Open Access

Random projections and Hotelling’s T2 statistics for change detection in high-dimensional data streams


Cite

Achlioptas, D. (2001 ). Database friendly random projections, Proceedings of the 20th ACM SIGMOD-SIGACT-SIGARTSymposium on Principles of Database Systems, Santa Barbara,CA, USA, pp. 274-281.10.1145/375551.375608Search in Google Scholar

Ailon, N. and Chazelle, B. (2006). Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform, Proceedings of the 38th Annual ACM Symposium on Theoryof Computing, Seattle, WA, USA, pp. 557-563.Search in Google Scholar

Arriaga, R. and Vempala, S.(1999). An algorithmic theory of learning: Robust concepts and random projection, Proceedings of the 40th Annual IEEE Symposium on theFoundations of Computer Science, New York, NY, USA, pp. 616-623.Search in Google Scholar

Biau, G. and Devroye, L. and Lugosi, G. (2008). On the performance of clustering in Hilbert spaces IEEE Transactionson Information Theory 54(2): 781-790.10.1109/TIT.2007.913516Search in Google Scholar

Bodnar, O. and Schmid, W. (2005). Multivariate control charts based on a projection approach Allgemeines StatistischesArchiv 89(1): 75-93.10.1007/s101820500193Search in Google Scholar

Chandola, V., Banerjee, A. and Kumar, V. (2009). Anomaly detection: A survey, ACM Computing Surveys41(3): 15:1-15:58.10.1145/1541880.1541882Search in Google Scholar

Cramer, H. and Wold, H.(1936). Some theorems on distribution functions, Journal of the London Mathematical Society11(2): 290-295.10.1112/jlms/s1-11.4.290Search in Google Scholar

Cuesta-Albertos, J.A., del Barrio, E., Fraiman, R. and Matran, C. (2007). The random projection method in goodness of fit for functional data, Computational Statistics and DataAnalysis 51(10): 4814-4831.10.1016/j.csda.2006.09.007Search in Google Scholar

Cuturi, M., Vert, J-P. and dAspremont, A. (2009). White functionals for anomaly detection in dynamical systems, in Y. Bengio, D. Schuurmans, J. Lafferty, C.K.I. Williams and A. Culotta (Eds.), Advances in Neural InformationProcessing Systems, Vol. 22, MIT Press, Vancouver, pp. 432-440.Search in Google Scholar

Dasgupta, S. and Gupta, A. (2003). An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structuresand Algorithms 22(1): 60-65.10.1002/rsa.10073Search in Google Scholar

Donoho D.L. (2000 ). High-dimensional data analysis: The curses and blessings of dimensionality, Technical report, Department of Statistics, Stanford University, Stanford, CA.Search in Google Scholar

Frankl, P. and Maehara, H. (1987). The Johnson-Lindenstrauss lemma and the sphericity of some graphs, Journal of CombinatorialTheory A 44(3): 355-362.10.1016/0095-8956(88)90043-3Search in Google Scholar

Forbes, C., Evans, M. and Hastings, N. and Peacock, B. (2011). Statistical Distributions, 4th Edn., John Wiley and Sons, Inc., Hoboken, NJ.Search in Google Scholar

Hyv¨arinen, A., Karhunen, J. and Oja, E. (2001). IndependentComponent Analysis, Wiley, New York, NY.10.1002/0471221317Search in Google Scholar

Hotelling, H. (1931). The generalization of Student’s ratio TheAnnals of Mathematical Statistics 2(3): 360-378.10.1214/aoms/1177732979Search in Google Scholar

Indyk, P. and Motwani, R. (1998). Approximate nearest neighbors: Towards removing the curse of dimensionality, Proceedings of the 30th Annual ACM Symposium on theTheory of Computing, Dallas, TX, USA, pp. 604-613.Search in Google Scholar

Indyk, P. and Naor, A.(2007). Nearest neighbor preserving embeddings, ACM Transactions on Algorithms 3(3): 31:1-31:12.10.1145/1273340.1273347Search in Google Scholar

Jolliffe, I.T. (1986). Principal Component Analysis, Springer-Verlag, New York, NY.10.1007/978-1-4757-1904-8Search in Google Scholar

Johnson, W.B. and Lindenstrauss, J.(1984). Extensions of Lipschitz mapping into Hilbert space, ContemporaryMathematics 26: 189-206.10.1090/conm/026/737400Search in Google Scholar

Korbicz, J., Ko´scielny, J.M., Kowalczuk, Z. and Cholewa, W. (Eds.) (2004). Fault Diagnosis. Models, ArtificialIntelligence, Applications. Springer Verlag, Berlin/Heidelberg/New York, NY.10.1007/978-3-642-18615-8Search in Google Scholar

Lee, J.A. and Verleysen, M. (2007). Nonlinear DimensionalityReduction, Springer, New York, NY.10.1007/978-0-387-39351-3Search in Google Scholar

Li, P., Hastie, T.J. and Church, K.W. (2006a). Nonlinear estimators and tail bounds for dimension reduction in L1 using Cauchy random projections, Technical report, Department of Statistics, Stanford University, Stanford, CA.Search in Google Scholar

Li, P., Hastie, T.J. and Church, K.W. (2006b). Sub-Gaussian random projections, Technical report, Department of Statistics, Stanford University, Stanford, CA.Search in Google Scholar

