

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7168
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Correlation Measure for Quadri-Partitioned Neutrosophic
Refined Sets and Its Application in Medical Diagnosis

A. A. Azzam1, B. Alreshidi1, M. Aldawood1, Mohamed M. Awad1, Abdelhalim
Hasnaoui2, Medhat Ahmed Abu-Tahon3, Ahmad M. Abdel-Mageed3, Arif Mehmood4,∗

1 Mathematics Department, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz
University, Al-Kharj 11942, Saudi Arabia
2 Department of Mathematics, College of Science, Northern Border University, Arar 91431,
Saudi Arabia
3 Department of Biological Sciences, College of Science, Northern Border University, Arar 91431,
Saudi Arabia
4 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail
Khan 29050, KPK, Pakistan

Abstract. This work introduces a new correlation measure of quadri-partitioned neutrosophic refined
sets (QPNRS) to address the problem of ambiguity and bias that may exist in the medical diagnostic.
The model is a patient-symptom-disease interaction model that uses four membership functions—truth,
relative truth, relative falsehood, and falsehood—to illustrate its interactions. Table III concludes the
results by demonstrating that there is clear disease and estimator variance and that correlations between
QPNRS measures Q and R are consistently strong and positive (0.743–0.901) across four medical condi-
tions. In particular, P2 is more extensively distributed, while P1 is constantly high in all environments,
reaching its maximum value of 0.901 in the case of tuberculosis. These findings support the combined
clinical use of Q and R by demonstrating a strong, primarily favorable reliance between them. We apply
a multifaceted unsupervised learning pipeline to two distinct datasets—the conventional Iris dataset and
patient disease-correlation profiles—in order to augment the correlation analysis. In both cases, the
consistently strongly separated clusters support the process and imply that the detected groupings are
not the result of labeling or modeling artifacts.
2020 Mathematics Subject Classifications: 03E72, 62H30, 68T37
Key Words and Phrases: Quadri-partitioned neutrosophic refined sets (QPNRS), correlations between
QPNRS measures (Q and R), patient-symptom-disease interaction model, patient disease-correlation
profiles

1. Introduction

Modeling human thought and handling uncertainty in complicated decision-making situa-
tions have been one of the main goals of computational intelligence. A paradigm shift occurred
when Zadeh [3] created fuzzy sets, and in order to address the vagueness, the idea of partial
membership was introduced. This was advanced by Atanassov’s intuitionistic fuzzy sets (IFS)
[4], which used a more nuanced method of expressing reluctance by adding a measure of non-
membership. Additionally, the model was extended to interval-valued IFS [5], which enhances
the model’s ability to capture imprecise data. The Neutrosophy, as forth by Smarandache [1],

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7168

Email addresses: aa.azzam@psau.edu.sa (A. A. Azzam), b.alreshidi@psau.edu.sa (B. Alreshidi),
m.aldawood@psau.edu.sa (M. Aldawood), m.abdelgalil@psau.edu.sa (M. M. Awad),
abdllhalim.hasanawa@nbu.edu.sa (A. H. Hasnaoui), medhatahon@gmail.com (M. A. Abu-Tahon),
ahmed.mohammed@nbu.edu.sa (A. M. Abdel-Mageed), mehdaniyal@gmail.com (A. Mehmood)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 2 of 26

is a philosophical foundation for handling the indeterminacy of real-world systems and rep-
resents an even more profound generalization of these concepts. A more reliable and flexible
way to describe incomplete, inconsistent, and uncertain information is to use neutrophilic sets
(NS), which explicitly treat truth-membership (T), indeterminacy-membership (I), and falsity-
membership (F) as distinct constituents [14]. Later developments into simpler, more practical
forms like interval neutrosophic sets (INS) [12] and single-valued neutrosophic sets (SVNS) [11]
have made neutrosophy extremely valuable in data science and engineering. The similarity as-
sessment is one of the most important aspects of these sets’ viability. A lot of work has gone
into creating a robust similarity metric for neutrosophic situations. A groundbreaking study
by Broumi and Smarandache [6] introduced a number of these measures, which Majumdar and
Samanta [7] subsequently improved and examined in relation to entropy and similarity. Ye
has made contributions in this area by delivering a variety of vector-based [2], interval-based
[8], and cosine similarity measures [13] to multicriteria decision-making (MCDM) in interval
neutrosophic and single-valued settings, even in cases where the weights have no knowledge at
all [9]. These are the cornerstones of modern MCDM techniques, along with other optimization
techniques as AHP’s cosine maximization method [10]. Additionally, recent advancements have
expanded to explore hybrid frameworks, such as the incorporation of neutrosophic sets into
plithogenic logic to process multi-dimensional attribute interactions [15] and the application of
such models in specific domains, such as supply chain management [17] and medical diagnosis
[16].
The usage of sophisticated fuzzy and neutrosophic sets for pattern identification and decision-
making has significantly increased, particularly in the medical industry. Researchers have devel-
oped more sophisticated metrics that are unique to complex contexts based on the preliminary
studies on the similarity of vectors to SVNS [2] and cosines to IFS [9]. In the MCDM problems
of uncertainty, for instance, the outranking relation approach proposed by Peng et al. [18] with
a novel single-valued neutrosophic similarity measure has demonstrated excellent performance.
In the specific instance of medical diagnosis, Abdel-Basset et al. [19] have developed a com-
prehensive viewpoint on medical diagnosis in a neutrophilic setting that successfully manages
ambiguity and inconsistency of symptoms and illnesses, resulting in a more realistic model than
traditional fuzzy or IFS approaches. Distance-based tactics were included as a result of the
hunt for more effective solutions. A number of generalized similarity metrics on SVNS were
introduced by Sahin and Kucuk [20], which significantly improved the discrimination capacity
of applications like medical diagnostics. Their work demonstrated that the degree of similar-
ity or distance that is employed has a significant impact on the diagnosis outcomes. Ye [21]
introduced the idea of cross-entropy on SVNS, which is an effective way to parametrize the dis-
crimination information that is crucial to distinguishing between identical medical situations,
adding to the methodological toolbox. The advancements have significantly improved the way
in which the uncertainties inherent in the medical diagnosis process are managed. Combining
neutrosophic sets with machine learning methods to carry out validation and enhance pattern
recognition is a recent and significant trend. While clustering techniques like K-Means and
fuzzy C-Means (FCM) are not new, it is more recent research that has used them to support
the findings of neutrosophic similarity metrics. The regular FCM algorithm [22] offered the
majority of the foundation for handling uncertainty throughout the clustering process, but it
was unable to handle indeterminacy, which served as the inspiration for the neutrosophic vari-
ations. Researchers have filled this gap by developing a novel neutrosophic C-means clustering
algorithm that incorporates truth, indeterminacy, and falsity into the clustering process. Guo
and Sengur [23] have also improved the clustering technique on noisy images. Although there
is no obvious connection between this concept and medical diagnoses, it is clear that machine
learning and neutrosophic logic techniques work well together. Despite their widespread usage
in visualization [24], dimensionality reduction techniques like PCA and t-SNE are now employed
to visually validate cluster elements inferred from neutrosophic correlation measurements. This


