Periodic Autoregressive Model Research Articles

Application of Time-Variant Systems Theory to the Unsteady Aerodynamics of Rotary Wings

This work employs the principles of time-variant systems theory to investigate the unsteady aerodynamics of rotary-wing configurations under periodic equilibrium conditions. Their application enables an extension of the pulse technique for system identification, as well as the adaptation of the linear-frequency-domain formulation commonly utilized in fixed-wing to rotary-wing scenarios. These methodologies effectively incorporate the aerodynamic nonlinearities associated with the equilibrium state into an efficient time-variant linearized representation of the unsteady aerodynamics. To promote its application in the context of rotary-wing aeroelasticity, a state-space realization based on a periodic autoregressive model with exogenous input is subsequently employed. Upon transformation from discrete to continuous time, the resulting aerodynamic model adopts a linear continuous-time periodic state-space formulation, offering compatibility for its coupling with a wide range of structural models. The proposed aerodynamic framework tailored to rotary-wing aeroelasticity holds applicability across a spectrum of aerodynamic models of arbitrary complexity, spanning from incompressible potential flow approximations to potentially more sophisticated methods. Showcasing the potential of this framework, the widely studied lossy Mathieu equation and the aerodynamic response to a flap perturbation about the periodic equilibrium condition of a prototypical rotor blade section, incorporating nonlinearities through an analytical dynamic stall model, are considered.

Empirical study of periodic autoregressive models with additive noise – estimation and testing

Periodic autoregressive (PAR) time series with finite variance is considered as one of the most common models of second-order cyclostationary processes. However, in the real applications, the signals with periodic characteristics may be disturbed by additional noise related to measurement device disturbances or to other external sources. Thus, the known estimation techniques dedicated for PAR models may be inefficient for such cases. When the variance of the additive noise is relatively small, it can be ignored and the classical estimation techniques can be applied. However, for extreme cases, the additive noise can have a significant influence on the estimation results. In this paper, we propose four estimation techniques for the noise-corrupted PAR models with finite-variance distributions. The methodology is based on Yule-Walker equations utilizing the autocovariance function. It can be used for any type of the finite-variance additive noise. The presented simulation study clearly indicates the efficiency of the proposed techniques, also for extreme case, when the additive noise is a sum of the Gaussian additive noise and additive outliers. The proposed estimation techniques are also applied for testing if the data correspond to noise-corrupted PAR model. This issue is strongly related to the identification of informative component in the data in case when the model is disturbed by additive non-informative noise. The power of the test is studied for simulated data. Finally, the testing procedure is applied for two real time series describing particulate matter concentration in the air.

Electricity consumption simulation using random coefficient periodic autoregressive model

The concept of a continuous-time conditional linear random process involves a random kernel being integrated stochastically by a process with independent increments, which is often called a “generative process”. In cases where the “generative process” is Poisson, the resulting model represents an investigated signal as a sum of numerous stochastically dependent random impulses each of which occurs according to some inhomogeneous Poisson arrival process. This model can be applied to represent various processes related to energy consumption, such as electricity loads of electrical power systems, gas and water consumption, and other energy resources, while also considering the signals’ cyclostationarity, which is usually caused by the rhythmic nature of consumer behaviour. A member of the discretetime conditional linear cyclostationary random processes class is the random coefficient periodic autoregressive (RCPAR) model, which is appropriate for use in energy informatics, including estimation, forecasting, and computer simulation purposes. The primary objective of the paper is to establish a procedure for simulating the hourly electricity consumption of small and medium-sized enterprises using the RCPAR model, which has periodic parameters and creates cyclostationary properties while also accounting for the investigated process conditional heteroscedasticity. The statistical estimation step of the proposed procedure uses the general methodology for estimating the parameters of the RCPAR model and the methods of statistical analysis of cyclostationary signals. This step is used to identify the simulation characteristics. The simulation step is based on the methods of cyclostationary white noise generation and its transformation by a digital linear filter with random parameters. The last ones are obtained using the Gaussian random vectors computer simulation methods, taking into account the cyclostaionarity property.

Some empirical results using block bootstrap in estimating the coefficients of a periodic autoregressive model

Abstract The bootstrap proposed by Efron (1979) resulted a useful method in estimating the distribution of an estimator or a test statistic by resampling the data in the case of independent and identically distributed observations. Although it was not as effective in the case of dependent data as in the case of independent and identically distributed data, an adaptation was obtained using the block bootstrap. The block bootstrap consists in dividing the data into blocks of observations and then resampling these blocks with replacement. When resampling periodic data, we must take in consideration the periodicity present. Periodically correlated time series and in particular those related with PAR processes have been object of many recent studies due to numerous applications in real data problems. The aim of this paper is to use a block bootstrap procedure proposed, Block Bootstrap of the Residuals, in the case of PAR (Periodic Autoregressive) models. The results obtained in the case of estimating the coefficients in a PAR model studied are very good and are characterized by small values of Bias, Mean Squared Error and Standard Deviation. Also the bootstrap estimations obtained are closer to the true values than the usual classic point estimations.

