Source-linked AI summary
A correlation coefficient of belief functions
Wen Jiang
TL;DR
Conflict management remains an open issue in Dempster-Shafer evidence theory, and existing conflict measures have shortcomings. The paper proposes a new correlation coefficient that considers focal-element non-intersection and difference, with numerical examples indicating effective conflict measurement among belief functions.
Problem
Conflict management remains an open issue because existing methods do not fully resolve how to measure conflict between evidences.
Method
The paper proposes a new correlation coefficient of belief functions that simultaneously considers non-intersection and differences among focal elements.
Results
Numerical examples and comparisons demonstrate that the proposed correlation coefficient effectively measures conflict among belief functions.
Takeaways & Limitations
The proposed coefficient offers a correlation-based approach for measuring conflict among belief functions and overcoming drawbacks of existing methods.
Takeaways & Limitations
The paper notes that a quantitative conflict measure is not always suitable, and the classical coefficient can incorrectly indicate conflict because it ignores focal-element differences.
Abstract
from arXiv · showhide
How to manage conflict is still an open issue in Dempster-Shafer evidence theory. The correlation coefficient can be used to measure the similarity of evidence in Dempster-Shafer evidence theory. However, existing correlation coefficients of belief functions have some shortcomings. In this paper, a new correlation coefficient is proposed with many desirable properties. One of its applications is to measure the conflict degree among belief functions. Some numerical examples and comparisons demonstrate the effectiveness of the correlation coefficient.
1. Introduction
Conflict management remains open in Dempster-Shafer evidence theory because existing measures do not fully capture conflict between evidences. The paper proposes a correlation coefficient that jointly considers focal-element non-intersection and difference, and applies it to conflict measurement.
- Conflict management remains an open issue in Dempster-Shafer evidence theory, despite hundreds of proposed methods.
- Measuring evidence conflict is a necessary first step because it determines whether evidence should be combined or conflict-management procedures applied.
- The classical conflict coefficient k can be misleading because it measures combined belief assigned to the empty set while ignoring focal-element differences.
- Effective conflict measurement must simultaneously account for non-intersection and differences among focal elements.
- The paper proposes a new correlation coefficient of belief functions incorporating both factors and uses it to define a conflict coefficient.
- Numerical examples and comparisons indicate that the proposed coefficient effectively measures conflict among belief functions and has desirable properties.
2. Preliminaries
The preliminaries define Dempster-Shafer evidence theory, its basic probability assignments, and Dempster’s combination rule with conflict coefficient k. They also introduce evidence distances, Liu’s two-dimensional conflict model, and Song et al.’s correlation coefficient, including reported limitations of these approaches.
- Dempster-Shafer evidence theory: Dempster-Shafer theory uses a finite discernment frame of mutually exclusive hypotheses and its power set as the domain for evidence assignments.The framework is introduced as Dempster-Shafer evidence theory and is widely used in uncertainty modeling and processing.
- Evidence distance and conflict models: Evidence similarity and conflict can be assessed with Jousselme distance, pignistic probability distance, Liu’s two-dimensional model, and correlation coefficients.Liu’s model unites pignistic betting distance with k, while Song et al.’s coefficient uses a Jaccard matrix to modify BPAs.
- Correlation coefficient of evidence: Song et al.’s BPA modification may repeatedly allocate belief values, so the modified BPA may violate a required condition; its correlation coefficient also lacks a stated property.The coefficient measures relevance between evidence bodies, with higher conflict corresponding to a lower coefficient value.
3. A new correlation coefficient
The paper proposes a correlation coefficient for belief functions that accounts for non-intersection and differences among focal elements, proves its desirable properties, and uses it to define conflict.
- It considers both non-intersection and differences among focal elements when measuring correlation.
- For the proposed coefficient, disjoint focal elements characterize the zero-correlation condition when focal-element masses are nonzero.
- The proposed coefficient measures relevance between two bodies of evidence, with larger values indicating higher relevance.
- A zero coefficient corresponds to absent relevance, whereas a coefficient of one implies complete relevance and identical belief functions.
- The paper mathematically proves that the proposed coefficient satisfies all desirable properties defined for correlation coefficients.
- The coefficient is used to define a conflict coefficient whose value increases with conflict, ranging from no conflict for identical evidence to complete contradiction.
4. Numerical examples
Numerical examples compare the proposed conflict coefficient with classical conflict, evidence-distance, and correlation measures across contradictory, identical, and varying-evidence BPAs. The proposed coefficient tracks intuitive conflict behavior and addresses shortcomings identified for the existing measures.
- Example 1: In Example 1, two highly contradictory BPAs are correctly reflected by k, d_BPA, and the proposed kr, whereas Song et al.’s cor = 0.3668 indicates relatively high correlation.The paper therefore describes Song et al.’s value as unreasonable for this case.
- Example 2: In Example 2, both BPA pairs totally contradict, but d_BPA and cor indicate similarity or relevance, including cor = 0.5606 for m3 and m4.The paper concludes that these measures cannot always provide correct conflict measurements.
- Example 4: In Example 4, kr and d_BPA decrease as set A approaches {1,2,3,4,5} and increase as A departs from it, while k fails to differentiate the changes.The authors judge the proposed coefficient and Jousselme’s evidence distance appropriate for measuring conflict in this example.
- Comparison: The proposed correlation coefficient considers both non-intersection and difference among focal elements, combining information absent from the classical coefficient and evidence distance.The paper concludes that the proposed measures correctly and effectively quantify relevance and conflict between belief functions.
5. Conclusions
The paper concludes that its new correlation coefficient of belief functions overcomes drawbacks of existing methods. Numerical examples illustrate its efficiency for conflict management.
- Conclusion: The paper presents a new correlation coefficient of belief functions designed to overcome drawbacks of existing methods.The coefficient is proposed as an approach to measuring conflict among belief functions.
- Conclusion: Numerical examples and comparisons illustrate the efficiency of the proposed correlation coefficient for conflict management.The conclusion frames conflict management as an application of the proposed coefficient.