Abstract
This chapter describes fuzzy clustering models. Fuzzy clustering models are typical examples of model-based clustering. The purpose of the model-based clustering is to obtain the optimal partition of objects by fitting the model to the observed similarity (or dissimilarity) of objects. The merit of model-based clustering is that we can obtain a mathematically clearer solution as the clustering result, because we know the mathematical features of the model. However, when we observe a large amount of complex data, it is difficult to fit the simple model to the data to obtain a useful result. In order to solve this problem, we have extended the model in the framework of fuzzy clustering models to adjust to the complexity caused from the recent variety of vast amounts of data. This chapter describes how we extend the fuzzy clustering models, along with the mathematical features of the several fuzzy clustering models. In particular, we describe the novel generalized aggregation operator defined on the product space of linear spaces and the generalized aggregation operator based nonlinear fuzzy clustering model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Wiley, New York (1958)
Ai-Ping, L., Yan, J., Quan-Yuan, W.: Harmonic triangular norm aggregation operators in multicriteria decision systems. J. Converg. Inf. Technol. 2(1), 83–92 (2007)
Banfield, J.D., Raftery, A.E.: Model-based Gaussian and non-Gaussian clustering. Biometrics 49, 803–821 (1993)
Beliakov, G.: How to build aggregation operators from data. Int. J. Intell. Syst. 18, 903–923 (2003)
Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981)
Bock, H.H.: Probabilistic models in cluster analysis. Comput. Stat. Data Anal. 23, 5–28 (1996)
Carroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via an N-generalization of the Eckart-Young decomposition. Psychometrika 35, 283–319 (1970)
Carroll, J.D., Arabie, P.: INDCLUS: an individual differences generalization of ADCLUS model and the MAPCLUS algorithm. Psychometrika 48, 157–169 (1983)
Clogg, C.C.: Some latent structure models for the analysis of Likert-type data. Soc. Sci. Res. 9, 287–301 (1979)
Clogg, C.C.: Latent structure models of mobility. Am. J. Sociol. 86(4), 836–868 (1981)
Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. Chapman & Hall (2001)
Friedman, H.P., Rubin, J.: On some invariant criteria for grouping data. J. Am. Stat. Assoc. 62, 1159–1178 (1967)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publications (2000)
Lazarsfeld, P.F.: The Interpretation and Mathematical Foundation of Latent Structure Analysis, Measurement and Prediction, pp. 413–472. Princeton University Press (1950)
Lazarsfeld, P.F., Henry, N.W.: Latent Structure Analysis. Houghton-Mifflin, Boston (1968)
Menger, K.: Statistical metrics. Proc. Natl. Acad. Sci. U.S.A. 28, 535–537 (1942)
Merigo, J.M., Casanovas, M.: Fuzzy generalized hybrid aggregation operators and its application in fuzzy decision making. Int. J. Fuzzy Syst. 12(1), 15–24 (2010)
Ruspini, E.H.: A new approach to clustering. Inf. Control 15, 22–32 (1969)
Sato, M., Sato, Y.: An additive fuzzy clustering model. Jpn. J. Fuzzy Theory Syst. 6, 185–204 (1994)
Sato, M., Sato, Y.: On a general fuzzy additive clustering model. Int. J. Intell. Autom. Soft Comput. 1(4), 439–448 (1995)
Sato-Ilic, M., Ito, S.: Kernel fuzzy clustering model. In: 24th Fuzzy System Symposium, pp. 153–154 (2008) (in Japanese)
Sato-Ilic, M., Ito, S., Takahashi, S.: Nonlinear kernel-based fuzzy clustering model. In: Viattchenin, D.A. (ed.) Developments in Fuzzy Clustering, pp. 56–73. VEVER, Minsk (Belarus) (2009)
Sato-Ilic, M.: Generalized aggregation operator based nonlinear fuzzy clustering model. In: Intelligent Engineering Systems Through Artificial Neural Networks, New York, USA, vol. 20, pp. 493–500 (2010)
Schonemann, P.H.: An algebraic solution for class of subjective metrics models. Psychometrika 37, 441–451 (1972)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Dover Publications (2005)
Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press (2004)
Shepard, R.N., Arabie, P.: Additive clustering: representation of similarities as combinations of discrete overlapping properties. Psychol. Rev. 86(2), 87–123 (1979)
Scott, A.J., Symons, M.J.: Clustering methods based on likelihood ratio criteria. Biometrics 27, 387–397 (1971)
Takane, Y., Kiers, H.A.L.: Latent class DEDICOM. J. Classif. 225–247 (1997)
Tardiff, R.M.: Topologies for probabilistic metric spaces. Pac. J. Math. 65(1), 233–251 (1976)
Tardiff, R.M.: On a functional inequality arising in the construction of the product of several metric spaces. Aequationes Math. 20, 51–58 (1980)
Zadeh, L.A.: Fuzzy Sets. Inf. Control 8, 338–353 (1965)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Sato-Ilic, M. (2021). Fuzzy Clustering Models and Their Related Concepts. In: Lesot, MJ., Marsala, C. (eds) Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications. Studies in Fuzziness and Soft Computing, vol 394. Springer, Cham. https://doi.org/10.1007/978-3-030-54341-9_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-54341-9_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-54340-2
Online ISBN: 978-3-030-54341-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)