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 3 of 26

approach provides realistic, unmonitored validation of theoretical framework-based diagnostic
classifications. Additionally, Smarandache [15] postulated the appearance of lithogenic sets,
which may offer fresh perspectives. Plithageny provides a clear explanation of the relationship
between the qualities’ values and conflicts, which makes its use particularly suitable to com-
plex healthcare systems. Plithogenic sets will be able to provide a more fine-grained knowledge
representation, which will enable the development of more accurate diagnostic algorithms, ac-
cording to some of the initial implementations created by Smarandache [25]. With machine
learning techniques representing both theory and practice, our suggested QPNRS model, which
combines lithogenic logic with neutrosophic principles, can be seen as a significant step toward
developing strong correlation measures to medical diagnosis. The investigation of supra open
soft sets and its various decompositions and applications, as well as soft set theory, are both
superseded by supra soft topology, according to the sources provided. Fundamental results on
decompositions of supra soft sets, together with the concept of soft continuity in this context,
were provided by El-Sheikh and Abd Ellatif’s basic contributions [28]. Supra soft strongly gen-
eralized closed sets [30], supra soft strongly extra generalized closed sets [31], and supra soft
strongly semi* generalized closed sets [33] are only a few of the specific classes of soft sets that
are now being studied in this field. Furthermore, the ideas have been crucial in defining and
researching different forms of soft connectedness [35] as well as related soft separation axioms
[29][32][34]. The research of supra soft compactness [37] and interactions with soft ideals [36]
have also contributed to the development of the theory.
Clinicians can eliminate redundant testing, expedite routine tasks, and focus on the more con-
venient or informative score without sacrificing accuracy when there is a strong and consistent
relationship between Q and R. It is risky to apply a single cross-condition cutoff because the
strength of this link varies by measurement and disease. By evaluating these changes, addi-
tional disease-specific cutoffs of control may be established, and metrics can be made comparable
across locations. In addition to generating hypotheses, quality checks, and early phenotyping
before annotation, which is expensive, unsupervised algorithms that employ clean labels where
none exist can also detect edge instances and natural clusterings in patient profiles. When
the same clustering pipeline is run on a recognized benchmark, like Iris, and clinical data, the
workflow gains external credibility and interpretability, making it easier to rely on and repeat.
Generally speaking, risk reduction techniques like predictive modeling, feature cutting, and pro-
tocol standardization are considerably safer and more reproducible when they exhibit strong
QR dependency and stable clusters. This also paves the way for future clinical applications.
We present the first cross, multi-metric analysis of NRS Q and R’s movement. Their as-
sociation across four medical disorders is measured using the three complementing measures
(P1, P2, andP3). Q and R always have a very strong and positive association (0.743-0.901),
albeit this varies based on the condition and the measure, for example. P1 stays high across
all conditions, but P2 varies with disease. We also apply similar patterns to two very different
datasets-the classical Iris dataset and patient disease-correlation profiles-to verify that they are
not artificial. In both cases, we discover clear, distinct clusters. Combining these findings yields
a more trustworthy picture that differentiates between behavior that is stable and behavior that
is condition-sensitive, going beyond a single correlation value. This is a clinically significant
extra detail that can help determine which metric to prioritize, reduce pointless measurements,
and enable better, situation-specific diagnostic regimens.

2. Preliminaries

This section presents the foundational concepts required for the subsequent sections.

Definition 1. [26]Let M be a space of points (objects), with a generic element in M denoted
by r. A neutrosophic set Ê in M is characterized by the truth membership function TÊ(r),
an indeterminacy membership function IÊ(r), and a falsity membership function FÊ(r), The


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 4 of 26

functions TÊ(r), IÊ(r), and FÊ(r) are real standard or non-standard subsets of ]−0, 1+[; that is
TÊ(r) : M −→]−0, 1+[, IÊ(r) : M −→]−0, 1+[ and FÊ(r) : M −→]−0, 1+[, with the condition
−0 ≤ TÊ(r) + IÊ(r) + FÊ(r) ≤ 3+. This definition of a neutrosophic set is difficult to apply in
the real world in scientific and engineering field.

Therefore, the concept of SVNS, which is an instance of a neutrosophic set, was introduced
by Wang et al. [27].

Definition 2. [27]Let M be a space of points (objects), with a generic element in M denoted by
r. An SVNS Ê in M is characterized by the truth membership function TÊ(r), an indeterminacy
membership function IÊ(r), and a falsity membership function FÊ(r). For each r ∈ M, there
is TÊ(r), IÊ(r), andFÊ(r) ∈ [0, 1] and 0 ≤ TÊ(r) + IÊ(r) +FÊ(r) ≤ 3. Therefore, an SVNS Ê
can be represented by

Ê = {⟨r, TÊ(r) + IÊ(r) + FÊ(r)⟩ : r ∈ M}

Definition 3. [27]Let M be a space of points (objects), with a generic element in M denoted
by r. For SVNS Ê and Ĝ. Ê ⊆ Ĝ if and only if TÊ(r) ≤ TĜ(r), IÊ(r) ≥ IĜ(r), FÊ(r) ≥ FĜ(r)

for any r ∈ M. Also Ê = Ĝ if and only if Ê ⊆ Ĝ and Ĝ ⊆ Ê.

3. Quadri-Partitioned Neutrosophic Refined Sets

This study presents a new set, quadri-partitioned neutrosophic refined sets (QPNRS), which
can be used as an extension of neutrosophic refined sets to express uncertainty, indeterminacy,
and dual-valued information in complete decision systems. The order and coherence of their
structural criteria set apart the four QPNRS membership sequences: truth, relative truth,
relative false, and falsity. Basic functions like intersection, equality, union, and subset are defined
as the classical set theory is expanded into the neutrosophic refined domain. Additionally, it is
proposed that the similarity between QPNRS be measured using a correlation coefficient, which
is symmetric, restricted, reflexive, and equal to unity only on sets of identical sets.

Definition 4. Let M be a universe. A quadri-partitioned neutrosophic refined set (QPNRs) Ê
on M can be defined as follows:

Ê =

(
⟨r, (T 1

Ê
(r), T 2

Ê
(r), ..., T p

Ê
(r)), (RT 1

Ê
(r), RT 2

Ê
(r), ..., RT p

Ê
(r)),

(RF 1
Ê
(r), RF 2

Ê
(r), ..., RF p

Ê
(r)), (F 1

Ê
(r), F 2

Ê
(r), ..., F p

Ê
(r))⟩ : r ∈ M

)

where (T 1
Ê
(r), T 2

Ê
(r), ..., T p

Ê
(r)) : M −→ [0, 1], (RT 1

Ê
(r), RT 2

Ê
(r), ..., RT p

Ê
(r)) : M −→ [0, 1],

(RF 1
Ê
(r), RF 2

Ê
(r), ..., RF p

Ê
(r)) : M −→ [0, 1] and (F 1

Ê
(r), F 2

Ê
(r), ..., F p

Ê
(r)) : M −→ [0, 1]. Such

that 0 ≤ supT i
Ê
(r) + supRT i

Ê
(r) + supRF i

Ê
(r) + supF i

Ê
(r) ≤ 3. (i = 1, 2, 3, .., p)

T 1
Ê
(r) ≤ T 2

Ê
(r)... ≤ T p

Ê
(r)), RT 1

Ê
(r) ≤ RT 2

Ê
(r)... ≤ RT p

Ê
(r)) for any r ∈ M. T 1

Ê
(r), T 2

Ê
(r), ..., T p

Ê
(r),

RT 1
Ê
(r), RT 2

Ê
(r), ..., RT p

Ê
(r), RF 1

Ê
(r), RF 2

Ê
(r), ..., RF p

Ê
(r), and F 1

Ê
(r), F 2

Ê
(r), ..., F p

Ê
(r) is the Truth-

membership sequence, Relative Truth-membership sequence, Relative False-membership sequence
and False-membership sequence of the element M, respectively. Also, P is called the dimension
(cardinality) of QPNRS Ê, denoted d(Ê). We arrange the Truth-membership, Relative Truth-
membership sequence in decreasing order but the corresponding Relative False-membership and
False-membership sequence may not be in decreasing or increasing order.

