Forecasting with prediction intervals for periodic autoregressive moving average models

Periodic autoregressive moving average (PARMA) models are indicated for time series whose mean, variance, and covariance function vary with the season. In this paper, we develop and implement forecasting procedures for PARMA models. Forecasts are developed using the innovations algorithm, along with an idea of Ansley. A formula for the asymptotic error variance is provided, so that Gaussian prediction intervals can be computed. Finally, an application to monthly river flow forecasting is given, to illustrate the method.

[1]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[2]  J. Salas,et al.  Conceptual Basis of Seasonal Streamflow Time Series Models , 1992 .

[3]  C. Ansley.  An algorithm for the exact likelihood of a mixed autoregressive-moving average process , 1979 .

[4]  A. I. McLeod,et al.  Parsimony, model adequacy and periodic correlation in time series forecasting , 1993, 1611.01535.

[5]  R H Jones,et al.  Time series with periodic structure. , 1967, Biometrika.

[6]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[7]  Mark M. Meerschaert,et al.  Modeling river flows with heavy tails , 1998 .

[8]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[9]  Mark M. Meerschaert,et al.  Innovations algorithm for periodically stationary time series , 1999 .

[10]  Richard A. Davis,et al.  Time Series: Theory and Methods (2Nd Edn) , 1993 .

[11]  K. Hipel,et al.  Time series modelling of water resources and environmental systems , 1994 .

[12]  Mark M. Meerschaert,et al.  PERIODIC MOVING AVERAGES OF RANDOM VARIABLES WITH REGULARLY VARYING TAILS , 1997 .

[13]  Mark M. Meerschaert,et al.  Identification of periodic autoregressive moving average models and their application to the modeling of river flows , 2006 .

[14]  Robert Lund,et al.  PARSIMONIOUS PERIODIC TIME SERIES MODELING , 2006 .

[15]  Robert Lund,et al.  Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models , 2000 .

[16]  Mark M. Meerschaert,et al.  Fourier-PARMA Models and Their Application to River Flows , 2007 .

[17]  Keith W. Hipel,et al.  Forecasting Quarter-Monthly Riverflow , 1985 .

[18]  H. Hurd,et al.  Periodically Correlated Random Sequences , 2007 .

[19]  Matthew C. Bowers,et al.  On the distributions of seasonal river flows: Lognormal or power law? , 2012 .

[20]  A. I. McLeod.  DIAGNOSTIC CHECKING OF PERIODIC AUTOREGRESSION MODELS WITH APPLICATION , 1994 .

[21]  Mark M. Meerschaert,et al.  Asymptotic results for Fourier‐PARMA time series , 2010 .

[22]  Robert Lund,et al.  Periodic Time Series Models , 2005 .