Probabilistic Shadowed Sets: Persistence and Transience in Fuzzy Uncertainty
Keywords:
Shadowed sets, Fuzzy sets, Approximation of fuzzy setsAbstract
Shadowed sets provide a three-way representation of uncertainty by separating positive, negative, and boundary evidence. However, the conventional shadow region is treated as a single homogeneous component, disregarding structural information associated with local variations in fuzziness. This paper introduces Probabilistic Shadowed Sets (PSS), a probabilistic refinement based on the principle that uncertainty possesses both magnitude and structural organization. Starting from a fuzzy membership function, we construct a decomposition into membership, non-membership, persistent uncertainty, and transient uncertainty. The distinction between the two forms of uncertainty is induced by a persistence function derived from local variations in fuzziness, yielding a four-state representation that preserves probability conservation while enriching the descriptive capacity of shadowed-set approximations. We establish fundamental properties, including structural non-invertibility and entropy refinement, and show that the probabilistic representation contains information unavailable in conventional shadowed sets. Experimental studies involving diverse continuous membership functions demonstrate that PSS consistently reveals structural differences that remain indistinguishable under classical shadowed-set approximations. These results position Probabilistic Shadowed Sets as a principled extension of shadowed-set theory for structure-aware uncertainty representation.
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Copyright (c) 2026 Tamunokuro Opubo William-West, Paul Augustine Ejegwa (Author)
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