Journal of Neutrosophic and Fuzzy Systems

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.

Three reconstructions of the same missing cell can each be defensible and still disagree materially. Let e=(eN,eS,eR) denote contextual-neighborhood, low-rank, and cross-variable ridge estimates. The central question is therefore not only which value should replace xi j, but how much coherent evidence supports that replacement. Neutrosophic–Fuzzy Multi-View Imputation (NFMVI) addresses this question by assigning each source a fuzzy reliability μs ∈ (0,1] and a Gaussian concordance as = exp[−(es−c)2/(2γ2)] around a reliability-weighted center c. These quantities generate the source state (Ts, Is,Fs) = (μsas,1−as,1− μs), from which ws = Ts/Σr Tr yields the convex reconstruction bx = Σswses. The same state produces a cell-level unresolved-evidence fraction U = 1−Σs Ts/Σs(Ts+Is+Fs), so reconstruction and uncertainty are generated by one mechanism rather than by separate post-processing. Evaluation uses two real multivariate datasets, MCAR, value-dependent MAR, and structured channel deletion at 10–30%, giving 144 common deterministic masks. The evidence is deliberately mixed. On the macroeconomic panel, NFMVI attains standardized RMSE 0.3425, below KNN (0.3776), Bayesian-ridge chained imputation (0.3794), and iterative SVD (0.4987); on the diabetes covariates, NFMVI (0.7641) and chained imputation (0.7659) are nearly indistinguishable overall, with chained imputation remaining better under MAR. The uncertainty signal is more distinctive: macro error-screening AUC averages 0.8156, and RMSE rises from 0.1717 in the lowest-U quartile to 0.6202 in the highest. Paired bootstrap contrasts, correlation-structure error, ablation, and parameter sensitivity therefore support a narrower conclusion than universal accuracy dominance: NFMVI provides an interpretable fuzzy–neutrosophic rule for reconciling heterogeneous imputers while retaining cell-level evidence conflict for downstream scrutiny.