Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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.
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.
Balcilar, M., & McLeod, A. I. (2011). pear: Package for periodic autoregression analysis. http://cran.r-project.org/web/packages/pear/.
Bloomfield, P., Hurd, H. L., & Lund, R. (1994). Periodic correlation in stratospheric ozone data. Journal of Time Series Analysis, 15, 127–150.
Bloomfield, P., Hurd, H. L., Lund, R. B., & Smith, R. (1995). Climatological time series with periodic correlation. Journal of Climate, 8, 2787–2809.
Brockwell, P. J., & Davis, R. A. (2002). Introduction to time series and forecasting. New York: Springer.
Broszkiewicz-Suwaj, E., Makagon, A., Weron, R., & Wyłomańska, A. (2004). On detecting and modeling periodic correlation in financial data. Physica A, 336, 196–205.
Cramér, H. (1961). Methods of mathematical statistics. New York: Princeton University Press.
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.
Dudek, A. E., Hurd, H., & Wójtowicz, W. perARMA: Package for periodic time series analysis. R package version 1.5, http://cran.r-project.org/web/packages/perARMA/.
Franses, P. H. (1996). Periodicity and stochastic trends in economic time series. Oxford: Oxford Press.
Gardner, W. A. (1994). Cyclostationarity in communications and and signal processing. New York: IEEE Press.
Gardner, W. A., Napolitano, A., & Paura, L. (2006). Cyclostationarity: Half a century of research. Signal Processing, 86, 639–697.
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.
Gladyshev, E. G. (1961). Periodically correlated random sequences. Soviet Mathematics, 2, 385–388.
Hurd, H. L., & Miamee, A. G. (2007). Periodically correlated random sequences: Spectral theory and practice. Hoboken: Wiley.
Jones, R., & Brelsford, W. (1967). Time series with periodic structure. Biometrika, 54, 403–408.
López-de-Lacalle, J. (2012). partsm: Periodic autoregressive time series models. R package version 1.1, http://cran.r-project.org/web/packages/partsm/.
Pagano, M. (1978). On periodic and multiple autoregressions. Annals of Statistics, 6(6), 1310–1317.
Parzen, E., & Pagano, M. (1979). An approach to modelling seasonally stationary time series. Journal of Econometrics, 9(1–2), 137–153.
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.
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.
Sakai, H. (1991). On the spectral density matrix of a periodic ARMA process. Journal of Time Series Analysis, 12(2), 73–82.
Smadi, A. A. (2009). Periodic auto-regression modeling of the temperature data of Jordan. Pakistan Journal of Statistics, 25(3), 323–332.
Swider, D. J., & Weber, C. (2007). Extended ARMA models for estimating price developments on day-ahead electricity markets. Electric Power Research, 77, 583–593.
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.
Vecchia, A. V. (1985). Maximum likelihood estimation for periodic autoregressive moving average models. Technometrics, 27, 375–384.
Vecchia, A. V. (1985). Periodic autoregressive moving average (PARMA) modeling with applications to water resources. Water Resources Bulletin, 21, 721–730.
Acknowledgments
Research of Anna E. Dudek was partially supported by the Polish Ministry of Science and Higher Education and AGH local grant.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
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. https://doi.org/10.1007/978-3-319-16330-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-16330-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16329-1
Online ISBN: 978-3-319-16330-7
eBook Packages: EngineeringEngineering (R0)