Definition 5. Let Ê, Ĝ ∈ QPNRS. Then, Ê is said to be QPNRS subset to Ĝ denoted by
Ê ⊆ Ĝ if T i

Ê
(r) ≤ T i

Ĝ
(r), RT i

Ê
(r) ≤ RT i

Ĝ
(r), RF i

Ê
(r) ≥ RF i

Ĝ
(r), F i

Ê
(r) ≥ F i

Ĝ
(r) for all r ∈ M

and (i = 1, 2, 3, .., p).


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 5 of 26

Definition 6. Let Ê, Ĝ ∈ QPNRS. Then, Ê is said to be QPNRS equal to Ĝ denoted by
Ê = Ĝ if T i

Ê
(r) = T i

Ĝ
(r), RT i

Ê
(r) = RT i

Ĝ
(r), RF i

Ê
(r) = RF i

Ĝ
(r), F i

Ê
(r) = F i

Ĝ
(r) for all r ∈ M

and (i = 1, 2, 3, .., p).

Definition 7. Let Ê, Ĝ ∈ QPNRS then the union of Ê and Ĝ is denoted by Ê ∪ Ĝ = C and
is defined by

C =

(
⟨r, (T 1

C(r), T
2
C(r), ..., T

p
C(r)), (RT 1

C(r), RT 2
C(r), ..., RT p

C(r)),

(RF 1
C(r), RF 2

C(r), ..., RF p
C(r)), (F

1
C(r), F

2
C(r), ..., F

p
C(r))⟩ : r ∈ M

)

T i
C = T i

Ê
(r) ∨ T i

Ĝ
(r), RT i

C = RT i
Ê
(r) ∨ RT i

Ĝ
(r), RF i

C = RF i
Ê
(r) ∧ RF i

Ĝ
(r), and F i

C = F i
Ê
(r) ∧

F i
Ĝ
(r) for all r ∈ M and (i = 1, 2, 3, .., p).

Definition 8. Let Ê, Ĝ ∈ QPNRS then the intersection of Ê and Ĝ is denoted by Ê ∩ Ĝ = D
and is defined by

D =

(
⟨r, (T 1

D(r), T
2
D(r), ..., T

p
D(r)), (RT 1

D(r), RT 2
D(r), ..., RT p

D(r)),

(RF 1
D(r), RF 2

D(r), ..., RF p
D(r)), (F

1
D(r), F

2
D(r), ..., F

p
D(r))⟩ : r ∈ M

)

T i
D = T i

Ê
(r) ∧ T i

Ĝ
(r), RT i

D = RT i
Ê
(r) ∧ RT i

Ĝ
(r), RF i

D = RF i
Ê
(r) ∨ RF i

Ĝ
(r), and F i

D = F i
Ê
(r) ∨

F i
Ĝ
(r) for all r ∈ M and (i = 1, 2, 3, .., p).

Definition 9. Let {M = r1, r2, r3, ..., rn} be the finite universe of discourse and

Ê = {⟨T j

Ê
(ri) +RT j

Ê
(ri) +RF j

Ê
(ri) + F j

Ê
(ri)⟩|ri ∈ M}

Ĝ = {⟨T j

Ĝ
(ri) +RT j

Ĝ
(ri) +RF j

Ĝ
(ri) + F j

Ĝ
(ri)⟩|ri ∈ M}

be two quadri-partitioned neutrosophic refined sets consisting of the T j: truth-membership, RT j:
membership, RF j: membership and F j: membership functions. Then the correlation coefficient
of Ê and Ĝ

QPNRS(Ê, Ĝ) =
CNRS(Ê, Ĝ)√

CQPRS(Ê, Ê)× CQPRS(Ĝ, Ĝ)

Where

CQPNRS(Ê, Ĝ) =
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ĝ
(ri)+RT j

Ê
(ri)RT j

Ĝ
(ri)+RF j

Ê
(ri)RF j

Ĝ
(ri)+F j

Ê
(ri)F

j

Ĝ
(ri)}

CQPNRS(Ê, Ê) =
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ê
(ri)+RT j

Ê
(ri)RT j

Ê
(ri)+RF j

Ê
(ri)RF j

Ê
(ri)+F j

Ê
(ri)F

j

Ê
(ri)}

and

CQPNRS(Ĝ, Ĝ) =
1

p

p∑
j=1

n∑
i=1

{T j

Ĝ
(ri)T

j

Ĝ
(ri)+RT j

Ĝ
(ri)RT j

Ĝ
(ri)+RF j

Ĝ
(ri)RF j

Ĝ
(ri)+F j

Ĝ
(ri)F

j

Ĝ
(ri)}

Proposition 1. The defined correlation measure between Ê and Ĝ satisfies the following prop-
erties:

(i) 0 ≤ S(Ê, Ĝ) ≤ 1.

(ii) S(Ê, Ĝ) = 1 iff Ê = Ĝ.


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 6 of 26

(iii) S(Ê, Ĝ) = S(Ĝ, Ê).

Proof.

(i) 0 ≤ S(Ê, Ĝ) ≤ 1. As T and RT membership, RF and F-membership functions of the
QPRS lies between 0 and 1, S(Ê, Ĝ) is also lies between 0 and 1.

(ii) S(Ê, Ĝ) = 1 iff Ê = Ĝ.
a. Let two QPNRS Ê and Ĝ be equal (i.e., Ê = Ĝ). Hence for any T j

Ê
(ri) = T j

Ĝ
(ri), RT j

Ê
(ri) =

RT j

Ĝ
(ri), RF j

Ê
(ri) = RF j

Ĝ
(ri), F

j

Ê
(ri) = T j

Ĝ
(ri). Then,

CQPNRS(Ê, Ê) = CQPNRS(Ĝ, Ĝ)

=
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ê
(ri) +RT j

Ê
(ri)RT j

Ê
(ri) +RF j

Ê
(ri)RF j

Ê
(ri) + F j

Ê
(ri)F

j

Ê
(ri)}

and

CQPNRS(Ê, Ĝ) =
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ĝ
(ri)+RT j

Ê
(ri)RT j

Ĝ
(ri)+RF j

Ê
(ri)RF j

Ĝ
(ri)+F j

Ê
(ri)F

j

Ĝ
(ri)}

=
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ê
(ri) +RT j

Ê
(ri)RT j

Ê
(ri) +RF j

Ê
(ri)RF j

Ê
(ri) + F j

Ê
(ri)F

j

Ê
(ri)}

= CQPNRS(Ê, Ê)

Hence

S(Ê, Ĝ) =
CNRS(Ê, Ĝ)√

CQPRS(Ê, Ê)× CQPRS(Ĝ, Ĝ)
=

CNRS(Ê, Ê)√
CQPRS(Ê, Ê)× CQPRS(Ê, Ê)

= 1

b. Let S(Ê, Ĝ) = 1 then the unit measure is possible only if CNRS(Ê,Ĝ)√
CQPRS(Ê,Ê)×CQPRS(Ĝ,Ĝ)

= 1.

