×

zbMATH — the first resource for mathematics

Spectral estimation for locally stationary time series with missing observations. (English) Zbl 1252.60034
Summary: Time series arising in practice often have an inherently irregular sampling structure or missing values, that can arise for example due to a faulty measuring device or complex time-dependent nature. Spectral decomposition of time series is a traditionally useful tool for data variability analysis. However, existing methods for spectral estimation often assume a regularly-sampled time series, or require modifications to cope with irregular or ‘gappy’ data. Additionally, many techniques also assume that the time series are stationary, which in the majority of cases is demonstrably not appropriate. This article addresses the topic of spectral estimation of a non-stationary time series sampled with missing data. The time series is modelled as a locally stationary wavelet process in the sense introduced by G. P. Nason et al. [“Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum”. J. R. Stat. Soc. B 62, No. 2, 271–292 (2000)] and its realization is assumed to feature missing observations. Our work proposes an estimator (the periodogram) for the process wavelet spectrum, which copes with the missing data whilst relaxing the strong assumption of stationarity. At the centre of our construction are second generation wavelets built by means of the lifting scheme, designed to cope with irregular data. We investigate the theoretical properties of our proposed periodogram, and show that it can be smoothed to produce a bias-corrected spectral estimate by adopting a penalized least squares criterion. We demonstrate our method with real data and simulated examples.

MSC:
60G10 Stationary stochastic processes
62M15 Inference from stochastic processes and spectral analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berger, A.L.: Long-term variations of daily insolation and quaternary climatic changes. J. Atmos. Sci. 35, 2362–2367 (1978) · doi:10.1175/1520-0469(1978)035<2362:LTVODI>2.0.CO;2
[2] Bos, R., de Waele, S., Broersen, P.M.T.: Autoregressive spectral estimation by application of the Burg algorithm to irregularly sampled data. IEEE Trans. Instrum. Meas. 51(6), 1289–1294 (2002) · doi:10.1109/TIM.2002.808031
[3] Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, Berlin (2009) · Zbl 1169.62074
[4] Broersen, P.M.T.: Automatic spectral analysis with missing data. Digit. Signal Process. 16(6), 754–766 (2006) · doi:10.1016/j.dsp.2006.01.001
[5] Broersen, P.M.T.: Time series models for spectral analysis of irregular data far beyond the mean data rate. Meas. Sci. Technol. 19(1), 015103 (2008). http://stacks.iop.org/0957-0233/19/i=1/a=015103
[6] Broersen, P.M.T., de Waele, S., Bos, R.: Autoregressive spectral analysis when observations are missing. Automatica 40(9), 1495–1504 (2004) · Zbl 1055.93553 · doi:10.1016/j.automatica.2004.04.011
[7] Cazelles, B., Chavez, M., Magny, G.C., Guégan, J., Hales, S.: Time-dependent spectral analysis of epidemiological time-series with wavelets. J. R. Soc. Interface 4(15), 625–636 (2007) · doi:10.1098/rsif.2007.0212
[8] Chatfield, C.: The Analysis of Time Series: An Introduction. Chapman &amp; Hall/CRC Press, London/Boca Raton (2004) · Zbl 1050.62089
[9] Clinger, W., Van Ness, J.W.: On unequally spaced time points in time series. Ann. Stat. 4(4), 736–745 (1976) · Zbl 0351.62066 · doi:10.1214/aos/1176343545
