Estimation of the significance of the Foster’s wavelet spectrum by means of a permutation test and its application for paleoclimate records
DOI:
https://doi.org/10.21701/bolgeomin.129.3.006Keywords:
irregular sampling, multi-scale phenomena, non-stationarity, permutation test, weighted wavelet Z-transformAbstract
Here we propose a permutation test -a non-parametric computing-intensive test- to evaluate the statistical significance of the Foster’s wavelet spectrum by means of Monte Carlo simulations. A procedure (algorithm) is introduced in order to carry out this aim. The Foster’s wavelet is an adequate method to cope directly with unevenly spaced paleoclimatic time series. We have conducted time series simulations to study the performance of the Foster’s wavelet spectrum and applied the permutation test to localize periodic signals known a priori by randomly increasing the fraction of missing data (from 25% to 75% of the total amount of data of an evenly spaced time series). We found that the periodic signals are progressively lost as larger amounts of data are removed. This loss becomes noticeable at circa 50% of the missing data and gets more evident when 75% of the data is removed. The signal loss is more pronounced in the high-frequency range, due to an intrinsic bias in the wavelet spectrum. Notwithstanding, the Foster’s wavelet spectrum and the permutation test succeeded in locating the periodic signals, even with a moderate amount of missing data. Finally, we applied our procedure to the oxygen isotope ratio (δ18O) data of the GISP2 deep ice core (Greenland) and we were able to detect the main spectral signature of the unevenly spaced time series of the GISP2 δ18O record, i.e., the spectral peak around 1,470 years.
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References
Andronov, I.L. 1998. Wavelet analysis of time series by the least-squares method with supplementary weights. Kinematics and Physics of Celestial Bodies, 14, 374-392.
Braun, H., Ditlevsen, P., Kurths, J. and Mudelsee, M. 2010. Limitations of red noise in analysing Dansgaard- Oeschger events. Climate of the Past, 6(1), 85-92. https://doi.org/10.5194/cp-6-85-2010
Cazelles, B, M. Chavez, D. Berteaux, F. Ménard, J.O. Vik, S. Jenouvrier, and N.C. Stenseth. 2008. Wavelet analysis of ecological time series. Oecologia, 156(2), 287-304. https://doi.org/10.1007/s00442-008-0993-2 PMid:18322705
Dansgaard, W, Johnsen, S.J., Clausen, H.B., Dahl-Jensen, D., Gundestrup, N.S., Hammer, C.U. Hvidberg, C.S., Steffensen, J.P., Sveinbjörnsdottir, A.E., Jouzel, J. and Bond, G. 1993. Evidence for general instability of past climate from a 250-kyr ice-core record. Nature, 364, 218-220. https://doi.org/10.1038/364218a0
Foster, G. 1996a. Time series analysis by projection. I. Statistical properties of Fourier analysis. The Astronomical Journal, 111, 541-554. https://doi.org/10.1086/117805
Foster, G. 1996b. Wavelets for period analysis of unevenly sampled time series. The Astronomical Journal, 112, 1709-1729. https://doi.org/10.1086/118137
Grinsted, A., Moore, J.C., and Jevrejeva, S. 2004. Application of the cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 11, 561-566. https://doi.org/10.5194/npg-11-561-2004
Grootes, P.M. and Stuiver, M. 1997. Oxygen 18/16 variability in Greenland snow and ice with 10−3- to 105-year time resolution. Journal of Geophysical Research: Oceans, 102(C12), 26455-26470. https://doi.org/10.1029/97JC00880
Grossman A. and Morlet, J. 1984. Decomposition of Hardy functions into square integrable wavelets of constant shape. The SIAM Journal on Mathematical Analysis 15, 723-736. https://doi.org/10.1137/0515056
Hasselmann, K. 1976. Stochastic climate models Part I. Theory. Tellus, 28(6), 473-485. https://doi.org/10.1111/j.2153-3490.1976.tb00696.x
Haubold, H.J. 1998. Wavelet analysis of the new solar neutrino capture rate data for the homestake experiment. Astrophysics and Space Science, 258(1-2), 201-218. https://doi.org/10.1016/S0375-9474(97)00268-6
Kirby, J.F. 2005. Which wavelet best reproduces the Fourier power spectrum? Computers and Geoscience, 31(7), 846-864. https://doi.org/10.1016/j.cageo.2005.01.014
Kumar, P. and Foufoula-Georgiou, E. 1994. Wavelet Analysis in Geophysics: an Introduction.Wavelets in Geophysics, London: Academic Press. https://doi.org/10.1016/B978-0-08-052087-2.50007-4
Liu, Y., Liang, X.S. and Weisberg, R.H. 2007. Rectification of the bias in the wavelet power spectrum. Journal of Atmospheric and Oceanic Technology., 24(12), 2093-2102. https://doi.org/10.1175/2007JTECHO511.1
Lomb, N.R. 1976. Least-squares frequency analysis of unequally spaced data. Astrophysics and Space Science, 39(2), 447-462. https://doi.org/10.1007/BF00648343
Mallat, S. 1999. A Wavelet Tour of Signal Processing. London: Academic press. https://doi.org/10.1016/B978-012466606-1/50008-8
Mann, M.E. and Lees, J. M. 1996. Robust estimation of background noise and signal detection in climatic time series. Climatic Change, 33(3), 409-445. https://doi.org/10.1007/BF00142586
Maraun, D.J. 2007. Sowas: Software for Wavelet Analysis and Synthesis, R package version 0.93,http://tocsy.pikpotsdam. de/wavelets/ (Last accessed 22 February 2016).
