EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6484 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mapping Love: A Heptapartitioned Neutrosophic Machine Learning Study of University Students’ Romantic Sensations Raed Hatamleh1, Nasir Odat1, Abdallah Alhusban2, Arif Mehmood3,∗, Alaa M. Abd El-latif4, Husham M. Attaalfadeel4, Walid Abdelfattah4, Ahmad. M. Abdel-Mageed5 1 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan 2 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan 3 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan 4 Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia 5 Department of Biological Sciences, College of Science Northern Border university, Arar 91431, Saudi Arabia Abstract. This paper introduces the novel concept of single valued heptapartitioned neutrosophic sets (SVHNSs) which is the generalized version of the neutrosophic sets. This set consists of seven mem- bership functions which are more sensitive to real-world problems. Membership functions are defined as an absolute true, relative true, absolute false, relative false, contradiction, unknown (undefined) and ignorance respectively. This scenario of indeterminacy provides a better accuracy. Moreover, several properties of this set are also addressed. This study focuses on the romantic sensations experienced by young boys and girls in a variety of contexts. The data set supporting this study comprises individuals aged 18-25, with data collected from the Psychology Department at Peshawar University, Pakistan. This data was critically analyzed using the Single-Valued Heptapartitioned Neutrosophic Set (SVHNS). For a real-world application involving the romantic feelings of young individuals across various dimensions, machine learning and graphical algorithms such as Encrypted K-Means Clustering, Encrypted K-Means Clustering Heat Map, Encrypted Elbow Method, Decrypted K-Means Clustering, Encrypted Correla- tion Matrix, and Decrypted Correlation Matrix were applied and visualized. These algorithms assist in examining and developing relationships among various factors that influence the romantic feelings of young men and women. The proposed techniques offer new dimensions not only for psychological studies in general but also specifically for understanding emotional disorders and breakups in romantic relationships among university students. 2020 Mathematics Subject Classifications: 68T05, 03E72, 62H30, 91F20 Key Words and Phrases: Neutrosophic Set, Single Valued Heptapartitioned Neutrosophic Set (SVHNS), Distance Measures, K-Means Algorithm, Machine Learning Techniques, Applications of SVHNS ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6484 Email addresses: raed@jadara.edu.jo (R. Hatamleh), nodat@jadara.edu.jo (N. Odat), dralhosban@inu.edu.jo (A. Alhusban), mehdaniyal@gmail.com (A. Mehmood), alaa.ali@nbu.edu.sa (A. M. Abd El-latif), husham.alhassan@nbu.edu.sa (H. M. Attaalfadeel), walid.abdelfattah@nbu.edu.sa (W. Abdelfattah), ahmad.mohammed@nbu.edu.sa (A. M. Abdel-Mageed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 2 of 23 1. Introduction Fuzzy set theory (FST) was for the first time established by computer scientist Zadeh in the year of 1965 in [1]. After that, an ample of theoretical achievement in a related topic was explored. Sine FST has flexible boundaries so it can be installed in a number of real- world problems that has got uncertainty. When the domain of study is wide then in that case FST can be applied but very carefully because chances of errors swelled up. To control such pieces of errors interval valued FST was initiated in [2]. The extension of FST was made in [3] by introducing a novel concept of intuitionistic fuzzy set theory (IFST). The beauty of IFST lies in fact that it governs the incomplete information by its truth and false candidature-ship values. This theory has one serious drawback which is that it cannot afford the existence of indeterminate data. This gap was unbridged for a piece of time but finally a strong researcher Smarandache [4] filled in this gap by introducing extremely a new theory which is known as neutrosophic set theory (NST).One effective mathematical paradigm for handling partial, ambiguous, uncertain, or confusing data is (NST). The generalization of IFST was made to NST in [5] and the authors developed number of examples for better understanding the situation. Single valued NST was addressed in [6] and in continuation, complex NST was studied in [7]. Soft set theory (SST) was initiated for the first time by polished researcher Molodtsov in [8]. This theory is actually generalization of FST. This is used to reduce the error that exists in the mathematical problems that can be handled by FST and IFST. This theory uses the concept of decision variables and these variables can be connected with the power set of the CST. The key point of this theory is that it does not demand grade of membership as in FST while studying the data in between the lines. Since the inception of this theory up to date, number of attempts is made. The combination of FST with SST was made in [9] and a new theory was given birth and is known as fuzzy soft set theory (FSST). The authors presented examples for clear understanding the notion. The study was stretched and the notion of vague soft set theory (VSST) in [10] was explored with the explanation of examples. For clear understanding the notion of IVVSST was installed in [11-13]. The notion of SEST was studied in [14]. The idea of SMST was reflected in [15]. It was noted and studied that FSST was lacking of false and impermanency possible state of problems and so this gap was bridged in [16] and new theory sprung off with the name NSST. The authors defined the basic operations, operators and established the basic results. Examples were