[10] Cranstoun, S.D., Ombao, H.C., von Sachs, R., Guo, W., Litt, B.: Time-frequency spectral estimation of multichannel EEG using the Auto-SLEX method. IEEE Trans. Biomed. Eng. 49(9), 988–996 (2002) · doi:10.1109/TBME.2002.802015
[11] Crucifix, M.: Global change: climate’s astronomical sensors. Nature 456(7218), 47–48 (2008) · doi:10.1038/456047a
[12] Crucifix, M., Rougier, J.: On the use of simple dynamical systems for climate predictions. Euro Phys. J. 174(1), 11–31 (2009)
[13] Crucifix, M., Loutre, M.F., Berger, A.: The climate response to the astronomical forcing. In: Calisesi, Y., Bonnet, R.M., Gray, L., Langen, J., Lockwood, M. (eds.) Solar Variability and Planetary Climates. Space Sciences Series of ISSI, vol. 23, pp. 213–226. Springer, New York (2007)
[14] Dahlhaus, R.: Fitting time series models to nonstationary processes. Ann. Stat. 25(1), 1–37 (1997) · Zbl 0871.62080 · doi:10.1214/aos/1034276620
[15] Dahlhaus, R., Subba Rao, S.: Statistical inference for time-varying ARCH processes. Ann. Stat. 34(3), 1075–1114 (2006) · Zbl 1113.62099 · doi:10.1214/009053606000000227
[16] Dahlhaus, R., Subba Rao, S.: A recursive online algorithm for the estimation of time-varying ARCH parameters. Bernoulli 13(2), 389–422 (2007) · Zbl 1127.62078 · doi:10.3150/07-BEJ5009
[17] Dilmaghani, S., Henry, I.C., Soonthornnonda, P., Christensen, E.R., Henry, R.C.: Harmonic analysis of environmental time series with missing data or irregular sample spacing. Environ. Sci. Technol. 41(20), 7030–7038 (2007) · doi:10.1021/es0700247
[18] Engle, R.F.: The econometrics of ultra-high-frequency data. Econometrica 68(1), 1–22 (2000) · Zbl 1056.91535 · doi:10.1111/1468-0262.00091
[19] Fryźlewicz, P.: Wavelet techniques for time series and Poisson data. Ph.D. thesis, University of Bristol, UK (2003)
[20] Fryźlewicz, P., Nason, G.P.: Haar-Fisz estimation of evolutionary wavelet spectra. J. R. Stat. Soc. B 68, 611–634 (2006) · Zbl 1110.62121 · doi:10.1111/j.1467-9868.2006.00558.x
[21] Fryźlewicz, P., Sapatinas, T., Rao, S.: A Haar-Fisz technique for locally stationary volatility estimation. Biometrika 93(3), 687 (2006) · Zbl 1109.62095 · doi:10.1093/biomet/93.3.687
[22] Hall, P., Fisher, N.I., Hoffmann, B.: On the nonparametric estimation of covariance functions. Ann. Stat. 22(4), 2115–2134 (1994) · Zbl 0828.62036 · doi:10.1214/aos/1176325774
[23] Jansen, M., Nason, G.P., Silverman, B.W.: Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients. In: Unser, M., Aldroubi, A. (eds.) Wavelet Applications in Signal and Image Processing IX. SPIE, vol. 4478, pp. 87–97 (2001)
[24] Jansen, M., Nason, G.P., Silverman, B.W.: Multidimensional nonparametric regression using lifting. Tech. Rep. 04:17, Statistics Group, Department of Mathematics, University of Bristol, UK (2004)
[25] Jansen, M., Nason, G.P., Silverman, B.W.: Multiscale methods for data on graphs and irregular multidimensional situations. J. R. Stat. B 71(1), 97–125 (2009) · Zbl 1231.62054 · doi:10.1111/j.1467-9868.2008.00672.x
[26] Jones, R.H.: Spectral analysis with regularly missed observations. Ann. Math. Stat. 33(2), 455–461 (1962) · Zbl 0114.34503 · doi:10.1214/aoms/1177704572
[27] Knight, M.I., Nason, G.P.: Improving prediction of hydrophobic segments along a transmembrane protein sequence using adaptive multiscale lifting. SIAM J. Multiscale Model. Simul. 5, 115–129 (2006) · Zbl 1236.62154
