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Fuzzy Clustering

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An Introduction to Clustering with R

Part of the book series: Behaviormetrics: Quantitative Approaches to Human Behavior ((BQAHB,volume 1))

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Abstract

Fuzzy clustering methods produce a soft partition of units. Unlike standard methods, each unit is assigned to a cluster according to a membership degree that takes value in the interval [0, 1]. Starting from the most known algorithm, the Fuzzy k-Means, in the last decades, several variants have been proposed. By relaxing the unit-sum constraints of the membership degrees, we move from the fuzzy to the possibilistic approach. In this case, the membership degrees are better identified as typicality degrees. In this chapter, fuzzy and possibilistic clustering methods will be first briefly introduced from a theoretical point of view, and after their application to benchmark case studies will be presented.

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References

  1. Abelson, R.P., Sermat, V.: Multidimensional scaling of facial expressions. J. Exp. Psychol. Gen. 63, 546–554 (1962)

    Article  Google Scholar 

  2. Anderson, E.: The irises of the Gaspe Peninsula. Bull. Am. Iris Soc. 59, 2–5 (1936)

    Google Scholar 

  3. Babuska, R., Van der Veen, P.J., Kaymak, U.: Improved covariance estimation for Gustafson-Kessel clustering. In: Proceedings of the 2002 IEEE International Conference on Fuzzy Systems, pp. 1081–1085 (2002)

    Google Scholar 

  4. Bagus, F.A.F., Pramana, S.: advclust: Object Oriented Advanced Clustering. R package version 0.4 (2016). https://CRAN.R-project.org/package=advclust

  5. Barni, M., Cappellini, V., Mecocci, A.: Comments on ‘A possibilistic approach to clustering’. IEEE T. Fuzzy Syst. 4, 393–396 (1996)

    Article  Google Scholar 

  6. Bezdek, J.C.: Cluster validity with fuzzy sets. J. Cybern. 3, 58–73 (1974)

    Article  MathSciNet  Google Scholar 

  7. Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithm. Plenum Press, New York (1981)

    Book  Google Scholar 

  8. Campello, R.J.G.B.: A fuzzy extension of the Rand index and other related indexes for clustering and classification assessment. Pattern Recognit. Lett. 28, 833–841 (2007)

    Article  Google Scholar 

  9. Campello, R.J.G.B., Hruschka, E.R.: A fuzzy extension of the silhouette width criterion for cluster analysis. Fuzzy Sets Syst. 157, 2858–2875 (2006)

    Article  MathSciNet  Google Scholar 

  10. Cebeci, Z.: Comparison of internal validity indices for fuzzy clustering. J. Agr. Inform. 10, 1–14 (2019)

    Google Scholar 

  11. Charytanowicz, M., Niewczas, J., Kulczycki, P., Kowalski, P.A., Lukasik, S., Zak, S.: A complete gradient clustering algorithm for features analysis of X-ray images. In: Pietka, E., Kawa, J. (eds.) Information Technologies in Biomedicine, pp. 15–24. Springer-Verlag, Berlin (2010)

    Chapter  Google Scholar 

  12. Chavent, M., Kuentz, V., Labenne, A., Liquet, B., Saracco, J.: PCAmixdata: Multivariate Analysis of Mixed Data. R package version 3.1 (2017). https://CRAN.R-project.org/package=PCAmixdata

  13. Chou, C.-H., Su, M.-C.: A modified version of the \(K\)-means algorithm with a distance based on cluster symmetry. IEEE T. Pattern Anal. Mach. Intell. 23, 674–680 (2001)

    Article  Google Scholar 

  14. Davé, R.N.: Characterization and detection of noise in clustering. Pattern Recognit. Lett. 12, 657–664 (1991)

    Article  Google Scholar 

  15. Davé, R.N.: Validating fuzzy partitions obtained through c-shells clustering. Pattern Recognit. Lett. 17, 613–623 (1996)

    Article  Google Scholar 

  16. Davé, R.N., Sen, S.: Robust fuzzy clustering of relational data. IEEE T. Fuzzy Syst. 10, 713–727 (2002)

    Article  Google Scholar 

  17. de Leeuw, J., Mair, P.: Multidimensional scaling using majorization: SMACOF in R. J. Stat. Softw. 31, 1–30 (2009)

