Volume 10 • Issue 2 • PP: 10 – 15 • 2025
Entropy–Remoteness Neutrosophic Fuzzy c-Means for Separating Boundary Ambiguity from Outlierness
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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A weak fuzzy assignment can mean two geometrically different things. A record may lie between otherwise legitimate clusters, so its membership vector ui = (ui1, . . . ,uiK) is genuinely ambiguous; or it may be remote from every prototype, in which case diffuse membership is a symptom of outlierness rather than a boundary. Treating both cases through one fuzziness scalar makes the prototype update unable to explain why a record should have reduced influence. This paper constructs Entropy–Remoteness Neutrosophic Fuzzy c-Means (ERN-FCM) from two separate quantities. Boundary indeterminacy is Ii = α[−Σk uik loguik]/logK, whereas robust remoteness is Fi = 1−exp(−τρi) with ρi = [(δi−q.90)/(q.90−q.50+ε)]+ and δi = mink ∥xi−vk∥2. Their conjunction gives truth Ti = (1−Ii)(1−Fi) and therefore Ti +Ii +Fi = 1+IiFi. Prototypes are updated by v+k = Σi Tium ikxi/Σi Tium ik, so remote or highly ambiguous records contribute less without being hard-deleted. The numerical study deliberately separates the two geometries. On Wine with 10% gross contamination, mean adjusted Rand index rises from 0.8665 for ordinary FCM to 0.8975 for ERN-FCM; on Breast Cancer the corresponding values are 0.7014 and 0.7074, whereas Iris remains a counterexample where FCM is slightly better (0.6328 versus 0.6179). The remoteness-aware uncertainty 1−Ti identifies gross outliers with AUC 0.9898–0.9956 at 10% contamination, and on a separate boundary benchmark Ii identifies bridge observations with AUC 0.9963. Paired bootstrap contrasts, component ablation, parameter sensitivity, and fixed-point diagnostics support a focused conclusion: ERN-FCM is most useful when cluster recovery and the type of uncertain observation must be distinguished, not as a universal replacement for FCM.
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