Identification and validation of periodic autoregressive model with additive noise: finite-variance case

In this paper, we address the problem of modeling data with periodic autoregressive (PAR) time series and additive noise. In most cases, the data are processed assuming a noise-free model (i.e., without additive noise), which is not a realistic assumption in real life. The first two steps in PAR model identification are order selection and period estimation, so the main focus is on these issues. Finally, the model should be validated, so a procedure for analyzing the residuals, which are considered here as multidimensional vectors, is proposed. Both order and period selection, as well as model validation, are addressed by using the characteristic function (CF) of the residual series. The CF is used to obtain the probability density function, which is utilized in the information criterion and for residuals distribution testing. To complete the PAR model analysis, the procedure for estimating the coefficients is necessary. However, this issue is only mentioned here as it is a separate task (under consideration in parallel). The presented methodology can be considered as the general framework for analyzing data with periodically non-stationary characteristics disturbed by finite-variance external noise. The original contribution is in the selection of the optimal model order and period identification, as well as the analysis of residuals. All these findings have been inspired by our previous work on machine condition monitoring that used PAR modeling.

Daily Streamflow Time Series Modeling by Using a Periodic Autoregressive Model (ARMA) Based on Fuzzy Clustering

The behavior of hydrological processes is periodic and stochastic due to the influence of climatic factors. Therefore, it is crucial to develop the models based on their periodicity and stochastic nature for prediction. Furthermore, forecasting the streamflow, as one of the main components of the hydrological cycle, is a primary subject. In this study, a statistical method, Fuzzy C-means clustering, was used to find the periodicity in the daily discharge time series, whereas autoregressive moving average, ARMA, was used in modeling every cluster. Dividing the daily stream flow time series into smaller groups based on their similar statistical behavior by using a statistical method for analyzing and a combination of Fuzzy C-means clustering and ARMA modeling is the innovation of this study. We draw on the daily discharge data of four different river stations in Hesse state in Germany. The collected data cover 18 years, from 2000 to 2017. Root mean square error (RMSE) was used to evaluate the accuracy. The results revealed that the performance of ARMA in four stations for predicting every cluster was reliable. In addition, it must be highlighted that by clustering the daily stream flow time series into smaller groups, forecasting different days of the year will be possible.

Battery Dispatching for End Users With On-Site Renewables and Peak Demand Charges—An Approximate Dynamic Programming Approach

Battery energy storage systems (BESSs) have the potential to reduce end users’ electricity bills by shifting their grid demand in response to price incentives. Some electricity retailers charge end users for their peak demand in the billing cycle regardless of the time of occurrence. Yet, using BESSs to reduce demand charges is challenging, since the net demand of end users can be highly uncertain due to low aggregation and on-site renewable generation. In this article, an approximate dynamic programming (ADP) methodology is developed to control a BESS to minimize an end user’s electricity bill, which includes both an energy charge and monthly peak demand charge. To address time-varying uncertainty, the net demand is modeled using a periodic autoregressive (PAR) model, which is then used to formulate a Markov Decision Process whose objective is to minimize the sum of energy, peak demand, and battery usage costs. A backward ADP strategy is developed which is enabled by new closed-form expressions for the probability distribution of the expected stage and tail costs when a radial basis function (RBF) value function approximation (VFA) is employed. The approach is applied to real net demand data from a small Australian residential community and compared to two benchmark policies: a lookup table VFA and a model predictive control (MPC) approach. The results show that the proposed approach reduces the average monthly peak demand by about 25% (yielding an 8% reduction in the electricity bill), whereas the best-performing benchmark policy reduced the peak demand by about 17%.

