Skip to main content
SpringerLink
Log in

Three-way k-means: integrating k-means and three-way decision

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

The traditional k-means, which unambiguously assigns an object precisely to a single cluster with crisp boundary, does not adequately show the fact that a cluster may not have a well-defined cluster boundary. This paper presents a three-way k-means clustering algorithm based on three-way strategy. In the proposed method, an overlap clustering is used to obtain the supports (unions of the core regions and the fringe regions) of the clusters and perturbation analysis is applied to separate the core regions from the supports. The difference between the support and the core region is regarded as the fringe region of the specific cluster. Therefore, a three-way explanation of the cluster is naturally formed. Davies–Bouldin index (DB), Average Silhouette index (AS) and Accuracy (ACC) are computed by using core region to evaluate the structure of three-way k-means result. The experimental results on UCI data sets and USPS data sets show that such strategy is effective in improving the structure of clustering results.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Afridi MK, Azam N, Yao JT, Alanazi E (2018) A three-way clustering approach for handling missing data using GTRS. Int J Approx Reason. https://doi.org/10.1016/j.ijar.2018.04.001

    Article  MathSciNet  Google Scholar 

  2. Bezdek J (1981) Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York

    Book  Google Scholar 

  3. Bezdek J, Pal N (1998) Some new indexes of cluster validity. IEEE Trans Syst Man Cybern B 28:301–315

    Article  Google Scholar 

  4. Blake CL, Merz CJ (2005) UCI machine learning repository. http://www.ics.uci.edu/mlearn/MLRepository.html

  5. Hao C, Li JH, Fan M, Liu WQ, Tsang ECC (2017) Optimal scale selection in dynamic multi-scale decision tables based on sequential three-way decisions. Inf Sci 415:213–232

    Article  Google Scholar 

  6. Huang CC, Li JH, Mei CL, Wu WZ (2017) Three-way concept learning based on cognitive operators: an information fusion viewpoint. Int J Approx Reason 83:218–242

    Article  MathSciNet  Google Scholar 

  7. Jain AK (2010) Data clustering: 50 years beyond k-means. Pattern Recognit Lett 31:651–666

    Article  Google Scholar 

  8. Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Comput Surv 31:264–323

    Article  Google Scholar 

  9. Kaufman L, Rousseeuw PJ (1990) Finding groups in data: an introduction to cluster analysis. Wiley, New York

    Book  Google Scholar 

  10. Lang GM, Miao DQ, Cai MJ (2017) Three-way decision approaches to conflict analysis using decision-theoretic rough set theory. Inf Sci 406:185–207

    Article  Google Scholar 

  11. LeCun Y, Bottou L, Bengio Y, Haffner P (1990) USPS zip code handwritten digits database. http://www.ics.uci.edu/mlearn/MLRepository.html

  12. Li CP, Li JH, He M (2016) Concept lattice compression in incomplete contexts based on k-medoids clustering. Int J Mach Learn Cybern 7:539–552

    Article  Google Scholar 

  13. Li HX, Zhou XZ (2011) Risk decision making based on decision-theoretic rough set: a three-way view decision model. Int J Comput Inf Sys 4:1–11

    Article  Google Scholar 

  14. Li JH, Huang CC, Qi JJ, Qia YH, Liu WQ (2017) Three-way cognitive concept learning via multi-granularity. Inf Sci 378:244–263

    Article  Google Scholar 

  15. Li W, Miao DQ, Wang WL, Zhang N (2010) Hierarchical rough decision theoretic framework for text classification. In: IEEE international conference on cognitive informatics, pp 484–489

  16. Li Y, Zhang C, Swan JR (2000) An information filtering model on the web and its application in jobagent. Knowl-Based Syst 13:285–296

    Article  Google Scholar 

  17. Liang DC, Liu D (2015) A novel risk decision-making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans Fuzzy Syst 23:237–247

    Article  Google Scholar 

  18. Lingras P, West C (2004) Interval set clustering of web users with rough k-means. J Intell Inf Syst 23:5–16

    Article  Google Scholar 

  19. Liu D, Li TR, Liang DC (2012) Three-way government decision analysis with decision-theoretic rough sets. Int J Uncertain Fuzz 20:119–132

    Article  Google Scholar 

  20. Liu D, Yao YY, Li TR (2011) Three-way investment decisions with decision-theoretic rough sets. Int J Comput Inf Sys 4:66–74

    Article  Google Scholar 

  21. Macqueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley symposium on mathematical statistics and probability, pp 281–197

  22. Maulik U, Bandyopadhyay S (2002) Performance evaluation of some clustering algorithms and validity indices. IEEE Trans Pattern Anal 24:1650–1654

    Article  Google Scholar 

  23. Mirkin B (1991) Mathematical classification and clustering. Kluwer, Boston

    MATH  Google Scholar 

  24. Mitra S, Banka H, Pedrycz W (2006) Rough-fuzzy collaborative clustering. IEEE Trans Syst Man Cybern B 36:795–805

    Article  Google Scholar 

  25. Mitra S, Pedrycz W, Barman B (2010) Shadowed c-means: integrating fuzzy and rough clustering. Pattern Recognit 43:1282–1291

