Periodic ARMA Models Research Articles

Holiday Load Power Forecasting Based on SOIF-PAMAM method

Focusing on the particularity of holiday load, in this paper, a periodic autoregressive moving average model (PAMAM) algorithm based on selecting optimal input features (SOIF) is proposed to predict the short-term holiday power load. In short-term load forecasting models, there are few researches on feature selection (FS). However, as more and more intelligent hybrid models are used in real-time load forecasting, FS has become a key factor affecting the forecasting accuracy. Based on the idea of SOIF, PAMAM model is proposed to improve the influence of FS factors, and the holiday equations are combined into periodic autoregressive moving average model, so as to improve the short-term forecasting. In order to simplify the calculation, in this paper, the probability distribution is used to calculate the FS, and the autoregressive spline algorithm is used to establish the nonlinear solar radiation and temperature effect model. Based on the statistics of solar radiation intensity, temperature and other data during the Spring Festival, in this paper we analyze the influence of the above factors on the short-term power load forecasting during holidays. Experimental results show that SOIF-PAMAM algorithm in which temperature and other weather conditions are considered can significantly improve the prediction accuracy, the average absolute error is 2.45%, and the root mean square error is 2.61%.

Portmanteau tests for periodic ARMA models with dependent errors

In this article, we derive the asymptotic distributions of residual and normalized residual empirical autocovariances and autocorrelations of (parsimonious) periodic autoregressive moving‐average (PARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We then deduce the modified portmanteau statistics. We establish the asymptotic behavior of the proposed statistics. It is shown that the asymptotic distribution of the modified portmanteau tests is that of a weighted sum of independent chi‐squared random variables, which can be different from the usual chi‐squared approximation used under independent and identically distributed assumption on the noise. We also propose another test based on a self‐normalization approach to check the adequacy of PARMA models. A set of Monte Carlo experiments and an application to financial data are presented.

Parsimonious time series modeling for high frequency climate data

Climate data often provides a periodically stationary time series, due to seasonal variations in the mean and covariance structure. Periodic ARMA models, where the parameters vary with the season, capture the nonstationary behavior. High frequency data collected weekly or daily results in a large number of model parameters. In this article, we apply discrete Fourier transforms to the parameter vectors, and develop a test for the statistically significant harmonics. An example of daily high temperatures illustrates the method, whereby a periodic autoregressive model with 1095 parameters is reduced to a parsimonious 12 parameter version without any apparent loss of fidelity.

Measures of Cross‐Dependence for Bidimensional Periodic AR(1) Model with α‐Stable Distribution

Discrete‐time models with periodic behavior are useful for the description of different phenomenon. The most popular time series taking into consideration the periodicity of the real data is the periodic autoregressive moving average (PARMA) model. The PARMA models were considered in the literature from a theoretical and practical point of view. Most of the considerations related to the PARMA models are based on the assumption of the Gaussian (or finite‐variance) distribution of the noise. However, in many applications, the Gaussian distribution seems to be inappropriate. Thus, generalized models are considered. The natural extension of the Gaussian distribution is the ‐stable one which is a perfect distribution for the modeling of real data with large observations. However, for the ‐stable‐based models the classical methods adequate to Gaussian‐based systems cannot be used. The main problem comes from the fact that, in general, for the ‐stable based models the covariance cannot be applied as a measure of dependence. Thus, alternative measures are used. In this article, we consider the generalization of the classical PARMA models and take into consideration the ‐stable PAR system. Moreover, we analyze the bidimensional version of the univariate model and examine its structure of cross‐dependence in the language of the alternative cross‐dependence measures appropriate for the infinite‐variance systems. As the main result, we prove that the ratio of two considered alternative cross‐dependence measures tends to the stability index of the noise distribution. This result is the continuation of the authors' previous research where a similar study was performed for one‐dimensional models based on the ‐stable distribution. Moreover, in the authors' recent papers the stationary bidimensional time series models were considered in the same direction. Finally, we propose a possible application of the introduced methodology.

Investigation of the accuracy of linear and nonlinear time series models in modeling and forecasting of pan evaporation in IRAN

Water evaporation process is one of the main components of the hydrological cycle, according to which the correct estimation of this phenomenon plays an important role in irrigation management and river flow forecasting. Statistical and random models in the form of time series analysis are commonly used methods of estimation and forecasting. In this study, ARMA (autoregressive moving average), ARIMA (autoregressive integrated moving average), PARMA (periodic autoregressive moving average), and BL (bilinear) time series models (TSMs) are used to predict the annual and monthly pan evaporation values from the evaporation stations in several provinces of Iran in a 20-year statistical period. Results showed that PARMA model’s accuracy in term of correlation coefficient and Nash–Sutcliffe efficiency for almost all 31 stations had a precise forecasting among other proposed TSMs. For PARMA model, the forecasting accuracy in term of NSE indicated that PARMA model almost was the promising model among other linear and nonlinear TSMs in prediction of Epan for all stations, except Arak (NSE = 0.94) and Qom (NSE = 0.93). The ARIMA model for Khorramabad and Bandar Abbas with NSE = 0.52 had the unreliable prediction for Epan compared with other stations. In addition, Arak station in term of RMSE had the least error, 23.79 mm/day and 10.90 mm/day, respectively, for training and testing stages among the other stations.

