Skip to main content
SpringerLink
Log in

Intuitionistic Fuzzy Proximal Support Vector Machines for Pattern Classification

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

Support vector machine is a powerful technique for classification and regression problems. In the binary data problems, it classifies the points by assigning them to one of the two disjoint halfspaces. However, this method fails to handle the noises and outliers present in the dataset and the solution of a large-sized quadratic programming problem is required to obtain the decision surface in input or in feature space. We propose the intuitionistic fuzzy proximal support vector machine (IFPSVM) which classifies the patterns according to its proximity with the two parallel planes that are kept as distant as possible from each other. These two parallel ‘proximal’ planes can be obtained by solving a system of linear equations only. There is an intuitionistic fuzzy number associated with each training point which is framed by its degree of membership and non-membership. The membership degree of a pattern considers its distance from the corresponding class center and the degree of non-membership of a pattern is given by the ratio of the number of heterogeneous points to the number of total points in its neighborhood. The proposed technique effectively reduces the impact of noises and distinguishes the edge support vectors and outliers. Computational simulations on an artificial and eleven UCI benchmark datasets using linear, polynomial and Gaussian kernel functions, show the effectiveness of the proposed IFPSVM method. The experiments prove that it can handle large datasets with less computational time and yields better accuracy.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42

Similar content being viewed by others

References

  1. Atanassov KT (1999) Intuitionistic fuzzy sets: theory and applications. Physica-Verlag, Heidelberg

    Book  Google Scholar 

  2. Blake CL, Merz CJ (1998) UCI repository for machine learning databases. Department of Information and Computer Sciences, University of California, Irvine. http://www.ics.uci.edu/~mlearn/MLRepository.html

  3. Bradley PS, Mangasarian OL (2000) Massive data discrimination via linear support vector machines. Optim Methods Softw 13(1):1–10

    Article  MathSciNet  Google Scholar 

  4. Burges CJ (1998) A tutorial on support vector machines for pattern recognition. Data Min Knowl Disc 2(2):121–167

    Article  Google Scholar 

  5. Cao L, Tay FEH (2001) Financial forecasting using support vector machines. Neural Comput Appl 10(2):184–192

    Article  Google Scholar 

  6. Chen SG, Wu XJ (2018) A new fuzzy twin support vector machine for pattern classification. Int J Mach Learn Cybern 9(9):1553–1564

    Article  Google Scholar 

  7. Cortes C, Vapnik VN (1995) Support vector networks. Mach Learn 20(3):273–297

    MATH  Google Scholar 

  8. Cristianini N, Taylor JS (2000) An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, Cambridge

    Book  Google Scholar 

  9. Fung G, Mangasarian OL (2001) Proximal support vector machine classifiers. In: Proceedings of conference on knowledge discovery and data mining, pp 77–86

  10. Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Maryland

    MATH  Google Scholar 

  11. Grzegorzewski P (2002) Nearest interval approximation of a fuzzy number. Fuzzy Set Syst 130(3):321–330

    Article  MathSciNet  Google Scholar 

  12. Guo B, Gunn SR, Damper RI, Nelson JDB (2008) Customizing kernel functions for SVM-based hyperspectral image classification. IEEE Trans Image Process 17(4):622–629

    Article  MathSciNet  Google Scholar 

  13. Guyon I, Matic N, Vapnik VN (1996) Discovering informative patterns and data cleaning. MIT Press, Cambridge, pp 181–203

    Google Scholar 

  14. Ha MH, Huang S, Wang C, Wang XL (2011) Intuitionistic fuzzy support vector machine. J Hebei Univ (Nat Sci Ed) 3:225–229

    Google Scholar 

  15. Ha M, Wang C, Chen J (2013) The support vector machine based on intuitionistic fuzzy number and kernel function. Soft Comput 17(4):635–641

    Article  Google Scholar 

  16. Jayadeva Khemchandani R, Chandra S (2004) Fast and robust learning through fuzzy linear proximal support vector machines. Neurocomputing 61:401–411

    Article  Google Scholar 

  17. Jiang X, Yi Z, Lv JC (2006) Fuzzy SVM with a new fuzzy membership function. Neural Comput Appl 15:268–276

    Article  Google Scholar 

  18. Joachims T (1998) Text categorization with support vector machines: learning with many relevant features. In: European conference on machine learning Springer Berlin Heidelberg, pp 137–142

  19. Khemchandani R, Sharma S (2016) Robust least squares twin support vector machine for human activity recognition. Appl Soft Comput 47:33–46

    Article  Google Scholar 

  20. Lin CF, Wang SD (2002) Fuzzy support vector machines. IEEE Trans Neural Netw 13(2):464–471

    Article  Google Scholar 

  21. Liu X, Li M, Wang L, Dou Y, Yin J, Zhu E (2017) Multiple kernel k-means with incomplete kernels. In: Proceeding of association for the advancement of artificial intelligence conference, pp 2259–2265

