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A Study of Cluster Validity Indices for Real-Life Data

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Artificial Intelligence and Soft Computing (ICAISC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10246))

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Abstract

In this paper a study of several cluster validity indices for real-life data sets is presented. Moreover, a new version of validity index is also proposed. All these indices can be considered as a measure of data partitioning accuracy and the performance of them is demonstrated for real-life data sets, where three popular algorithms have been applied as underlying clustering techniques, namely the Complete–linkage, Expectation Maximization and K-means algorithms. The indices have been compared taking into account the number of clusters in a data set. The results are useful to choose the best validity index for a given data set.

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References

  1. Arbelaitz, O., Gurrutxaga, I., Muguerza, J., Pérez, J.M., Perona, I.: An extensive comparative study of cluster validity indices. Pattern Recogn. 46, 243–256 (2013)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Bilski, J., Smoląg, J.: Parallel architectures for learning the RTRN and Elman dynamic neural networks. IEEE Trans. Parallel Distrib. Syst. 26(9), 2561–2570 (2015)

    Article  Google Scholar 

  4. Bilski, J., Wilamowski, B.M.: Parallel learning of feedforward neural networks without error backpropagation. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS, vol. 9692, pp. 57–69. Springer, Cham (2016). doi:10.1007/978-3-319-39378-0_6

    Google Scholar 

  5. Bilski, J., Kowalczyk, B., Żurada, J.M.: Application of the givens rotations in the neural network learning algorithm. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS, vol. 9692, pp. 46–56. Springer, Cham (2016). doi:10.1007/978-3-319-39378-0_5

    Google Scholar 

  6. Bradley, P., Fayyad, U.: Refining initial points for k-means clustering. In: Proceedings of the Fifteenth International Conference on Knowledge Discovery and Data Mining, New York, pp. 9–15. AAAI Press (1998)

    Google Scholar 

  7. Cpałka, K., Rebrova, O., Nowicki, R., Rutkowski, L.: On design of flexible neuro-fuzzy systems for nonlinear modelling. Int. J. Gen. Syst. 42(6), 706–720 (2013)

    Article  MATH  Google Scholar 

  8. Cpałka, K., Rutkowski, L.: Flexible Takagi-Sugeno fuzzy systems. In: Proceedings of the 2005 IEEE International Joint Conference on IJCNN Neural Networks (2005)

    Google Scholar 

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

    Article  Google Scholar 

  10. Duch, W., Korbicz, J., Rutkowski, L., Tadeusiewicz, R. (eds.): Biocybernetics and Biomedical Engineering 2000. Neural Networks, vol. 6. Akademicka Oficyna Wydawnicza EXIT (2000)

    Google Scholar 

  11. Dunn, J.C.: Well separated clusters and optimal fuzzy partitions. J. Cybernetica 4, 95–104 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ester, M., Kriegel, H.-P., Sander, J., Xu, X.: A density-based algorithm for discovering clusters in large spatial data sets with noise. In: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, Portland, pp. 226–231 (1996)

    Google Scholar 

  13. Fränti, P., Rezaei, M., Zhao, Q.: Centroid index: cluster level similarity measure. Pattern Recogn. 47(9), 3034–3045 (2014)

    Article  Google Scholar 

  14. Gabryel, M.: A bag-of-features algorithm for applications using a NoSQL database. Inf. Softw. Technol. 639, 332–343 (2016)

    Article  Google Scholar 

  15. Gabryel, M., Grycuk, R., Korytkowski, M., Holotyak, T.: Image indexing and retrieval using GSOM algorithm. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2015. LNCS, vol. 9119, pp. 706–714. Springer, Cham (2015). doi:10.1007/978-3-319-19324-3_63

    Chapter  Google Scholar 

  16. Guha, S., Rastogi, R., Shim, K.: CURE: an efficient clustering algorithm for large databases. In: Proceedings of the 1998 ACM-SIGMOD International Conference Management of Data (SIGMOD 1998), pp. 73–84 (1998)

    Google Scholar 

  17. Guha, S., Rastogi, R., Shim, K.: ROCK: a robust clustering algorithm for categorical attributes. In: The Proceedings of the IEEE Conference on Data Engineering (1999)

    Google Scholar 

  18. Gustafson, E., Kessel, W.: Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings of IEEE CDC (1978). doi:10.1109/CDC.1978.268028

  19. Halkidi, M., Batistakis, Y., Vazirgiannis, M.: Clustering validity checking methods: part II. ACM SIGMOD Record 31(3), 19–27 (2002)

    Article  MATH  Google Scholar 

  20. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Data Mining, Inference and Prediction. Springer, New York (2001)

    MATH  Google Scholar 

  21. Hinneburg, A., Keim, D.A.: An efficient approach to clustering in large multimedia databases with noise. In: Knowledge Discovery and Data Mining (1998)

    Google Scholar 

  22. Jain, A., Dubes, R.: Algorithms for Clustering Data. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  23. Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, New York (1990)

    Book  MATH  Google Scholar 

  24. Lago-Fernández, L.F., Corbacho, F.: Normality-based validation for crisp clustering. Pattern Recogn. 43(3), 782–795 (2010)

    Article  MATH  Google Scholar 

  25. Lichman, M.: UCI machine learning repository. University of California, School of Information and Computer Science, Irvine, CA (2013). http://archive.ics.uci.edu/ml