also generated for almost all results for clear understanding. Since the inception to date NSST has got big attention of the researchers because it is used in every walk of life keeping in view this the authors made further study on NSST in [17-24]. SST was engaged with other theories in [25,26]. All the basic operations and operators are also defined and addressed on the behest of these said theories. The n-valued refined NSST was studied in [27] and complex NSEST was studied in [28.29]. The time-NSS was installed in [30]. The authors provided examples for explanation of this new idea. Much more study was made in [31-36] on NSST and its engagements with number of operators namely aggregation and dombi weighted etc. These operators are effective techniques to handle NSST. The Q- FSST and multi Q-FSST were addressed in [37,38]. The notion of Q-IFSST and its basic were given in [39]. Finally, the notion of Q-NSST was studied in [40]. The similarity measures etc. were discussed in [41-46]. 1.1. Literature review The more generalization and refinement of NST was addressed in [47] while introducing extremely a new theory which is known as quadripartitioned NST. On the basis of this con- cept, basic operations were introduced and better examples were installed to understand it. The authors, actually divided the indeterminacy that happened in [4] into two parts which are named as contradiction and ignorance. The authors also proposed the definitions of distance, similarity measure (SM) and entropy. Finally, applications of the proposed situation are used R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 3 of 23 in problems concerning pattern recognition etc. quadripartitioned NST was further generliased excellently into pentapartitioned NST, in this case the indeterminacy was stretched in [48] into three possibilities i.e. contradiction, unknown and ignorance. Then on the basis of this novel idea, basic operations and related results were studied and examples were established for almost all results. In continuation, further it was stretched in to heptapartitioned NST in [49]. The authors also discussed the basic results and properties. Since polarity makes big contribution in the field of science and since the inception of refined NST no touch had been given so in this direction the first attempt was made in [50] and introduced the notion of QSVBNST. The au- thors addressed some practical problems that are used in day to day life while using the notion of distance approach. The study of imaginative play in children was beautifully addressed in [51] and since imaginative play involves a lot of imaginary elements, the research of imaginative pretend play in children aged 1 to 10 years was chosen for SVRNS analysis. When compared to other neutrosophic sets, SVRNS will be a better fit for describing these facts. For a practical use involving child psychology, machine learning algorithms like K-means, parallel axes coor- dinate, etc., were implemented and visualized. The suggested algorithms assist in analyzing a child’s mental capacity based on creative play. These algorithms facilitate the establishment of a relationship between a number of factors influencing imaginative play and a child’s cognitive capacities, allowing for the derivation of logical conclusions. There is also a quick comparison of the various algorithms that were employed. These two distinct indeterminate values char- acterize the Double-Valued Neutrosophic Set (DVNS), a modified version of a neutrosophic set introduced by the authors [52]. This excellent work defines and illustrates its associated properties and axioms. In a number of scientific domains, including data mining, machine learning, and pattern recognition, clustering is crucial. FSs and IFSs cannot handle ambiguous and inconsistent data as accurately as DVNS can. This is in contrast to SVNS. A clustering approach is built upon the definition of a generalized distance measure between DVNSs and the associated distance matrix. In order to cluster the data represented by DVNI, this article suggests using the double-valued neutrosophic minimum spanning tree (DVN-MST) clustering technique. The uses and efficiency of this clustering technique are illustrated with illustrative cases. The DVN-MST clustering algorithm is compared to various clustering algorithms, such as fuzzy minimum spanning tree, intuitionistic fuzzy minimum spanning tree, and single valued neutrosophic minimum spanning Tree. In a multicriteria decision-making situation where the criteria values for alternatives are taken into account in a DRINS environment, this suggested cross entropy is employed [53]. Likewise, a cross entropy based on indeterminacy and employing DRINS is suggested. The DRINW cross entropy and the indeterminacy based cross entropy between the ideal alternative and an alternative are derived in order to rank the alternatives corresponding to the cross entropy values. During the process of making decisions, the choice or options that are most desired are selected. A practical example is given to show how the suggested approach can be used. A quick comparison between the suggested approach and the current approaches is done. This paper introduces the triple refined indeterminate neutrosophic set (TRINS), a case of the refined neutrosophic set [54]. It offers the added ability to sensitively and accurately depict the ambiguous, imprecise, inconsistent, and partial information that ex- ists in the real world. Further study can be seen in [55-57]. New research on fuzzy set and soft set theory has also been made by new models of decision-making and new algebra structures based on interval-valued and neutrosophic sets. The trigonometric ℘-rung and neutrosophic interval-valued approaches were used by Hatamleh et al. to