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PARMA Models with Applications in R

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Cyclostationarity: Theory and Methods - II (CSTA 2014)

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Periodic autoregressive–moving-average models (periodic ARMA models, PARMA models) are used to model non-stationary time series with periodic structure. They are similar to ARMA except the coefficients that are periodic in time with a common period. They are widely applied in climatology, hydrology, meteorology and economics data. In this paper we want to familiarize the readers with all the essential steps of PARMA model fitting. We present in detail the non-parametric spectral analysis, model identification, parameter estimation, diagnostic checking (model verification) and prediction on the real data example. Our aim is to provide appropriate tool for the complete analysis of periodic time series using PARMA modelling and to popularize this approach among non-specialists.

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  1. Anderson, P. L., Meerschaert, M. M., & Zhang, K. (2013). Forecasting with prediction intervals for periodic autoregressive moving average models. Journal of Time Series Analysis, 34(2), 187–193.

    Google Scholar 

  2. Anderson, P. L., Tesfaye, Y. G., & Meerschaert, M. M. (2007). Fourier-PARMA models and their application to river flows. Journal of Hydrologic Engineering, 12(5), 462–472.

    Google Scholar 

  3. Balcilar, M., & McLeod, A. I. (2011). pear: Package for periodic autoregression analysis.

  4. Bloomfield, P., Hurd, H. L., & Lund, R. (1994). Periodic correlation in stratospheric ozone data. Journal of Time Series Analysis, 15, 127–150.

    Google Scholar 

  5. Bloomfield, P., Hurd, H. L., Lund, R. B., & Smith, R. (1995). Climatological time series with periodic correlation. Journal of Climate, 8, 2787–2809.

    Google Scholar 

  6. Brockwell, P. J., & Davis, R. A. (2002). Introduction to time series and forecasting. New York: Springer.

    Book  MATH  Google Scholar 

  7. Broszkiewicz-Suwaj, E., Makagon, A., Weron, R., & Wyłomańska, A. (2004). On detecting and modeling periodic correlation in financial data. Physica A, 336, 196–205.

    Google Scholar 

  8. Cramér, H. (1961). Methods of mathematical statistics. New York: Princeton University Press.

    Google Scholar 

  9. Dehay, D., & Hurd, H. L. (1994). Representation and estimation for periodically and almost periodically correlated random processes. In W. A. Gardner (Ed.), Cyclostationarity in communications and signal processing. New York: IEEE Press.

    Google Scholar 

  10. Dudek, A. E., Hurd, H., & Wójtowicz, W. perARMA: Package for periodic time series analysis. R package version 1.5,

  11. Franses, P. H. (1996). Periodicity and stochastic trends in economic time series. Oxford: Oxford Press.

    MATH  Google Scholar 

  12. Gardner, W. A. (1994). Cyclostationarity in communications and and signal processing. New York: IEEE Press.

    MATH  Google Scholar 

  13. Gardner, W. A., Napolitano, A., & Paura, L. (2006). Cyclostationarity: Half a century of research. Signal Processing, 86, 639–697.

    Google Scholar 

  14. Gerr, N. L., & Hurd, H. L. (1991). Graphical methods for determining the presence of periodic correlation in time series. Journal of Time Series Analysis, 12, 337–350.

    Google Scholar 

  15. Gladyshev, E. G. (1961). Periodically correlated random sequences. Soviet Mathematics, 2, 385–388.

    Google Scholar 

  16. Hurd, H. L., & Miamee, A. G. (2007). Periodically correlated random sequences: Spectral theory and practice. Hoboken: Wiley.

    Book  Google Scholar 

  17. Jones, R., & Brelsford, W. (1967). Time series with periodic structure. Biometrika, 54, 403–408.

    Google Scholar 

  18. López-de-Lacalle, J. (2012). partsm: Periodic autoregressive time series models. R package version 1.1,

  19. Pagano, M. (1978). On periodic and multiple autoregressions. Annals of Statistics, 6(6), 1310–1317.

    Google Scholar 

  20. Parzen, E., & Pagano, M. (1979). An approach to modelling seasonally stationary time series. Journal of Econometrics, 9(1–2), 137–153.

    Google Scholar 

  21. Rasmussen, P. F., Salas, J., Fagherazzi, L., Rassam, J. C., & Bobée, B. (1996). Estimation and validation of contemporaneous PARMA models for streamflow simulation. Water Resources Research, 32(10), 3151–3160.

    Google Scholar 

  22. Sabri, K., Badaoui, M., Guillet, F., Belli, A., Millet, G., & Morin, J. B. (2010). Cyclostationary modelling of ground reaction force signals. Signal Processing, 90, 1146–1152.

    Article  MATH  Google Scholar 

  23. Sakai, H. (1991). On the spectral density matrix of a periodic ARMA process. Journal of Time Series Analysis, 12(2), 73–82.

    Google Scholar 

  24. Smadi, A. A. (2009). Periodic auto-regression modeling of the temperature data of Jordan. Pakistan Journal of Statistics, 25(3), 323–332.

    Google Scholar 

  25. Swider, D. J., & Weber, C. (2007). Extended ARMA models for estimating price developments on day-ahead electricity markets. Electric Power Research, 77, 583–593.

    Google Scholar 

  26. Tesfaye, Y., Meerschaert, M. M., & Anderson, P. L. (2006). Identification of periodic autoregressive moving average models and their application to the modeling of river flows. Water Resources Research, 42, W01419.

    Google Scholar 

  27. Vecchia, A. V. (1985). Maximum likelihood estimation for periodic autoregressive moving average models. Technometrics, 27, 375–384.

    Google Scholar 

  28. Vecchia, A. V. (1985). Periodic autoregressive moving average (PARMA) modeling with applications to water resources. Water Resources Bulletin, 21, 721–730.

    Google Scholar 

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Research of Anna E. Dudek was partially supported by the Polish Ministry of Science and Higher Education and AGH local grant.

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Correspondence to Anna E. Dudek .

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Dudek, A.E., Hurd, H., Wójtowicz, W. (2015). PARMA Models with Applications in R. In: Chaari, F., Leskow, J., Napolitano, A., Zimroz, R., Wylomanska, A., Dudek, A. (eds) Cyclostationarity: Theory and Methods - II. CSTA 2014. Applied Condition Monitoring, vol 3. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16329-1

  • Online ISBN: 978-3-319-16330-7

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