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Optimizing the C Index Using a Canonical Genetic Algorithm

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Applications of Evolutionary Computation (EvoApplications 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11454))

Abstract

Clustering is an important family of unsupervised machine learning methods. Cluster validity indices are widely used to assess the quality of obtained clustering results. The C index is one of the most popular cluster validity indices. This paper shows that the C index can be used not only to validate but also to actually find clusters. This leads to difficult discrete optimization problems which can be approximately solved by a canonical genetic algorithm. Numerical experiments compare this novel approach to the well-known c-means and single linkage clustering algorithms. For all five considered popular real-world benchmark data sets the proposed method yields a better C index than any of the other (pure) clustering methods.

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References

  1. Jain, A.K., Dubes, R.C.: Algorithms for Clustering Data. Prentice Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  2. Duda, R.O., Hart, P.E.: Pattern Classification and Scene Analysis. Wiley, New York (1973)

    MATH  Google Scholar 

  3. Sneath, P., Sokal, R.: Numerical Taxonomy. Freeman, San Francisco (1973)

    MATH  Google Scholar 

  4. MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Berkeley Symposium on Mathematical Statistics and Probability, vol. 14, pp. 281–297 (1967)

    Google Scholar 

  5. Hruschka, E.R., Campello, R.J.G.B., Freitas, A.A., de Carvalho, A.C.P.L.F.: A survey of evolutionary algorithms for clustering. IEEE Trans. Syst. Man Cybern. Part C (Appl. Rev.) 39(2), 133–155 (2009)

    Article  Google Scholar 

  6. Handl, J., Knowles, J., Dorigo, M.: Strategies for the increased robustness of ant-based clustering. In: Di Marzo Serugendo, G., Karageorgos, A., Rana, O.F., Zambonelli, F. (eds.) ESOA 2003. LNCS (LNAI), vol. 2977, pp. 90–104. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24701-2_7

    Chapter  Google Scholar 

  7. Kanade, P.M., Hall, L.O.: Fuzzy ants as a clustering concept. In: NAFIPS International Conference, Chicago, pp. 227–232, July 2003

    Google Scholar 

  8. Ji, C., Zhang, Y., Gao, S., Yuan, P., Li, Z.: Particle swarm optimization for mobile ad hoc networks clustering. In: International Conference on Networking, Sensing and Control, Taipeh, Taiwan, pp. 372–375, March 2004

    Google Scholar 

  9. Omran, M.G.H., Salman, A., Engelbrecht, A.P.: Dynamic clustering using particle swarm optimization with application in image segmentation. Pattern Anal. Appl. 8(4), 332 (2006)

    Article  MathSciNet  Google Scholar 

  10. Tillett, J., Rao, R., Sahin, F., Rao, T.M.: Particle swarm optimization for the clustering of wireless sensors. Digit. Wirel. Commun. 5100, 73–83 (2003)

    Google Scholar 

  11. Xiao, X., Dow, E.R., Eberhart, R., Miled, Z.B., Oppelt, R.J.: Gene clustering using self-organizing maps and particle swarm optimization. In: International Parallel and Distributed Processing Symposium, Nice, France, April 2003

    Google Scholar 

  12. Dalrymple-Alford, E.C.: Measurement of clustering in free recall. Psychol. Bull. 74(1), 32 (1970)

    Article  Google Scholar 

  13. Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985)

    Article  Google Scholar 

  14. Hubert, L.J., Levin, J.R.: A general statistical framework for assessing categorical clustering in free recall. Psychol. Bull. 83(6), 1072 (1976)

    Article  Google Scholar 

  15. Bezdek, J.C., Moshtaghi, M., Runkler, T.A., Leckie, C.: The generalized C index for (internal) fuzzy cluster validity. IEEE Trans. Fuzzy Syst. 24(6), 1500–1512 (2017)

    Article  Google Scholar 

  16. Dunn, J.C.: Well-separated clusters and optimal fuzzy partitions. J. Cybern. 4(1), 95–104 (1974)

    Article  MathSciNet  Google Scholar 

  17. Rathore, P., Ghafoori, Z., Bezdek, J.C., Palaniswami, M., Leckie, C.: Approximating Dunn’s cluster validity indices for partitions of big data. IEEE Trans. Cybern. 9, 1–13 (2019)

    Google Scholar 

  18. Davies, D.L., Bouldin, D.W.: A cluster separation measure. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-1 2, 224–227 (1979)

    Article  Google Scholar 

  19. Rousseeuw, P.J.: Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20, 53–65 (1987)

    Article  Google Scholar 

  20. Ibrahim, O.A., Keller, J.M., Bezdek, J.C.: Analysis of streaming clustering using an incremental validity index. In: IEEE International Conference on Fuzzy Systems, Rio de Janeiro, Brazil, pp. 1–8 (2018)

