Abstract
Time series data with periodic trends like daily temperatures or sales of seasonal products can be seen in periods fluctuating between highs and lows throughout the year. Generalized least squares estimators are often computed for such time series data as these estimators have minimum variance among all linear unbiased estimators. However, the generalized least squares solution can require extremely demanding computation when the data is large. This paper studies an efficient algorithm for generalized least squares estimation in periodic trended regression with autoregressive errors. We develop an algorithm that can substantially simplify generalized least squares computation by manipulating large sets of data into smaller sets. This is accomplished by coining a structured matrix for dimension reduction. Simulations show that the new computation methods using our algorithm can drastically reduce computing time. Our algorithm can be easily adapted to big data that show periodic trends often pertinent to economics, environmental studies, and engineering practices.
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References
Bergland, G.D.: A fast Fourier transform algorithm for real-valued series. Commun. ACM 11, 703–710 (1968)
Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991)
Ghysels, E., Osborn, D.: The Econometric Analysis of Seasonal Time Series. Cambridge University Press, Cambridge (2001)
Grenander, U., Rosenblatt, M.: Statistical Analysis of Stationary Time Series. Wiley, New York (1957)
Judge, G.G., Hill, R.C., Griffiths, W.E., Lütkepohl, H., Lee, T.C.: Introduction to the Theory and Practice of Econometrics, 2nd edn. Wiley, New York (1988)
Kay, S.M.: Maximum likelihood estimation of signals in autoregressive noise. Amer. Statist. 42, 88–101 (1994)
Lee, J.: A reformulation of weighted least squares estimators. Amer. Statist. 63, 49–55 (2009)
Li, S., Lund, R.: Multiple change point detection via Genetic Algorithms. J. Climate 25, 674–686 (2012)
Malik, W.Q., Schummers, J., Sur, M., Brown, E.N.: Denoising two-photon calcium imaging data. PLoS ONE 6, e20490 (2011). doi:10.1371/journal.pone.0020490
Mann, M.E., Lees, J.M.: Robust estimation of background noise and signal detection in climate time series. Clim. Chang. 33, 409–445 (1996)
Piegorsch, W.W., Bailer, A.J.: Analyzing Environmental Data. Wiley, Chichester (2005)
Purdon, P.L., Solo, V., Weisskoff, R.M., Brown, E.N.: Locally regularized spatiotemporal modeling and model comparison for functional MRI. NeuroImage 14, 912–923 (2001)
Shumway, R.H., Stoffer, D.S.: Time Series Analysis and its Applications with R Examples, 3rd edn. Springer, New York (2011)
Sorensen, H.V., Jones, D.L., Heideman, M.T., Burrus, C.S.: Real-valued fast Fourier transform algorithms. IEEE Trans. Acoust. Speech Signal Processing 35, 849–863 (1987)
Yang, R., Su, Z.: Analyzing circadian expression data by harmonic regression based on autoregressive spectral estimation. Bioinformatics 26, i168–i174 (2010). doi:10.1093/bioinformatics/btq189
Zinde-Walsh, V., Galbraith, J.W.: Estimation of a linear regression model with stationary ARMA(p,q) errors. J. Economet. 47, 333–357 (1991)
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Lee, J., Dini, A. & Negri, W. An efficient generalized least squares algorithm for periodic trended regression with autoregressive errors. Numer Algor 71, 59–75 (2016). https://doi.org/10.1007/s11075-015-9984-7
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DOI: https://doi.org/10.1007/s11075-015-9984-7