This refers that T j

Ê
(ri) = T j

Ĝ
(ri), RT j

Ê
(ri) = RT j

Ĝ
(ri), RF j

Ê
(ri) = RF j

Ĝ
(ri), F

j

Ê
(ri) =

T j

Ĝ
(ri) for all i, j. Hence Ê = Ĝ.

(iii) If S(Ê, Ĝ) = S(Ĝ, Ê), it obvious that

CNRS(Ê, Ĝ)√
CQPRS(Ê, Ê)× CQPRS(Ĝ, Ĝ)

=
CNRS(Ĝ, Ê)√

CQPRS(Ĝ, Ĝ)× CQPRS(Ê, Ê)
= CQPNRS(Ĝ, Ê)

As

CQPNRS(Ê, Ĝ) =
1

p

p∑
j=1

n∑
i=1

{T j

Ê
(ri)T

j

Ĝ
(ri)+RT j

Ê
(ri)RT j

Ĝ
(ri)+RF j

Ê
(ri)RF j

Ĝ
(ri)+F j

Ê
(ri)F

j

Ĝ
(ri)}

=
1

p

p∑
j=1

n∑
i=1

{T j

Ĝ
(ri)T

j

Ê
(ri) +RT j

Ê
(ri)RT j

Ĝ
(ri) +RF j

Ĝ
(ri)RF j

Ê
(ri) + F j

Ĝ
(ri)F

j

Ê
(ri)}

= CQPNRS(Ĝ, Ê)


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 7 of 26

4. A Symptom-Disease Correlation Model based on Double-Valued
Neutrosophic Set Theory

Numerous uncertainties in medical diagnosis make it challenging to unite different symptom
sets under a single illness term. This difficulty is exacerbated by the increased amount of
information that doctors can now access thanks to modern technologies.
Furthermore, there are instances in the real world when each component can be categorized
based on its true, relative truth, relative false, and false membership. It is better to undertake a
multi-time inspection (sampling a single patient at different times) in order to make a diagnosis
because a single check-up could lead to an error. A quadri-partitioned neutrosophic refined
set (QPNRS)-based methodology is presented in this work to improve diagnostic accuracy by
modeling the patient-symptom-disease linkages. The given measure of connection between the
diseases and the symptoms of the patients provides a suitable medical diagnosis. This method is
distinct in that it takes into account truth membership, relative truth membership, relative false
membership, and false membership functions. Three times during the day, at 8:00, 12:00, and
16:00, the symptoms of three patients (P1, P2, andP3) were noted. Temperature variations,
coughing, headaches, sore throats, and body aches were all recorded. A quadri-partitioned
neutrosophic refined matrix Q, which displays the evolution of patients and symptoms over
time, was created by organizing them. A second R matrix was developed to fit symptoms
and four prevalent diseases: viral fever, TB, throat infection, and typhoid. The diagnostic
result for each patient was determined using the highest correlation score between Q and R.
The results of the study show that patient P1 is most likely to have tuberculosis (correlation
= 0.901), and patients P2 and P3 are most likely to have typhoid (correlations = 0.878 and
0.869, respectively). This uncertainty-friendly QPNRS-based diagnostic methodology is quite
beneficial when it comes to ambiguity in symptom rating and a systematic, data-driven approach
to supporting medical decision-making. This approach has been demonstrated using a medical
diagnosis.

Example 1. The framework defines three sets: Let
S = {Temperature, cough, throatpain, headache, bodypain} be a group of symptoms, P =
{P1, P2, P3} be a group of patients, and D = {V iralFever, Tuberculosis, Typhoid, Throatdisease}
be a group of diseases. As part of the recommended remedy, patients will be observed three times
a day. This temporal refinement generates unique Truth, Re-Truth, Re-False, and False mem-
bership values for each patient, resulting in a more complex dataset for diagnosis.

Table 1: Q (The Relation between Patient and Symptoms)
Q Temperature Cough Throat Pain Headache Body pain

P1

(0.4,0.2,0.1,0.4)
(0.3,0.2,0.2,0.6)
(0.2,0.3.0.2,0.5)

(0.5,0.2,0.2,0.4)
(0.4,0.1,0.0,0.3)
(0.3,0.3,0.1,0.5)

(0.3,0.3,0.2,0.5)
(0.2,0.3,0.3,0.4)
(0.1,0.3,0.3,0.3)

(0.5,0.2,0.1,0.4)
(0.5,0.2,0.2,0.7)
(0.3,0.2,0.1,0.6)

(0.5,0.1,0.1,0.4)
(0.2,0.2,0.1,0.5)
(0.1,0.2,0.2,0.3)

P2

(0.6,0.2,0.1,0.5)
(0.5,0.3,0.2,0.2)
(0.4,0.2,0.2,0.5)

(0.6,0.2,0.1,0.7)
(0.4,0.3,0.1,0.2)
(0.2,0.2,0.2,0.5)

(0.6,0.2,0.1,0.3)
(0.3,0.3,0.2,0.4)
(0.1,0.2,0.2,0.5)

(0.6,0.2,0.1,0.1)
(0.4,0.3,0.1,0.8)
(0.2,0.2,0.2,0.3)

(0.4,0.2,0.2,0.5)
(0.3,0.1,0.2,0.7)
(0.1,0.3,0.2,0.5)

P3

(0.8,0.1,0.2,0.5)
(0.7,0.3,0.2,0.4)
(0.6,0.2,0.2,0.4)

(0.5,0.3,0.2,0.3)
(0.3,0.3,0.1,0.3)
(0.1,0.4,0.2,0.4)

(0.3,0.2,0.1,0.6)
(0.2,0.3,0.2,0.7)
(0.1,0.2,0.2,0.5)

(0.6,0.1,0.1,0.5)
(0.5,0.2,0.1,0.6)
(0.2,0.1,0.1,0.6)

(0.6,0.2,0.2,0.5)
(0.3,0.2,0.1,0.4)
(0.2,0.1,0.1,0.6)

Let the samples be taken at three different timings in a day (in 08:00, 12:00, 16: 00)


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Table 2: R (The relation among Symptoms and Diseases)

