Abstract
Whenever we classify a dataset into some clusters, we need to consider how to handle the uncertainty included into data. In those days, the ability of computers are very poor, and we could not help handling data with uncertainty as one point. However, the ability is now enough to handle the uncertainty of data, and we hence believe that we should handle the uncertain data as is. In this paper, we will show some clustering methods for uncertain data by two concepts of “tolerance” and “penalty-vector regularization”. The both concepts are more useful to model and handle the uncertainty of data and more flexible than the conventional methods for handling the uncertainty. By the way, we construct a clustering algorithm by putting one objective function. Hence, we can say that the whole clustering algorithm depends on its objective function. In this paper, we will thereby introduce various types of objective functions for uncertain data with the concepts of tolerance and penalty-vector regularization, and construct the clustering algorithms for uncertain data.
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References
Bezdek, J. C. (1981). Pattern Recognition with Fuzzy Objective Function Algorithms. New York: Plenum.
Endo, Y., Hasegawa, Y., Hamasuna, Y., & Miyamoto, S. (2008). Fuzzy c-means for data with rectangular maximum tolerance range. Journal of Advanced Computational Intelligence and Intelligent Informatics, 12(5), 461–466.
Endo, Y., Murata, R., Haruyama, H., & Miyamoto, S. (2005). Fuzzy c-means for data with tolerance. In Proc. 2005 International Symposium on Nonlinear Theory and Its Applications, pp. 345–348.
Hasegawa, Y., Endo, Y., & Hamasuna, Y. (2008). On fuzzy c-means for data with uncertainty using spring modulus, SCIS&ISIS 2008.
Jajuga, K. (1991). L 1-norm based fuzzy clustering. Fuzzy Sets and Systems, 39, 43–50.
Kanzawa, Y., Endo, Y., & Miyamoto, S. (2007). Fuzzy c-means algorithms for data with tolerance based on opposite criterions. IEICE Transactions Fundamentals, E90-A(10), 2194–2202.
Kanzawa, Y., Endo, Y., & Miyamoto, S. (2008). Fuzzy c-means algorithms for data with tolerance using kernel functions. IEICE Transactions Fundamentals, E91-A(9), 2520–2534.
Mercer, J. (1909). Functions of positive and negative type and their connection with the theory of integral equations. Philosophical Transactions of the Royal Society A, 209, 415–446.
Miyamoto, S., Kawasaki, Y., & Sawazaki, K. (2009). An explicit mapping for fuzzy c-means using kernel function and application to text analysis, IFSA/EUSFLAT 2009.
Miyamoto, K., & Mukaidono, M. (1997). Fuzzy c-means as a regularization and maximum entropy approach. In Proc. of the 7th International Fuzzy Systems Association World Congress (IFSA’97), Vol. 2, pp. 86–92.
Miyamoto, S., Umayahara, K., & Mukaidono, M. (1998). Fuzzy classification functions in the methods of fuzzy c-means and regularization by entropy. Journal of Japan Society for Fuzzy Theory and Systems, 10(3), 548–557.
Murata, R., Endo, Y., Haruyama, H., & Miyamoto, S. (2006). On fuzzy c-means for data with tolerance. Journal of Advanced Computational Intelligence and Intelligent Informatics, 10(5), 673–681.
Schölkopf, B., Smola, A., & Müller, K.-R. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10, 1299–1319.
Vapnik, V. N. (1998). Statistical Learning Theory. New York: Wiley.
Vapnik, V. N. (2000). The Nature of Statistical Learning Theory, 2nd edn. New York: Springer.
Acknowledgements
This study is partly supported by the Grant-in-Aid for Scientific Research (C) (Project No.21500212) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Endo, Y., Miyamoto, S. (2012). Various Types of Objective Functions of Clustering for Uncertain Data. In: Ermoliev, Y., Makowski, M., Marti, K. (eds) Managing Safety of Heterogeneous Systems. Lecture Notes in Economics and Mathematical Systems, vol 658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22884-1_12
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DOI: https://doi.org/10.1007/978-3-642-22884-1_12
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