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On the convergence of some possibilistic clustering algorithms

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Abstract

In this paper, an analysis of the convergence performance is conducted for a class of possibilistic clustering algorithms (PCAs) utilizing the Zangwill convergence theorem. It is shown that under certain conditions the iterative sequence generated by a PCA converges, at least along a subsequence, to either a local minimizer or a saddle point of the objective function of the algorithm. The convergence performance of more general PCAs is also discussed.

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References

  • Bezdek, J. C. (1980). A Convergence theorem for the fuzzy ISODATA clustering algorithms. IEEE Transactions on Pattern Analysis and Machine Intellegence, PAMI-2(1), 1–8.

    Google Scholar 

  • Dave, R. N., & Krishnapuram, R. (1997). Robust clustering methods: a unified view. IEEE Transactions on Fuzzy Systems, 5(2), 270–293.

    Article  Google Scholar 

  • Dey, V., Pratihar, D. K., & Datta, G. L. (2011). Genetic algorithm-tuned entropy-based fuzzy C-means algorithm for obtaining distinct and compact clusters. Fuzzy Optimization and Decision Making, 10(2), 153–166.

    Article  MathSciNet  Google Scholar 

  • Hathaway, R. J., Bezdek, J. C., & Tucker, W. T. (1987). An improved convergence theory for the fuzzy ISODATA clustering algorithms, the analysis of fuzzy information (Vol. 3, pp. 123–132). Boca Raton: CRC Press.

  • Höppner, F., & Klawonn, F. (2003). A contribution to convergence theory of fuzzy \(c\)-means and derivatives. IEEE Transactions on Fuzzy Systems, 11(5), 682–694.

    Article  Google Scholar 

  • Krishnapuram, R., & Keller, J. M. (1993). A possibilistic approach to clustering. IEEE Transactions on Fuzzy Systems, 1(2), 98–110.

    Article  Google Scholar 

  • Krishnapuram, R., & Keller, J. M. (1996). The possibilistic \(c\)-means algorithm: insights and recommendations. IEEE Transactions on Fuzzy Systems, 4(3), 385–393.

    Article  Google Scholar 

  • Krishnapuram, R., Frigui, H., & Nasraoui, O. (1995). Fuzzy and possibilistic shell clustering algorihm and their application to boundary detection and surface approximation. IEEE Transactions on Fuzzy Systems, 3, 29–60.

    Article  Google Scholar 

  • Oussalah, M., & Nefti, S. (2008). On the use of divergence distance in fuzzy clustering. Fuzzy Optimization and Decision Making, 7(2), 147–167.

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, M.-S., & Wu, K.-L. (2006). Unsupervised possibilistic clustering. Pattern Recognition, 39(1), 5–21.

    Article  Google Scholar 

  • Zangwill, W. (1969). Nolinear programming: a unified approach. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Zhang, Y., & Chi, Z.-X. (2008). A fuzzy support vector classifier based on Bayesian optimization. Fuzzy Optimization and Decision Making, 7(1), 75–86.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, J., & Hung, C. C. (2007). A generalized approach to possibilistic clustering algorithms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15(2), 117–138.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the Shanghai Philosophy and Social Science Planning Project grant (2012BGL006), Australian Research Council Discovery Grants (DP1096218 and DP130102691) and Linkage Grants (LP100200774 and LP120100566).

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Authors and Affiliations

  1. School of Management, Shanghai University, Shanghai, 200444, China

    Jian Zhou

  2. Faculty of Engineering and Information Technology, University of Technology, Sydney, Australia

    Longbing Cao

  3. School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Nan Yang

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  1. Jian Zhou
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  2. Longbing Cao
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  3. Nan Yang
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Correspondence to Nan Yang.

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Zhou, J., Cao, L. & Yang, N. On the convergence of some possibilistic clustering algorithms. Fuzzy Optim Decis Making 12, 415–432 (2013). https://doi.org/10.1007/s10700-013-9159-8

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  • DOI: https://doi.org/10.1007/s10700-013-9159-8

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