    Google Scholar 

  18. Di Lorenzo, P.: usmap: US Maps including Alaska and Hawaii. R package version 0.5.0 (2019). https://CRAN.R-project.org/package=usmap

  19. Engen, B., Levy, N., Schlossberg, H.: The dimensional analysis of a new series of facial expressions. J. Exp. Psychol. Gen. 55, 454–458 (1958)

    Article  Google Scholar 

  20. Ferraro, M.B., Giordani, P.: A toolbox for fuzzy clustering using the R programming language. Fuzzy Sets Syst. 279, 1–16 (2015)

    Article  MathSciNet  Google Scholar 

  21. Ferraro, M.B., Giordani, P., Serafini, A.: fclust: an R package for fuzzy clustering. R J. 11(1), 205–233 (2019)

    Article  Google Scholar 

  22. Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eugen. 7, 179–188 (1935)

    Google Scholar 

  23. Fukuyama, Y., Sugeno, M.: A new method of choosing the number of clusters for the fuzzy \(c\)-means method. In: Proceedings of the 5th Fuzzy System Symposium, pp. 247-250 (1989) (in Japanese)

    Google Scholar 

  24. Gath, I., Geva, A.B.: Unsupervised optimal fuzzy clustering. IEEE T. Pattern Anal. Mach. Intell. 11, 773–781 (1989)

    Article  Google Scholar 

  25. Giordani, P., Ferraro, M.B., Martella, F.: datasetsICR: Datasets from the Book “An Introduction to Clustering with R”, R package version 1.0 (2020). https://CRAN.R-project.org/package=datasetsICR

  26. Gustafson, D.E., Kessel, W.C.: Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings of the 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes, pp. 761–766 (1979)

    Google Scholar 

  27. Hathaway, R.J.: Another interpretation of the EM algorithm for mixture distributions. Stat. Probab. Lett. 4, 53–56 (1986)

    Article  MathSciNet  Google Scholar 

  28. Henderson, H.V., Velleman, P.F.: Building multiple regression models interactively. Biometrics 37, 391–411 (1981)

    Article  Google Scholar 

  29. Kaufman, L., Rousseuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York (1990)

    Book  Google Scholar 

  30. Klawonn, F., Chekhtman, V., Janz, E.: Visual inspection of fuzzy clustering results. In: Benitez, J.M., Cordon, O., Hoffmann, F., Roy, R. (eds.) Advances in Soft Computing - Engineering Design and Manufacturing, pp. 65–76. Springer, London (2003)

    Google Scholar 

  31. Klawonn, F., Höppner, F.: An alternative approach to the fuzzifier in fuzzy clustering to obtain better clustering. In: Proceedings of Eusflat Conference, pp. 730–734 (2003)

    Google Scholar 

  32. Klawonn, F., Kruse, R., Winkler, R.: Fuzzy clustering: more than just fuzzification. Fuzzy Sets Syst. 281, 272–279 (2015)

    Article  MathSciNet  Google Scholar 

  33. Krishnapuram, R., Joshi, A., Nasraoui, O., Yi, L.: Low-complexity fuzzy relational clustering algorithms for web mining. IEEE T. Fuzzy Syst. 9, 595–607 (2001)

    Article  Google Scholar 

  34. Krishnapuram, R., Keller, J.M.: A possibilistic approach to clustering. IIEEE T. Fuzzy Syst. 1, 98–110 (1993)

    Article  Google Scholar 

  35. Krishnapuram, R., Keller, J.M.: The possibilistic \(c\)-means algorithm: insights and recommendations. IEEE T. Fuzzy Syst. 4, 385–393 (1996)

    Article  Google Scholar 

  36. Kwon, S.H.: Cluster validity index for fuzzy clustering. Electron. Lett. 34, 2176–2177 (1998)

    Article  Google Scholar 

  37. Li, R.-P., Mukaidono, M.: A maximum-entropy approach to fuzzy clustering. In: Proceedings of 1995 IEEE International Conference on Fuzzy Systems, pp. 2227–2232 (1995)

