Autoregressive Moving-average Processes Research Articles

Modelling history-dependent parameters in the SMPS format for stochastic programming

This paper proposes two extensions to the SMPS format for stochastic programs to permit modelling of autoregressive-moving average (ARMA) processes. Sampling-based algorithms can thus proceed independently of any underlying modelling system, increasing efficiency. An illustrative example demonstrates the power of the new constructs.

Another Look at Z-transform Technique for Deriving Unit Impulse Response Function

This paper presents a technique to derive the unit impulse response functions (UIRF) used for determination of unit hydrograph by employing the Z-transform technique to the response function derived from the Auto Regressive Moving Average (ARMA) process of order (p, q). The proposed approach was applied to reproduce direct surface runoff for single storm event data registered over four watersheds of area ranging from 0.42 to 295 km2. It is observed that the UIRF based on ARMA (1, 2) and ARMA (2, 2) provides a better representation of the watershed response. Further, to test the superiority of the developed impulse response function form ARMA process, the direct runoff hydrographs were computed using the simple ARMA process and optimized Nash’s two parameter model and compared with the results obtained from UIRF’s of ARMA model. The performance of the models based on the graphical presentation as well as from the test statistics viz. RMSE and MAPE indicates that UIRF-ARMA (p, q) performs better than optimized Nash Model and mostly similar to simple ARMA (p,q) model. Further more, the ARMA process of order p ≤ 2 and q ≤ 2 is generally sufficient and less cumbersome than the Argand diagram based approach for UIRF derivation.

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.

A Note on Non‐Negative Arma Processes

Abstract. Recently, there has been much research on developing models suitable for analysing the volatility of a discrete‐time process. Since the volatility process, like many others, is necessarily non‐negative, there is a need to construct models for stationary processes which are non‐negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non‐negative kernel by non‐negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non‐negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non‐negative almost surely.

ARMA model parameter estimation based on the equivalent MA approach

The paper investigates the relation between the parameters of an autoregressive moving average (ARMA) model and its equivalent moving average (EMA) model. On the basis of this relation, a new method is proposed for determining the ARMA model parameters from the coefficients of a finite-order EMA model. This method is a three-step approach: in the first step, a simple recursion relating the EMA model parameters and the cepstral coefficients of an ARMA process is derived to estimate the EMA model parameters; in the second step, the AR parameters are estimated by solving the linear equation set composed of EMA parameters; then, the MA parameters are obtained via simple computations using the estimated EMA and AR parameters. Simulations including both low- and high-order ARMA processes are given to demonstrate the performance of the new method. The end results are compared with the existing method in the literature over some performance criteria. It is observed from the simulations that our new algorithm produces the satisfactory and acceptable results.

Correlated Errors in the Parameters Estimation of the ARFIMA Model: A Simulated Study

Processes with correlated errors have been widely used in economic time series. The fractionally integrated autoregressive moving-average processes—ARFIMA(p, d, q)—(Hosking, 1981) have been explored to model stationary and non stationary time series with long-memory property. This work uses the Monte Carlo simulation method to evaluate the performance of some parametric and semiparametric estimators for long and short-memory parameters of the ARFIMA model with conditional heteroskedastic (ARFIMA-GARCH model). The comparison is based on the empirical bias and the mean squared error of each estimator.

Automated classification of evoked quantal events

We provide both theoretical and computational improvements to the analysis of synaptic transmission data. Theoretically, we demonstrate the correlation structure of observations within evoked postsynaptic potentials (EPSP) are consistent with multiple random draws from a common autoregressive moving-average (ARMA) process of order (2, 2). We use this observation and standard time series results to construct a statistical hypothesis testing procedure for determining whether a given trace is an EPSP. Computationally, we implement this method in R, a freeware statistical language, which reduces the amount of time required for the investigator to classify traces into EPSPs or non-EPSPs and eliminates investigator subjectivity from this classification. In addition, we provide a computational method for calculating common functionals of EPSPs (peak amplitude, decay rate, etc.). The methodology is freely available over the internet. The automated procedure to index the quantal characteristics greatly facilitates determining if any one or multiple parameters are changing due to experimental conditions. In our experience, the software reduces the time required to perform these analyses from hours to minutes.