propose weighted and geometric operators and, further on, to propose the concept of interaction operators to Pythagorean neu- trosophic sets [59, 61, 62]. In Mahase and Shihadeh [60], the authors studied (α, β)-intuitionistic fuzzy ideals of ordered ternary semigroups. AI-assisted fuzzy soft relations were also used in health monitoring through wearable devices [63]. El-Sheikh and Abd El-Latif [64] introduced fundamental research on soft topological structures, and these were further developed in re- search on supra compactness and separation axioms [65, 66]. The authors addressed number R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 4 of 23 of examples for better understanding the results and their applications. Neutrosophic cognitive maps (NCMs), a model based on neutrosophic logic, are used in this study to examine young children’s imaginative play [58]. In order to create connections between the various ideas asso- ciated with children’s imaginative play in the age range of 1 to 10 years old, who come from socially, economically, and educationally disadvantaged backgrounds, NCMs are built with the assistance of expert advice. The NCMs play a crucial role in removing the obstacle caused by the intricate and frequently imprecise nature of social or psychological data. Expert interpre- tations and video recordings of kids playing were used to gather data. Experts said that 15 characteristics / concepts associated with kids using the same toy were seen, and while some of the relationships between the concepts were unclear, it was permissible to apply NCMs. These NCMs were constructed based on the opinions of five experts, and one of their concealed pat- terns turned out to be a fixed point. This section reviews some important concepts pertaining to the theory of SVNSs. 2. Preliminaries In this section, some basic definitions are reviewed pertaining to the theory of single valued neutrosophic set (SVNS) which are necessary for the upcoming sections. Definition 1. [59] A neutrosophic set (NS) A on the universe of discourse X is categorized by truth membership function MF- TA(x), an indeterminacy MF- IA(x), and a false MF-FA(x). The functions TA(x), IA(x) and FA(x) are real standard or non-standard subsets of ]−0,+1[; that is TA(x) : X −→]−0,+1[, IA(x) : X −→]−0,+1[ and FA(x) : X −→]−0,+1[ with the condition −0 ≤ supTA(x) + supIA(x) + supFA(x) ≤ 3+. It was challenging to use this definition of a neutrosophic set in practical scientific and technical domains. As a result, [6] established the idea of a single valued neutrosophic set. Definition 2. [6] A single valued neutrosophic set (SVNS) A on the universe of discourse X is characterize by truth MF- TA(x), an indeterminacy MF- IA(x), and a false MF-FA(x). The functions TA(x), IA(x) and FA(x) are subsets of ]0, 1[; that is TA(x) : X −→]0, 1[, IA(x) : X −→ ]0, 1[ and FA(x) : X −→]0, 1[ with the condition 0 ≤ supTA(x) + supIA(x) + supFA(x) ≤ 3. So SVNS can be written as A = {⟨x, TA(x), IA(x), FA(x)⟩ : x ∈ X} Definition 3. [6] A SVNS A is contained in the other SVNS B, if TA(x) ≤ TA(x), IA(x) ≥ IA(x), FA(x) ≥ FA(x) for each x ∈ X, B ⇔ A ⊆ B&B ⊆ A.Ac = {⟨x, TA(x), 1−IA(x), FA(x)⟩ : x ∈ X} Definition 4. [6] Let A and B be two SVNSs on the universe of discourse X then (i) . A ∪B = {x, ⟨max(TA, TB),min(IA, IB),max(FA, FB)⟩;x ∈ X} (ii) . A ∩B = {x, ⟨min(TA, TB),max(IA, IB),min(FA, FB)⟩;x ∈ X} 3. Single Valued Heptapartitioned Neutrosophic Sets Neutrosophic set theory (NST) is one of the most excellent and interesting theory which is used fully in pure and applied mathematics as well. This theory considers three possible membership values and these are true, false and indeterminacy membership values respectively. The two values are clear crystal. The third one is extremely interesting because it deals with uncertainty that happened everywhere in our daily life. The indeterminacy has got infinite number of refinement. The accuracy can be improved and uncertainty can be possibly reduced into certainty and the error can be griped that may happened in the calculation due to purely R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 5 of 23 using the indeterminacy value as it as. The research can be made more realistic and sensible if the refinement of the indeterminacy is done. In the real applications the indeterminacy can be divided into seven possible values as absolute true, relative true, absolute false, relative false, contradiction, unknown (undefined) and ignorance. This scenario of indeterminacy provides a better accuracy. Definition 5. A single valued heptapartitioned neutrosophic set (SVHNS) A on the universe of discourse X is characterize by absolute true MF- TA(x), relative true MF- RTA(x),an un- known UA(x), a contradiction CA(x), an ignorance GA(x), an absolute false MF-FA(x) and relative false RFA(x). The functions TA(x),RTA(x), UA(x), CA(x), GA(x), FA(x) and RFA(x) are subsets of ]0, 1[; that is TA(x) : X −→]0, 1[, RTA(x) : X −→]0, 1[, UA(x) : X −→]0, 1[, CA(x) : X −→]0, 1[, GA(x) : X −→]0, 1[, FA(x) : X −→]0, 1[ and RFA(x) : X −→]0, 1[ with the condition 0 ≤ supTA(x) + supRTA(x) + supUA(x) + supCA(x) + supGA(x) + supFA(x) + supRFA(x) ≤ 7. So SVHNS can be written as A = {⟨x, TA(x), RTA(x), UA(x), CA(x), GA(x), FA(x), RFA(x)⟩ : x ∈ X} Definition 6. A SVHNS A is contained in the other SVHNS B, if TA(x) ≤ TA(x), RTA(x) ≤ RTA(x), UA(x) ≤ UA(x), CA(x) ≥ CA(x), GA(x) ≥ GA(x), FA(x) ≥ FA(x), RFA(x) ≥ RFA(x) for each x ∈ X, A = B ⇔ A ⊆ B&B ⊆ A. Ac = {⟨x, FA(x), RFA(x), GA(x), 1− UA(x), CA(x), TA(x), RTA(x)⟩ : x ∈ X} Definition 7. Let A and B be two SVHNSs on the universe of discourse X then (i) . A ∪B = {x, [ max(TA, TB),max(RTA, RTB),min(CA, CB),min(UA, UB), max(GA, GB),min(FA, FB),min(RFA, RFB) ];x ∈ X} (ii) . A ∩B = {x, [ min(TA, TB),min(RTA, RTB),max(CA, CB),max(UA, UB), min(GA, GB),max(FA, FB),max(RFA, RFB) ];x ∈ X} 4. Characterization of Single Valued Heptapartitioned Neutrosophic Sets in terms of Distance Measures In this section, the concept of general distance is defined between two SVHNSs and the different types of distance measures are discussed and these are weighted Hamming distance and weighted Euclidean distance. In addition to this, algorithms are designed and the flowcharts are also picture out for clear understanding the situation. Definition 8. Let us consider two SVHNSs A and B on the universe of discourse on X = x1, x2, ....xn which are represented by A = {⟨xi, TA(xi), RTA(xi), CA(xi), UA(xi), GA(xi), RFA(xi), FA(xi)⟩ : xi ∈ X} and B = {⟨xi, TB(xi), RTB(xi), CB(xi), UB(xi), GB(xi), RFB(xi), FB(xi)⟩ : xi ∈ X} such that TA(xi), RTA(xi), CA(xi), UA(xi), GA(xi), RFA(xi), FA(xi) ∈ [0, 1] and TB(xi), RTB(xi), CB(xi), UB(xi), GB(xi), RFB(xi), FB(xi) ∈ [0, 1] for every xi ∈ X. Let wi(i = 1, 2, ....n) is weight of elements ξi(i = 1, 2, ....n), wi ≥ 0(i = 1, 2, ....n) and ∑n i=1wi = 1. Then, the generalized heptapartitioned neu- trosophic soft set weighted distance is defined as dλ(A,B) = [17 ∑n i=1wi{|TA(xi)−TB(xi)|λ+|RTCA(xi)−RTCB(xi)|λ+|CA(xi)−CB(xi)|λ+ |UA(xi)− UB(xi)|λ + |GA(xi)−GB(xi)|λ + |RFA(xi)−RFB(xi)|λ + |FA(xi)− FB(xi)|λ}] 1 λ with, λ > 0. The above equation reduces to the SVHNS weighted Hamming distance and the SVHNS weighted Euclidean distance, by replacing λ = 1, 2 respectively. The SVHNS weighted Hamming R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 6 of 23 distance is given as: dλ(A,B) = [17 ∑n i=1wi{|TA(xi)− TB(xi)|+ |RTCA(xi)−RTCB(xi)|+ |CA(xi)− CB(xi)|+ |UA(xi)− UB(xi)|+ |GA(xi)−GB(xi)|+ |RFA(xi)−RFB(xi)|+ |FA(xi)− FB(xi)|}] where, λ = 1. The following is the SVHNS weighted Euclidean distance: dλ(A,B) = [17 ∑n i=1wi{|TA(xi)−TB(xi)|2+ |RTCA(xi)−RTCB(xi)|2+ |CA(xi)−CB(xi)|2+ |UA(xi)− UB(xi)|2 + |GA(xi)−GB(xi)|2 + |RFA(xi)−RFB(xi)|2 + |FA(xi)− FB(xi)|2}] 1 2 where, λ = 2. The algorithm to obtain the generalized SVHNS weighted distance dλ(A,B) between two SVHNS A and B is given in Algorithm 1. Generalized SVHNS Algorithm: |1.P rocedureGeneralizedSV HNSDistance(a, b, weights, λ)| |2.distance −→ 0| |3.For(i ∈ ⟨a⟩)| |4.distance −→ distance+ weights[i] ∗ (| |5.(a[i].truth− b[i].truth) + | |6.(a[i].truetendingcomplex− b[i].truetendingcomplex) + | |7....+ ||8.(a[i].indeterminacy − Confusion− b[i].indeterminacy − Confusion)| |9.)λ| |10.Returndistance/7| The related flowchart is given in Figure 1. Figure 1: Flow Chart for SVHNS 5. Characterization of K-Means algorithm in Terms of Single Valued Heptapartitioned Neutrosophic Sets Algorithm 2. K-mean algorithm using SVHNS data set is given as 1. Initialize SVHNS instances a and b with respective truth values, complex memberships, etc. 2. Create a 2D array of SVHNS values with the attributes of a and b. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 7 of 23 3. Initialize K (number of clusters) and max iterations. 4. Randomly initialize centroids using SVHNS values. 5. Initialize empty arrays beta and nj for centroid updates. 6. Create an empty array prev clusters to track previous clusters. 7. Loop for max iterations: a. Create empty clusters for each iteration. b. Assign data points to the closest clusters. c. Update beta and nj values for centroid calculations. d. Update centroids based on beta and nj. e. Check for convergence by comparing current clusters with previous clusters. f. If convergence is reached, exit the loop. g. Update previous clusters. 8. Print the final clusters and centroids. Example Usage: a. Create SVHNS instances a and b. b. Define SVHNS values using the attributes of a and b. c. Create an instance of the SVHNS class with initialized values. d. Run k-means SVHNS with SVHNS values, K = 2, and print the final clusters and centroids. The related flow chart is given in Figure 2 Figure 2: Flow Chart for K-Mean Clustering of SVHNS Data Set After gathering and processing the data, we employed the following machine learning ap- proaches in this research. 6. Characterization of Machine Learning Techniques in Terms of Single Valued Heptapartitioned Neutrosophic Sets Encrypted K-Mean Clustering method is a novel method for K-mean clustering on a data which is encrypted and this method has the ability to preserve the confidential data of sensitive information. Heat map is supposed to be very informative structure and as we know that heat R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 8 of 23 maps helps us in understanding the correlation in data (the graphical illustration of data where values are represented by colors) and in particular here encrypted correlation heat map has been reflected for the given data in future section. A method for determining the proper value of K (number of clusters) in K-means clustering is the elbow method. The Elbow method is a strategy for determining the appropriate value of K (number of clusters). It creates consistency in the cluster analysis design. The Elbow Method is used to determine the optimal number of clusters within the dataset. This helps reveal natural groupings of individuals with similar data patterns, contributing to a better understanding of the diversity in experiences represented in the data. Implementing the Elbow Method on the encrypted SVHNS dataset enhances the robustness and reliability of our analysis. Decrypted K-Mean clustering is very useful in the field of security measures. When K-Means clustering is applied to decrypted datasets, the data becomes accessible not only to the intended recipient but also potentially to others along the transmission path, raising security concerns. Encrypted and decrypted correlation matrices are used to analyze inter-component relationships while maintaining data security, with the encrypted version ensuring a secure environment during analysis. 