Advertisement

Point Symmetry-Based Distance Measures and Their Applications to Clustering

  • Sanghamitra Bandyopadhyay
  • Sriparna Saha

Abstract

This chapter presents some symmetry-based distances used for clustering a given data set. In recent years some symmetry-based similarity measurements have been developed. The definitions of these measures and their advantages and disadvantages are elaborately described in the first part of this chapter. In the second part, a recently developed genetic algorithm-based clustering technique, named GAPS, that uses a symmetry-based distance for assignment of points to different clusters and for fitness computation is elaborately described. Kd-tree-based nearest neighbor search is further used to reduce the time complexity of computing symmetry-based distance. A time complexity analysis of this algorithm is provided. The convergence proof of the GAPS clustering technique is also established in the present chapter. Results on a wide range of data sets show that GAPS is able to detect any type of clusters, irrespective of their geometrical shape and overlapping nature, as long as they possess the characteristic of symmetry. GAPS is compared with some existing symmetry-based clustering techniques for several artificial and real-life data sets.

Keywords

Cluster Center Cluster Technique Query Point Cluster Assignment Stochastic Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    UC Irvine Machine Learning Repository. http://www.ics.uci.edu/~mlearn/MLRepository.html
  2. 4.
    Anderberg, M.R.: Cluster Analysis for Application. Academic Press, New York (1973) Google Scholar
  3. 5.
    Anderson, T.W., Sclove, S.L.: Introduction to the Statistical Analysis of Data. Houghton Mifflin, Boston (1978) zbMATHGoogle Scholar
  4. 12.
    Attneave, F.: Symmetry information and memory for pattern. Am. J. Psychol. 68, 209–222 (1995) CrossRefGoogle Scholar
  5. 20.
    Bandyopadhyay, S., Maulik, U.: Genetic clustering for automatic evolution of clusters and application to image classification. Pattern Recognit. 35(6), 1197–1208 (2002) zbMATHCrossRefGoogle Scholar
  6. 27.
    Bandyopadhyay, S., Saha, S.: GAPS: A clustering method using a new point symmetry based distance measure. Pattern Recognit. 40(12), 3430–3451 (2007) zbMATHCrossRefGoogle Scholar
  7. 28.
    Bandyopadhyay, S., Saha, S.: A point symmetry based clustering technique for automatic evolution of clusters. IEEE Trans. Knowl. Data Eng. 20(11), 1–17 (2008) CrossRefGoogle Scholar
  8. 33.
    Bensaid, A.M., Hall, L.O., Bezdek, J.C., Clarke, L.P., Silbiger, M.L., Arrington, J.A., Murtagh, R.F.: Validity-guided (re)clustering with applications to image segmentation. IEEE Trans. Fuzzy Syst. 4(2), 112–123 (1996) CrossRefGoogle Scholar
  9. 34.
    Bentley, J.L., Weide, B.W., Yao, A.C.: Optimal expected-time algorithms for closest point problems. ACM Trans. Math. Softw. 6(4), 563–580 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 58.
    Chou, C.H., Su, M.C., Lai, E.: Symmetry as a new measure for cluster validity. In: 2nd WSEAS Int. Conf. on Scientific Computation and Soft Computing, Crete, Greece, pp. 209–213 (2002) Google Scholar
  11. 60.
    Chung, K.L., Lin, J.S.: Faster and more robust point symmetry-based K-means algorithm. Pattern Recognit. 40(2), 410–422 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 61.
    Chung, K.L., Lin, K.S.: An efficient line symmetry-based K-means algorithm. Pattern Recognit. Lett. 27(7), 765–772 (2006) CrossRefGoogle Scholar
  13. 100.
    Fisher, R.A.: The use of multiple measurements in taxonomic problems. Annu. Eugen. 3, 179–188 (1936) Google Scholar
  14. 103.
    Friedman, J.H., Bently, J.L., Finkel, R.A.: An algorithm for finding best matches in logarithmic expected time. ACM Trans. Math. Softw. 3(3), 209–226 (1977) zbMATHCrossRefGoogle Scholar
  15. 112.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, New York (1989) zbMATHGoogle Scholar
  16. 129.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) Google Scholar
  17. 136.
    Hu, M.K.: Visual pattern recognition by moment invariants. IEEE Trans. Inf. Theory 8(2), 179–187 (1962) zbMATHCrossRefGoogle Scholar
  18. 145.
    Jain, A.K., Murty, M.N., Flynn, P.J.: Data clustering: A review. ACM Comput. Surv. 31(3), 264–323 (1999) CrossRefGoogle Scholar
  19. 146.
    Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley, New York (1971) zbMATHGoogle Scholar
  20. 167.
    Krishna, K., Murty, M.N.: Genetic K-means algorithm. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 29(3), 433–439 (1999) CrossRefGoogle Scholar
  21. 178.
    Lin, J.Y., Peng, H., Xie, J.M., Zheng, Q.L.: Novel clustering algorithm based on central symmetry. In: Proceedings of 2004 International Conference on Machine Learning and Cybernetics, 26–29 August 2004, vol. 3, pp. 1329–1334 (2004) Google Scholar
  22. 188.
    Maulik, U., Bandyopadhyay, S.: Genetic algorithm based clustering technique. Pattern Recognit. 33(9), 1455–1465 (2000) CrossRefGoogle Scholar
  23. 199.
    Milligan, G.: An algorithm for generating artificial test clusters. Psychometrika 50(1), 123–127 (1981) MathSciNetCrossRefGoogle Scholar
  24. 213.
    Pal, P., Chanda, B.: A symmetry based clustering technique for multi-spectral satellite imagery. In: ICVGIP (2002) Google Scholar
  25. 236.
    Rudolph, G.: Convergence analysis of canonical genetic algorithms. IEEE Trans. Neural Netw. 5(1), 96–101 (1994) CrossRefGoogle Scholar
  26. 243.
    Saha, S., Bandyopadhyay, S.: Application of a new symmetry based cluster validity index for satellite image segmentation. IEEE Geosci. Remote Sens. Lett. 5(2), 166–170 (2008) CrossRefGoogle Scholar
  27. 262.
    Srinivas, M., Patnaik, L.M.: Adaptive probabilities of crossover and mutation in genetic algorithms. IEEE Trans. Syst. Man Cybern. 24(4), 656–667 (1994) CrossRefGoogle Scholar
  28. 266.
    Su, M.C., Chou, C.H.: A modified version of the K-means algorithm with a distance based on cluster symmetry. IEEE Trans. Pattern Anal. Mach. Intell. 23(6), 674–680 (2001) CrossRefGoogle Scholar
  29. 300.
    Zabrodsky, H., Peleg, S., Avnir, D.: Symmetry as a continuous feature. IEEE Trans. Pattern Anal. Mach. Intell. 17(12), 1154–1166 (1995) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sanghamitra Bandyopadhyay
    • 1
  • Sriparna Saha
    • 2
  1. 1.Machine Intelligence UnitIndian Statistical InstituteKolkataIndia
  2. 2.Dept. of Computer ScienceIndian Institute of TechnologyPatnaIndia

Personalised recommendations