Estimation for periodic ARMA models with unspecified noises

This paper presents a constructive neural network model for seasonal streamflow forecasting. This surface water hydrology is basic to the design and operation of the reservoir. A good example is the operation of a reservoir with an uncontrolled inflow but having a means of regulating the outflow. If information on the nature of the inflow is determinable in advance, then the reservoir can be operated by some decision rule to minimize downstream flood damage. For this reasons, several companies in the Brazilian Electrical Sector use the linear time-series models such as PARMA (Periodic Auto regressive Moving Average) models developed by Box-Jenkins. This paper provides for river flow prediction a numerical comparison between neural networks, called nonlinear sigmoidal regression blocks networks (NSRBN) and PARMA models. The model was implemented to forecast weekly average inflow on an step-ahead basis. It was tested on four hydroelectric plants located in different river basins in Brazil. The results obtained in the evaluation of the performance of NSRBN were better than the results obtained with PARMA models.

For autoregressive moving average (ARMA) models with infinite variance innovations, quasi-likelihood-based estimators (such as Whittle estimators) suffer from complex asymptotic distributions depending on unknown tail indices. This makes statistical inference for such models difficult. In contrast, the least absolute deviations estimators (LADE) are more appealing in dealing with heavy tailed processes. In this paper, we propose a weighted least absolute deviations estimator (WLADE) for ARMA models. We show that the proposed WLADE is asymptotically normal, is unbiased, and has the standard root-n convergence rate even when the variance of innovations is infinity. This paves the way for statistical inference based on asymptotic normality for heavy-tailed ARMA processes. For relatively small samples numerical results illustrate that the WLADE with appropriate weight is more accurate than the Whittle estimator, the quasi-maximum-likelihood estimator (QMLE), and the Gauss–Newton estimator when the innovation variance is infinite and that the efficiency loss due to the use of weights in estimation is not substantial.The authors thank the two referees for their valuable suggestions. The work was partially supported by an EPSRC research grant (UK) and the Natural Science Foundation of China (grant 10471005).

Lower order periodic stochastic models such as periodic autoregressive (PAR) and periodic autoregressive moving average (PARMA) models with and without harmonics are fitted to the monthly streamflows into Krishnarajasagara (KRS) reservoir located at the upper reaches of the river Cauvery in South India and a comparison is done between these models in terms of preservation of periodic statistics as well as over-year drought characteristics. Further, the comparison has been extended to the Thomas-Fiering lognormal 3-parameter (TFLN3) model which is supposed to be quite simple in structure and at the same time efficient in preservation of low flows. It is found that TFLN3 is able to reproduce the first two moments very well but, not the skewness or periodic correlations. On the other hand, PAR(2) and PARMA(1,1) models are able to preserve the periodic correlations. The periodic models which are not subjected to Fourier fitting, report lower relative root mean square error in overall standard deviation and correlations, compared to the ones which are subjected to Fourier fitting. From the over-year drought analysis, it is observed that all the stochastic models fitted underestimate severe drought durations corresponding to higher truncation levels. The over-year drought characteristics are computed using the method suggested by Wang and Salas (1989), based on one hundred periodic synthetic flow sequences generated from the different models. A comparative study of the preservation of drought characteristics corresponding to various truncation levels and return periods is presented.

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.

An autoregressive moving average (ARMA) is a time series model that is applied in everyday life for pattern recognition and forecasting. The ARMA model contains a noise which is assumed to have a specific distribution. The noise is often considered to have a Gaussian distribution. However in applications, the noise is sometimes found that does not have a Gaussian distribution. The first objective is to develop the ARMA model in which noise has a Laplacian distribution. The second objective is to estimate the parameters of the ARMA model. The ARMA model parameters include ARMA model orders, ARMA model coefficients, and noise variance. The parameter estimation of the ARMA model is carried out in the Bayesian framework. In the Bayesian framework, the ARMA model parameters are treated as a variable that has a prior distribution. The prior distribution for the ARMA model parameters is combined with the likelihood function for the data to get the posterior distribution for the parameter. The posterior distribution for parameters has a complex form so that the Bayes estimator cannot be determined analytically. The reversible jump Markov chain Monte Carlo (MCMC) algorithm was adopted to determine the Bayes estimator. The first result, the ARMA model can be developed by assuming Laplacian distribution noise. The second result, the performance of the algorithm was tested using simulation studies. The simulation shows that the reversible jump MCMC algorithm can estimate the parameters of the ARMA model correctly.

A useful class of models for seasonal time series that cannot be filtered or standardized to achieve second-order stationarity is that of periodic autoregressive moving average (PARMA) models, which are extensions of ARMA models that allow periodic (seasonal) parameters. An approximation to the exact likelihood for Gaussian PARMA processes is developed, and a straightforward algorithm for its maximization is presented. The algorithm is tested on several periodic ARMA(l, 1) models through simulation studies and is compared to moment estimation via the seasonal Yule–Walker equations. Applicability of the technique is demonstrated through an analysis of a seasonal stream-flow series from the Rio Caroni River in Venezuela.

Smooth least absolute deviation estimators for outlier-proof identification

This note considers a three‐step non‐Gaussian quasi‐maximum likelihood estimation (TS‐NGQMLE) of the double autoregressive model with its asymptotics, which improves efficiency of the GQMLE and circumvents inconsistency of the NGQMLE when the innovation is heavy‐tailed. Under mild conditions, the estimator not only can achieve consistency and asymptotic normality regardless of density misspecification of the innovation, but also outperforms the existing estimators, such as the GQMLE and the (weighted) least absolute deviation estimator, when the innovation is indeed heavy‐tailed.