R Viral Fever Tuberculosis Typhoid Throat dis-
ease

Temperature (0.2,0.3,0,2,0.6) (0.4,0.4,0.2,0.5) (0.6,0.3,0.1,0.5) (0.3,0.4,0.2,0.8)
Cough (0.6,0.2,0.2,0.6) (0.8,0.1,0.1,0.3) (0.3,0.1,0.1,0.6) (0.2,0.2,0.2,0.1)
Throat Pain (0.5,0.1,0.1,0.3) (0.4,0.3,0.2,0.3) (0.4,0.3,0.2,0.5) (0.2,0.4,0.2,0.2)
Headache (0.6,0.5,0.3,0.2) (0.2,0.2,0.1,0.6) (0.1,0.3,0.3,0.3) (0.2,0.3,0.2,0.5)
Body Pain (0.7,0.3,0.2,0.4) (0.2,0.2,0,1,0.4) (0.2,0.2,0.1,0.4) (0.2,0.1,0.1,0.3)

Table 3: The Correlation Measure between NRS Q and R
Correlation Measure Viral Fever Tuberculosis Typhoid Throat disease

P1 0.872 0.901 0.864 0.869
P2 0.845 0.743 0.878 0.787
P3 0.792 0.839 0.869 0.811

The optimal correlation value in Table 3 provides the accurate medical diagnosis. Thus,
although patients P2 and P3 have typhoid, patient P1 has tuberculosis. The suggested QPNRS
correlation measure can effectively convert difficult, multi-time, and ambiguous symptom data
into a measurable and practical diagnostic tool, as demonstrated in this table.

5. Result and Discussion

The K-Means clustering figure (k=2) in Figure 1 illustrates how patient profiles (P1 − P3)
group according to their correlational patterns. Point color indicates cluster membership, while
the axes (PCA Components 1 and 2) provide a low-dimensional perspective of the correlation
features. The centroids of the two clusters are indicated by large black ”X” markings. Two
distinct clusters are present: P2 and P3 in Cluster 0 (red dots) indicate that the correlation
patterns of the two profiles for illnesses are similar. Their closeness to one another in PCA
space indicates that their illness connection patterns are more comparable than P1. Because
Cluster 1 (green dot) solely contains P1, it may be distinguished from P2 and P3. This divide
demonstrates P1’s unique correlation behavior, especially its strong and distinctive link to tu-
berculosis. This separation is supported by the cluster centers, where one centroid is between
P2 and P3, suggesting their shared structure informed by typhoid, and the other is closer to P1,
reflecting its profile saturated with tuberculosis. Generally speaking, the clustering provides
an unsupervised confirmation of the earlier correlation analysis: P2 and P3 appear to be cor-
related partners primarily influenced by typhoid, while P1 stands out due to its preponderant
association with tuberculosis, which clearly separates it from the other profiles.


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Figure 1

When projected onto two main components, the three patient profiles (P1 − P3) cluster
according to their patterns of correlation, as shown in Fig. 2’s K-Means++ clustering map
(k=2). The centroids of the two clusters are shown by the black ”X” marks, and each point
represents a profile. There are two clear clusters: Cluster 1 (green point) contains only P1, while
Cluster 0 (red points) contains both P2 and P3. P2 and P3 are closely positioned, indicating
that their correlation tendencies are identical, particularly because to their high association
with typhoid. However, P1’s partitioning into a discrete cluster suggests a unique correlation
structure, which is dominated by its significant connection with tuberculosis. These cluster
centers-one with the unique pattern of P1 and another with the shared characteristics of P2 and
P3-support this separation. Generally speaking, the clustering finds two natural patient groups:
P1 as an outlier with a unique diagnostic profile and P2 and P3 as a similarity group.


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Figure 2

The three patient profiles (P1 − P3) are grouped based on their correlation patterns after
normalizing and dimensionality reduction into two principal components, as shown in Fig. 3,
the K-Means++ clustering diagram (k=2) on normalized data. The cluster centroids are indi-
cated by the black ”X” markers, and each point represents a profile. Two clusters are readily
apparent: P2 and P3 are found in Cluster 0 (red dots), whereas P1 is the only one found in
Cluster 1 (green dot). The fact that P2 and P3 are located in the same cluster indicates that
these two profiles have comparable normalized correlation patterns, with the strength of these
correlations being dominated by their shared relationship with typhoid. Contrarily, P1’s place-
ment in another cluster reveals another unique normalized profile with a strong Tuberculosis
correlation as its core, setting it apart from the rest. The confirmed cluster centers also support
this separation: one centroid matches closely with P2 and P3’s common characteristics, while
the other identifies P1’s singular pattern. In total, the clustering shows two inherent groupings
of patients in normalized conditions P2 and P3 in a similarity pair, and P1 as an outlier with a
unique diagnostic profile.


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Figure 3

In Figure 4 the flower samples are grouped into three groups based on feature similarities
is shown by the K-Means clustering plot on the PCA-reduced Iris data. Each point is a single
sample that is projected onto the two major components for viewing, and the titles of the
groups are represented by the enormous black dots with a black X. There are three distinct
clusters found. Cluster 1 (green points) is tightly packed on the left, forming a tight, well-
spaced cluster associated with a single species of iris (likely Iris setosa). Cluster 2, which has
an intermediate nature (often Iris versicolor), partially overlaps Cluster 0 (red points), which
is in the center of the field and includes more widely distributed samples. The blue dots in
Cluster 2, which is located on the right, are dispersed around the cluster’s centroid and partially
overlap with Cluster 0, indicating both a degree of separation and resemblance (typically Iris
virginica). The cluster centers that have one centroid inside the thick green cluster, another
centroid lying at the center of the red samples, and a third centroid balancing the blue cluster
serve as validation for these groupings. While the green cluster is an autonomous cluster with
few cross-sections, the red and blue cross-sections highlight the dataset’s transitional patterns.
Intersected, each clearly unique. Overall, the Iris dataset’s nature can be effectively depicted
using PCA’s reduced clustering: one species is represented by a tight and distinct clustering,
while the other two exhibit more gradual boundary parting with some overlap, which is normal
given their biological similarity.


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Figure 4

The normalized and PCA-reduced Iris data’s K-Means++ clustering plot is shown in Fig.
5, which also shows how the samples are grouped into three distinct clusters based on how
similar they are on the normalized features. Each flower is described by the points, and the
cluster centroids are indicated by the enormous black X of the points. The presence of three
clusters is clear. Cluster 1 (green points), which represents a single species (often Iris setosa), is
located on the far left and is compact and dense with little overlap. Cluster 2 (blue points) lies
halfway between the two and partially overlaps with Cluster 0, while Cluster 0 (red points) is on
the far right and has a more extensive distribution.Given the biological resemblances between
Iris versicolor and Iris virginica, this overlap suggests characteristics of transition between the
two families. The cluster centers support this distribution: the centroid in the green cluster is
firmly integrated, the centroid in the red cluster is at the center, and the third centroid is at the
center of the blue group. With the green cluster becoming even more noticeable and the red
and blue clusters’ partial boundary overlap still discernible, the normalized scaling improves
the separation.
Overall, the normalized data’s K-Means++ clustering provides a great and well-structured view
of the Iris dataset. Two of them exhibit a similarity gradient, while one is compact and highly
segregated. These examples demonstrate how normalization can highlight natural species splits
while preserving overlaps where features are convergent.


