Comparison of Cluster Representations from Partial Second- to Full Fourth-Order Cross Moments for Data Stream Clustering

Under seven external clustering evaluation measures, a comparison is made for cluster representations from the partial second order to the fourth order in data stream clustering. Two external clustering evaluation measures, purity and cross entropy, adopted for data stream clustering performance evaluation in the past, penalize the performance of an algorithm when each hypothesized cluster contains points in different target classes or true clusters, while ignoring the issue of points in a target class falling into different hypothesized clusters. The seven measures will address both sides of the clustering performance. The represented geometry by the partial second-order statistics of a cluster is non-oblique ellipsoidal and cannot describe the orientation, asymmetry, or peakedness of a cluster. The higher-order cluster representation presented in this paper introduces the third and fourth cross moments, enabling the cluster geometry to be beyond an ellipsoid. The higher-order statistics allow two clusters with different representations to merge into a multivariate normal cluster, using normality tests based on multivariate skewness and kurtosis. The clustering performance under the seven external clustering evaluation measures with a synthetic and two real data streams demonstrates the effectiveness of the higher-order cluster representations.

[1]  Paul S. Bradley,et al.  Scaling Clustering Algorithms to Large Databases , 1998, KDD.

[2]  M. Meilă.  Comparing clusterings---an information based distance , 2007 .

[3]  B. Dom,et al.  Research Report an Information-theoretic External Cluster-validity Measure an Information-theoretic External Cluster-validity Measure 2. the Evaluation Problem , 2007 .

[4]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[5]  Tian Zhang,et al.  BIRCH: an efficient data clustering method for very large databases , 1996, SIGMOD '96.

[6]  Aoying Zhou,et al.  Density-Based Clustering over an Evolving Data Stream with Noise , 2006, SDM.

[7]  Sudipto Guha,et al.  Streaming-data algorithms for high-quality clustering , 2002, Proceedings 18th International Conference on Data Engineering.

[8]  H. Hotelling.  The Generalization of Student’s Ratio , 1931 .

[9]  G. Karypis,et al.  Criterion Functions for Document Clustering ∗ Experiments and Analysis , 2001 .

[10]  Philip S. Yu,et al.  A Framework for Clustering Evolving Data Streams , 2003, VLDB.

[11]  Eyke Hüllermeier,et al.  Online clustering of parallel data streams , 2006, Data Knowl. Eng..

[12]  Olivier Ledoit,et al.  Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size , 2002 .

[13]  Adrian E. Raftery,et al.  MCLUST Version 3 for R: Normal Mixture Modeling and Model-Based Clustering † , 2007 .

[14]  Julia Hirschberg,et al.  V-Measure: A Conditional Entropy-Based External Cluster Evaluation Measure , 2007, EMNLP.