This paper studies the asymptotic properties of a smoothed least absolute deviations estimator in a nonlinear parametric model with multiple change-points occurring at the unknown times with independent and identically distributed errors. The model is nonlinear in the sense that between two successive change-points the regression function is nonlinear into respect to parameters. It is shown via Monte Carlo simulations that its performance is competitive with that of least absolute deviations estimator and it is more efficient than the least squares estimator, particularly in the presence of the outlier points. If the number of change-points is unknown, an estimation criterion for this number is proposed. Interest of this method is that the objective function is approximated by a differentiable function and if the model contains outliers, it detects correctly the location of the change-points.

Abstract. The aim of this article is to give a complete description of the spectral measure of periodic autoregressive moving‐average (PARMA) system in terms of its coefficients. In the analysis we use the spectral theory of strongly harmonizable sequences presented in Hurd [Journal of Multivariate Analysis(1989) Vol. 29, pp. 53–67] and the form of the unique bounded solution of ARMA model with periodic coefficients. As an application of the theoretical results, we present some examples of the spectral measures for PARMA models.

For analysis and design of water resource systems, it is sometimes useful to generate high-resolution (e.g., weekly) synthetic river flows. Periodic autoregressive moving average (PARMA) time series models provide a powerful tool for generating synthetic flows. Periodically stationary models are indicated when the basic statistics (mean, variance, and autocorrelation) of the time series exhibit significant seasonal variations. Parameter estimation for high-resolution PARMA models involves numerous parameters, which can lead to overfitting. Thus, this paper develops a parsimonious method of parameter fitting for high-resolution PARMA models, using discrete Fourier transforms to represent the set of periodic autoregressive and moving average model coefficients. Model parameters are computed via the innovations algorithm, and the asymptotic distributions of the discrete Fourier transform coefficients are obtained. Those asymptotic results are useful to determine the statistically significant Fourier coeffici...

Three important parametric models for describing the correlation functions and spectra of stationary stochastic processes are the autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models. Quite recently, the MATLAB toolbox “ARMASA” has been made publicly available. This toolbox provides state-of-the-art algorithms to perform automatic identification and selection between the models based on the estimated prediction error. ARMASA works on a single segment of data, whereas in some applications, the data are available as multiple segments. We could process each segment independently and average the estimated autocorrelation functions or spectra afterward. Better performance, however, can be expected when all segments are processed simultaneously, for two reasons. Initially, the bias in the estimated model parameters depends on the number of observations in a segment. Averaging models from segments does not reduce this bias, in contrast to identification from all the data at once. In either case, the variance of the estimated model parameters will decrease. By considering all the data at once this variance reduction can be considered in the order selection stage, decreasing the selection bias. In this paper, we extend the parameter estimation algorithms of the ARMASA toolbox to the case of multiple segments. AR parameter estimation from segments has been considered before, but not for the MA and ARMA models. In addition, in the AR case, model order selection can be done easily when the segments are independent. But, in the application we are primarily interested in, the segments are formed by lines of scanning transmission electron microscopy (STEM) images, which can be correlated with each other. Because the correlation among data segments influences the effective number of independent observations available it becomes necessary to estimate that number. This is not an easy problem in general. For observations from a stochastic process with an isotropic 2-D autocorrelation function we can, however, formulate an accurate estimator in terms of the line autocorrelation function. This situation arises, for example, in STEM images of amorphous specimens. From the AR model identification for segments, the extension to the MA and ARMA models is rather straightforward because MA and ARMA algorithms are based on Durbin's methods, whose first step is the estimation of a long AR model. We compare the segment-based algorithms to conventional ARMASA with simulations of stationary stochastic processes. An illustrative example involving simulated STEM images shows large potential benefits.

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.

The increase in the producer prices limits the activities of the companies and leads to the deterioration of the national product, employment and consumer prices. In this study, the relations between the oil price, exchange rate, interest rate, wages and the producer prices for the period of 2002Q01-2022Q03 in Türkiye were examined using autoregressive moving averages (ARIMA) and artificial neural network (ANN) methods. The employed ANN structure consists of an input layer, a hidden layer with 100 neurons and an output layer. The ANN is trained and then the modelling and forecasting performances of the traditional ARIMA and nonlinear ANN methods are compared. RMSE, MAE, MAPE and R2 criteria were used to evaluate the predictive power of the ARIMA and ANN models. As a result of automatic ARIMA model estimation, it has been determined that the producer prices can be modelled using an ARMA(4.4) model, which is a subset of the ARIMA modelling. MAE, RMSE, MAPE and R2 values of ARIMA and ANN models show that the ARMA(4,4) model has slightly better accuracy compared to the ANN model. In addition, according to the ARMA(4.4) model, it is shown that the interest rate, exchange rate, oil prices and wages affect producer prices. In this context, our policy recommendations are to follow a low interest policy and to encourage the use and production of electric vehicles to reduce the use of fossil fuels in order to reduce producer prices. Keywords: Producer price index, artificial neural network ARIMA, economic modelling.

Weighted least absolute deviations estimation for ARFIMA time series with finite or infinite variance