ARMA Model Identification

Journal of Optimization Theory and Applications, 1987

In view of recent results on the asymptotic behavior of the prediction error covariance for a state variable system (see Ref. I), an identification scheme for autoregressive moving average (ARMA) processes is proposed. The coefficients of the d-step predictor determine asymptotically the system moments Uo,..., Ua-1. These moments are also nonlinear functions of the coefficients of the successive 1-step predictors. Here, we estimate the state variable parameters by the following scheme. First, we use the Burg technique (see Ref. 2) to find the estimates of the coefficients of the successive t-step predictors. Second, we compute the moments by substitution of the estimates provided by the Burg technique for the coefficients in the nonlinear functions relating the moments with the 1-step predictor coefficients. Finally, the Hankel matrix of moment estimates is used to determine the coefficients of the characteristic polynomial of the state transition matrix (see Refs. 3 and 4).

Mathematical Problems in Engineering, 2012

Bezerra et al. 2008 proposed a new method, based on Yule-Walker equations, to estimate the ARMA spectral model. In this paper, a Bayesian approach is developed for this model by using the noninformative prior proposed by Jeffreys 1967 . The Bayesian computations, simulation via Markov Monte Carlo MCMC is carried out and characteristics of marginal posterior distributions such as Bayes estimator and confidence interval for the parameters of the ARMA model are derived. Both methods are also compared with the traditional least squares and maximum likelihood approaches and a numerical illustration with two examples of the ARMA model is presented to evaluate the performance of the procedures.

A new ARMA estimation algorithm is proposed. It is based on a fundamental relationship which shows that the AR polynomial of an ARMA(N, M) model belongs to the linear space spanned by the forward and backward linear predictors. This relationship allows us to construct an equivalent linear system with two inputs and the same output of the ARMA system. The inputs of this new system are the forward and backward linear prediction errors. As in this case the inputs and output are known, a least-squares identification algorithm is used to obtain the parameters of the system. These parameters define three polynomials. One of them is the AR polynomial. The other two converge asymptotically to the MA polynomial and to zero. Simple recursions are available to perform such a limiting operation. Zusammenfassung. Ein neuer Algorithmus zur Spektralschatzung mit ARMA-Modellen wird vorgeschlagen. Er beruht auf einer grundlegenden Beziehung, die besagt, dap das Nennerpolynom eines ARMA(N, M)-Modells zum linearen Raum gehort, der durch den Vorwarts-und den Ruckwartspradiktor aufgespannt wird. Dieser Zusammenhang erlaubt es, ein aquivalentes lineares System mit zwei Eingangen und einem unveranderten Ausgang zu kontruieren: Die Eingangssignale des neuen Systems sind dabei die Fehlersignale der beiden Pradiktoren. Da in diesem Fall Ein-und Ausgange bekannt sind, kann ein Minimal-Fehlerquadrat-Schatzalgorithmus benutzt werden, um die Systemparameter zu bestimmen. Diese Kenngropen definieren die Polynome. Eines davon ist das AR-Polynom; die beiden ubrigen konvergieren asymptotisch gegen das MA-Polygon und gegen Null. Einfache Rekursionsbeziehungen fur diesen Grenziibergang werden vorgelegt. RCsurne. Un nouveau algorithme d'estimation ARMA est propost. I1 est bast sur une rtlation fondamentale qui montre que le polynome AR d'un modtle ARMA(N, M) appartient a I'espace lintaire engendrt par les predicteurs lintaires direct et retrograde. Cette rtlation nous permet de construire un systbme lintaire tquivalent,avec deux entrtes et la mbme sortie que le systbme ARMA. Les entrtes du nouveau systtme sont les erreurs de prediction lintaires directe et rttrograde. Comme les entries et sorties sont connues, un algorithme &identification des moindres carrts est utilist pour obtenir les paramttres du systbme. Ces paramttres definissent trois polyn8mes. L'un est le polyn8me AR. Les deux autres convergent asymptotiquement vers le polyn8me MA et vers ztro. De simples rtcursions sont disponibles pour effectuer le passage a la limite.

Automatica, 1996

An improvement of a batch algorithm for parameter estimation of an ARMA process is presented. A fast and iterative algorithm is presented, and provides estimates with the same asymptotic distribution as the maximum likelihood estimates. This algorithm is a modified version of the method suggested by Mayne and Firoozan [(1982). Linear identification of ARMA processes. Auromatica, 18, 461-4661 and requires one less regression than the previous method, if only one pass is executed.

Journal of Time Series Analysis, 2000

Subset models are often useful in the analysis of stationary time series. Although subset autoregressive models have received a lot of attention, the same attention has not been given to subset autoregressive moving-average (ARMA) models, as their identi®cation can be computationally cumbersome. In this paper we propose to overcome this disadvantage by employing a genetic algorithm. After encoding each ARMA model as a binary string, the iterative algorithm attempts to mimic the natural evolution of the population of such strings by allowing strings to reproduce, creating new models that compete for survival in the next population. The success of the proposed procedure is illustrated by showing its ef®ciency in identifying the true model for simulated data. An application to real data is also considered.

Journal of Time Series Analysis, 1984

This paper reviews several different methods for identifying the orders of autoregressive-moving average models for time series data. The case is made that these have a common basis, and that a unified approach may be found in the analysis of a matrix G, defined to be the covariance matrix of forecast values. The estimation of this matrix is considered, emphasis being placed on the use of high order autoregression to approximate the predictor coefficients. Statistical procedures are proposed for analysing G, and identifying the model orders. A simulation example and three sets of real data are used to illustrate the procedure, which appears to be a very useful tool for order identification and preliminary model estimation.

IEEE Transactions on Signal Processing, 1993

2000

This work presents the application of Evolutionary Computation techniques to the identi cation (order selection and parameter estimation) of an AutoRegressive Moving Average model (ARMA). Our method combines the e ectiveness of the Multi Model Partitioning (MMP) theory with the robustness of the Genetic Algorithms (GAs) in order to give optimum estimations of the noise sequence embedded to the moving average terms of the model. Although the noise sequence's coding is very complicated, the proposed algorithm succeeds better results compared to the classical methods, since it has the ability to search the whole values' range. This is because, in contradiction with all the known classical methods, our algorithm is able to estimate with high precision the unknown parameters even in the case of large order in the moving average terms of the model.

Decision Support Systems, 2000

Recently, several researchers have attempted to use neural network approaches in conjunction with the extended sample Ž . autocorrelation function ESACF to automatically identify ARMA models. The work to date appears promising, but generalizations are limited by the fact that the test and training sets for the neural networks were generated from random perturbations of prototype ESACF tables. This paper develops test and training sets by varying the parameters of actual Ž . ARMA processes. The results show that the ability of neural networks to accurately identify the order of an ARMA p,q model from its transformed ESACF is much lower than reported by previous researchers, and is especially low for time series with fewer than 100 observations. q

Econometric Theory, 1998