7. Data set Visualization and Structural Characterization Researchers conducted sessions with young adults of varying backgrounds to explore ro- mantic emotions in young adults. During the sessions, held in educational institutions and community settings, participants explored romantic feelings and experiences. A professional psychologist facilitated each session, fostering a comfortable environment for open communi- cation. In the course of collecting data, participants discussed their favorite romantic themes, relationships, and everyday experiences. Participants were treated with small gestures like sharing their favorite sharing-gifts snacks or chocolates to create a relaxed atmosphere. Subse- quently, participants were prompted to engage in simulated romantic conversations, simulating phone calls with their imagination. A total of 10 sessions were conducted in educational in- stitutions, and 2 in community settings, ensuring a varied dataset. To enhance diversity, an additional 7 videos were sourced from online platforms, showcasing young adults engaging in similar romantic scenarios. The recorded descriptions from these sessions and videos were cru- cial in assigning values to seven membership functions, forming the basis for constructing the romantic sentiment analysis system (RSAS). Table 1 outlines the parameters employed in analyzing romantic feelings, with the first 11 parameters drawn from existing literature and an additional 5 parameters introduced by the expert to capture the nuances of romantic emotions among young adults. Table 1: Parameter Description for Romantic Feelings S.No Parameter Name Description 1. Shared Dreams (SD) The shared dreams and aspirations within the context of romantic feelings, whether based on real or imaginative situations or settings. 2. Expressive Dance (ED) Movements expressing romantic feelings, indi- cating cognitive patterns and emotional involve- ment. 3. Non-verbal Communi- cation (NVC) Non-verbal expressions using body parts (hands, head) to convey romantic ideas or emotions. 4. Emotional Expression (EE) Movement of facial muscles for non-verbal com- munication, reflecting romantic emotions. Continued from previous page R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 9 of 23 Table 1 – continued from previous page S.No Parameter Name Description 5. Quality Time Together (QTT) Duration and nature of social interactions dur- ing romantic moments, influence the depth of emotional connection. 6. Play Materials Used(PMU) Objects or symbols representing romantic ele- ments in the context of romantic feelings. 7. Collaborative Creativ- ity (CC) Partner’s approach to using provided elements, offering insights into shared imaginative experi- ences in romantic feelings. 8. Expressing Feelings (EF) Vocal or non-vocal expression of romantic feel- ings and emotions by both partners. 9. Emotional Tone (ET) Tone reflecting the mood and state of mind of both partners during romantic interactions. 10. Mutual Roles (MR) The roles both partners assume and assign to each other within the context of romantic feel- ings. 11. Shared Enthusiasm (SE) Extent of the partners’ involvement and shared excitement during romantic activities. 12. Emotional Gaze (EG) Movement of the eyes expressing emotions and connection during romantic moments. 13. Associative Thinking (AT) Mental process by which both partners form as- sociations and create shared romantic experi- ences. 14. Grammaticality Cor- rect Expressions (GCE) Ability to construct grammatically correct sen- tences expressing romantic feelings with proper structure and syntax. 15. Connected Conversa- tions (CC) Whether sentences formed during romantic in- teractions are related to each other, enhances the depth of connection. 16. Gift Sharing (GS) The act of sharing symbolic or imaginary gifts within the context of romantic feelings. 8. Method of Romantic Feeling Assessment The expert views on the evaluation parameters for romantic feelings are provided below: 1. Shared Dreams (SD): An imaginative theme based on real situations increases the truth membership function. Degrees of complexity and indeterminacy are considered within the [0, 1] range. 2. Expressive Dance (ED): Truth membership increases with physical movements expressing romantic feelings. Complex and indeterminate values from [0, 1] if movements are challenging to interpret. 3. Non-verbal Communication (NVC): Truth membership increases with non-verbal expres- sions. Body parts conveying romantic ideas contribute. Complex and indeterminate values are assigned within [0, 1] . 4. Emotional Expression (EE): Facial movements reflecting romantic emotions increase truth membership. Complex and indeterminacy values are assigned within [0, 1] for difficult interpre- tations. 5. Quality Time Together (QTT): Truth membership increases with social interactions during romantic moments. Indeterminate and complex values from [0, 1] for challenging interpreta- tions. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 10 of 23 6. Play Materials Used (PMU): No specific expert views provided. The dataset represents objects or symbols representing romantic elements. 7. Collaborative Creativity (CC): Truth membership increases with a partner’s imaginative use of elements. Indeterminate and complex values from [0, 1] for difficult interpretations. 8. Expressing Feelings (EF): Truth membership increases with vocal/non-vocal expressions. Indeterminate and complex values from [0, 1] for difficult interpretations. 9. Emotional Tone (ET): Truth membership increases with the tone reflecting the mood. In- determinate and complex values from [0, 1] for challenging interpretations. 