Periodic ARMA Models Research Articles

Identification of periodic autoregressive moving average models and their application to the modeling of river flows

The generation of synthetic river flow samples that can reproduce the essential statistical features of historical river flows is useful for the planning, design, and operation of water resource systems. Most river flow series are periodically stationary; that is, their mean and covariance functions are periodic with respect to time. This article develops model identification and simulation techniques based on a periodic autoregressive moving average (PARMA) model to capture the seasonal variations in river flow statistics. The innovations algorithm is used to obtain parameter estimates. An application to monthly flow data for the Fraser River in British Columbia is included. A careful statistical analysis of the PARMA model residuals, including a truncated Pareto model for the extreme tails, produces a realistic simulation of these river flows.

Calculation of the Fisher Information Matrix for Periodic ARMA Models

ABSTRACT The present article is mainly concerned with the calculation of the Fisher information matrix associated to a periodic autoregressive moving average model (P-ARMA). We provide a computation algorithm based on the conditional likelihood function expression. The established algorithm extends Klein and Mélard's algorithm (1989) elaborated for the classical ARMA models, to the case of periodic autoregressive moving average models. Moreover, for the application of this algorithm, we provide a procedure to compute the theoretical periodic autocovariance function in terms of the parameters of the periodic model. In addition, we give a necessary and sufficient condition for non singular Fisher information matrix of a periodic ARMA model.

Influence asymptotique de la correction par la moyenne sur l'estimation d'un modèle AR(1) périodique

This Note studies asymptotic influence of mean-correction on the parameter least squares estimation for a periodic AR(1) model. Unlike the stationary ARMA case, we show that fitting a periodic ARMA model with intercepts to the observed series can provide substantial gains in terms of asymptotic accuracy for the parameter least squares estimators compared with fitting a periodic ARMA model without intercepts to the mean-corrected series. To cite this article: A. Gautier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Testing and modelling autoregressive conditional heteroskedasticity of streamflow processes

Abstract. Conventional streamflow models operate under the assumption of constant variance or season-dependent variances (e.g. ARMA (AutoRegressive Moving Average) models for deseasonalized streamflow series and PARMA (Periodic AutoRegressive Moving Average) models for seasonal streamflow series). However, with McLeod-Li test and Engle's Lagrange Multiplier test, clear evidences are found for the existence of autoregressive conditional heteroskedasticity (i.e. the ARCH (AutoRegressive Conditional Heteroskedasticity) effect), a nonlinear phenomenon of the variance behaviour, in the residual series from linear models fitted to daily and monthly streamflow processes of the upper Yellow River, China. It is shown that the major cause of the ARCH effect is the seasonal variation in variance of the residual series. However, while the seasonal variation in variance can fully explain the ARCH effect for monthly streamflow, it is only a partial explanation for daily flow. It is also shown that while the periodic autoregressive moving average model is adequate in modelling monthly flows, no model is adequate in modelling daily streamflow processes because none of the conventional time series models takes the seasonal variation in variance, as well as the ARCH effect in the residuals, into account. Therefore, an ARMA-GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) error model is proposed to capture the ARCH effect present in daily streamflow series, as well as to preserve seasonal variation in variance in the residuals. The ARMA-GARCH error model combines an ARMA model for modelling the mean behaviour and a GARCH model for modelling the variance behaviour of the residuals from the ARMA model. Since the GARCH model is not followed widely in statistical hydrology, the work can be a useful addition in terms of statistical modelling of daily streamflow processes for the hydrological community.

Computation and Characterization of Autocorrelations and Partial Autocorrelations in Periodic ARMA Models

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving‐average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.

Modeling plant growth using regression with periodically correlated errors

Recent developments in the analysis of periodically correlated time series are used to analyze data in a new approach to modeling plant growth. The data—carbon dioxide (CO2) exchange rates in plants—were gathered using a novel and innovative system for near-continuous measurement of CO2 exchange. Least squares estimates are derived for a piecewise polynomial trend, which carries over in an obvious fashion to a general linear model. The residuals from fitting this trend have seasonal components that are well modeled using a periodic autoregressive moving average (PARMA) model. Standard errors for the trend estimates are then derived based on the fitted PARMA model.

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving‐average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one‐step‐ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.

On Periodic Structures and Testing for Seasonal Unit Roots

Abstract The standard testing procedures for seasonal unit roots developed so far have been based mainly on time-invariant autoregressive integrated moving average (ARIMA) processes with AR polynomials involving seasonal differencing. One attractive alternative is to use periodic ARMA models. Convenient procedures are presented for testing for the presence of unit roots at the zero and seasonal frequencies in periodic time series. The limiting distributions are derived and tabulated simulation evidence illustrates the advantages of allowing for periodicity. The tests are illustrated via applications to macroeconomic and ozone level data.

