Entropy–Remoteness Neutrosophic Fuzzy c-Means for

Separating Boundary Ambiguity from Outlierness

Das Suman Das1,* Ajoy Kanti2

1 Assistant Professor Grade II (Mathematics), Department of Education, National Institute of Technology Calicut, Kozhikode–673601,

Kerala, India

2 Associate Professor, Department of Mathematics, Tripura University, Agartala–799022, Tripura, India

Received: November 12, 2024 Revised: December 16, 2024 Accepted: January 17, 2025 ⋆ Corresponding author

ABSTRACT

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.

Keywords: Fuzzy c-means Single-valued neutrosophic information Clustering Entropy Outlier detection

Boundary ambiguity Robust prototypes

1. INTRODUCTION

Consider a fuzzy partition U = (uik) ∈ [0,1]n×K with

ΣKk

=1 uik = 1. If ui ≈ (1/2,1/2,0, . . .), the observation may

be a legitimate boundary record. If instead all prototype

distances are large and similar, the same diffuse pattern can

arise because the record is remote from every cluster. In both

cases the ordinary fuzzy c-means update assigns a nonzero

contribution um

ik to prototype k. The numerical appearance of

uncertainty is similar, but the geometry is not. The central

question of this paper is therefore not simply how to make

clustering “more robust,” but how to distinguish ambiguity

from outlierness before they enter the prototype update.

Fuzzy sets formalize graded membership μ ∈ [0,1] rather