Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 14 1 2007 10.5194/npg-14-79-2007 http://www.nonlin-processes-geophys.net/14/79/2007/ http://www.nonlin-processes-geophys.net/14/79/2007/npg-14-79-2007.html http://www.nonlin-processes-geophys.net/14/79/2007/npg-14-79-2007.pdf 79 88 2007-02-12 Bayesian modeling and significant features exploration in wavelet power spectra D. V. Divine dmitry.divine@npolar.no F. Godtliebsen Department of Mathematics and Statistics, Faculty of Science, University of Troms\o, 9037, Norway This study proposes and justifies a Bayesian approach to modeling wavelet coefficients and finding statistically significant features in wavelet power spectra. The approach utilizes ideas elaborated in scale-space smoothing methods and wavelet data analysis. We treat each scale of the discrete wavelet decomposition as a sequence of independent random variables and then apply Bayes' rule for constructing the posterior distribution of the smoothed wavelet coefficients. Samples drawn from the posterior are subsequently used for finding the estimate of the true wavelet spectrum at each scale. The method offers two different significance testing procedures for wavelet spectra. A traditional approach assesses the statistical significance against a red noise background. The second procedure tests for homoscedasticity of the wavelet power assessing whether the spectrum derivative significantly differs from zero at each particular point of the spectrum. Case studies with simulated data and climatic time-series prove the method to be a potentially useful tool in data analysis. Berger, J. (Ed.): Statistical decision theory and Bayesian analysis, Springer Verlag, 1985. Box, E., Jenkins, G., and Reinsel, G.: Time series analysis: Forecasting and control, Prentice-Hall, Englewood Cliffs, 1994. Chaudhuri, P. and Marron, J.: SiZer for exploration of structures in curves, J. Am. Statist. Assoc., 94, 807–823, 1999. Dansgaard, W., Johnsen, S., Clausen, H., Dahl-Jensen, D., Gundestrup, N., Hammer, C., Hvidberg, C., Steffensen, J., Sveinbjornsdottir, A., Jouzel, J., and Bond, G.: Evidence for general instability of past climate from a 250-kyr ice-core record, Nature, 364, 218–220, doi:10.1038/364218a0, 1993. Dong, B., Sutton, R., and Scaife, A.: Multidecadal modulation of El Niño–Southern Oscillation (ENSO) variance by Atlantic Ocean sea surface temperatures, Geophys. Res. Lett., 33, L08 705, doi:10.1029/2006GL025766, 2006. Fisher, D A., Reeh, N., and Clausen, H B.: Stratigraphic noise in the time series derived from ice cores, Ann. Glaciol., 7, 76–83, 1985. Godtliebsen, F. and Ă˜igård, T.: A Visual Display Device for Significant Features in Complicated Signals, Comput. Statist. Data Analysis, 48, 317–343, 2005. Grinsted, A., Moore, J., and Jevrejeva, S.: Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, 2004. Grootes, P. and Stuiver, M.: Oxygen 18/16 variability in Greenland snow and ice with $10^-3-10^-5$-year time resolution, J. Geophys. Res., 102(C12), 26 455–26 470, 1997. Kaiser, G.: A friendly guide to wavelets, Birkhauser Boston Inc., Cambridge, MA, USA, 1994. Maraun, D. and Kurths, J.: Cross wavelet analysis: significance testing and pitfalls, Nonlin. Processes Geophys., 11, 505–514, 2004. Maraun, D., Kurths, J., and Holschneider, M.: Nonstationary Gaussian Processes in Wavelet Domain: Synthesis, Estimation and Significance Testing, Phys. Rev. E, 75(1), 016707, 2007. Nason, G P., von Sachs, R., and Kroisandt, G.: Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum, J. Roy. Statist. Soc., 62(B), 271–300, 2000. Ă˜igård, T.: Statistical analysis and representation of non-trivial signals and random processes, Ph.D. thesis, University of Troms\o, Troms\o, Norway, 2004. Park, C., Marron, J., and Rondonotti, V.: Dependent SiZer: Goodness-of-fit tests for tune series models, J. Appl. Statist., 8, 999–1017, 2004. Percival, D. and Walden, A.: Wavelet methods for time-series analysis, Cambridge University Press, Cambridge, 2000. Rue, H.: Fast Sampling of Gaussian Markov Random Fields, J. Roy. Statist. Soc., 63, 325–338, 2001. Schulz, M. and Mudelsee, M.: REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series, Computers & Geosciences, 28, 421–426, 2002. Torrence, C. and Compo, G.: A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79(1), 61–78, 1998.