Alternative dependency measures-based approach for estimation of the α–stable periodic autoregressive model

The paper deals with the estimation methodology for the α-stable periodic autoregressive (PAR) models. For the classical (Gaussian) PAR time series is cyclostationary systems the periodicity is manifested in the model characteristics. One of the most common characteristics used in this context is the autocovariance function. One of the methods of the PAR model estimation utilizes the Yule-Walker equations that contain the autocovariance function of the given process. However, for the infinite variance version of the PAR model (i.e., α-stable-distributed) it is expected that the autocovariance function is not properly defined. Thus, alternative measures need to be used. In this paper, we present the general idea of the modified Yule-Walker equations for the considered model. It is proposed to replace the classical dependency measure by the dependency measures properly defined for infinite variance models. We demonstrate two possible modifications based on the covariation and fractional lower order covariance. The first approach was recently proposed in the literature while the latter is a novel algorithm. We compare the effectiveness of both estimation methods for the considered model. Finally, the results are compared with the classical Yule-Walker approach. We demonstrate the limitations of the classical method for the considered model. The possible application for real data analysis is demonstrated.

Modelling and forecasting for solar irradiance from solarimetric station

This paper proposes two approaches for modeling solar irradiance time series. The first is an exploratory analysis and second is a periodic forecast model for solar irradiance data from solarimetric station in a country city of Bahia, Brazil. Two normality hypothesis tests were applied, Anderson-Darling and Shapiro-Wilk. The shape and symmetry of the data were also analyzed using boxplot and histograms to investigate the extreme points, as well as great asymmetry in the distribution of data at different times. Through adherence tests, the normal distribution of solar irradiance in most hours within the annual periodic is rejected. A study by season was carried out, which showed a different behavior in relation to the symmetry of the data, with autumn being the most uniform and spring being the most stochastic. Spring and summer presented better conditions for installation of photovoltaic plates, due to the high solar irradiance rates, and autumn and winter presented satisfactory solar irradiance to maintain this form of generation throughout the year.The second approach is a forecast to solar irradiance in photovoltaic generation systems. The importance of this forecast type is to favor the routine of planning, operation and maintenance of these types of systems, in addition to serving as a basis for feasibility studies and expansion of solar generation. Given a stochastic resource and the periodic behavior of solar irradiance data, a periodic autoregressive model was considered; the statistics were used by the maximum likelihood method, based on hourly measurements of irradiance over a period of one year.

PAR(1) model analysis: a web-based shiny application for analysing periodic autoregressive models

In this paper, a web-based shiny application called the ‘PAR(1) Model Analysis’— that allows the modelling, estimation and prediction of a periodic autoregressive time series with scale mixtures of skew-normal innovations, a general and quite flexible class of error distributions—is presented. The class of scale mixtures of skew-normal distributions is often used for statistical procedures of analysing symmetrical and asymmetrical data. The formulation of the scale of a mixture of skew-normal periodic autoregressive models and the estimating methods, which will be applied in the web application, is briefly explained. The embedded tools in the web application and their applications on data analysis are fully described, and finally, a real data example is completely analysed by using this app to illustrate its handling.

Periodic autoregressive models in flow regulating reservoirs design

Historical streamflow series may not present critical drought periods. As a result, volumes of flow regulating reservoirs estimated from historical data can lead to reservoirs design underestimation. Periodic stochastic models can be used to generate multiple different sequences of flows that present statistical properties close to those corresponding to the historical series but including more critical dry periods. The different volume values estimates allow probabilistic analysis and definition of regulating reservoir volumes based on periodic flows synthetic sequences occurrence risks. This study aims at assessing the performance of different periodic autoregressive models in generating sequences of flows and reproducing regulating reservoirs volumes. Periodic multiplicative autoregressive moving average models (PMIX), complete or parsimonious, showed better performances than traditional periodic autoregressive and autoregressive moving average models, for most historical streamflow series considered. The consideration of synthetic series generated by using PMIX multiplicative models can reduced risks of underestimation of reservoirs volumes.

New estimation method for periodic autoregressive time series of order 1 with additive noise

The periodic behavior of real data can be manifested in the time series or in its characteristics. One of the characteristics that often manifests the periodic behavior is the sample autocovariance function. In this case, the periodically correlated (PC) behavior is considered. One of the main models that exhibits PC property is the periodic autoregressive (PARMA) model that is considered as the generalization of the classical autoregressive moving average (ARMA) process. However, when one considers the real data, practically the observed trajectory corresponds to the “pure” model with the additional noise which is a result of the noise of the measurement device or other external forces. Thus, in this paper we consider the model that is a sum of the periodic autoregressive (PAR) time series and the additive noise with finite-variance distribution. We present the main properties of the considered model indicating its PC property. One of the main goals of this paper is to introduce the new estimation method for the considered model’s parameters. The novel algorithm takes under consideration the additive noise in the model and can be considered as the modification of the classical Yule–Walker algorithm that utilizes the autocovariance function. Here, we propose two versions of the new method, namely the classical and the robust ones. The effectiveness of the proposed methodology is verified by Monte Carlo simulations. The comparison with the classical Yule–Walker method is presented. The approach proposed in this paper is universal and can be applied to any finite-variance models with the additive noise.

