Neutrosophic–Fuzzy Multi-View Imputation for Incomplete

Multivariate Measurement Data

Ahmed Hatip1,*

1 Department of Mathematics, Gaziantep University, Gaziantep, Turkey

Received: November 20, 2024 Revised: December 30, 2024 Accepted: January 03, 2025 ⋆ Corresponding author

ABSTRACT

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.

Keywords: Fuzzy reliability Single-valued neutrosophic information Missing data Multivariate imputation

Evidence fusion Uncertainty quantification Incomplete data

1. INTRODUCTION

A missing entry is often treated as though it had a single natural

replacement. In a multivariate matrix X ∈ Rn×d observed

only on Ω, however, one cell (i, j) /∈ Ω may support several

incompatible reconstructions. A local estimator may give

eN, a low-rank model eS, and a conditional model eR. When

maxs,r |es−er| ≈ 0, their agreement itself is evidence; when

the spread is large, reporting only bxi j discards information

about epistemic conflict. This paper takes that conflict as

part of the imputation problem rather than as an incidental

by-product of model choice.

The statistical context is well established. Rubin’s framework

distinguishes missingness according to how the probability

of being unobserved depends on observed and unobserved

quantities [1]; Little’s MCAR test addresses the narrower