Mason, R.L., Tracy, N.D. and Young, J.C., (1992). Multivariate control charts for individual observations, Journal ofQuality Technology 24(2): 88-95.10.1080/00224065.1992.12015232Search in Google Scholar

Mason, R.L. and Young, J.C. (2002). Multivariate StatisticalProcess Control with Industrial Application, SIAM, Philadelphia, PA.10.1137/1.9780898718461Search in Google Scholar

Mathai, A.M. and Provost, S.B. (1992). Quadratic Formsin Random Variables: Theory and Applications, Marcel Dekker, New York, NY.Search in Google Scholar

Matouˆsek, J.(2008). On variants of the Johnson-Lindenstrauss lemma, Random Structures and Algorithms 33(2): 142-156.10.1002/rsa.20218Search in Google Scholar

Milman, V.(1971). A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Functional Analysis and ItsApplications 5(4): 28-37, (English translation).Search in Google Scholar

Montgomery, D.C. (1996 ). Introduction to Statistical QualityControl, 3rd Edn., John Wiley and Sons, New York, NY.Search in Google Scholar

Qin, S.J.(2003). Statistical process monitoring: Basics and beyond Journal of Chemometrics 17(8-9): 480-502.10.1002/cem.800Search in Google Scholar

Rao, C.R. (1973). Linear Statistical Inference andIts Applications, John Wiley and Sons, New York, NY/London/Sydney/Toronto.Search in Google Scholar

Runger, G.C. (1996). Projections and the U-squared multivariate control chart, Journal of Quality Technology28(3): 313-319.10.1080/00224065.1996.11979681Search in Google Scholar

Runger, G., Barton, R., Del Castillo, E. and Woodall, W.H. (2007). Optimal monitoring of multivariate data for fault patterns, Journal of Quality Technology 39(2): 159-172.10.1080/00224065.2007.11917683Search in Google Scholar

Skubalska-Rafajłowicz, E. (2006). RBF neural network for probability density function estimation and detecting changes in multivariate processes, in L. Rutkowski, R. Tadeusiewicz, L.A. Zadeh and J. ˙Zurada (Eds.), Artificial Intelligence and Soft Computing, Lecture Notes in Computer Science, Vol. 4029, Springer-Verlag, Berlin/Heidelberg, pp. 133-141.10.1007/11785231_15Search in Google Scholar

Skubalska-Rafajłowicz, E. (2008). Random projection RBF nets for multidimensional density estimation, InternationalJournal of Applied Mathematics and Computer Science18(4): 455-464, DOI: 10.2478/v10006-008-0040-9. 10.2478/v10006-008-0040-9Search in Google Scholar

Skubalska-Rafajłowicz, E. (2009). Neural networks with sigmoidal activation functions dimension reduction using normal random projection, Nonlinear Analysis 71(12): e1255-e1263.10.1016/j.na.2009.01.124Search in Google Scholar

Skubalska-Rafajłowicz, E. (2011). Fast and efficient method of change detection in statistically monitored high-dimensional data streams, Proceedings of the 10thInternational Science and Technology Conference on Diagnosticsof Processes and Systems, Zamo´s´c, Poland, pp. 256-260.Search in Google Scholar

Srivastava,M.S. (2009). A review of multivariate theory for high dimensional data with fewer observations, in A. SenGupta (Ed.), Advances in Multivariate Statistical Methods, Vol. 9, World Scientific, Singapore, pp. 25-52.10.1142/9789812838247_0002Search in Google Scholar

Sulliva, J.H. and Woodall, W.H. (2000). Change-point detection of mean vector or covariance matrix shifts using multivariate individual observations, IIE Transactions32(6): 537-549.10.1080/07408170008963929Search in Google Scholar

Tsung F. and Wang K. (2010). Adaptive charting techniques: Literature review and extensions, in H.-J. Lenz, P.-T. Wilrich and W. Schmid (Eds.), Frontiers in StatisticalQuality Control, Vol. 9, Springer-Verlag, Berlin/Heidelberg, pp. 19-35.10.1007/978-3-7908-2380-6_2Search in Google Scholar

Vempala, S. (2004). The Random Projection Method, American Mathematical Society, Providence, RI.10.1090/dimacs/065/01Search in Google Scholar

Wang, K. and Jiang, W. (2009). High-dimensional process monitoring and fault isolation via variable selection, Journalof Quality Technology 41(3): 247-258.10.1080/00224065.2009.11917780Search in Google Scholar

Wang, J. (2012). Geometric Structure of High-DimensionalData and Dimensionality Reduction, Higher Education Press, Beijing/Springer-Verlag, Berlin/Heidelberg.10.1007/978-3-642-27497-8_3Search in Google Scholar

Wold, H. (1966). Estimation of principal components and related models by iterative least squares in P. Krishnaiaah (Ed.), Multivariate Analysis, Academic Press, New York, NY, pp. 391-420.Search in Google Scholar

Zorriassatine, F., Tannock, J.D.T. and O‘Brien, C. (2003). Using novelty detection to identify abnormalities caused by mean shifts in bivariate processes, Computers and Industrial Engineering44(3): 385-408.10.1016/S0360-8352(02)00215-2Search in Google Scholar

eISSN:
2083-8492
ISSN:
1641-876X
Language:
English
Publication timeframe:
4 times per year
Journal Subjects:
Mathematics, Applied Mathematics