Fourier transform of basic probability assignment

Ruolan Cheng

doi:10.1016/j.ins.2025.122818

2025-11-01

PRE

Ruolan Cheng

Yong Deng *

DOI：10.1016/j.ins.2025.122818

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摘要

En 中文

The Fourier transform constitutes a crucial foundation for frequency-domain analysis. In probability theory, its counterpart for probability distributions, known as the characteristic function, has been extensively studied and applied. As a generalization of the probability distribution, the basic probability assignment (BPA) in Dempster-Shafer theory (DST) models uncertainty by assigning belief masses to the power set of basic events. However, there is still no relevant research involving the frequency-domain perspective of such power set information distribution. To address this research gap, this paper conducts the first exploration of the Fourier transform in DST. A Fourier transform method for BPA is proposed, and its theoretical properties are rigorously examined, including proofs of uniform continuity, positive semi-definiteness, and uniqueness theorems. Some numerical examples are presented to visualize the BPAs in the frequency domain, and the statistical nature of their moments is further analyzed. The proposed Fourier transform can not only degenerate into the characteristic function by setting specific frequency parameters, but also reveal a correspondence with the commonality and implicability functions. It enables the four combination rules in DST to be implemented in the frequency domain, thereby opening a new avenue for processing and updating uncertain information.

Keyword:

Fourier transform

Dempster-Shafer theory

Basic probability assignment

Frequency-domain visualization

Moments

Combination rules