EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS

KATERINA G. TSAKIRI; IGOR G. ZURBENKO

EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS

KATERINA G. TSAKIRI¹*, IGOR G. ZURBENKO²
¹Division of Math, Science and Technology, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale-Davie, FL 33314-7796, USA
²Department of Biometry and Statistics, School of Public Health, State University of New York at Albany, One University Place, Rensselaer, NY 12144, USA
* Corresponding Author : ktsakiri@gmail.com

Received : 16-11-2010 Accepted : 19-10-2011 Published : 15-12-2011
Volume : 2 Issue : 2 Pages : 40 - 48
J Stat Math 2.2 (2011):40-48

Cite - MLA : KATERINA G. TSAKIRI and IGOR G. ZURBENKO "EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS." Journal of Statistics and Mathematics 2.2 (2011):40-48.

Cite - APA : KATERINA G. TSAKIRI, IGOR G. ZURBENKO (2011). EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS. Journal of Statistics and Mathematics, 2 (2), 40-48.

Cite - Chicago : KATERINA G. TSAKIRI and IGOR G. ZURBENKO "EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS." Journal of Statistics and Mathematics 2, no. 2 (2011):40-48.

Copyright : © 2011, KATERINA G. TSAKIRI and IGOR G. ZURBENKO, Published by Bioinfo Publications. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Abstract

This paper demonstrates the effect of independent noise in principal components of k normally distributed random variables defined by a population covariance matrix. We prove that the principal components determined by a joint distribution of the original sample affected by noise can be essentially different in comparison with those determined from the original sample. However when the differences between the eigenvalues of the population covariance matrix are sufficiently large compared to the level of the noise, the effect of noise in principal components proved to be negligible. We support the theoretical results by using simulation study and examples. We also compare the results about the eigenvalues and eigenvectors in the two dimensional case with other models examined before. This theory can be applied in any field for the decomposition of the time series in multivariate analysis.

References

[1] Bickel P.J., Levina E. (2008) Ann. of Stat., 36, 199-227
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[2] Davis C. and Kahan W.M. (1970) III SIAM J. of Numerical Anal., 70, 1-47
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[3] Demmel J., Veselic K. (1992) SIAM J. of Matrix Anal. and Applications, 13, 1204-1245
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[4] Martin J.F. (1988) Component and Correspondence Analysis. Dimension Reduction by Functional Approximation, J.L.A. van Rijckevorsel and J. de Leeuw, eds., Wiley, Chichester, 103-114
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[5] Mathias R. (1997) SIAM J. of Matrix Anal. and Applications, 18, 959-980.
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[6] Nadler B. (2008) Ann. of Stat., 36, 2791-2817.
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[7] Parlett B.N. (1980) The symmetric eigenvalue problem, Prentice-Hall, New Jersey
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[8] Tsakiri K.G. and Zurbenko I.G. (2008) JSM proceedings, Stat. Comput. Section, American Statistical Association, Alexandria, VA: American, 569-576.
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[9] Webb A.R. (1996) Stat. and Comput., 6, 159-168
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[10] Zurbenko I.G. and Sowizral M. (1999) Far East J. of Theor. Stat., 3, 139-157
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[11] Jolliffe I.T. (2002) Principal Component Analysis, Springer-Verlang, New York
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[12] Zurbenko I.G. (1986) The Spectral Anal. of Time Series, North Holland Series in Statistics and Probability, Amsterdam
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[13] Yang W. and Zurbenko I. (2010) Wiley Interdisciplinary Revives: In Comput. Stat., Wiley, 2, 107-115.
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[14] Tsakiri K.G. and Zurbenko I.G. (2010) J. of Air Qual., Atmos. and Health, DOI: 10.1007/s11869-010-0084-5
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[15] Shores T.S. (2007) Appl. Linear Algebra and Matrix Anal., Springer, New York
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[16] Johnson R.A., Wichern D.W. (2002) Appl. Multivariate Stat. Anal., Prentice Hall, New Jersey
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[17] Tsakiri K.G. and Zurbenko I.G. (2010) Meteorol. and Atmos. Phys., 109, 129-137
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[18] Kelly N.A., Ferman M.A., and Wolff G.T. (1986) J. of the Air Pollut. Control Assoc., 36, 150-158
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[19] Korsog P.E. and Wolff G.T. (1991) Atmos. Environ., 25B, 47-57
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[20] Salazar-Ruiz E., Ordieres J.B., Vergara E.P., Capuz-Rizo S.F. (2008) J. of Environ. Model. and Soft., 23, 1056-1069
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

[21] Yang W. and Zurbenko I. (2010) Wiley interdisciplinary revives in comput. Stat., WIREs in Comput. Stat., 2, 340-351.
» CrossRef » Google Scholar » PubMed » DOAJ » CAS » Scopus

Licence

ISSN & EISSN

Scan QR Code

Journal Details

Special Issues

Publishing Ethics

Share

EFFECT OF NOISE IN PRINCIPAL COMPONENT ANALYSIS

Translate Article

Article Category

Article Statistics

Downloads

Citations

Cited By

Cite

Import Cite

Share

Related Article

Google Scholar

PubMed