[28] Knight, M.I., Nason, G.P.: A nondecimated lifting transform. Stat. Comput. 19(1), 1–16 (2009) · doi:10.1007/s11222-008-9062-2
[29] Knight, M.I., Nunes, M.A.: nlt: a nondecimated lifting scheme algorithm. R package version 2.1-1 (2010)
[30] Lüthi, D., Le Floch, M., Bereiter, B., Blunier, T., Barnola, J.M., Siegenthaler, U., Raynaud, D., Jouzel, J., Fischer, H., Kawamura, K., et al.: High-resolution carbon dioxide concentration record 650,000–800,000 years before present. Nature 453(7193), 379–382 (2008) · doi:10.1038/nature06949
[31] Mikosch, T., Starica, C.: Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Rev. Econ. Stat. 86(1), 378–390 (2004) · doi:10.1162/003465304323023886
[32] Mondal, D., Percival, D.B.: Wavelet variance analysis for gappy time series. Ann. Inst. Stat. Math. 62(5), 943–966 (2010) · Zbl 1432.62309 · doi:10.1007/s10463-008-0195-z
[33] Nason, G.P.: Wavelet Methods in Statistics with R. Springer, Berlin (2008) · Zbl 1165.62033
[34] Nason, G.P., Von Sachs, R.: Wavelets in time-series analysis. Philos. Trans. R. Soc. Lond. A 357(1760), 2511–2526 (1999) · Zbl 1054.62583 · doi:10.1098/rsta.1999.0445
[35] Nason, G.P., Von Sachs, R., Kroisandt, G.: Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. B 62(2), 271–292 (2000) · Zbl 04558571 · doi:10.1111/1467-9868.00231
[36] Nason, G.P., Sapatinas, T., Sawczenko, A.: Wavelet packet modelling of infant sleep state using heart rate data. Sankhyā B 63(2), 199–217 (2001) · Zbl 1192.94034
[37] Nason, G.P., Kovac, A., Maechler, M.: Wavethresh: Software to perform wavelet statistics and transforms. R package version 4.2-1 (2008)
[38] Nunes, M.A., Knight, M.I.: Adlift: an adaptive lifting scheme algorithm. R package version 1.2-3
[39] Nunes, M.A., Knight, M.I., Nason, G.P.: Adaptive lifting for nonparametric regression. Stat. Comput. 16(2), 143–159 (2006) · doi:10.1007/s11222-006-6560-y
[40] Ombao, H., Raz, J., Von Sachs, R., Guo, W.: The SLEX model of a non-stationary random process. Ann. Inst. Stat. Math. 54(1), 171–200 (2002) · Zbl 0993.62082 · doi:10.1023/A:1016130108440
[41] Percival, D.B., Walden, A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge (2000) · Zbl 0963.62079
[42] Priestley, M.B.: Spectral Analysis and Time Series. Academic Press, San Diego (1981) · Zbl 0537.62075
[43] Sanderson, J.: Wavelet methods for time series with bivariate observations and irregular sampling grids. Ph.D. thesis, University of Bristol, UK (2010)
[44] Stoica, P., Sandgren, N.: Spectral analysis of irregularly-sampled data: Paralleling the regularly-sampled data approaches. Digit. Signal Process. 16(6), 712–734 (2006) · doi:10.1016/j.dsp.2006.08.012
[45] Sweldens, W.: The lifting scheme: a new philosophy in biorthogonal wavelet construction. In: Laine, A., Unser, M. (eds.) Wavelet Applications in Signal and Image Processing III. Proc. SPIE, vol. 2569, pp. 68–79 (1995)
[46] Van Bellegem, S., Von Sachs, R.: Locally adaptive estimation of evolutionary wavelet spectra. Ann. Stat. 36, 1879–1924 (2008) · Zbl 1142.62067 · doi:10.1214/07-AOS524
[47] Witt, A., Schumann, A.Y.: Holocene climate variability on millennial scales recorded in Greenland ice cores. Nonlinear Process. Geophys. 12(3), 345–352 (2005) · doi:10.5194/npg-12-345-2005
[48] Wolff, E.W.: Understanding the past-climate history from Antarctica. Antarct. Sci. 17(04), 487–495 (2005) · doi:10.1017/S0954102005002919
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.