Skip to main content
SpringerLink
Log in

Globally automatic fuzzy clustering for probability density functions and its application for image data

  • Published:
Applied Intelligence Aims and scope Submit manuscript

Abstract

Clustering for probability density functions (CDF) can be categorized as non-fuzzy and fuzzy approaches. Regarding the second approach, the iterative refinement technique has been used for searching the optimal partition. This method could be easily trapped at a local optimum. In order to find the global optimum, a meta-heuristic optimization (MO) algorithm must be incorporated into the fuzzy CDF problem. However, no research utilizing MO to solve the fuzzy CDF problem has been proposed so far due to the lack of a reasonable encoding for converting a fuzzy clustering solution to a chromosome. To address this shortcoming, a new definition called Gaussian prototype is defined first. This type of prototype is capable of accurately representing the cluster without being overly complex. As a result, prototypes’ information can be easily integrated into the chromosome via a novel prototype-based encoding method. Second, a new objective function is introduced to evaluate a fuzzy CDF solution. Finally, Differential Evolution (DE) is used to determine the optimal solution for fuzzy clustering. The proposed method, namely DE-AFCF, is the first to propose a globally automatic fuzzy CDF algorithm, which not only can automatically determine the number of clusters k but also can search for the optimal fuzzy partition matrix by taking into account both clustering compactness and separation. The DE-AFCF is also applied in some image clustering problems, such as processed image detection, and traffic image recognition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Jain AK, Dubes RC (1988) Algorithms for clustering data. Prentice-Hall, Inc.

  2. Li L, Zhou X, Li Y, Gu J, Shen S (2020) An improved genetic algorithm with lagrange and density method for clustering. Concurr Comput Pract Exp 32(24):5969

    Article  Google Scholar 

  3. Cai L, Zhu L, Jiang F, Zhang Y, He J (2021) Research on multi-source poi data fusion based on ontology and clustering algorithms. Appl Intell :1–17

  4. Chen R, Tang Y, Tian L, Zhang C, Zhang W (2021) Deep convolutional self-paced clustering. Appl Intell :1–15

  5. Fränti P, Sieranoja S (2018) K-means properties on six clustering benchmark datasets. Appl Intell 48(12):4743–4759

    Article  MATH  Google Scholar 

  6. Fukunaga K (2013) Introduction to statistical pattern recognition. Academic Press Inc., San Diego

    MATH  Google Scholar 

  7. MacQueen J et al (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth berkeley symposium on mathematical statistics and probability, vol. 1, Oakland, pp 281–297

  8. Paul D, Saha S, Mathew J (2020) Improved subspace clustering algorithm using multi-objective framework and subspace optimization. Expert Syst Appl 158:113487

    Article  Google Scholar 

  9. Vo-Van T, Nguyen-Hai A, Tat-Hong M, Nguyen-Trang T (2020) A new clustering algorithm and its application in assessing the quality of underground water. Sci Program :2020

  10. Chen Y, Hu X, Fan W, Shen L, Zhang Z, Liu X, Du J, Li H, Chen Y, Li H (2020) Fast density peak clustering for large scale data based on KNN. Knowl-Based Syst 187:104824

    Article  Google Scholar 

  11. Chen J-H, Hung W-L (2021) A jackknife entropy-based clustering algorithm for probability density functions. J Stat Comput Simul 91(5):861–875

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen J, Chang Y, Hung W (2018) A robust automatic clustering algorithm for probability density functions with application to categorizing color images. Commun Stat-Simul Comput 47(7):2152–2168

    Article  MathSciNet  MATH  Google Scholar 

  13. Goh A, Vidal R (2008) Unsupervised riemannian clustering of probability density functions. In: Joint European conference on machine learning and knowledge discovery in databases, Springer, pp 377–392

  14. Montanari A, Calò DG (2013) Model-based clustering of probability density functions. ADAC 7(3):301–319

    Article  MathSciNet  MATH  Google Scholar 

  15. Vo Van T, Pham-Gia T (2010) Clustering probability distributions. J Appl Stat 37(11):1891–1910

    Article  MathSciNet  MATH  Google Scholar 

  16. VoVan T, NguyenTrang T (2018) Similar coefficient for cluster of probability density functions. Commun Stat-Theory Methods 47(8):1792–1811