    Google Scholar 

  38. Li, R.-P., Mukaidono, M.: Gaussian clustering method based on maximum-fuzzy-entropy interpretation. Fuzzy Sets Syst. 102, 253–258 (1999)

    Article  MathSciNet  Google Scholar 

  39. MacQueen, J.: Some methods for classification and analysis of multivariate observations. Proceedings of the Fifth Berkeley Symposium on Mathematics, Statistics and Probability, vol. 1, pp. 281–298 (1967)

    Google Scholar 

  40. Maechler, M., Rousseeuw, P., Struyf, A., Hubert, M., Hornik, K.: cluster: Cluster Analysis Basics and Extensions. R package version 2.1.0 (2019)

    Google Scholar 

  41. Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., Leisch, F., Chang, C.-C., Lin, C.-C.: e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.7-3 (2019). https://CRAN.R-project.org/package=e1071

  42. Pal, N.R., Pal, K., Bezdek, J.C.: A mixed \(c\)-means clustering model. In: Proceedings of FUZZ-IEEE’97, pp. 11–21 (1997)

    Google Scholar 

  43. Pal, N.R., Pal, K., Keller, J.M., Bezdek, J.C.: A possibilistic fuzzy \(c\)-means clustering algorithm. IEEE T. Fuzzy Syst. 13, 517–530 (2005)

    Article  Google Scholar 

  44. R Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna (2020). https://www.R-project.org

  45. Ruspini, E.H.: Numerical methods for fuzzy clustering. Inf. Sci. 2, 319–350 (1970)

    Article  Google Scholar 

  46. Saad, M.F., Alimi, A.M.: Modified fuzzy possibilistic \(c\)-means. In: Proceedings of the International Multiconference of Engineers and Computer Scientists, pp. 18–20 (2009)

    Google Scholar 

  47. Sarkar, D., Andrews, F., Wright, K., Klepeis, N., Murrell, P.: lattice: Trellis graphics for R, R package version 0.20-41 (2020). https://CRAN.R-project.org/package=lattice

  48. Tang, Y.G., Sun, F.C., Sun, Z.Q.: Improved validation index for fuzzy clustering. In: American Control Conference, June 8–10, Portland, OR, USA (2005)

    Google Scholar 

  49. Timm, H., Borgelt, C., Döring, C., Kruse, R.: An extension to possibilistic fuzzy cluster analysis. Fuzzy Sets Syst. 147, 3–16 (2004)

    Article  MathSciNet  Google Scholar 

  50. Wachs, J., Shapira, O., Stern, H.: A method to enhance the ‘possibilistic \(C\)-means with repulsion’ algorithm based on cluster validity index. In: Abraham, A., de Baets, B., Köppen, M., Nickolay, B. (eds.) Applied Soft Computing Technologies: The Challenge of Complexity. Advances in Soft Computing, vol. 34. Springer, Berlin (2006)

    Google Scholar 

  51. Wang, W., Zhang, Y.: On fuzzy cluster validity indices. Fuzzy Sets Syst. 158, 2095–2117 (2007)

    Article  MathSciNet  Google Scholar 

  52. Wickham, H.: ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag, New York (2016)

    Book  Google Scholar 

  53. Winkler, R., Klawonn, F., Kruse, R.: Fuzzy clustering with polynomial fuzzifier function in connection with \(m\)-estimators. Appl. Comput. Math. 10, 146–163 (2011)

    MathSciNet  MATH  Google Scholar 

  54. Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE T. Pattern Anal. Mach. Intell. 13, 841–847 (1991)

    Article  Google Scholar 

  55. Yang, M.-S., Wu, K.-L.: Unsupervised possibilistic clustering. Pattern Recognit. 39, 5–21 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Rome, Italy

    Paolo Giordani, Maria Brigida Ferraro & Francesca Martella

Authors
  1. Paolo Giordani
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  2. Maria Brigida Ferraro
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  3. Francesca Martella
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Correspondence to Paolo Giordani .

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Giordani, P., Ferraro, M.B., Martella, F. (2020). Fuzzy Clustering. In: An Introduction to Clustering with R. Behaviormetrics: Quantitative Approaches to Human Behavior, vol 1. Springer, Singapore. https://doi.org/10.1007/978-981-13-0553-5_5

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