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Figure 5

The distribution of the three species (setosa, versicolor, and virginica) in the lower dimen-
sions (TSNE-1, TSNE-2, and TSNE-3) is shown by the parallel coordinates plot of the 3D t-SNE
visualization of the Iris data in Fig. 6. Each colored line’s value corresponds to a single flower
sample and connects the values of the three components. It is possible to observe particular
grouping patterns. The upper band of the TSNE-2 values is where the red lines (setosa) are
mostly concentrated, creating a fairly uniform and continuous band. This demonstrates Setosa’s
well-known distinctiveness from the other two species. Versicolor, the green lines’ intermediate
band, is scattered across TSNE-1 and TSNE-3 before descending in TSNE-2. The transitional
traits between setosa and virginica are depicted by their placement. The lower regions of TSNE-
1 and TSNE-2 exhibit the dominance of the blue lines (virginica), while TSNE-3 shows the lines
going upward to the higher side. Compared to setosa, this wider distribution is more erratic.
These two species are not as easily separated as setosa because of their biological similarities,
which are highlighted by the junction of the green (versicolor) and blue (virginica) lines.
The setosa red lines, on the other hand, are clearly distinguished, and the dataset clearly shows
that it has a unique characteristic. Generally speaking, the parallel coordinates map provides
a multivariate view of the Iris dataset’s separability: versicolor and virginica are more broadly
distributed, suggesting a stronger link in the feature space, while setosa is a small, discrete
cluster.


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Figure 6

The distribution of the three species (setosa, versicolor, and virginica) in the normalized
components (TSNE-1, TSNE-2, and TSNE-3) of the data is depicted in the parallel coordinates
plot of the normalized 3D t-SNE representation of the iris data in Fig. 7. Each line represents
opposition to a sample, and normalization conditionalizes the values of 0 and 1, allowing direct
comparisons between species samples. Various patterns of structure are created. In a strong
band around 1.0, the red lines (setosa) are concentrated in the upper part of the TSNE-2 values.
This indicates that setosa, a unique feature space of the other species, is stable during separa-
tion. They are unique on the dimensions because the green lines (virginica) are more common
on the lower values of TSNE-1 and rise strongly on TSNE-3, producing an upward line. The
blue (versicolor) lines that separate the setosa and virginica extremes are situated in the middle
of each of the three axes. The range between the mid-range values of TSNE-1 and TSNE-2 is
where the overlap between the blue (versicolor) and green (virginica) curves is most noticeable.
Their biological proximity, where feature features are less clearly divided, is the basis for this.
The red (setosa) lines, on the other hand, are comparatively well separated, demonstrating its
individuality. Because the scales are aligned, normalized parallel coordinate plots are generally
easier to comprehend and make cross-species comparisons more visually appealing. It empha-
sizes that whereas versicolor and virginica have transitional and overlapping features in the
t-SNE components and partially overlap, setosa always constitute a separate cluster.


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Figure 7

The iris data is shown in 3-D t-SNE format in Fig. 8, which converts the high-dimensional
flower data to three dimensions (t-SNE1, t-SNE2, t-SNE3) so that the species may be dis-
tinguished in two dimensions. Each point represents a single sample, which is colored green
(Virginica), blue (Versicolor), and red (Setosa). The species’ numerical range of distribution
across the embedding is shown. Because setosa (red) is concentrated on the positive end, ver-
sicolor (blue) is distributed between -40 and +80, and virginica (green) is concentrated at the
negative end (-60 to 20), the t-SNE1 values vary from +90 to -60. Setosa is more concentrated
on the upper half (0 to +60), virginica is more focused on the middle range (-20 to +40), and
versicolor is more focused on the lower half (-60 to +30) of the t-SNE2 axis, which covers a
region of roughly -70 to +70. Versicolor is concentrated at the lower end (-60 to 0), virginica
is in the middle (-20 to +40), and setosa reemerges at a higher level (10 to +60) along the
t-SNE3 axis, which runs between -70 and +65. Cluster separation is visible but not perfect. A
very compact cluster, t-SNE2 (= +20 - +60) and t-SNE3 (= 10 - +60), which are unique to
its morphological features, are also taken by Setosa. With wide extension over low values of
t-SNE3 (∼ -60 to 0) and negative and positive values of t-SNE1 (∼ -40 to +80), Versicolor is
partially overlapping Virginica. Virginica shares with setosa and versicolor the central areas,
covering t-SNE1 (c. t -60 t +20) and t-SNE3 (c. t -20 t +40) in the middle. On the whole, the
3D t-SNE plot proves that the most separable species is setosa, which makes a distinct cluster
in t-SNE2 and t-SNE3. Conversely, versicolor and virginica also overlap in ranges of t-SNE1
-40 t-SNE1 20 and t-SNE3 -20 t-SNE3 40 as is expected given the biological similarity between
them. The numerical frequencies represent the distribution of the dataset in the embedding
with setosa being separate whereas the other two species have boundaries which are shared.


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Figure 8

In order to demonstrate the separation at the species level, Fig. 9 presents the t-SNE
visualization of the Iris data set in 3D, which comprises the high-dimensional floral features
into three reduced components (t-SNE1, t-SNE2, and t-SNE3). Per point, the species-specific
flowers are green (virginica), blue (versicolor), and red (setosa).
Numerical ranges are used to disperse clusters. Along the t-SNE1 axis, the values fall between
negative and positive -70 and +90. The concentration curves of versicolor (blue) are widely
distributed (between -40 and +80), which overlap with both others, setosa (red) are more
concentrated in the central-to-positive range (0 to +70), and virginica (green) are concentrated
on the negative side (-70 to 0). Setosa scores are higher (0 to +70), virginia scores are in the
center (-20 to +40), and versicolor scores are lower (-60 to +30) along the t-SNE2 spectrum,
which has scores ranging from -70 to +80. The t-SNE3 axis, which spans a range of roughly -70
to +65, provides an additional separation of the species, with virginica between the two (-20
to +40), versicolor on the negative (-60 to 0), and setosa at the top (10 to +65). According to
cluster patterns, setosa produces the clearest cluster with less overlap and higher t-SNE2 and
t-SNE3 bands. On the other hand, versicolor and virginica exhibit significant overlap, especially
in the middle regions of t-SNE1 (-40 +20) and t-SNE3 (-20 +40).Given that they share more
physical traits than setosa, this relationship shows that they are biologically comparable. In
general, 3D t-SNE projection shows a layered picture of separability: virginica and versicolor are
more in contact with each other and form intermediate zones in the embedding space, whereas
setosa appears as a small, distinct cluster. In the reduced dimensions, each species can be found
in distinct subregions that partially overlap, as shown by the numerical ranges.