  26. Meng, X., van Dyk, D.: The EM algorithm - An old folk-song sung to a fast new tune. J. R. Stat. Soc. Ser. B (Methodol.) 59(3), 511–567 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Murtagh, F.: A survey of recent advances in hierarchical clustering algorithms. Comput. J. 26(4), 354–359 (1983)

    Article  MATH  Google Scholar 

  28. Pakhira, M.K., Bandyopadhyay, S., Maulik, U.: Validity index for crisp and fuzzy clusters. Pattern Recogn. 37(3), 487–501 (2004)

    Article  MATH  Google Scholar 

  29. Pal, N.R., Bezdek, J.C.: On cluster validity for the fuzzy c-means model. IEEE Trans. Fuzzy Syst. 3(3), 370–379 (1995)

    Article  Google Scholar 

  30. Pascual, D., Pla, F., Sánchez, J.S.: Cluster validation using information stability measures. Pattern Recogn. Lett. 31(6), 454–461 (2010)

    Article  Google Scholar 

  31. Pelleg, D., Moore, A.W.: X-means: extending k-means with efficient estimation of the number of clusters. In: Proceedings of the Seventeenth International Conference on Machine Learning, pp. 727–734 (2000)

    Google Scholar 

  32. Rohlf, F.: Single-link clustering algorithms. In: Krishnaiah, P.R., Kanal, L.N. (eds.) Handbook of Statistics, vol. 2, pp. 267–284 (1982)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Rutkowski, L., Cpałka, K.: Compromise approach to neuro-fuzzy systems. In: Sincak, P., Vascak, J., Kvasnicka, V., Pospichal, J. (eds.) Intelligent Technologies - Theory and Applications. New Trends in Intelligent Technologies. Frontiers in Artificial Intelligence and Applications, vol. 76, pp. 85–90 (2002)

    Google Scholar 

  35. Rutkowski, L., Przybył, A., Cpałka, K., Er, M.J.: Online speed profile generation for industrial machine tool based on neuro-fuzzy approach. In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2010. LNCS, vol. 6114, pp. 645–650. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13232-2_79

    Chapter  Google Scholar 

  36. Rutkowski, L., Cpałka, K.: A neuro-fuzzy controller with a compromise fuzzy reasoning. Control Cybern. 31(2), 297–308 (2002)

    MATH  Google Scholar 

  37. Saha, S., Bandyopadhyay, S.: Some connectivity based cluster validity indices. Appl. Soft Comput. 12(5), 1555–1565 (2012)

    Article  Google Scholar 

  38. Sheikholeslami, G., Chatterjee, S., Zhang, A.: Wave cluster: a multiresolution clustering approach for very large spatial databases. In: Proceedings of the 1998 International Conference on Very Large Data Bases (VLDB 1998), pp. 428–439 (1998)

    Google Scholar 

  39. Shieh, H.-L.: Robust validity index for a modified subtractive clustering algorithm. Appl. Soft Comput. 22, 47–59 (2014)

    Article  Google Scholar 

  40. Starczewski, A.: A new validity index for crisp clusters. Pattern Anal. Appl. (2015). doi:10.1007/s10044-015-0525-8

  41. Starczewski, A., Krzyżak, A.: A modification of the silhouette index for the improvement of cluster validity assessment. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS, vol. 9693, pp. 114–124. Springer, Cham (2016). doi:10.1007/978-3-319-39384-1_10

    Google Scholar 

  42. Wang, W., Yang, J., Muntz, M.: STING: a statistical information grid approach to spatial data mining. In: Proceedings of the 1997 International Conference on Very Large Data Bases (VLDB 1997), pp. 186–195 (1997)

    Google Scholar 

  43. Weka 3: Data Mining Software in Java. University of Waikato, New Zealand. http://www.cs.waikato.ac.nz/ml/weka/

  44. Zhao, Q., Fränti, P.: WB-index: a sum-of-squares based index for cluster validity. Data Knowl. Eng. 92, 77–89 (2014)

    Article  Google Scholar 

  45. Zhang, T., Ramakrishnan, R., Linvy, M.: BIRCH: an efficient data clustering method for very large data sets. Data Min. Knowl. Discov. 1(2), 141–182 (1997)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Computational Intelligence, Częstochowa University of Technology, Al. Armii Krajowej 36, 42-200, Częstochowa, Poland

    Artur Starczewski

  2. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

    Adam Krzyżak

Authors
  1. Artur Starczewski
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  2. Adam Krzyżak
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Correspondence to Artur Starczewski .

Editor information

Editors and Affiliations

  1. Częstochowa University of Technology, Częstochowa, Poland

    Leszek Rutkowski

  2. Częstochowa University of Technology, Częstochowa, Poland

    Marcin Korytkowski

  3. Częstochowa University of Technology, Częstochowa, Poland

    Rafał Scherer

  4. AGH University of Science and Technology, Kraków, Poland

    Ryszard Tadeusiewicz

  5. University of California, Berkeley, California, USA

    Lotfi A. Zadeh

  6. University of Louisville, Louisville, Kentucky, USA

    Jacek M. Zurada

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Starczewski, A., Krzyżak, A. (2017). A Study of Cluster Validity Indices for Real-Life Data. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L., Zurada, J. (eds) Artificial Intelligence and Soft Computing. ICAISC 2017. Lecture Notes in Computer Science(), vol 10246. Springer, Cham. https://doi.org/10.1007/978-3-319-59060-8_15

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  • DOI: https://doi.org/10.1007/978-3-319-59060-8_15

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