A New Similarity Measure for Picture Fuzzy Sets and Its Various Applications
Academic Background
In fields such as decision analysis, pattern recognition, and medical diagnosis, fuzzy set theory provides essential mathematical tools for handling uncertainty and ambiguity. Traditional fuzzy sets (Fuzzy Set, FS) and intuitionistic fuzzy sets (Intuitionistic Fuzzy Set, IFS) have certain limitations when dealing with complex data, especially when neutrality needs to be considered. Picture fuzzy sets (Picture Fuzzy Set, PFS), as an extension of fuzzy set theory, introduce the dimension of neutrality, enabling a more comprehensive description of fuzzy information in the real world. However, existing PFS similarity measures often yield unreasonable results in certain scenarios, such as failing to meet axiomatic requirements, producing contradictions when calculating similarities between different PFS, and performing poorly in pattern classification. To address these issues, this paper proposes a novel PFS similarity measure based on the inverse tangent function and demonstrates its applications in classification and medical diagnosis.
Source of the Paper
This paper is co-authored by Wathek Chammam (Majmaah University, Saudi Arabia), Abdul Haseeb Ganie (Thapar Institute of Engineering and Technology, India), Maha Mohammed Saeed (King Abdulaziz University, Saudi Arabia), Amira M. Sief (Future High Institute of Engineering in Fayoum, Egypt), and Mohammad M. Khalaf (Mustaqbal University, Saudi Arabia). It was published in 2025 in the journal Cognitive Computation with the DOI 10.1007/s12559-025-10449-7.
Research Process
1. Research Objectives and Method Design
The primary objective of this paper is to propose a new PFS similarity measure and validate its effectiveness in pattern analysis and medical diagnosis. The research is divided into the following steps:
- Similarity Measure Design: Based on the inverse tangent function, a new PFS similarity measure formula is proposed, and its compliance with axiomatic requirements is proven.
- Numerical Experiments: Through multiple numerical cases, the performance of the new method is compared with existing methods to verify its superiority.
- Application Validation: The new method is applied to pattern classification and medical diagnosis problems to demonstrate its practical value.
- Decision Method Improvement: An improved TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) multi-criteria decision-making method is proposed to address the shortcomings of the traditional TOPSIS method.
2. Similarity Measure Design
The similarity measure formula proposed in this paper is based on the inverse tangent function, with the specific form:
[
\text{sm}_g(A_1, A2) = 1 - \frac{1}{3} \left( \tan^{-1} \mu{A1} - \tan^{-1} \mu{A2} + \tan^{-1} \nu{A1} - \tan^{-1} \nu{A2} + \tan^{-1} \pi{A1} - \tan^{-1} \pi{A_2} \right)
]
where (\mu), (\nu), and (\pi) represent membership, non-membership, and neutrality degrees, respectively. Theoretical proofs show that this method satisfies the four axioms of similarity measures: non-negativity, symmetry, reflexivity, and monotonicity.
3. Numerical Experiments
To validate the effectiveness of the new method, multiple numerical cases are designed to compare the performance of the new method with existing methods in calculating PFS similarity. The experimental results show that existing methods sometimes yield unreasonable results, such as failing to distinguish similarities between different PFS or being unable to calculate similarity in certain extreme cases. In contrast, the new method performs excellently in all cases, accurately calculating similarities between different PFS, with results aligning with intuition.
4. Application Validation
Pattern Classification
The new method is applied to pattern classification problems, using the Iris plant dataset for validation. By converting the dataset into PFS form, similarities between different categories are calculated. The experimental results show that the new method performs excellently in classification tasks, accurately identifying the categories of unknown patterns.
Medical Diagnosis
In medical diagnosis applications, PFS is used to represent patient symptoms and disease symptoms, and the similarity between patients and diseases is calculated to determine possible diseases. The experimental results show that the new method can effectively assist medical diagnosis, with results consistent with existing methods but offering a more stable calculation process.
5. Improved TOPSIS Method
The traditional TOPSIS method selects the optimal solution by only considering the maximum similarity to the Positive Ideal Solution (PIS) while ignoring the minimum similarity to the Negative Ideal Solution (NIS). This paper proposes an improved TOPSIS method that ensures the optimal solution performs best in both aspects by simultaneously considering similarity to PIS and NIS. The experimental results show that the improved method is more reasonable in selecting the optimal solution.
Main Results and Conclusions
1. Superiority of the Similarity Measure
The PFS similarity measure based on the inverse tangent function proposed in this paper performs excellently in all numerical cases, accurately calculating similarities between different PFS, with results aligning with intuition. Compared to existing methods, the new method is more stable when handling extreme cases and complex data.
2. Application Value
The application validation in pattern classification and medical diagnosis demonstrates the broad practical value of the new method. Particularly in medical diagnosis, the new method can effectively assist doctors in determining possible diseases, improving diagnostic accuracy.
3. Improved TOPSIS Method
The improved TOPSIS method is more reasonable in selecting the optimal solution by simultaneously considering similarity to PIS and NIS, ensuring the optimal solution performs best in both aspects. This improvement provides a more reliable solution for multi-criteria decision-making problems.
Research Highlights
- Novel Similarity Measure: The PFS similarity measure designed based on the inverse tangent function addresses the limitations of existing methods in handling complex data.
- Broad Application Validation: Practical applications in pattern classification and medical diagnosis demonstrate the practical value of the new method.
- Improved Decision Method: The proposed improved TOPSIS method addresses the shortcomings of the traditional method, providing a more reliable solution for multi-criteria decision-making problems.
Research Significance and Value
The PFS similarity measure and its application research proposed in this paper not only enrich the content of fuzzy set theory but also provide new tools and methods for solving practical problems. In fields such as pattern classification, medical diagnosis, and multi-criteria decision-making, the new method has broad application prospects, effectively improving the accuracy and reliability of decisions. Future research can further explore the application of this method in other fields, such as image processing and bidirectional approximate reasoning.