Limiting Spectral Distribution of Large-Dimensional Sample Covariance Matrices Generated by the Periodic Autoregressive Model

The explicit representation for the limiting spectral moments of sample covariance matrices generated by the periodic autoregressive model (PAR) is established. We propose to use the moment-constrained maximum entropy method to estimate the spectral density function. The experiments show that the maximum entropy spectral density function curve obtained based on the fourth-order limiting spectral moment can match histograms of the eigenvalues of the covariance matrices very well.

Comparative study of forecasting approaches in monthly streamflow series from Brazilian hydroelectric plants using Extreme Learning Machines and Box & Jenkins models

Abstract Several activities regarding water resources management are dependent on accurate monthly streamflow forecasting, such as flood control, reservoir operation, water supply planning, hydropower generation, energy matrix planning, among others. Most of the literature is focused on propose, compare, and evaluate the forecasting models. However, the decision on forecasting approaches plays a significant role in such models’ performance. In this paper, we are focused on investigating and confront the following forecasting approaches: i) use of a single model for the whole series (annual approach) versus using 12 models, each one responsible for predicting each month (monthly approach); ii) for multistep forecasting, the use of direct and recursive methods. The forecasting models addressed are the linear Autoregressive (AR) and Periodic Autoregressive (PAR) models, from the Box & Jenkins family, and the Extreme Learning Machines (ELM), an artificial neural network architecture. The computational analysis involves 20 time series associated with hydroelectric plants indicated that the monthly approach with the direct multistep method achieved the best overall performances, except for the cases in which the coefficient of variation is higher than two. In this case, the recursive approach stood out. Also, the ELM overcame the linear models in most cases.

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.

Modelling Tourism in Australia Based on Periodic Autoregressive Time Series Models

Tourism has become very important in international commerce and also represents one of the main sources of income for some developing countries in the world. It is the act of people travelling and staying in a place outside their usual environment for leisure, business or other purposes, and this may include sightseeing, camping, retreats, etc Tourism is one of the key economic growth contributors and has contributed towards complete growth and development of Australia by bringing numerous economic value and benefits to her, and also building her brand value, image and identity. This paper seeks to generate a periodic autoregressive PAR model that could be used to make reliable forecast for tourism in Australia. Periodic autoregressive models are for seasonally observed data, particularly quarterly and monthly. Its’ parameters take different values across the seasons. The data used in this work was extracted from the official website of the Australian Bureau of Statistics (ABS), (www.abs.gov.au). It consists of monthly historical data of the number of short term visitors in Australia from January 1998 to December 2017 and was analysed using R- statistical software. The result revealed the order of the PAR model, and also verified that there is periodic variation (periodicity) in the tourism data. It was also verified that there is existence of single unit root and periodic integration which led to the fitting of periodic integrated autoregressive (PIAR(2)) model as the suitable model for the Australian tourism data. Finally, the residual generated from the model was subjected to statistical test and it showed a white noise behavior. Based on these findings, it is concluded that periodic autoregressive time series model can be used to generate reliable forecast for Australian tourism data. Further research on the stochastic nature and seasonality of Australian tourism data was recommended.

Ability Assessment of the Stationary and Cyclostationary Time Series Models to Predict Drought Indices

Drought forecasting and monitoring play a significant role in reducing the negative effects of global meteorological droughts caused by different intensities at different temporal and spatial scales in different regions, especially in regions with high dependency on rainwater. The present study tries to compare the accuracy of stationary time series (ST) models including autoregressive moving average (ARMA), moving average (MA) and autoregressive (AR) and cyclostationary time series (CT) models including periodic autoregressive moving average (PARMA), periodic moving average (PMA) and periodic autoregressive (PAR) to predict drought index (i.e. monthly reconnaissance drought index (RDI)) in periodic data series considering that CT models are more powerful and efficient than ST models by using data series of 8 synoptic stations with different climate conditions in Iran from 1967 to 2017. According to the results the monthly RDI was significantly periodic in all selected stations. The PAR (25) model was the best fitted CT model in data series at all stations and on the other hand, the following models were the best-fitted ST models in data series: the AR models at Babolsar and Rasht AR (25) and at Gorgan AR (24) and ARMA models at Tehran ARMA (2, 3), at Zahedan and Shiraz ARMA (2, 4) and at Esfahan and Shahre Kord ARMA (2, 5). Based on the best fitted CT and ST models, the results showed that the correlation coefficients (R) between observed and simulated RDI vary from 0.882 to 0.946 and from 0.693 to 0.874, respectively from January 1967 to December 2017. According to the best fitted CT and ST models, the validation test of the best fitted models indicated that the R between observed and simulated RDI vary from 0.634 to 0.883 and 0.585 to 0.847, respectively from January 2012 to December 2017. In total, it can be concluded that that the accuracy and capability of CT models in predicting the RDI were more than those of the ST models at all stations and the hypothesis of the study was confirmed.

