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Significance Tests for the Wavelet Cross Spectrum and Wavelet Linear Coherence : Volume 26, Issue 12 (02/12/2008)

By Ge, Z.

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Book Id: WPLBN0004001965
Format Type: PDF Article :
File Size: Pages 11
Reproduction Date: 2015

Title: Significance Tests for the Wavelet Cross Spectrum and Wavelet Linear Coherence : Volume 26, Issue 12 (02/12/2008)  
Author: Ge, Z.
Volume: Vol. 26, Issue 12
Language: English
Subject: Science, Annales, Geophysicae
Collections: Periodicals: Journal and Magazine Collection, Copernicus GmbH
Historic
Publication Date:
2008
Publisher: Copernicus Gmbh, Göttingen, Germany
Member Page: Copernicus Publications

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Ge, Z. (2008). Significance Tests for the Wavelet Cross Spectrum and Wavelet Linear Coherence : Volume 26, Issue 12 (02/12/2008). Retrieved from http://members.worldlibrary.net/


Description
Description: Ecosystems Research Division, NERL, USEPA, 960 College Station Road, Athens, GA 30605, USA. This work attempts to develop significance tests for the wavelet cross spectrum and the wavelet linear coherence as a follow-up study on Ge (2007). Conventional approaches that are used by Torrence and Compo (1998) based on stationary background noise time series were used here in estimating the sampling distributions of the wavelet cross spectrum and the wavelet linear coherence. The sampling distributions are then used for establishing significance levels for these two wavelet-based quantities. In addition to these two wavelet quantities, properties of the phase angle of the wavelet cross spectrum of, or the phase difference between, two Gaussian white noise series are discussed. It is found that the tangent of the principal part of the phase angle approximately has a standard Cauchy distribution and the phase angle is uniformly distributed, which makes it impossible to establish significance levels for the phase angle. The simulated signals clearly show that, when there is no linear relation between the two analysed signals, the phase angle disperses into the entire range of [−Π,Π] with fairly high probabilities for values close to ±Π to occur. Conversely, when linear relations are present, the phase angle of the wavelet cross spectrum settles around an associated value with considerably reduced fluctuations. When two signals are linearly coupled, their wavelet linear coherence will attain values close to one. The significance test of the wavelet linear coherence can therefore be used to complement the inspection of the phase angle of the wavelet cross spectrum.

The developed significance tests are also applied to actual data sets, simultaneously recorded wind speed and wave elevation series measured from a NOAA buoy on Lake Michigan. Significance levels of the wavelet cross spectrum and the wavelet linear coherence between the winds and the waves reasonably separated meaningful peaks from those generated by randomness in the data set. As with simulated signals, nearly constant phase angles of the wavelet cross spectrum are found to coincide with large values in the wavelet linear coherence between the winds and the waves. Not limited to geophysics, the significance tests developed in the present work can also be applied to many other quantitative studies using the continuous wavelet transform.


Summary
Significance tests for the wavelet cross spectrum and wavelet linear coherence

Excerpt
Elsayed, M. A. K.: Nonlinear wave-wave interactions, J. Coastal Res., 24, 798–803, 2008.; Foufoula-Georgiou, E. and Kumar, P. (Eds.): Wavelets in Geophysics, Academic, San Diego, 1994.; Ge, Z.: Significance tests for the wavelet power and the wavelet power spectrum, Ann. Geophys., 25, 2259–2269, 2007.; Ge, Z. and Liu, P. C.: A time-localized response of wave growth process under turbulent winds, Ann. Geophys., 25, 1253–1262, 2007.; Ge, Z. and Liu, P. C.: Long-term wave growth and its linear and nonlinear interactions with wind fluctuations, Ann. Geophys., 26, 747–758, 2008.; Grinsted, A., Moore, J. C., and Jevrejeva, S.: Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, 2004.; Jenkins, G. M. and Watts, D. G.: Spectral analysis and its applications, Holden-Day, San Francisco, 1968.; Kotz, S. and Srinivasan, R.: Distribution of product and quotient of Bessel function variates, Ann. I. Stat. Math., 21, 201–210, 1969.; Liu, P. C.: Wavelet spectrum analysis and ocean wind waves, Wavelets in Geophysics, edited by: Foufoula-Georgiou, E. and Kumar, P., Academic, San Diego, p. 151–166, 1994.; Lundstedt, H., Liszka, L., Lundin, R., and Muscheler, R.: Long-term solar activity explored with wavelet methods, Ann. Geophys., 24, 769–778, 2006.; Papoulis, A.: Probability, random variables, and stochastic processes, McGraw-Hill, New York, 1965.; Rigozo, N. R., da Silva, H. E., Nordemann, D. J. R., Echer, E., Echer, M. P. D., and Prestes, A.: The medieval and modern maximum solar activity imprints in tree ring data from Chile and stable isotope records from Antarctica and Peru, J. Atmos. Sol.-Terr. Phys., 70, 1012–1024, 2008.; Torrence, C. and Compo, G. P.: A practical guide to wavelet analysis, B. Am. Meteorol. Soc., 79, 61–78, 1998.; Torrence, C. and Webster, P. J.: Interdecadal changes in the ENSO-Monsoon system, J. Climate, 12, 2679–2690, 1999.; Van Milligen, B. Ph., Sánchez, E., Estrada, T., Hidalgo, C., Brañas, B., Carreras, B., and García, L.: Wavelet bicoherence: A new turbulence analysis tool, Phys. Plasmas, 2, 3017–3032, 1995.; Wells, W. T., Anderson, R. L., and Cell, J. W.: The distribution of the product of two central or non-central chi-square variates, Ann. Math. Stat., 33, 1016–1020, 1962.

 

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