MCMC for Integer-Valued ARMA processes

Abstract. The classical statistical inference for integer-valued time-series has primarily been restricted to the integer-valued autoregressive (INAR) process. Markov chain Monte Carlo (MCMC) methods have been shown to be a useful tool in many branches of statistics and is particularly well suited to integer-valued time-series where statistical inference is greatly assisted by data augmentation. Thus in this article, we outline an efficient MCMC algorithm for a wide class of integer-valued autoregressive moving-average (INARMA) processes. Furthermore, we consider noise corrupted integer-valued processes and also models with change points. Finally, in order to assess the MCMC algorithms inferential and predictive capabilities we use a range of simulated and real data sets.

Computation of the exact cramer-rao lower bound for 2-D ARMA parameter Estimation-I: the quarter-plane case

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper

The estimation of the order of an ARMA process using third-order statistics

The paper proposes a new approach to find an autoregressive moving average (ARMA) model order. The basic idea is to extend the previous approach proposed by Liang et al. to third order statistics (TOS). The algorithm uses data matrices rather than calculating cumulants of the observed signal. Hence, we avoid the non-stationary effects, which is due to finite-length observations. The system is driven by a zero-mean independent and identically distributed (i.i.d.) non-Gaussian process. The input signal is unobservable. The observed sequence is corrupted by a zero-mean additive Gaussian noise. Examples are given to demonstrate the performance of the proposed algorithm.

The asymptotic distribution of the constant behavior of the generalized partial autocorrelation function of an ARMA process

The two-way array of the generalized partial autocorrelations (GPAC's) of an autoregressive moving-average (ARMA) model shows a constant behavior and a zero behavior, which are useful for ARMA model identification. In this paper the asymptotic joint distribution of the GPAC estimators of the constant behavior is derived, which shows the corresponding asymptotic variance increases geometrically as the lag does.

Provenance of correlations in psychological data

Two distinct families of statistical processes are considered in the production of psychophysical time series data (Gilden, 1997, 2001; Gilden, Thornton, & Mallon, 1995). We inquire whether the spectral signatures of the underlying dynamics are better described in terms of short-range autoregressive moving-average (ARMA) processes or long-range fractal processes. A thorough presentation of both families is given so as to clarify the scope and generalizability of the models as descriptions of choice reaction time data. Analyses of data are supplemented by the construction of a spectral likelihood classifier that discriminates between the two families of processes. The classifier has sufficient sensitivity to ensure that fractals are correctly identified and that ARMA processes will rarely be misconstrued as belonging to the fractal family. Spectral likelihood classification illustrates an extremely general framework for testing competing spectral hypotheses and is offered for use in measuring the specific character of fluctuations in designed experiments.

Adaptive Blind Multiuser Detection over Flat Fast Fading Channels Using Particle Filtering

We propose a method for blind multiuser detection (MUD) in synchronous systems over flat and fast Rayleigh fading channels. We adopt an autoregressive-moving-average (ARMA) process to model the temporal correlation of the channels. Based on the ARMA process, we propose a novel time-observation state-space model (TOSSM) that describes the dynamics of the addressed multiuser system. The TOSSM allows an MUD with natural blending of low-complexity particle filtering (PF) and mixture Kalman filtering (for channel estimation). We further propose to use a more efficient PF algorithm known as the stochastic M -algorithm (SMA), which, although having lower complexity than the generic PF implementation, maintains comparable performance.

The Biasing Effects of Unmodeled ARMA Time Series Processes on Latent Growth Curve Model Estimates

The purpose of this study was to evaluate the robustness of estimated growth curve models when there is stationary autocorrelation among manifest variable errors. The results suggest that when, in practice, growth curve models are fitted to longitudinal data, alternative rival hypotheses to consider would include growth models that also specify autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes. AR (i.e., simplex) processes are commonly found in longitudinal data and may diminish the ability of a researcher to detect growth if not explicitly modeled. MA and ARMA processes do not affect the fit of growth models, but do notably bias some of the parameters.

Probabilistic Properties of a Nonlinear ARMA Process with Markov Switching

ABSTRACT We consider a nonlinear autoregressive moving average (ARMA) process with Markov switching and find sufficient conditions for strict stationarity, geometric ergodicity, and the existence of moments of the process with respect to the stationary distribution. Functional central limit theorem is also obtained.

