1 more sections

Figures (8)

  • Figure 1: Regardless of the value of $p$, the final distribution of $\mathbb{C}om({{x}_{1}})$, $\mathbb{C}om({{x}_{2}})$ and $\mathbb{C}om({{X}})$ is stable and tends to the same value.
  • Figure 2: The process of $x_{1}$, $x_{2}$ allocation to singletons in Example 1 is self-similar.
  • Figure 3: As the number of elements in FoD increases, the maximum value of most entropies is greater than the maximum value of Shannon entropy.
  • Figure 4: The change trend of different entropies with the increase of negation times.
  • Figure 5: The relationship between the three parts.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 19 more