Abstract
In this study, a Bezdek-type fuzzified possibilistic clustering algorithm for spherical data (bPCS), its kernelization (K-bPCS), and spectral clustering approach (sK-bPCS) are proposed. First, we propose the bPCS by setting a fuzzification parameter of the Tsallis entropy-based possibilistic clustering optimization problem for spherical data (tPCS) to infinity, and by modifying the cosine correlation-based dissimilarity between objects and cluster centers. Next, we kernelize bPCS to obtain K-bPCS, which can be applied to non-spherical data with the help of a given kernel, e.g., a Gaussian kernel. Furthermore, we propose a spectral clustering approach to K-bPCS called sK-bPCS, which aims to solve the initialization problem of bPCS and K-bPCS. Furthermore, we demonstrate that this spectral clustering approach is equivalent to kernelized principal component analysis (K-PCA). The validity of the proposed methods is verified through numerical examples.
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References
Bezdek, J.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981)
MacQueen, J.B.: Some methods of classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 281–297 (1967)
Miyamoto, S., Mukaidono, M.: Fuzzy \(c\)-means as a regularization and maximum entropy approach. In: Proceedings of the 7th International Fuzzy Systems Association World Congress (IFSA 1997), vol. 2, pp. 86–92 (1997)
Ménard, M., Courboulay, V., Dardignac, P.: Possibilistic and probabilistic fuzzy clustering: unification within the framework of the non-extensive thermostatistics. Pattern Recogn. 36, 1325–1342 (2003)
Krishnapuram, R., Keller, J.M.: A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst. 1, 98–110 (1993)
Krishnapuram, R., Keller, J.M.: The possibilistic \(c\)-means algorithm: insights and recommendations. IEEE Trans. Fuzzy Syst. 4, 393–396 (1996)
Strehl, A., Ghosh, J., Mooney, R.: Impact of similarity measures on web-page clustering. In: Proceedings of the AAAI2000, pp. 58–64 (2000)
Dhillon, I.S., Modha, D.S.: Concept decompositions for large sparse text data using clustering. Mach. Learn. 42, 143–175 (2001)
Miyamoto, S., Mizutani, K.: Fuzzy multiset model and methods of nonlinear document clustering for information retrieval. In: Torra, V., Narukawa, Y. (eds.) MDAI 2004. LNCS (LNAI), vol. 3131, pp. 273–283. Springer, Heidelberg (2004)
Mizutani, K., Inokuchi, R., Miyamoto, S.: Algorithms of nonlinear document clustering based on fuzzy set model. Int. J. Intell. Syst. 23(2), 176–198 (2008)
Kanzawa, Y.: A maximizing model of Bezdek-like spherical fuzzy \(c\)-means. Int. J. Intell. Syst. 19(5), 662–669 (2015)
Kanzawa, Y.: On kernelization for a maximizing model of bezdek-like spherical fuzzy c-means clustering. In: Torra, V., Narukawa, Y., Endo, Y. (eds.) MDAI 2014. LNCS, vol. 8825, pp. 108–121. Springer, Heidelberg (2014)
Kanzawa, Y.: Fuzzy clustering based on \(\alpha \)-divergence for spherical data and for categorical multivariate data. In: Proceedings of the FUZZ-IEEE2015, #15091 (2015)
Kanzawa, Y.: On possibilistic clustering methods based on Shannon/Tsallis-entropy for spherical data and categorical multivariate data. In: Torra, V., Narukawa, T. (eds.) MDAI 2015. LNCS, vol. 9321, pp. 115–128. Springer, Heidelberg (2015)
Scholkopf, B., Smola, A., Muller, K.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)
Hornik, K., Feinerer, I., Kober, M., Buchta, C.: Spherical k-means clustering. J. Stat. Softw. 50(10), 1–22 (2012)
Zelnik-Manor, L., Perona, P.: Self-tuning spectral clustering. Adv. Neural Inf. Process. Syst. 17, 1601–1608 (2005)
Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2, 193–218 (1985)
Kanzawa, Y.: Sequential cluster extraction using power-regularized possibilistic \(c\)-means. JACIII 19(1), 67–73 (2015)
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Kanzawa, Y. (2016). On Bezdek-Type Possibilistic Clustering for Spherical Data, Its Kernelization, and Spectral Clustering Approach. In: Torra, V., Narukawa, Y., Navarro-Arribas, G., Yañez, C. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2016. Lecture Notes in Computer Science(), vol 9880. Springer, Cham. https://doi.org/10.1007/978-3-319-45656-0_15
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