  22. Lu J, Zhang E (2007) Gait recognition for human identification based on ICA and fuzzy SVM through multiple views fusion. Pattern Recognit Lett 28(16):2401–2411

    Article  Google Scholar 

  23. Mercer J (1909) Functions of positive and negative type and their connection with the theory of integral equations. Philos Trans R Soc Lond A 209:415–446

    Article  Google Scholar 

  24. Prakash KA, Suresh M, Vengataasalam S (2016) A new approach for ranking of intuitionistic fuzzy numbers using a centroid concept. Math Sci 10(4):177–184

    Article  MathSciNet  Google Scholar 

  25. Rezvani S, Wang X, Pourpanah F (2019) Intuitionistic fuzzy twin support vector machines. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2019.2893863

    Article  Google Scholar 

  26. Schölkopf B, Tsuda K, Vert JP (2004) Support vector machine applications in computational biology. MIT Press, Cambridge

    Book  Google Scholar 

  27. Song Q, Hu W, Xie W (2002) Robust support vector machine with bullet hole image classification. IEEE Trans Syst Man Cybern 32(4):440–448

    Article  Google Scholar 

  28. Sun Z, Sun Y (2003) Fuzzy support vector machine for regression estimation. IEEE Int Conf Syst Man Cybern 4:3336–3341

    Google Scholar 

  29. Suykens JAK, Vandewalle J (1999) Least squares support vector machine classifiers. Neural Process Lett 9(3):293–300

    Article  Google Scholar 

  30. Tang WM (2011) Fuzzy SVM with a new fuzzy membership function to solve the two-class problems. Neural Process Lett 34(3):209–219

    Article  Google Scholar 

  31. Vapnik VN (1995) The nature of statistical learning theory. Springer, Berlin

    Book  Google Scholar 

  32. Wang Y, Wang S, Lai KK (2005) A new fuzzy support vector machine to evaluate credit risk. IEEE Trans Fuzzy Syst 13:820–831

    Article  Google Scholar 

  33. Wu K, Yap KH (2006) Fuzzy SVM for content-based image retrieval: a pseudo-label support vector machine framework. IEEE Comput Intell Mag 1(2):10–16

    Article  Google Scholar 

  34. Xian GM (2010) An identification method of malignant and benign liver tumors from ultrasonography based on GLCM texture features and fuzzy SVM. Expert Syst Appl 37(10):6737–6741

    Article  Google Scholar 

  35. Yan X, Bai Y, Fang SC, Luo J (2018) A proximal quadratic surface support vector machine for semi-supervised binary classification. Soft Comput 22:6905–6919

    Article  Google Scholar 

  36. Yu R, Qiao L, Chen M, Lee SW, Fei X, Shen D (2019) Weighted graph regularized sparse brain network construction for MCI identification. Pattern Recognit 90:220–231

    Article  Google Scholar 

  37. Yu X (2014) Blurred trace infrared image segmentation based on template approach and immune factor. Infrared Phys Technol 67:116–120

    Article  Google Scholar 

  38. Yu X, Zhou Z, Gao Q, Li D, Riha K (2018) Infrared image segmentation using growing immune field and clone threshold. Infrared Phys Technol 88:184–193

    Article  Google Scholar 

  39. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  Google Scholar 

  40. Zhou MM, Li L, Lu YL (2009) Fuzzy support vector machine based on density with dual membership. In: Proceedings of the eighth international conference on machine learning and cybernetics. IEEE, Baoding, pp 674–678

Download references

Acknowledgements

The authors sincerely thank the reviewers for the recommendation, valuable comments and the interesting suggestions which have considerably improved the presentation of the paper. The first author is grateful to the Ministry of Human Resource Development, India, for financial support, to carry out this work.

Author information

Authors and Affiliations

  1. Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand, India

    Scindhiya Laxmi & Shiv Kumar Gupta

Authors
  1. Scindhiya Laxmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiv Kumar Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiv Kumar Gupta.

Ethics declarations

Conflict of interest

Authors declare that they do not have any conflict of interest.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

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

Appendix A

Appendix A

Let H be any rectangular matrix of order \(m \times (n+1),~I\) be an identity matrix of order \((n+1)\times (n+1)\) and \(C>0.\) Then the matrix \(\displaystyle \left( H^TH+\frac{I}{C}\right) \) is always invertible.

Proof

Let x be a vector of order \((n+1) \times 1\). Then

$$\begin{aligned} x^TH^THx= & {} (Hx)^T(Hx)\\= & {} ||Hx||^2\\\ge & {} 0. \end{aligned}$$

This implies that \(H^TH\) is a positive semi definite matrix.

This further yields \(\displaystyle \bigg (H^TH+\frac{I}{C}\bigg ),\) where \(C>0\) is a positive real number, is positive definite matrix and hence it will be invertible. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Laxmi, S., Gupta, S.K. Intuitionistic Fuzzy Proximal Support Vector Machines for Pattern Classification. Neural Process Lett 51, 2701–2735 (2020). https://doi.org/10.1007/s11063-020-10222-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-020-10222-x

Keywords

Search

Navigation