10. Mutual Roles (MR): Truth membership increases with realistic role identification. Indeter- minate and complex values from [0, 1] for difficult interpretations. 11. Shared Enthusiasm (SE): Truth membership increases with partners’ involvement and ex- citement. No specific views on indeterminacy or complexity. 12. Emotional Gaze (EG): Truth membership increases with eye movements expressing emo- tions. Complex and indeterminacy values are assigned within [0, 1] for difficult interpretations. 13. Associative Thinking (AT): Truth membership increases with the mental process of forming associations. Indeterminate and complex values from [0, 1] for challenging interpretations. 14. Grammatically Correct Expressions (GCE): Truth membership increases with grammati- cally correct expressions. Indeterminate and complex values from [0, 1] for challenging linguis- tics. 15. Connected Conversations (CC): Truth membership increases with related sentences during romantic interactions. Indeterminate and complex values from [0, 1] for challenging coherence. 16. Gift sharing (GS): No specific expert views provided. The dataset represents the act of sharing symbolic or imaginary gifts within romantic feelings. These expert views guide the evaluation process, incorporating truth, indeterminacy, and com- plexity considerations for each romantic parameter. 9. Characterization of 16 feeling in Term of Example This section is devoted to an example based on young couple of 19-years-old and the following observations and interview about their romantic relationship were made by the physicalists while moving in domain of 16 decision variables. The developed Table 2 is given below. Table 2: Discription of 16 Feelings T RT U C G RF F Feeling Shared Dreams 0.4 0.2 0.15 0 0.25 0 0 Expressive Dance 0.3 0.3 0 0 0.25 0.15 0 Non-verbal Communication 0 0 0 0 0.25 0.75 0 Emotional Expression 0 0.75 0.25 0 0 0 0 Quality Time Together 0.2 0.2 0.2 0.3 0.1 0 0 Play Materials Used 0.15 0 0.3 0 0.25 0 0.3 Collaborative Creativity 0.15 0.3 0 0 0.25 0 0.3 Expressing Feeling 0.4 0.2 0.2 0.1 0.1 0 0 Emotional Tone 0.2 0.25 0.25 0 0.2 0 0.1 Mutual Roles 0.5 0 0.25 0 0.25 0 0 Shared Enthusiasm 0.5 0.25 0.25 0 0 0 0 Emotional Gaze 0 0 0.5 0 0.5 0 0 Associative Thinking 0.75 0 0 0 0.25 0 0 Continued on next page R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 11 of 23 Table2 – continued from previous page T RT U C G RF F Feeling Grammatically Correct Expressions 0.75 0 0.25 0 0 0 0 Connected Conversations 0.15 0 0.3 0.25 0 0 0.3 Gift sharing 0.15 0 0.3 0.25 0 0.2 0.1 Table 2 explains the description of the discussed parameters. The provided information presents a set of parameters for evaluating romantic feelings, each associated with a description and a corresponding dataset represented as SVHNS. Below is a concise summary in Table 3 Table 3: SVHNS for Example S.No Parameter Name Description SVHNS 1. Shared Dreams (SD) Dreams and aspirations shared within romantic feelings, real or imaginative [0.4,0.2,0.15, 0,0.25,0,0] 2. Expressive Dance (ED) Movements expressing roman- tic feeling, indicating cognitive patterns and emotional involvement [0.3,0.3,0, 0,0.25,0.15,0] 3. Non-verbal Communi- cation (NVC ) Non-verbal expressions using body parts to convey romantic ideas or emotions [0,0,0, 0,0.25,0.75,0] 4. Emotional Expression (EE) Movements of facial muscles for non- verbal communication, reflect- ing romantic emotions. [0,0.75, 0.25,0,0,0,0] 5. Quality Time Together (QTT) Duration and nature of social inter- actions during romantic moments [0.2,0.20,0.20,0.3,0.1,0,0] 6. Play Materials Used (PMU) Objects or symbols representing ro- mantic elements [0.15,0,0.3, 0,0.25,0 .0.3] 7. Collaborative Creativ- ity ( CC) Partner’s approach to using pro- vided elements, offering insights into shared imaginative experiences [0.15,0.3, 0,0,0.25,0 .0.3] 8. Expressing Feeling (EF) Vocal or non-vocal expression of ro- mantic feelings and emotions [0.4,0.20, 0.20,0.1,0.1,0 ,0] 9. Emotional Tone ( ET) Tone reflects the mood and state of mind during romantic interactions [0.2,0.25, 0.25,0,0.2,0 .0.1] 10. Mutual Roles ( MR) Roles both partners assume within romantic feelings [0.5,0, 0.25, 0,0.25,0,0] 11. Shared Enthusiasm (SE) Extent of partners involvement and shared excitement during romantic activities [0.5,0.25, 0.25,0,0 ,0,0] 12. Emotional Gaze ( EG) Movement of the eyes expressing emotions and connection during romantic moments [0,0.0.5, 0,0,0.5 ,0,0] 13. Associative Thinking (AT) Mental process by which both part- ners from associations and create shared romantic experiences [0.75,0, 0,0 ,0.25,0,0] 14. Grammatically Correct Expressions(GCE) Ability to construct grammatically correct sentences expressing roman- tic feelings [0.75,0, 0.25,0,0 ,0,0] continued on next page R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 12 of 23 Table3 – continued from previous page S.No Parameter Name Description SVHNS 15. Connected Conversa- tions (CC) Whether sentences formed during romantic interactions are related [0.15,0, 0.3,0.25,0 ,0,0.3] 16. Gift Sharing The data set represents the act of sharing symbolic or imaginary gifts within romantic feelings. [0.15, 0, 0.3, 0.25, 0, 0.2, 0.1] Likewise the SVHNS tuples for the other data sets was done with the help of the expert. Then these SVHNS sets are used for analysis using machine learning algorithms. 