On mixed PARMA modeling of epidemiological time series data

A rich class of traditional statistical models and methods is currently available for the early detection of epidemic activity in epidemiological surveillance systems. Real time surveillance though, is often difficult to be fully achieved because of the seasonality involved in the series. Indeed, whenever the correlation structure of a series depends on the season, the time series involved fails to reach stationarity with all the associated modeling consequences. In such situations, a useful class of models is that of periodic auto-regressive moving average (PARMA) models allowing parameters that depend on season. In this work, for the modeling of influenza-like syndrome morbidity, the general form as well as special cases of PARMA models are considered, and via model selection identification and likelihood-based techniques, the optimal model is selected. Climatological and meteorological covariates associated with influenza-like syndrome are also incorporated into the model structure. The derived results are satisfactory since the selected model succeeds in identifying the epidemic waves, and in estimating accurately the influenza-like syndrome morbidity burden in the case of Greece (for the period 2014–2016).

Estimation for periodic ARMA models with unspecified noises

In this paper we study the weighted least absolute deviations estimator (WLADE) for Periodic ARMA (PARMA) models with unspecified noises, asymptotic normality of the estimator is derived. The random weighting approach is proposed to estimate the asymptotic covariance matrices of WLADE. Simulation study is carried out to examine the performance of the proposed procedure.

Proof of Local Convergence of a New Algorithm for Covariance Matching of Periodic ARMA Models

In a companion paper, the authors have proposed a new algorithm for the partial stochastic realization of vector discrete-time processes from finite covariance data, based on a nonlinear generalization of the classical Yule-Walker equations. In particular the algorithm provides solutions of the covariance matching problem for periodic ARMA models on a finite interval. In this letter, we provide a proof of convergence of the algorithm for scalar periodic ARMA models based on Lyapunov stability theory.

A Univariate Time Varying Analysis of Periodic ARMA Processes

The standard approach for studying the periodic ARMA model with coefficients that vary over the seasons is to express it in a vector form. In this paper we introduce an alternative method which views the periodic formulation as a time varying univariate process and obviates the need for vector analysis. The specification, interpretation, and solution of a periodic ARMA process enable us to formulate a forecasting method which avoids recursion and allows us to obtain analytic expressions of the optimal predictors. Our results on periodic models are general, analogous to those for stationary specifications, and place the former on the same computational basis as the latter.

TIME SERIES MODELING OF MONTHLY RAINFALL IN ARID AREAS: CASE STUDY FOR SAUDI ARABIA

Stochastic techniques are essential in planning and management of water resources systems especially in arid and semi-arid areas where water is scarce. The forecasting of future events requires identifying proper stochastic models to be used in this process. For this purpose, a Periodic ARMA (PARMA) model and a temporal disaggregation models were used in this study to investigate weather they are appropriate for modeling the monthly rainfall data in Saudi Arabia. Results showed PARMA and temporal disaggregation models performed well in modeling the monthly rainfalls in Saudi Arabia. These models were able to preserve the basic seasonal statistics of the observed data well as preserving the seasonal correlation structure observed in the historical data. However, the PARMA model did not perform well at the annual level. In contrast, the disaggregation model performed well in preserving the correlation structure of the historical data at the annual level. Thus, these models can be used in modeling and forecasting of monthly rainfall in Arid and semi-arid areas.

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 study, 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.

Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models

The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. It extends Thm 3.1 of Basawa and Lund (2001) on least squares estimation of PARMA models with independent errors. It is seen that the asymptotic covariance matrix of the WLS estimators obtained under dependent errors is generally different from that obtained with independent errors. The impact can be dramatic on the standard inference methods based on independent errors when the latter are dependent. Examples and simulation results illustrate the practical relevance of our findings. An application to financial data is also presented.

Asymptotic results for Fourier-PARMA time series

Periodically stationary times series are useful to model physical systems whose mean behavior and covariance structure varies with the season. The Periodic Auto-Regressive Moving Average (PARMA) process provides a powerful tool for modelling periodically stationary series. Since the process is non-stationary, the innovations algorithm is useful to obtain parameter estimates. Fitting a PARMA model to high-resolution data, such as weekly or daily time series, is problematic because of the large number of parameters. To obtain a more parsimonious model, the discrete Fourier transform (DFT) can be used to represent the model parameters. This article proves asymptotic results for the DFT coefficients, which allow identification of the statistically significant frequencies to be included in the PARMA model.