Periodic autoregressive models for time series with integrated seasonality

A comprehensive seasonally integrated periodic autoregressive model is suggested which is shown to be flexible enough to include both the stochastic seasonal integrated and random trigonometric polynomial-based models. The demonstration of the equivalence between the two approaches is the objective of two theorems that are stated and proved in some details. A nice advantage of our model building procedure is that it is able to provide the user not only with a detailed model for data description and forecasting purpose but in addition with a hint at the presence of seasonal unit roots. The model which is illustrated in the present paper may be considered parsimonious, i.e. the number of estimated parameters, for a given goodness-of-fit criterion, is taken as low as possible, in two ways. First, by imposing unit roots and seasonal unit roots so that some estimated parameters are replaced by a differencing operator with fixed coefficients, and, second, by adopting a subset periodic autoregressive model, so that some parameters do not need to be estimated as they are constrained to equal zero. The effectiveness of our model is highlighted by an extensive simulation experiment that supports our claim that the model building procedure described here may be of good use as well for checking the existence of seasonal unit roots. Applications to real-world time series data sets are reported, and promising results are obtained that allow us to suggest that the seasonally integrated periodic autoregressive model may be safely used for modelling a wide range of seasonal time series data. In addition, well-known widely used tests, such as HEGY and the Taylor variance ratio test, are shown to provide us with results that generally agree with our findings.

Low-dimensional representations of Ni\xf1o 3.4 evolution and the spring persistence barrier

The El Niño-Southern Oscillation (ENSO) is the dominant mode of climate variability on interannual time scales, and its temporal evolution can be summarized using the Niño 3.4 index. Here we used EOF analysis to construct low-dimensional representations of the 12-month evolution of Niño 3.4 during different times of the year. The leading EOF explains more than 90% of the variance of the Niño 3.4 evolution from June to May, which means that the differences in evolution from one year to another are essentially differences in amplitude. Two EOFs explained 94% or more of the evolution variance for other 12-month periods of the year. The two-dimensional nature of the Niño 3.4 trajectories is a direct expression of the spring persistence barrier since the first EOF describes wintertime ENSO events, and the second EOF describes independent behavior during the antecedent spring. A periodic second-order autoregressive (AR2) model reproduced the observed properties, but a first-order model did not. Niño 3.4 EOFs in ocean-atmosphere coupled model forecasts matched observed EOFs with varying levels of fidelity depending on model and time of year. Forecast models with more accurate climatological covariance also have lower mean-squared error (MSE). Low-dimensional EOF-based statistical corrections reduced forecast MSE.

Synthetic scenario generation of monthly streamflows conditioned to the El Niño–Southern Oscillation: application to operation planning of hydrothermal systems

The Brazilian Interconnected Power System is hydro dominated and characterized by large reservoirs presenting multi-year regulation capability, arranged in complex cascades over several river basins. In this way, the expansion and operation planning should take into account the uncertainties about the future inflows to hydroplants reservoirs. Currently, a stochastic model for synthetic scenarios generation of monthly streamflow, based on Periodic Auto-Regressive formulation, is used to address the uncertainty. This is the official model used in the Brazilian energy operation planning by the Ministry of Mines and Energy, the National Operator of Electrical System, the Chamber of Electric Energy Commercialization and the Energy Planning Company. Recently, a great scientific effort has been made to include relevant climatic information in stochastic streamflow models. Among several important climatic phenomena in the Brazilian hydrological cycles, El Nino–Southern Oscillation has been pointed as one of the most important. Although the stochastic models that include exogenous variables or that use wavelets present good results, they have limitations for long-term horizon projections or are not suitable for applications that use stochastic dual dynamic programming, which is the case of the Brazilian electrical system. This work proposes an improvement to the current scenario generation model, in order to consider the climate information, but still being suitable to be applied in SDDP algorithms. To achieve this goal, a Markov-Switching Periodic Auto-Regressive model is presented. It is demonstrated that the methodology is able to generate synthetic scenarios which better resembles the observed streamflow, mainly during periods when the streamflow are below-average.