ESTIMATING LINEAR DYNAMICAL SYSTEMS USING SUBSPACE METHODS

This paper provides a survey on a class of so-called subspace methods whose main proponent is CCA proposed by Larimore (1983, Proceedings of the 1983 American Control Conference 2). Because they are based on regressions these methods for the estimation of ARMAX systems are attractive as a result of their conceptual simplicity and their numerical advantages as compared to traditional estimators based on criterion optimization. Under the assumption of correct specification the methods provide consistent and asymptotically normal estimates for stationary ARMAX processes where the innovations may be conditionally heteroskedastic and the exogenous inputs are strictly stationary of sufficiently short memory. For stationary autoregressive moving average (ARMA) processes with independent and identically distributed (i.i.d.) Gaussian innovations the estimates are even asymptotically efficient. For I(1) ARMA processes the estimates of both the long-run and the short-run dynamics are consistent without using the knowledge that the data are integrated in the algorithm. Additionally the algorithms provide easily accessible information on the appropriateness of the chosen model complexity. The algorithms include a number of design parameters that have to be set by the user. These include the order of the estimated system. This paper collects up-to-date knowledge on the effects of these design parameters, leading to a number of suggested automated choices to obtain a fully automated estimation procedure.

Necessary and Sufficient Restrictions for Existence of a Unique Fourth Moment of a Univariate Garch(P,Q) Process

A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m, eigenvalue restrictions have been proposed as necessary and sufficient restrictions for existence of a unique mth moment of the output of a univariate GARCH process or, equivalently, the 2mth moment of the underlying variable. However, proofs in the literature that an eigenvalue restriction is necessary and sufficient for existence of unique 4th or higher even moments of the underlying variable, are either incorrect, incomplete, or unecessarily long. Thus, the paper contains a short and general proof that an eigenvalue restriction is necessary and sufficient for existence of a unique 4th moment of the underlying variable of a univariate GARCH process. The paper also derives an expression for computing the 4th moment in terms of the GARCH parameters, which immediately implies a necessary and sufficient inequality restriction for existence of the 4th moment. Because the inequality restriction is easily computed in a finite number of basic arithmetic operations on the GARCH parameters and does not require computing eigenvalues, it provides an easy means for computing by hand the 4th moment and for checking its existence for low-dimensional GARCH processes. Finally, the paper illustrates the computations with some GARCH(1,1) processes reported in the literature.

ARMA Model order estimation using third order computations

A novel approach for estimating the order of a non-Gaussian autoregressive moving-average process is proposed. The proposed method uses third order computations, and is based on the minimum eigenvalue of a family of covariance matrices derived from the observed output data. The algorithm uses data matrices rather than calculated cumulants. Hence, we avoid the non-stationary effects, which is due to finite-length observations. Examples are given to demonstrate the performance of the proposed algorithm.

Spectral Factorization of the Autocorrelation Sequence of an IIR Filter Using Patterns of Three Functions

A 3-step spectral factorization method of the autocorrelation sequence of an IIR filter is presented, which is based on a 3-pattern identification method of an autoregressive moving-average process.

04/00801 Utility charges for extensions: pay us now or pay us later: Switzer, S. The Electricity Journal, 2003, 16, (8), 39–46

This paper evaluates different financial price and time series models, such as mean reversion, autoregressive moving average (ARMA), integrated ARMA (ARIMA) and general autoregressive conditional heteroscedasticity (GARCH) process, usually applied for electricity price simulations. However, as these models are developed to describe the stochastic behaviour of electricity prices, they are extended by a separate data treatment for the deterministic components (trend, daily, weekly and annual cycles) of electricity spot prices. Furthermore price jumps are considered and implemented within a regime-switching model. Since 2008 market design allows for negative prices at the European Energy Exchange, which also occurred for several hours in the last years. Up to now, only a few financial and time series approaches exist, which are able to capture negative prices. This paper presents a new approach incorporating negative prices.The evaluation of the different approaches presented points out that the mean reversion and the ARMA models deliver the lowest mean root square error between simulated and historical electricity spot prices gained from the European Energy Exchange. These models posses also lower mean average errors than GARCH models. Hence, they are more suitable to simulate well-fitting price paths. Furthermore it is shown that the daily structure of historical price curves is better captured applying ARMA or ARIMA processes instead of mean-reversion or GARCH models. Another important outcome of the paper is that the regime-switching approach and the consideration of negative prices via the new proposed approach lead to a significant improvement of the electricity price simulation.