Finding non-redundant, statistically significant regions in high dimensional data: a novel approach to projected and subspace clustering

ABSTRACT

Projected and subspace clustering algorithms search for clusters of points in subsets of attributes. Projected clustering computes several disjoint clusters, plus outliers, so that each cluster exists in its own subset of attributes. Subspace clustering enumerates clusters of points in all subsets of attributes, typically producing many overlapping clusters. One problem of existing approaches is that their objectives are stated in a way that is not independent of the particular algorithm proposed to detect such clusters. A second problem is the definition of cluster density based on user-defined parameters, which makes it hard to assess whether the reported clusters are an artifact of the algorithm or whether they actually stand out in the data in a statistical sense.

We propose a novel problem formulation that aims at extracting axis-parallel regions that stand out in the data in a statistical sense. The set of axis-parallel, statistically significant regions that exist in a given data set is typically highly redundant. Therefore, we formulate the problem of representing this set through a reduced, non-redundant set of axis-parallel, statistically significant regions as an optimization problem. Exhaustive search is not a viable solution due to computational infeasibility, and we propose the approximation algorithm STATPC. Our comprehensive experimental evaluation shows that STATPC significantly outperforms existing projected and subspace clustering algorithms in terms of accuracy.

References

1. E. Achtert, C. Böhm, H.-P. Kriegel, P. Kröger, I. Müller-Gorman, and A. Zimek. Detection and visualization of subspace clusters hierarchies. In DASFAA, 2007.
2. D. Agarwal, A. McGregor, J. Phillips, S. Venkatasubramanian, and Z. Zhu. Spatial scan statistics: approximations and performance study. In KDD, 2006.
3. C. C. Aggarwal, C. Procopiuc, J. L. Wolf, P. S. Yu, and J. S. Park. Fast algorithms for projected clustering. In SIGMOD, 1999.
4. R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. In SIGMOD, 1998.
5. R. Agrawal and R. Srikan. Fast algorithms for mining association rules. In VLDB, 1994.
6. I. Assent, R. Krieger, E. Müller, and T. Seidl. DUSC: Dimensionality unbiased subspace clustering. In ICDM, 2007.
7. A. Baddeley. Spatial point processes and their applications. Lecture Notes in Mathematics, 1892: 1--75, 2007..
8. Y. Benjamini and Y. Hochberg. Controlling the false discovery rate: a practical and powerful approach to multiple testing. JRSS-B, 57:289--200, 1995..
9. K. Beyer, J. Goldstein, R. Ramakrishnan, and U. Shaft. When is nearest neighbor meaningful? LNCS, 1540:217--235, 1999.
10. C. Böhm, K. Kailing, H.-P. Kriegel, and P. Kröger. Density connected clustering with local subspace preferences. In ICDM, 2004.
11. C. H. Cheng, A. W. Fu, and Y. Zhang. Entropy-based subspace clustering for mining numerical data. In KDD, 1999.
12. J. Friedman and N. Fisher. Bump hunting in high-dimensional data. Statistics and Computing, 9:123--143, 1999.
13. K. Kailing, H. P. Kriegel, and P. Kröger. Density-connected subspace clustering for high-dimensional data. In SDM, 2004.
14. H. P. Kriegel, P. Kröger, M. Renz, and S. Wurst. A generic framework for efficient subspace clustering of high-dimensional data. In ICDM, 2005.
15. G. Liu, J. Li, K. Sim, and L. Wong. Distance based subspace clustering with flexible dimension partitioning. In ICDE, 2007.
16. G. Moise and J. Sander. TR08-03. Technical report, University of Alberta, http://www.cs.ualberta.ca/research/techreports/2008/TR08-03.php, 2008.
17. G. Moise, J. Sander, and M. Ester. P3C: A robust projected clustering algorithm. In ICDM, 2006.
18. H. Nagesh, S. Goil, and A. Choudhary. Adaptive grids for clustering massive data sets. In SDM, 2001.
19. K. Ng, A. Fu, and C.-W. Wong. Projective clustering by histograms. IEEE TKDE, 17(3):369--383, 2005.
20. L. Parsons, E. Haque, and H. Liu. Subspace clustering for high dimensional data: a review. SIGKDD Explorations Newsletter, 6(1):90--105, 2004.
21. C. M. Procopiuc, M. Jones, P. K. Agarwal, and T. Murali. A Monte Carlo algorithm for fast projective clustering. In SIGMOD, 2002.
22. K. Sequeira and M. Zaki. SCHISM: a new approach for interesting subspace mining. In ICDM, 2004.
23. G. W. Snedecor and W. G. Cochran. Statistical Methods. Iowa State University Press, 1989.
24. K. Yip, D. Cheung, and M. Ng. HARP: a practical projected clustering algorithm. IEEE TKDE, 16(11):1387--1397, 2004.
25. K. Yip, D. Cheung, and M. Ng. On discovery of extremely low-dimensional clusters using semi-supervised projected clustering. In ICDE, 2005.
26. M. L. Yiu and N. Mamoulis. Iterative projected clustering by subspace mining. IEEE TKDE, 17(2):176--189, 2005.

Index Terms

• Information systems

  • Information retrieval

    • Retrieval tasks and goals

      • Clustering and classification

  • Information systems applications

    • Data mining

      • Clustering