    Article  Google Scholar 

  26. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:314–356

    Article  Google Scholar 

  27. Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer, Boston

    Book  Google Scholar 

  28. Pawlak Z (2004) Some issues on rough sets. Trans Rough Sets I 3100:1–58

    Article  Google Scholar 

  29. Pedrycz W (1998) Shadowed sets: representing and processing fuzzy sets. IEEE Trans Syst Man Cybern B 28:103–109

    Article  Google Scholar 

  30. Qi JJ, Qian T, Wei L (2016) The connections between three-way and classical concept lattices. Knowl-Based Syst 91:143–151

    Article  Google Scholar 

  31. Rousseeuw P (1987) Silhouettes: a graphical aid to the interpreta-tion and validation of cluster analysis. J Comput Math Appl Math 20:53–65

    Article  Google Scholar 

  32. Shivhare R, Cherukuri AK (2017) Three-way conceptual approach for cognitive memory functionalities. Int J Mach Learn Cybern 8:21–34

    Article  Google Scholar 

  33. Singh PK (2016) Three-way fuzzy concept lattice representation using neutrosophic set. Int J Mach Learn Cybern 8:1–11

    Google Scholar 

  34. Singh PK (2017) Interval-valued neutrosophic graph representation of concept lattice and its (\(\alpha,\beta,\gamma\))-decomposition. Arab J Sci Eng 43:1–18

    Google Scholar 

  35. Singh PK (2018) Similar vague concepts selection using their Euclidean distance at different granulation. Cogn Comput 10:228–241

    Article  Google Scholar 

  36. Singh PK (2018) Concept learning using vague concept lattice. Neural Process Lett 48:31–52

    Article  Google Scholar 

  37. Singh PK (2017) Medical diagnoses using three-way fuzzy concept lattice and their euclidean distance. Comp Appl Math 3:1–24

    Article  Google Scholar 

  38. Tou JT, Gonzalez RC (1974) Pattern recognition principles. Addison-Wesley, London

    MATH  Google Scholar 

  39. Wang PX, Yao YY (2018) CE3: A three-way clustering method based on mathematical morphology. Knowl-Based Syst 155:54–65

    Article  Google Scholar 

  40. Xu R, Wunsch DC (2005) Survey of clustering algorithms. IEEE Trans Neural Netw 16:645–678

    Article  Google Scholar 

  41. Yang X, Li TR, Fujita H, Liu D, Yao YY (2017) A unified model of sequential three-way decisions and multilevel incremental processing. Knowl-Based Syst 134:172–188

    Article  Google Scholar 

  42. Yao JT (2015) Web-based medical decision support systems for three-way medical decision making with game-theoretic rough sets. IEEE Trans Fuzzy Syst 23:3–15

    Article  Google Scholar 

  43. Yao YY (2009) Three-way decision: an interpretation of rules in rough set theory. In: Proceedings of RSKT’09, vol 5589, pp 642–649

    Chapter  Google Scholar 

  44. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353

    Article  MathSciNet  Google Scholar 

  45. Yao YY (2011) The superiority of three-way decisions in probabilistic rough set models. Inf Sci 181:1080–1096

    Article  MathSciNet  Google Scholar 

  46. Yao YY (2012) An outline of a theory of three-way decisions. In: Proceedings of RSCTC’12, vol 7413, pp 1–17

    Chapter  Google Scholar 

  47. Yao YY (2016) Three-way decisions and cognitive computing. Cogn Comput 8:543–554

    Article  Google Scholar 

  48. Yao YY (2017) Interval sets and three-way concept analysis in incomplete contexts. Int J Mach Learn Cybern 8:3–20

    Article  Google Scholar 

  49. Yu H (2017) A framework of three-way cluster analysis. In: Proceedings of international joint conference on rough sets, pp 300–312

    Chapter  Google Scholar 

  50. Yu H, Jiao P, Yao YY, Wang GY (2016) Detecting and refining overlapping regions in complex networks with three-way decisions. Inf Sci 373:21–41

    Article  Google Scholar 

  51. Yu H, Wang XC , Wang GY, Zeng XH (2018) An active three-way clustering method via low-rank matrices for multi-view data. Inf Sci. https://doi.org/10.1016/j.ins.2018.03.009

    Article  MathSciNet  Google Scholar 

  52. Yu H, Zhang C, Wang GY (2016) A tree-based incremental overlapping clustering method using the three-way decision theory. Knowl-Based Syst 91:189–203

    Article  Google Scholar 

  53. Zhang QH, Lv GX, Chen YH, Wang GY (2018) A dynamic three-way decision model based on the updating of attribute values. Knowl-Based Syst 142:71–84

    Article  Google Scholar 

  54. Zhang QH, Xia DY, Wang GY (2017) Three-way decision model with two types of classification errors. Inf Sci 420:431–453

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for their constructive and valuable comments. This work was supported in part by National Natural Science Foundation of China (nos. 61503160, 61773012 and 61572242), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 15KJB110004).

Author information

Authors and Affiliations

  1. School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, Jiangsu, People’s Republic of China

    Pingxin Wang

  2. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Pingxin Wang & Jusheng Mi

  3. School of Computer Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, People’s Republic of China

    Hong Shi & Xibei Yang

Authors
  1. Pingxin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xibei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jusheng Mi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pingxin Wang.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, P., Shi, H., Yang, X. et al. Three-way k-means: integrating k-means and three-way decision. Int. J. Mach. Learn. & Cyber. 10, 2767–2777 (2019). https://doi.org/10.1007/s13042-018-0901-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-018-0901-y

Keywords

Search

Navigation