    Article  MathSciNet  MATH  Google Scholar 

  17. Vovan T (2019) Cluster width of probability density functions. Intell Data Anal 23(2):385–405

    Article  MathSciNet  Google Scholar 

  18. Xu L, Hu Q, Hung E, Chen B, Tan X, Liao C (2015) Large margin clustering on uncertain data by considering probability distribution similarity. Neurocomputing 158:81–89

    Article  Google Scholar 

  19. Bezdek JC, Ehrlich R, Full W (1984) Fcm: The fuzzy c-means clustering algorithm. Comput Geosci 10(2-3):191–203

    Article  Google Scholar 

  20. Lei T, Liu P, Jia X, Zhang X, Meng H, Nandi AK (2019) Automatic fuzzy clustering framework for image segmentation. IEEE Trans Fuzzy Syst 28(9):2078–2092

    Article  Google Scholar 

  21. Lin M, Huang C, Chen R, Fujita H, Wang X (2021) Directional correlation coefficient measures for pythagorean fuzzy sets: their applications to medical diagnosis and cluster analysis. Compl Intell Syst 7(2):1025–1043

    Article  Google Scholar 

  22. Rubio E, Castillo O, Valdez F, Melin P, Gonzalez CI, Martinez G (2017) An extension of the fuzzy possibilistic clustering algorithm using type-2 fuzzy logic techniques. Adv Fuzzy Syst :2017

  23. Ruspini EH, Bezdek JC, Keller JM (2019) Fuzzy clustering: a historical perspective. IEEE Comput Intell Mag 14(1):45–55

    Article  Google Scholar 

  24. Nguyentrang T, Vovan T (2017) Fuzzy clustering of probability density functions. J Appl Stat 44(4):583–601

    Article  MathSciNet  MATH  Google Scholar 

  25. Phamtoan D, Vovan T (2020) Improving fuzzy clustering algorithm for probability density functions and applying in image recognition. Model Assist Stat Appl 15(3):249–261

    Google Scholar 

  26. Tran NT, Dao T-P, Nguyen-Trang T, Ha C-N (2021) Prediction of fatigue life for a new 2-dof compliant mechanism by clustering-based anfis approach. Math Probl Eng :2021

  27. Zheng L, Chao F, Mac Parthaláin N, Zhang D, Shen Q (2021) Feature grouping and selection: a graph-based approach. Inf Sci 546:1256–1272

    Article  MathSciNet  MATH  Google Scholar 

  28. Abd Elaziz M, Heidari AA, Fujita H, Moayedi H (2020) A competitive chain-based harris hawks optimizer for global optimization and multi-level image thresholding problems. Appl Soft Comput 95:106347

    Article  Google Scholar 

  29. Nguyen-Trang T, Nguyen-Thoi T, Nguyen-Thi K-N, Vo-Van T (2022) Balance-driven automatic clustering for probability density functions using metaheuristic optimization. International Journal of Machine Learning and Cybernetics :1–16

  30. Thao N-T (2019) An improved fuzzy time series forecasting model using the differential evolution algorithm. J Intell Fuzzy Syst 36(2):1727–1741

    Article  Google Scholar 

  31. Ochoa P, Castillo O, Soria J (2020) High-speed interval type-2 fuzzy system for dynamic crossover parameter adaptation in differential evolution and its application to controller optimization. Int J Fuzzy Syst 22(2):414–427

    Article  Google Scholar 

  32. Diem HK, Trung VD, Trung NT, Van Tai V, Thao NT (2018) A differential evolution-based clustering for probability density functions. IEEE Access 6:41325–41336

    Article  Google Scholar 

  33. Hu Z, Xiong S, Su Q, Zhang X (2013) Sufficient conditions for global convergence of differential evolution algorithm. J Appl Math :2013

  34. Chen J-H, Hung W-L (2015) An automatic clustering algorithm for probability density functions. J Stat Comput Simul 85(15):3047–3063