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Figure 9

The three patient profiles (P1, P2, and P3) are projected into two dimensions in Fig. 10,
which displays the t-SNE view of patient disease correlations. Each point in the plot represents
the overall correlation structure of a patient disease-wise. The abstract t-SNE dimensions,
shown by the horizontal axis (Component 1) and the vertical axis (Component 2), show the
similarity in relative terms; the closer the distances, the more similar the correlation patterns.
The numerical positions are used to highlight the grouping. P1 (blue), which is located at the
far left at around (-460, 140), indicates that this correlation profile differs greatly from others.
P2 (orange), which is located at roughly (190, -160) on the far right, forms a unique cluster
with a low vertical position. Positioned in the upper-right quadrant, around (140, 550), and
significantly above P2 and P1, P3 (green) shows a totally different correlation pattern. The
large spatial dispersion of the three dots indicates that patients’ patterns of correlation are
less comparable to one another. While P2 and P3 are significantly different on Component 2
(about 700 units), P1 and P2 are separated on Component 1 (approximately 650 units). P1 and
P3 have a wide (almost 750 units) diagonal spacing, which emphasizes the fact that the two
patients are not closely packed. Overall, each patient has a unique correlation signature across
diseases, according to the t-SNE projection. This graphic shows pure divergence as opposed
to the previous clustering trends that included overlap or pairings (like P2 and P3). P1, P2,
and P3 are separate, non-overlapping points in the reduced space, showing highly differentiated
disease-correlation patterns.


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Figure 10

The three patient profiles (P1, P2, and P3) are projected into a two-dimensional space by
the first two principal components, which indicate the most diversity in their correlation struc-
tures, in the PCA plot of the correlation measurements in Fig. 11. Every point represents a
single patient’s total disease-correlation pattern in reduced dimensions. Defined differentiation
is indicated by numerical position. With a value of around (2.3, 0.55), P1 is located in the
upper-right quadrant, suggesting that Component 2 has a mediocrely positive connotation and
Principal Component 1 has a strong positive implication.
With Component 1 showing a strong negative indication and Component 2 showing the max-
imum positive signal, P2 is located in the first quadrant, around (-2.0, 0.8). P3 has a mildly
negative bias for Component 1 and a strong negative bias for Component 2, and it is situ-
ated in the lower-center region of roughly (-0.3, -1.35). The distances between points show
the structural differences. The fact that P1 and P2 are big in Component 1 (with an average
separation of roughly 4.3 units) suggests that their correlation structure is opposing in this
key direction. While P1 and P3 are considerably differed on both axes (about 2.6 units along
PC1 and 1.9 units along PC2), P2 and P3 are different based on Component 2 (a difference of
about 2.1 units). The PCA projection generally shows that the three profiles are quite distinct,
with P1 having strong positive PC1 values (because to tuberculosis), P2 having strong positive
PC2 values (due to typhoid), and P3 having negative PC2 values (due to neither). This split
suggests that there is no meaningful grouping of the patients in the condensed space and that
each patient’s correlation pattern is unique.


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Figure 11

Plotting the three patient profiles (P1, P2, and P3) in three principal components, which
show the greatest variation in the disease-correlation structure of the three patients, is how the
3D PCA plot of correlation measurements is shown in Fig. 12. With an emphasis on decreased
spatial separability, all three axes (Principal Components 1, 2, and 3) are weighted sums of the
original variables. Numerical places exhibit distinct clustering. At around (2.2, 0.6, -9.0), P1

is on the right extreme of Principal Component 1, meaning that PC 1 has a positive impact
of about 1 and PC 2 has a modest impact. There is a significant negative PC1 and a positive
PC2, as shown by P2, which is at the far left at roughly (-2.0, 0.5, -8.6). It is marginally higher
on PC3 than the others. P3’s primary feature is the negative PC 2 value and its higher location
on PC 3, which places it even closer to the origin on PC1 and PC2 at roughly (-0.3, -1.2, -9.3).
There are notable structural variations in the dissemination. With a difference of almost 4.2
units, the P1 and P2 are the most different on PC1 and tend to show opposite correlations.
Additionally, P1 and P3 are separated along PC1 (about 2.5 units) and PC2 (about 1.8 units),
while P2 and P3 are sharp along PC2 (about 1.7 units). The three lie close to one other on
PC3 (around -9) which suggests that this component is less important in their separation than
PC1 and PC2. The 3D PCA often confirms that each of the three profiles is unique. P2 is
a high PC2 loading with negative PC1 (also typical of typhoid-related), P3 is a negative PC2
loading (different in both situations), and P1 is a strong positive PC1 loading (also typical of its
TB-dominated connection). The fact that the first two variables determine the main separation
and that P1, P2, and P3 have different signatures of association with disease is highlighted by
the modest format of the distribution along PC3.


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Figure 12

The three patient profiles (P1, P2, and P3) are projected into three non-linear components,
similarity patterns in their correlation patterns, in Fig. 13, which displays a 3D t-SNE depiction
of patient disease correlations. Individual patients are represented by the points, and the points’
positions show how similar or different the correlation patterns between the patients and the
diseases are. Numbers are used to highlight separation. P1 (blue) indicates a very strong
divergent correlation profile in that direction because it is on the extreme negative side of
Component 1, about between -1050 and -250 and 8.4.Since P2 (orange) is at the other extreme,
somewhere between 850 and 900 and 8.6, it is fundamentally distinct from P1. P3 (green),
which is identified by its Component 2 value, which is the most negative, is situated in a central
but isolated location close to (300, -1200, 8.5). Each patient profile is unique, as shown by the
axes’ dispersion. The correlation patterns of P1 and P2 are the least similar, as evidenced by
the biggest distance between them-nearly 1900 units. Another line of separation is highlighted
on Component 2, where P2 and P3 are separated by an average of 2100 units. Additionally,
Component 3 values are maintained very close to all three (≈ 8.4 -8.6), indicating that the
third axis is not experiencing separation. The 3D t-SNE plot generally shows a large number of
unique patient signatures that do not overlap on one another. P3 is distinguished by its strong
negative position on Component 2, P2 dominates in the positive of Components 1 and 2, and
P1 is kept on the negative of Component 1. This validates earlier PCA and 2D t-SNE results of
individual profiles and demonstrates that each patient’s disease-correlation structure is distinct.


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Figure 13

The three patient profiles (P1, P2, and P3) are mapped into reduced space by the 3D t-
SNE representation of normalized disease correlations in Fig. 14, where normalization ensures
that values from various patients are comparable. Each point represents the general correlation
structure of a single patient, while the plot’s separation represents the variation in each patient’s
normalized structure. The use of numbers highlights distinct and well-expressed grouping.
P1 (blue), which is located at roughly 1050, -1800, and 1.02 on the far right, shows a significant
divergence on Components 1 and 2. On the left, P2 (orange) has a value of roughly (-600,
700, 1.11), and P3 (green) is close to (200, 150, 1.08). Component 3’s vertical axis is incredibly
narrow, measuring 1.02 to 1.11, suggesting that Components 1 and 2 account for the majority of
the significant variance. The geographical distances highlight relative similarity. With a distance
of roughly 800 units on Component 1 and 550 on Component 2, P2 and P3 are closer to one
another, suggesting that their normalized correlation structure is rather comparable. However,
P1 is much farther away: P1 and P2 are separated by over 1600 units while on Component 2 and
about 1950 when on Component 2. Because of this, P1 is the most distinctive profile, whereas P2

and P3 are a fairly close pair. Overall, the 3D data’s normalized t-SNE depiction supports earlier
findings: While P2 and P3 have more comparable, though not identical, structures, P1 has a
unique disease-correlation signature. In order to concentrate interpreterability on Components
1 and 2 as the primary axes of separation, normalization removes variation in Component 3.