Stochastic water balance simulation for Lake Balaton (Hungary) under climatic pressure

Lake Balaton--due to its remarkable shallowness--is deemed to be sensitive to climatic variations. Historical records suggest, that the water level naturally fluctuated within significantly broader boundaries than the present regulation interval. We made a detailed, dynamic water balance simulations relying on a watershed model and two climatic scenarios. A periodic ARMA model simulated the natural water balance of the lake. The results confirmed, that the lake is not endangered under the expected climate changes. The probability of low levels may increase in the future by up to an order of magnitude. Uneven distribution and increased deviation of precipitation was found to be an important climatic factor from the aspect of low waters. A sensitivity analysis showed, that the sensitivity of the lake increases with the amplitude of climate change. The parameter uncertainty analysis revealed, that even with so long history of observations, the uncertainty of statistical estimates has larger effect than the anticipated climate change. Unfortunately, the time horizon of climate change is not far enough to make up for the necessary data collection, so possible interventions must dispense with a sound scientific justification.

Aggregation and systematic sampling of periodic ARMA processes

The aim of this work is to investigate the effects of temporal aggregation and systematic sampling on periodic autoregressive moving average (PARMA) time series. Firstly, it is shown that the class of weak PARMA processes, i.e. with uncorrelated but possibly dependent errors, is closed under a particular class of linear transformations that include both temporal aggregation and systematic sampling. This extends a similar result for autoregressive moving average processes; see [Wei, W.W.S., 2006. Time Series Analysis: Univariate and Multivariate Methods, second ed. Addison-Wesley, New York (Chapter 20)] for a review on the subject. Secondly, the properties of the noise of the transformed process are investigated. A sufficient condition is given under which aggregation and systematic sampling of a strong PARMA process, i.e. with independent errors, give rise in general to a weak PARMA process. Under that condition, the noise of the transformed process is neither strong nor a martingale difference. This result points out that the assumption of strong PARMA should not be used without careful considerations when analyzing aggregated time series that naturally occur in many scientific fields. The sufficient condition for non-independent errors is illustrated with the PARMA(1,1) model. A simulation study underlines the practical relevance of our findings and the importance of taking into account the dependence of the errors when fitting a PARMA model to an aggregated time series.

Fourier-PARMA Models and Their Application to River Flows

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...

Spectral measures of PARMA sequences

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.

TU‐C‐M100J‐09: Application of a PARMA Algorithm to Predict Intrafraction Respiratory Motion

Purpose: Accurate motion tracking can decrease normal tissue dose, PTV margins, and shorten treatment times required for radiation therapy. A Periodic Autoregressive Moving Average (PARMA) model was developed to predict one‐dimensional respiratory motion in human subjects. Methods and Materials: A PARMA prediction algorithm was applied to clinical respiration signals from three categories: free‐breathing, audio instruction, and audio‐visual coaching. The PARMA model of the respiration motion is a dual‐component signal, a partially correlated time‐series superimposed onto a periodic pattern. The periodic component was estimated by averaging over the inhale‐exhale cycles in a 60‐second sample of the respiration signal, and the autocorrelation in the residual signal was modeled by linear autoregressive moving average equations. The PARMA predictions were compared to the system latency errors produced when tumor tracking is done without forward predictions. Results: For audio‐visual coaching, the PARMA prediction errors (1σ) were 0.09 mm, 0.23 mm and 0.99 mm at 0.1 s, 0.2 s and 0.5 s respectively. A prediction error of 0.99 mm is equivalent to 7.2% of the average end‐inhale to end‐exhale motion. By comparison, the errors were 0.60 mm, 1.20, and 2.90 mm when the lagged signal is used as an estimate of current position at 0.1 s, 0.2 s and 0.5 s respectively. For free‐breathing respiration, PARMA predictions were accurate to within 0.10 mm (1σ) at 0.1 seconds, 0.24 mm at 0.2 seconds, and 1.07mm at 0.5 seconds. The system latency errors were also similar to the coached signals. Conclusion: Tumor tracking with dynamic MLC can theoretically achieve system latency times of 0.16 seconds with accuracy of about 1 mm. The prediction errors of the PARMA algorithm was well under one‐third of the calculated system latency error for all lag times investigated.

Dual-component model of respiratory motion based on the periodic autoregressive moving average (periodic ARMA) method

A new approach to the problem of modelling and predicting respiration motion has been implemented. This is a dual-component model, which describes the respiration motion as a non-periodic time series superimposed onto a periodic waveform. A periodic autoregressive moving average algorithm has been used to define a mathematical model of the periodic and non-periodic components of the respiration motion. The periodic components of the motion were found by projecting multiple inhale–exhale cycles onto a common subspace. The component of the respiration signal that is left after removing this periodicity is a partially autocorrelated time series and was modelled as an autoregressive moving average (ARMA) process. The accuracy of the periodic ARMA model with respect to fluctuation in amplitude and variation in length of cycles has been assessed. A respiration phantom was developed to simulate the inter-cycle variations seen in free-breathing and coached respiration patterns. At ±14% variability in cycle length and maximum amplitude of motion, the prediction errors were 4.8% of the total motion extent for a 0.5 s ahead prediction, and 9.4% at 1.0 s lag. The prediction errors increased to 11.6% at 0.5 s and 21.6% at 1.0 s when the respiration pattern had ±34% variations in both these parameters. Our results have shown that the accuracy of the periodic ARMA model is more strongly dependent on the variations in cycle length than the amplitude of the respiration cycles.

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.