PARAMETER ESTIMATION FOR PERIODIC ARMA MODELS

Abstract. In time series analysis of data sequences, the estimation of the parameters of an identified autoregressive moving‐average (ARMA) model is a well‐known and straightforward exercise. However, if the parameters of the model are periodic (i.e. a periodic ARMA (PARMA) model) then the estimation process becomes more difficult. This paper describes an on‐line parameter estimation technique, based on methods from automatic control, which is demonstrated to provide consistent estimates of PARMA model parameters.

DIAGNOSTIC CHECKING OF PERIODIC AUTOREGRESSION MODELS WITH APPLICATION

Abstract. An overview of model building with periodic autoregression (PAR) models is given emphasizing the three stages of model development:identification, estimation and diagnostic checking. New results on the distribution of residual autocorrelations and suitable diagnostic checks are derived. The validity of these checks is demonstrated by simulation. The methodology discussed is illustrated with an application. It is pointed out that the PAR approach to model development offers some important advantages over the more general approach using periodic autoregressive moving‐average models.

Multichannel ARMA modeling by least squares circular lattice filtering

The paper makes an attempt to develop least squares lattice algorithms for the ARMA modeling of a linear, slowly time-varying, multichannel system employing scalar computations only. Using an equivalent scalar, periodic ARMA model and a circular delay operator, the signal set for each channel is defined in terms of circularly delayed input and output vectors corresponding to that channel. The orthogonal projection of each current output vector on the subspace spanned by the corresponding signal set is then computed in a manner that allows independent AR and MA order recursions. The resulting lattice algorithm can be implemented in a parallel architecture employing one processor per channel with the data flowing amongst them in a circular manner. The evaluation of the ARMA parameters from the lattice coefficients follows the usual step-up algorithmic approach but requires, in addition, the circulation of certain variables across the processors since the signal sets become linearly dependent beyond certain stages. The proposed algorithm can also be used to estimate a process from two correlated, multichannel processes adaptively allowing the filter orders for both the processes to be chosen independently of each other. This feature is further exploited for ARMA modeling a given multichannel time series with unknown, white input. >

Modeling of streamflow processes at different time scales

The analysis and modeling of streamflow processes has attracted the attention of water resources specialists for several decades. A number of models have been suggested in the past for representing seasonal and annual streamflow processes. The topic addressed in this paper centers around the compatibility of stochastic models of streamflow at different time scales. More specifically, given a model for monthly flows, the models for the processes obtained by aggregation, i.e., models for bimonthly, quarterly, etc., time scales, are derived. Likewise, parameter space and covariance properties of such derived processes as well as the relationship of such properties of different time scales are given. These concepts are applied to modeling streamflow of the Niger River. The developments are restricted to the family of periodic autoregressive moving average (PARMA) processes. For instance, it was found that monthly flows based on the PARMA(2, 1) process leads to PARMA(2, 2) bimonthly flows and stationary ARMA(2, 2) annual flows. Furthermore, applications to modeling the Niger River flows suggest that one can reproduce the seasonal and annual second‐order statistics without using disaggregation if PARMA models are used for modeling the seasonal flows.

Spectral factorization of linear periodic systems with application to the optimal prediction of periodic ARMA models

A cyclostationary process is a stochastic process whose statistical parameters, such as mean and autocorrelation, exhibit suitable periodicity. In this paper, we consider the cyclospectral factorization problem which consists of finding a Markovian (state-space) realization of a given cyclostationary process. It is shown that a significant class of periodic state-space representations is in a one-to-one correspondence with the periodic solutions of a difference periodic Riccati equation. This result is applied to the solution of the prediction problem for ARMA models with periodically varying coefficients. If the periodic ARMA model is minimum-phase, the optimal predictor is given a simple input-output expression that generalizes the well-known one for time-invariant ARMAs. Otherwise, the computation of the optimal predictor calls for the solution of a cyclospectral factorization problem.

ESTIMATION IN MULTIPLE AUTOREGRESSIVE‐MOVING AVERAGE MODELS USING PERIODICITY

Abstract. An estimation procedure for multiple autoregressive‐moving average models is suggested which takes advantage of the possibility of transferring these multiple models to univariate periodic autoregressive‐moving average models. The procedure is simple enough to be practicable using less efficient computers and it should give acceptable estimates for longer time series. By using this method for shorter time series, the obtained estimates may serve as the initial estimates for more complicated estimation procedures.

PERIODIC AUTOREGRESSIVE‐MOVING AVERAGE (PARMA) MODELING WITH APPLICATIONS TO WATER RESOURCES1

ABSTRACTResults involving correlation properties and parameter estimation for autoregressive‐moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum‐likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.

Aggregation and estimation for low‐order periodic ARMA models

The aggregated time series resulting from summing over the seasons of a seasonal time series, which is assumed to follow either an AR(1) or ARMA(1, 1) model having periodic (seasonal) parameters, is shown to follow an ARMA(1, 1) model. Parameter estimation for such models is investigated analytically and via simulation. Significant gain in parameter estimation efficiency at the aggregated level is demonstrated when the seasonal data and their model is utilized rather than the aggregated (annual) data and their model.