Maraun, D., Kurths, J., and Holschneider, M. 2007. Nonstationary Gaussian processes in wavelet domain: synthesis, estimation, and significance testing. Physical Review E, 75(1), 16707 (1-14). https://doi.org/10.1103/PhysRevE.75.016707 PMid:17358292
Meyers, S.D., Kelly, B.G. and O'Brien, J.J. 1993. An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves. Monthly Weather Review, 121(10), 2858-2866. https://doi.org/10.1175/1520-0493(1993)121<2858:AITWAI>2.0.CO;2
Mi, X., Ren, H., Ouyang, Z., Wei, W. and Ma, K. 2005. The use of the Mexican Hat and the Morlet wavelets for detection of ecological patterns. Plant Ecology, 179(1), 1-19. https://doi.org/10.1007/s11258-004-5089-4
Mudelsee, M. 2002. TAUEST: a computer program for estimating persistence in unevenly spaced weather/climate time series. Computers & Geosciences, 28(1), 69-72. https://doi.org/10.1016/S0098-3004(01)00041-3
Mudelsee, M. 2014. Climate Time Series Analysis: Classical Statistical and Bootstrap Methods. Second Edition. Berlin: Springer. https://doi.org/10.1007/978-3-319-04450-7
Pardo-Igúzquiza, E. and Rodríguez-Tovar, F. J. 2000. The permutation test as a non-parametric method for testing the statistical significance of power spectrum estimation in cyclostratigraphic research. Earth and Planetary Science Letters, 181(1), 175-189. https://doi.org/10.1016/S0012-821X(00)00191-6
Pardo-Igúzquiza, E. and Rodríguez-Tovar, F. J. 2005. MAXENPER: a program for maximum entropy spectral estimation with assessment of statistical significance by the permutation test. Computers & Geosciences, 31(5), 555-567. https://doi.org/10.1016/j.cageo.2004.11.010
Pardo-Igúzquiza, E. and Rodríguez-Tovar, F. J. 2012. Spectral and cross-spectral analysis of uneven time series with the smoothed Lomb-Scargle periodogram and Monte Carlo evaluation of statistical significance. Computers & Geosciences, 49,f 207-216. https://doi.org/10.1016/j.cageo.2012.06.018
Pardo-Igúzquiza, E. and Rodríguez-Tovar, F. J. 2013. Análisis espectral de series temporales de variables geológicas con muestreo irregular. Boletín Geológico y Minero, 124(2), 319-333.
Percival, D.B. and Walden, A.T.. 2006. Wavelet Methods for Time Series Analysis. Cambridge: Cambridge Univ. Press.
Polanco, J., Ganzedo, U., Sáenz, J., Caballero-Alfonso, A.M. and Castro-Hernandez, J.J. 2011. Wavelet analysis of correlation among canary islands octopus captures per unit effort, sea-surface temperatures and the north Atlantic oscillation. Fisheries Research, 107(1), 177-183. https://doi.org/10.1016/j.fishres.2010.10.019
Polanco-Martínez, J.M. 2012. Aplicación de Técnicas Estadísticas en el Estudio de Fenómenos Ambientales y Ecosistémicos. PhD thesis (in Spanish). Leioa: University of Basque Country (UPV/EHU).
Polanco-Martínez, J.M. and Fernández-Macho, J. 2014. The package W2CWM2C: description, features and applications. Computing in Science & Engineering, 16(6), 68-78. https://doi.org/10.1109/MCSE.2014.96
Polanco-Martínez, J.M. and Faria, S.H. 2014. Hunting spectro- temporal information in unevenly spaced paleoclimate time series. BC3 Working Paper Series 2014-07.
Basque Centre for Climate Change (BC3). Bilbao, Spain. Polanco-Martínez, J.M. and Faria, S.H. 2015. Towards a new statistical tool for analyzing unevenly space paleoclimate time series. Proceedings of the International Work- Conference on Time Series Analysis ITISE 2015, pp. 98-107.
R. Development Core Team. 2011. R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.
Scargle, J.D. 1982. Studies in astronomical time series analysis. II- Statistical aspects of spectral analysis of unevenly spaced data. The Astrophysical Journal, 263, 835-853. https://doi.org/10.1086/160554
Scargle, J.D. 1989. Studies in astronomical time series analysis. III- Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data. The Astrophysical Journal, 343, 874-887. https://doi.org/10.1086/167757
Schulz, M. and Mudelsee, M.. 2002. REDFIT: estimating rednoise spectra directly from unevenly spaced paleoclimatic time series. Computers and Geosciences, 28(3), 421-426. https://doi.org/10.1016/S0098-3004(01)00044-9
Schulz, M. and Stattegger, K. 1997. SPECTRUM: spectral analysis of unevenly spaced paleoclimatic time series. Computers and Geosciences, 23(9), 929-945. https://doi.org/10.1016/S0098-3004(97)00087-3
Torrence C. and Compo, G.P. 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61-78. https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2
Weedon, G.P. 2003. Time-Series Analysis and Cyclostratigraphy: Examining Stratigraphic Records of Environmental Cycles. Cambridge: Cambridge Univ Press. https://doi.org/10.1017/CBO9780511535482
Witt, A. and Schumann, A.Y. 2005. Holocene climate variability on millennial scales recorded in Greenland ice cores. Nonlinear Processes in Geophysics, 12(3), 345-352. https://doi.org/10.5194/npg-12-345-2005
Zhang, Z. and Moore, J.C. 2011. Improved significance testing of wavelet power spectrum near data boundaries as applied to polar research. Advances in polar science, 22(3), 192-198. https://doi.org/10.3724/SP.J.1085.2011.00192
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Funding data
Eusko Jaurlaritza
Grant numbers POS_2015_1_0006
Ministerio de Economía y Competitividad
Grant numbers RYC-2012-12167
