Electronic Journal of Statistics

Spectral correction for locally stationary Shannon wavelet processes

Idris A. Eckley and Guy P. Nason

Full-text: Open access

Abstract

It is well-known that if a time series is not sampled at a fast enough rate to capture all the high frequencies then aliasing may occur. Aliasing is a distortion of the spectrum of a series which can cause severe problems for time series modelling and forecasting. The situation is more complex and more interesting for nonstationary series as aliasing can be intermittent. Recent work has shown that it is possible to test for the absence of aliasing in nonstationary time series and this article demonstrates that additional benefits can be obtained by modelling a series using a Shannon locally stationary wavelet (LSW) process. We show that for Shannon LSW processes the effects of dyadic-sampling-induced aliasing can be reversed. We illustrate our method by simulation on Shannon LSW processes and also a time-varying autoregressive process where aliasing is detected. We present an analysis of a wind power time series and show that it can be adequately modelled by a Shannon LSW process, the absence of aliasing can not be inferred and present a dealiased estimate of the series.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 184-200.

Dates
First available in Project Euclid: 18 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1392733138

Digital Object Identifier
doi:10.1214/14-EJS880

Mathematical Reviews number (MathSciNet)
MR3178543

Zentralblatt MATH identifier
1282.62210

Keywords
Time series aliasing wavelets local stationarity

Citation

Eckley, Idris A.; Nason, Guy P. Spectral correction for locally stationary Shannon wavelet processes. Electron. J. Statist. 8 (2014), no. 1, 184--200. doi:10.1214/14-EJS880. https://projecteuclid.org/euclid.ejs/1392733138


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References

  • Brown, B. G., Katz, R. W. and Murphy, A. H. (1984). Time series models to simulate and forecast wind speed and wind power. J. Clim. Appl. Meteorol. 23 1184–1195.
  • Cardinali, A. and Nason, G. P. (2010). Costationarity of locally stationary time series. J. Time Ser. Econom. 2 Article 1.
  • Chui, C. K. (1997). Wavelets: A Mathematical Tool for Signal Analysis. SIAM, Philadelphia.
  • Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
  • Eckley, I. A. and Nason, G. P. (2005). Efficient computation of the discrete autocorrelation wavelet inner product matrix. Stat. Comput. 15 83–92.
  • Eckley, I. A. and Nason, G. P. (2013). A test for the absence of aliasing for locally stationary wavelet time series (submitted for publication).
  • Fryzlewicz, P. and Nason, G. P. (2006). Haar-Fisz estimation of evolutionary wavelet spectra. J. R. Statist. Soc. B 68 611–634.
  • Genton, M. and Hering, A. (2007). Blowing in the wind. Significance 4 11–14.
  • Gott, A. N. and Eckley, I. A. (2013). A note on the effect of wavelet choice on the estimation of the evolutionary wavelet spectrum. Commun. Stat. Simulat. 42 393–406.
  • Hannan, E. J. (1960). Time Series Analysis. Chapman and Hall, London.
  • Harris, F. J. (2004). Multirate Signal Processing for Communication Systems. Prentice Hall, Upper Saddle River, NJ, USA.
  • Hinich, M. J. and Messer, H. (1995). On the principal domain of the discrete bispectrum of a stationary signal. IEEE Trans. Sig. Proc. 43 2130–2134.
  • Hinich, M. J. and Wolinsky, M. A. (1988). A test for aliasing using bispectral analysis. J. Am. Statist. Ass. 83 499–502.
  • Huang, M. and Chalabi, Z. S. (1995). Use of time-series analysis to model and forecast wind speed. J. Wind Eng. Ind. Aerod. 56 311–322.
  • Hunt, K. and Nason, G. P. (2001). Wind speed modelling and short-term prediction using wavelets. Wind Eng. 25 55-61.
  • Landberg, L., Giebel, G., Nielsen, H. A., Nielsen, T. and Madsen, H. (2003). Short-term prediction – An overview. Wind Energy 6 273–280.
  • Nason, G. P. (2013). A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. J. R. Statist. Soc. B 75 879–904.
  • Nason, G. P. and Sapatinas, T. (2002). Wavelet packet transfer function modelling of nonstationary time series. Stat. Comput. 12 45–56.
  • Nason, G. P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Statist. Soc. B 62 271-292.
  • Neumann, M. andvon Sachs, R. (1995). Wavelet thresholding: Beyond the Gaussian iid situation. In Wavelets and Statistics, ( A. Antoniadis and G. Oppenheim, eds.). Lecture Notes in Statistics 103 Springer-Verlag, New York.
  • Ombao, H. C., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX model of non-stationary random processes. Ann. Inst. Statist. Math. 54 171–200.
  • Priestley, M. B. (1983). Spectral Analysis and Time Series. Academic Press, London.
  • Sfetsos, A. (2002). A novel approach for the forecasting of mean hourly wind speed time series. Renew. Energ 27 163–174.