Fractal Mining

  • Daniel Barbara
  • Ping Chen


Self-similarity is the property of being invariant with respect to the scale used to look at the data set. Self-similarity can be measured using the fractal dimension. Fractal dimension is an important charactaristics for many complex systems and can serve as a powerful representation technique. In this chapter, we present a new clustering algorithm, based on self-similarity properties of the data sets, and also its applications to other fields in Data Mining, such as projected clustering and trend analysis. Clustering is a widely used knowledge discovery technique. The new algorithm which we call Fractal Clustering (FC) places points incrementally in the cluster for which the change in the fractal dimension after adding the point is the least. This is a very natural way of clustering points, since points in the same clusterhave a great degree of self-similarity among them (and much less self-similarity with respect to points in other clusters). FC requires one scan of the data, is suspendable at will, providing the best answer possible at that point, and is incremental. We show via experiments that FC effectively deals with large data sets, high-dimensionality and noise and is capable of recognizing clusters of arbitrary shape.


self-similarity clustering projected clustering trend analysis 


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  1. E. Backer. Computer-Assisted Reasoning in Cluster Analysis. Prentice Hall, 1995.Google Scholar
  2. A. Belussi and C. Faloutsos. Estimating the Selectivity of Spatial Queries Using the ‘Correlation’ Fractal Dimension. In Proceedings of the International Conference on Very Large Data Bases, pages 299–310, September 1995.Google Scholar
  3. P.S. Bradley, U. Fayyad, and C. Reina. Scaling Clustering Algorithms to Large Databases (Extended Abstract). In Proceedings of the ACM SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery, June 1998.Google Scholar
  4. CDIA. U.S. Historical Climatology Network Data, /epubs/ndpO19/ ushcn_r3.html.Google Scholar
  5. H. Chernoff. A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations. Annals of Mathematical Statistics, pages 493–509, 1952.Google Scholar
  6. C. Domingo, R. Gavaldá, and O. Watanabe. Practical Algorithms for Online Selection. In Proceedings of the first International Conference on Discovery Science, 1998.Google Scholar
  7. C. Domingo, R. Gavaldá, and O. Watanabe. Adaptive Sampling Algorithms for Scaling Up Knowledge Discovery Algorithms. In Proceedings of the second International Conference on Discovery Science, 2000.Google Scholar
  8. P. Domingos and G. Hulten. Mining High-Speed Data Streams. In Proceedings of the Sixth ACM-SIGKDD International Conference on Knowledge Discovery and Data Mining, Boston, MA, 2000.Google Scholar
  9. C. Faloutsos and V. Gaede. Analysis of the Z-ordering Method Using the hausdorff Fractal Dimension. In Proceedings of the International Conference on Very Large Data Bases, pages 40–50, September 1996.Google Scholar
  10. C. Faloutsos and I. Kamel. Relaxing the Uniformity and Independence Assumptions, Using the Concept of Fractal Dimensions. Journal of Computer and System Sciences, 55(2):229–240, 1997.MathSciNetCrossRefGoogle Scholar
  11. C. Faloutsos, Y. Matias, and A. Silberschatz. Modeling Skewed Distributions Using Multifractals and the ‘80-20 law’. In Proceedings of the International Conference on Very Large Data Bases, pages 307–317, September 1996.Google Scholar
  12. K. Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San Diego, California, 1990.Google Scholar
  13. P. Grassberger. Generalized Dimensions of Strange Attractors. Physics Letters, 97A:227–230, 1983.MathSciNetGoogle Scholar
  14. P. Grassberger and I. Procaccia. Characterization of Strange Attractors. Physical Review Letters, 50(5):346–349, 1983.Google Scholar
  15. S. Guha, R. Rastogi, and K. Shim. CURE: An Efficient Clustering Algorithm for Large Databases. In Proceedings of the ACM SIGMOD Conference on Management of Data, Seattle, Washington, pages 73–84, 1998.Google Scholar
  16. A. Jain and R. C. Dubes. Algorithms for Clustering Data. Prentice Hall, En-glewood Cliffs, New Jersey, 1988.Google Scholar
  17. L.S. Liebovitch and T. Toth. A Fast Algorithm to Determine Fractal Dimensions by Box Countig. Physics Letters, 141A(8), 1989.Google Scholar
  18. R.J. Lipton and J.F Naughton. Query Size Estimation by Adaptive Sampling. Journal of Computer Systems Science, pages 18–25, 1995.Google Scholar
  19. R.J. Lipton, J.F. Naughton, D.A. Schneider, and S. Seshadri. Efficient Sampling Strategies for Relational Database Operations. Theoretical Computer Science, pages 195–226, 1993.Google Scholar
  20. B.B. Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman, New York, 1983.Google Scholar
  21. D.A. Menascé, V.A. Almeida, R.C. Fonseca, and M.A. Mendes. A Methodology for Workload Characterizatoin for E-commerce Servers. In Proceedings of the ACM Conference in Electronic Commerce, Denver, CO, November 1999.Google Scholar
  22. J. Sarraille and P. DiFalco. FD3. Scholar
  23. E. Schikuta. Grid clustering: An efficient hierarchical method for very large data sets. In Proceedings of the 13th Conference on Pattern Recognition, IEEE Computer Society Press, pages 101–105, 1996.Google Scholar
  24. M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New York, 1991.Google Scholar
  25. S.Z. Selim and M.A. Ismail. K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(1), 1984.Google Scholar
  26. G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A Multi-Resolution Clustering Approach for Very Large Spatial Databases. In Proceedings of the 24th Very Large Data Bases Conference, pages 428–439, 1998.Google Scholar
  27. W. Wang, J. Yand, and R. Muntz. STING: A statistical information grid approach to spatial data mining. In Proceedings of the 23rd Very Large Data Bases Conference, pages 186–195, 1997.Google Scholar
  28. O. Watanabe. Simple Sampling Techniques for Discovery Science. IEICE Transactions on Information and Systems, January 2000.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Daniel Barbara
    • 1
  • Ping Chen
    • 2
  1. 1.George Mason UniversityFairfaxUSA
  2. 2.University of Houston-DowntownHoustonUSA

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