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Figure 14

6. Analysis of Patient Disease Correlation and Clustering Results

The effectiveness of a QPNRS correlation metric in medical diagnostics is demonstrated by
this discussion. According to Table III Analysis, the approach can diagnose P1 with tuberculosis
and P2 and P3 with typhoid. Additionally, P1 is always isolated whereas P2 and P3 are always
clustered by the clustering techniques (K-Means and K-Means++), confirming the commonal-
ities in patient profiles. Dimensionality reduction (PCA and t-SNE) is used to visualize these
particular groupings, while 3D projections provide nuanced information. The use of correlation
measurements in patient diagnosis and categorization is firmly based on this multifaceted study.

6.1. Aspect-based Analysis
The diagnostic strength of the correlation measures and the underlying patient profile group-

ings are the two main sections of the analysis that can be distinguished.
Diagnostic Accuracy: As Table 3 demonstrates, the primary outcome should be the successful
application of the QPNRS correlation measure for diagnosis. The disease with the highest cor-
relation rate for each patient is selected to provide an unambiguous diagnosis: TB in P1 (0.901),
typhoid in P2 and P3 (0.878 and 0.869, respectively). This demonstrates how the technique can
be applied to de-identify (de-ambiguate) complex symptom data into a particular, measurable
diagnostic tool. Profile Similarity and Distinction: Internal structures are indicated in part
by the correlation table itself. Typhoid has the highest values, but the correlation patterns of
P2 and P3 are quite similar across all disease cases. In contrast to the other two profiles, P1


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 23 of 26

is particularly connected with tuberculosis, which makes it stand out. The further clustering
techniques validate the initial notion, which is this observation in terms of aspect.

6.2. Method-based Analysis
To verify the diagnostic findings and familiarize themselves with the data structure, a variety

of unsupervised learner types were employed. Figures 1-3 show K-Means Clustering and K-
Means++ Clustering: Two strong clusters, such as one of P2, P3, and another of P1, were
consistently found in both raw and adjusted data by both non-optimized and optimized K-
Means algorithms. This provides excellent unsupervised evidence, showing that P2 and P3 are
substantially similar to each other and fundamentally different from P1. The strength of this
classification is demonstrated by the consistency of the algorithms and the data pre-processing.
PCA (Figs. 11, 12): t-SNE (Figs. 10, 13, 14): It is clear from the way the patient profiles
occupy distinct and diverse positions in the component space that the PCA provided a linear
projection to maximize variance. It was discovered that PC1 was significantly impacted by the
TB-Typhoid contrast (between P1 and P2), whereas PC2 was impacted by another component
of the correlation structure (between P3 and P2). A non-linear projection that placed emphasis
on local similarities was provided by t-SNE. It showed that each of the three patients was
completely different in 2D (Fig. 10). However, the 3D normalized t-SNE (Fig. 14) gave a more
detailed picture, showing that P2 and P3 were more closely associated to one another than to
P1, which validated the clusters’ findings.

6.3. Dimensional-based Analysis
The inquiry has used a range of perspectives in order to completely understand the results.

Table 3-2D Tabular Data: The patient vs. disease table in two dimensions was the subject of
the initial analysis. Although this is simple (and directly diagnosable), it is less successful in
illustrating the global pattern of patient similarity. Figures 1, 2, 10, and 11 show 2D projections:
For the data to be shown, it had to be reduced to two dimensions. High-dimensional correlations
are easily explained by intuition, and it was possible to show the main separation (P1/P2/P3

in clustering; the three are discrete in PCA/t-SNE). Figure 12, Figure 13, Figure 14: 3D
projections. The inclusion of a third dimension added greater detail. It was clear from 3D PCA
(Fig. 12) that the third component (PC3) did not significantly increase patient separation,
confirming that the first two components provide the primary discriminatory information. The
richest to examine were the 3D t-SNE plots (Figs. 13, 14), and it was intriguing to see how
normalization (Fig. 14) might highlight the relationship between P2 and P3 and how, in the
absence of the outlier P1, they obviously constitute a cohesive subgroup of their own.

7. Conclusion

In summary, our multi-dimensional analysis has successfully demonstrated that the pro-
posed QPNRS correlation measure is a useful tool for medical diagnosis since it may transform
ambiguous symptoms into specific diagnostic findings. Furthermore, K-Means clustering has
shown reliable and consistent findings in a variety of settings, providing strong support for un-
supervised patient stratification. In addition to confirming the distinct nature of the patient P1

diagnostic profile, the 2D and 3D visualizations created by integrating PCA and t-SNE offered a
more comprehensive and multifaceted understanding of the relative similarities and differences
among the three patient profiles. A logical and validated analytical pathway is produced by the
entire analysis process, which includes clustering, correlation calculation, and visualization.
Despite the positive results, this study had the following limitations: Small sample size: Three
patient profiles (P1, P2, and P3) are used in the analysis. Producing statistical significance or
drawing broad inferences about the method’s performance at the population level is insufficient.


A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7168 24 of 26

Limited Disease scope: Only four diseases were used in the testing of the diagnostic model. It
has not yet been proven to be accurate or applicable in a clinical practice setting when dozens
or even hundreds of conditions need to be taken into account. Dependency on Correlation
Thresholds: The diagnostic rule would just employ the highest correlation value and be ba-
sic in nature. It ignores situations when the two largest correlations are quite similar, which
could indicate comorbidity or a rare case. Interpretability of Reduced Dimensions: Principal
components and t-SNE dimensions are abstract and cannot be readily converted into specific,
clinically actionable symptoms or traits, even when PCA and t-SNE plots demonstrate clear
separations.
The following approaches are recommended for use in subsequent study in order to get around
the constraints and advance this research: Validation on an Extensive Data Set: Testing this
method on a sizable real-world data set of patient records is the next crucial step in ensuring
its external validity, strength, and diagnostic quality. Illness Taxonomy: To assess the model’s
scalability and discriminatory power in challenging diagnostic situations, it must be compared
to a more comprehensive and realistic illness collection. Enhancement of Advanced Diagnostic
Rules: To improve the diagnostic outcome, future research should investigate more complex
decision rules, such as those that apply machine learning classifiers to the correlation data,
incorporate a confidence interval, or take into account the shape of the correlation vector. Clin-
ical Integration and Interpretation: Research on how to improve the model’s interpretability for
clinicians should be conducted. This could involve attempting to map high component loadings
onto the initial symptoms or developing a hybrid system that combines expert knowledge with
this quantitative approach. Other Methods Investigation: To see if they provide any new infor-
mation about patient subgroups, different clustering techniques (like DBSCAN and Hierarchical
Clustering) and dimensionality reduction techniques (like UMAP) may be investigated. For fu-
ture work, neutrosophic soft sets could model the indeterminacy in non-Hermitian topological
phase transitions and the robustness of Majorana modes [38].

Acknowledgements

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for fund-
ing this research work through the project number (PSAU/2025/01/34111).
Authors’ Contributions: All authors read and approved the final manuscript.
Funding: Prince Sattam bin Abdulaziz University through the project number (PSAU/2025/01/34111).
Conflicts of Interest: The authors declare that they have no conflicts of interest to report
regarding the present study.

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