    Article  MathSciNet  MATH  Google Scholar 

  35. Vo-Van T, Nguyen-Thoi T, Vo-Duy T, Ho-Huu V, Nguyen-Trang T (2017) Modified genetic algorithm-based clustering for probability density functions. J Stat Comput Simul :1–16

  36. Tran DN, Vinant T, Colombani TM, Ho-Kieu D (2018) An r code for implementing non-hierarchical algorithm for clustering of probability density functions. J Adv Eng Comput 2(3):174–187

    Article  Google Scholar 

  37. Pham-Toan D, Vo-Van T, Pham-Chau A, Nguyen-Trang T, Ho-Kieu D (2019) A new binary adaptive elitist differential evolution based automatic k-medoids clustering for probability density functions. Math Probl Eng :2019

  38. Park S, Han S, Kim S, Kim D, Park S, Hong S, Cha M (2021) Improving unsupervised image clustering with robust learning. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 12278–12287

  39. Niu C, Zhang J, Wang G, Liang J (2020) Gatcluster: self-supervised gaussian-attention network for image clustering. In: European conference on computer vision, Springer, pp 735–751

  40. Ren Y, Wang N, Li M, Xu Z (2020) Deep density-based image clustering. Knowl-Based Syst 197:105841

    Article  Google Scholar 

  41. Wu J, Long K, Wang F, Qian C, Li C, Lin Z, Zha H (2019) Deep comprehensive correlation mining for image clustering. In: Proceedings of the IEEE/CVF international conference on computer vision, pp 8150–8159

  42. Zhao J, Lu D, Ma K, Zhang Y, Zheng Y (2020) Deep image clustering with category-style representation. In: European conference on computer vision, Springer, pp 54–70

  43. Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis: the kernel approach with S-Plus Illustrations vol. 18 OUP Oxford

  44. Che-Ngoc H, Nguyen-Trang T, Nguyen-Bao T, Nguyen-Thoi T, Vo-Van T (2020) A new approach for face detection using the maximum function of probability density functions. Ann Oper Res: 1–21

  45. Bezdek JC (2013) Pattern recognition with fuzzy objective function algorithms. Springer

  46. Dunn JC (1973) A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters. J Cybernet 3(3):32–57

    Article  MathSciNet  MATH  Google Scholar 

  47. Storn R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359

    Article  MathSciNet  MATH  Google Scholar 

  48. Elsisi M (2019) Future search algorithm for optimization. Evol Intel 12(1):21–31

    Article  Google Scholar 

  49. Holland JH, et al. (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT press

  50. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN’95-international conference on neural networks, vol. 4, IEEE, pp 1942–1948

  51. Nguyen-Trang T, Nguyen-Thoi T, Truong-Khac T, Pham-Chau A, Ao H (2019) An efficient hybrid optimization approach using adaptive elitist differential evolution and spherical quadratic steepest descent and its application for clustering. Sci Program: 2019

  52. Otsu N (1979) A threshold selection method from gray-level histograms. IEEE Trans Syst Man Cybernet 9(1):62–66

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam

    Thao Nguyen-Trang & Trung Nguyen-Thoi

  2. Faculty of Basic Sciences, Van Lang University, Ho Chi Minh City, Vietnam

    Thao Nguyen-Trang

  3. Faculty of Mechanical-Electrical and Computer Engineering, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Trung Nguyen-Thoi

  4. College of Natural Science, Can Tho University, Can Tho City, Vietnam

    Tai Vo-Van

Authors
  1. Thao Nguyen-Trang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Trung Nguyen-Thoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tai Vo-Van
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tai Vo-Van.

Ethics declarations

Conflict of Interests

No potential conflict of interest was reported by the authors.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen-Trang, T., Nguyen-Thoi, T. & Vo-Van, T. Globally automatic fuzzy clustering for probability density functions and its application for image data. Appl Intell 53, 18381–18397 (2023). https://doi.org/10.1007/s10489-023-04470-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10489-023-04470-2

Keywords

Search

Navigation