10. Results and Discussions Many Python libraries, including pandas, numpy, matplotlib, sklearn, seaborn, and pylab, were utilized for the purpose of graphical representation and data visualization. Python pro- gramming was used to visualize the previously stated techniques, and K-means clustering was performed based on the elbow curve result. These graphics have led to logical conclusions, and it has also been addressed how sixteen different aspects contribute to young boys’ and girls’ romantic feelings. A heat map’s color scale, which clearly displays the correlation and associa- tivity aspects, shows blue correlation for darker shades and reddish correlation for lighter shades. When two variables show a correlation with each other, it means they are connected. Reddish correlation is the term used to describe the relationship between an increase in one attribute and an increase in another. Blue connection occurs when one quality increases while another de- creases. The heat map clearly shows the correlation and associativity aspects with a color scale where darker shades indicate a blue correlation and lighter ones indicate a reddish link. The Elbow Method is strategically applied to determine optimal clusters, significantly contributing to a richer understanding of diversity in romantic feelings. This nuanced methodology aims to; comprehend 16 distinct feelings by considering absolute true, relative true, absolute false, relative false, contradiction, unknown (undefined) and ignorance. The application of K-means clustering on the data set revealed intriguing patterns in romantic feelings. Distinct clusters emerged, each characterized by individuals with similar romantic preferences. The correlation analysis provided deeper insights into the emotional landscape, particularly highlighting the relationships between dimensions such as ”quality time together” and ”expressing feelings.” These findings underscore the complexity and interconnectedness inherent in romantic experi- ences. The Elbow Method, strategically employed, determined optimal clusters, contributing significantly to our understanding of the diverse nature of romantic feelings. The results affirm the effectiveness of advanced machine-learning techniques in unravelling the complexities of hu- man emotions, especially in the context of romantic experiences. Figure 3, while testing Encrypted K-Mean clustering method on feature T for the param- eters “shared dreams” on the y − axis against “expressive dance” on the x − axis, it was found that higher concentration of points lies near x = 0.2 and y = 0.3. Initial centroid c1atx = 0.00andy = 0.0 and an Initial centroid c1atx = 0.00andy = 0.0, second centroid c2atx = 0.00andy = 0.19 and third centroid c3atx = 0.05andy = 0.9 After bring in action K-mean clustering we got closed to c3 as shown in the Figure 3. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 13 of 23 Figure 3: Encrypted K-Mean Clustering Method The Figure 4 has been devoted to heat map and as we know that heat maps helps us in understanding the correlation in data (the graphical illustration of data where values are repre- sented by colors) and in particular here encrypted correlation heat map has been reflected for the given data. The picture represents 32 combined feeling of male and female. Here we see that the dominant colour is dark blue and also at second number the influential colour is sky colour. Encrypted correlation heat map offering a visual representation of correlations while keeping the security of the data. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 14 of 23 Figure 4: Encrypted K-Mean Clustering Heat Map Figure 5 reflects encrypted Elbow method (this method is used to decide how many clusters it should consider and this method is actually the graph between K and distortion WCSS). Here the K-values are taken along x-axis and the distortion (WCSS) is taken along y-axis. We see that there are number of K-values but we encounter the business value of K which is at 4 that is at k = 4 the optimal value is 4 because here we see that the drastic change in y-axis value occurred at k = 4 the value of y = 2.4 as shown in the Figure 5. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 15 of 23 Figure 5: Encrypted Elbow Method Figure 6 , decrypted K-Mean clustering on feature T(where the data is already decrypted) for the parameters ”Shared dreams” on the y − axis against ”Gift sharing” on the x− axis, it was found that higher concentration of points lies near x = 0.15andy = 0.18. Initial centroid c1atx = 0.00andy = 0.0 and an Initial centroid c1atx = 0.00andy = 0.0 and second centroid c2atx = 0.00andy = 0.8. After bring in action K-mean clustering we got closed to centroid c3atx = 0.25andy = 0.18 as shown in the figure. The decrypted showcasing the versatility and security of the data. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 16 of 23 Figure 6: Decrypted K-Mean Clustering In Figure 7 Encrypted correlation matrix for feature T(where the data is already encrypted) is developed and we see that the off diagonal is filled with single element one and moreover we see that dark green shows positive correlation and the midnight blow shows negative correlation. The positive and negative correlations are symmetric about the off diagonal which is reflected by dark red colour. Here we notice that the highest positive value is 0.89 so this means that there is strong relationship between gift sharing and emotional tone and similarly the others. We notice that there are alsonegative values and the most negative value is −0.69 which indicates that there is strong negative relationship between emotional gaze and consummate love and associative thinking similarly the others. The correlation of variables to itself can be seen on the diagonal from bottom right to left. The correlation matrices highlighting the interior- component relationships in a secure environment. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 17 of 23 Figure 7: Encrypted Correlation Matrix Figure 8 decrypted correlation matrix has been plotted against the data. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 18 of 23 Figure 8: Decrypted Correlation Matrix 11. The Comparative Analysis in Table 4 The application of K-means clustering on the data set reveals intriguing patterns in romantic feelings. Distinct clusters emerge, each characterized by individuals with similar romantic pref- erences. Correlation analysis provides deeper insights into the emotional landscape, highlighting relationships between dimensions such as “Quality time together” and “Expressing feelings.” Visibility: The Elbow Method determines optimal clusters, contributing significantly to our understanding of the diverse nature of romantic feelings. Associativity: The results affirm the effectiveness of advanced machine-learning techniques in unravelling the complexities of human emotions, especially in the context of romantic expe- riences. The multidimensional analysis provides a holistic view, emphasizing the need for a nuanced approach to exploring and understanding intricate emotional states. Dynamicity: Medium Dynamicity: The data set exhibits a moderate level of dynamicity, indi- cating some variability or changes in patterns over the observed dimensions or features. Strong Dynamicity Analysis: Through a robust analysis, it is evident that the data set show- cases a strong level of dynamicity. This suggests significant variations and complexities in the relationships within the data. Scalability: Medium Scalability: The data set demonstrates a moderate level of scalability, indicating a reasonable ability to handle an increase in size or complexity without a significant decrease in performance. Strong Scalability Analysis: A thorough scalability analysis indicates that the data set possesses strong scalability. This suggests the data set’s capability to efficiently handle increased volume or complexity, making it versatile for diverse analytical tasks. R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 19 of 23 Table 4: Comparative Analysis Factors Heat Map K-Means Correlation Strong correlation visualization. Weak correlation analysis dynamic- ity. Visibility Weak Visibility in terms of factor Medium visibility in terms of factor. Associativity Strong associa- tivity representa- tion. Strong associativ- ity analysis. Dynamacity Medium dynam- icity. Strong dynamic- ity analysis. Scalability Medium scalabil- ity. Strong scalability analysis. Graphical Representation of Table 4, in terms of Factors and Methods are given in Figure 9 Figure 9: Analysis Level 12. Conclusion In conclusion, this study presents a comprehensive exploration of romantic feelings, leverag- ing advanced machine-learning techniques. The application of K-means clustering, correlation analysis, and innovative visualization methods has deepened our understanding of the emo- tional landscape. The results underscore the intricate nature of romantic experiences and the interconnectedness of various emotional dimensions. The dual-layered approach, combining machine-learning techniques with the single valued hep- tapartitioned neutrosophic set (SVHNS), has proven effective in providing a nuanced under- standing of 16 distinct feelings. Expert evaluations further enriched our exploration, shedding light on the interplay between physical expressions and emotional authenticity. On behalf of single valued heptapartitioned neutrosophic set (SVHNS) which is the generalized version of the neutrosophic set is utilized to make this study successful and this set consists of seven membership functions which are more sensitive to real-world problems. Membership functions are defined as an absolute true, relative true, absolute false, relative false, contradic- tion, unknown (undefined) and ignorance respectively. Moreover, several properties of this set were addressed. This study focuses on the romantic sensations experienced by young boys and girls in a variety of contexts. The data set supporting this research includes individuals aged 18-25, with data collected from the Psychology Department at Peshawar University, Pakistan. This data set was R. Hatamleh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6484 20 of 23 critically analyzed using the Single-Valued Heptapartitioned Neutrosophic Set (SVHNS).For real-world application, the study examines romantic feelings across various dimensions using machine learning and graphical algorithms. These include Encrypted K-Means Clustering, En- crypted K-Means Clustering Heat Map, Encrypted Elbow Method, Decrypted K-Means Clus- tering, Encrypted Correlation Matrix, and Decrypted Correlation Matrix. These algorithms are applied and visualized to uncover patterns and groupings within the data.The proposed methods help examine and establish relationships among several factors that influence roman- tic feelings among young men and women. These techniques offer new dimensions not only for psychological studies in general but also specifically for understanding emotional disorders and relationship breakups among young university couples. Future research endeavors should focus on expanding emotional data sets to enhance the robust- ness of emotion analysis methodologies. Refining SVHNS parameters and integrating sentiment analysis would further contribute to a more nuanced understanding of human feelings. Ex- ploring demographic factors is essential to uncover variations in emotional experiences across diverse populations. Advanced emotion detection methods and clustering techniques should be explored to provide a more holistic understanding of human emotional experiences. Addi- tionally, incorporating real-time data and dynamic factors could capture the evolving nature of emotions, ensuring the relevance and applicability of emotion analysis methodologies in dy- namic social contexts. This study lays the foundation for future investigations into the intricate fabric of human emotions, opening promising avenues for continued research and exploration. 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