    Google Scholar 

  21. Moshtaghi, M., Bezdek, J.C., Erfani, S.M., Leckie, C., Bailey, J.: Online cluster validity indices for performance monitoring of streaming data clustering. Int. J. Intell. Syst. 34, 541–563 (2019)

    Article  Google Scholar 

  22. Holland, J.H.: Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  23. Hartigan, J.A.: Clustering Algorithms. Wiley, New York (1975)

    MATH  Google Scholar 

  24. Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-hardness of Euclidean sum-of-squares clustering. Mach. Learn. 75(2), 245–248 (2009)

    Article  Google Scholar 

  25. Brucker, P.: On the complexity of clustering problems. In: Henn, R., Korte, B., Oettli, W. (eds.) Optimization and Operations Research. LNE, vol. 157, pp. 45–54. Springer, Heidelberg (1978). https://doi.org/10.1007/978-3-642-95322-4_5

    Chapter  Google Scholar 

  26. Bezdek, J.C., Hathaway, R.J.: Convergence of alternating optimization. Neural Parallel Sci. Comput. 11(4), 351–368 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Runkler, T.A., Bezdek, J.C., Hall, L.O.: Clustering very large data sets: The complexity of the fuzzy c-means algorithm. In: European Symposium on Intelligent Technologies, Hybrid Systems and their implementation on Smart Adaptive Systems, Albufeira, Portugal, pp. 420–425, September 2002

    Google Scholar 

  28. Arthur, D., Vassilvitskii, S.: k-means++: The advantages of careful seeding. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1027–1035 (2007)

    Google Scholar 

  29. Rose, K., Gurewitz, E., Fox, G.: A deterministic annealing approach to clustering. Pattern Recogn. Lett. 11(9), 589–594 (1990)

    Article  Google Scholar 

  30. Krishna, K., Murty, M.N.: Genetic k-means algorithm. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 29(3), 433–439 (1999)

    Article  Google Scholar 

  31. Chen, C.Y., Ye, F.: Particle swarm optimization and its application to clustering analysis. In: International Conference on Networking, Sensing and Control, Taipeh, Taiwan, pp. 789–794, March 2004

    Google Scholar 

  32. Cui, X., Potok, T.E., Palathingal, P.: Document clustering using particle swarm optimization. In: IEEE Swarm Intelligence Symposium, Pasadena, pp. 185–191, June 2005

    Google Scholar 

  33. Runkler, T.A., Katz, C.: Fuzzy clustering by particle swarm optimization. In: IEEE International Conference on Fuzzy Systems, Vancouver, pp. 3065–3072, July 2006

    Google Scholar 

  34. van der Merwe, D.W., Engelbrecht, A.P.: Data clustering using particle swarm optimization. IEEE Congr. Evol. Comput. 1, 215–220 (2003)

    Google Scholar 

  35. Runkler, T.A.: Ant colony optimization of clustering models. Int. J. Intell. Syst. 20(12), 1233–1261 (2005)

    Article  Google Scholar 

  36. Runkler, T.A.: Wasp swarm optimization of the c-means clustering model. Int. J. Intell. Syst. 23(3), 269–285 (2008)

    Article  Google Scholar 

  37. Tambouratzis, G., Tambouratzis, T., Tambouratzis, D.: Clustering with artificial neural networks and traditional techniques. Int. J. Intell. Syst. 18(4), 405–428 (2003)

    Article  Google Scholar 

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

    Book  Google Scholar 

  39. Krishnapuram, R., Keller, J.M.: A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst. 1(2), 98–110 (1993)

    Article  Google Scholar 

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

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Siemens AG, Corporate Technology, 80200, Munich, Germany

    Thomas A. Runkler

  2. University of Melbourne, Parkville, VIC, 3010, Australia

    James C. Bezdek

Authors
  1. Thomas A. Runkler
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  2. James C. Bezdek
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Corresponding author

Correspondence to Thomas A. Runkler .

Editor information

Editors and Affiliations

  1. University of Mainz, Mainz, Germany

    Paul Kaufmann

  2. University of Granada, Granada, Spain

    Pedro A. Castillo

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Runkler, T.A., Bezdek, J.C. (2019). Optimizing the C Index Using a Canonical Genetic Algorithm. In: Kaufmann, P., Castillo, P. (eds) Applications of Evolutionary Computation. EvoApplications 2019. Lecture Notes in Computer Science(), vol 11454. Springer, Cham. https://doi.org/10.1007/978-3-030-16692-2_19

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  • DOI: https://doi.org/10.1007/978-3-030-16692-2_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-16691-5

  • Online ISBN: 978-3-030-16692-2

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