An efficient and order-recursive algorithm for estimating stationary ARMA models (original) (raw)

Recursive covariance ladder algorithms for ARMA system identification

IEEE Transactions on Acoustics, Speech, and Signal Processing, 1988

The recently developed method of pure-order recursive ladder algorithms (PORLA) is extended to facilitate the identification of autoregressive moving-average (ARMA) models. Since the time recursion in this method is limited in the calculation of the input data covariance matrix, roundoff errors cannot propagate in time in higher stages of the pure-order recursively constructed ladder form. Thus, the superior least-squares tracking and fast start-up capability of the proposed algorithms is not corrupted by roundoff error. Furthermore, the algorithms allow the use of higher-order recursive windows on the data (e.g., recursive Hanning), which again significantly improves the tracking as well as the steady-state behavior. A computer program, an instructive example for implementation of the method on a massively parallel processor, and several experimental results which confirm the superior properties of the PORLA method over conventional techniques are shown

A THREE STEP ALGORITHM FOR ARMA MODELING

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.

Model Distance Criterion for Order Determination of ARMA Models

IFAC Proceedings Volumes, 1996

This rapcr addrc•sscs the issue of onJcr determination I"or ARMA models. A novel u mcept (;allcd thc lIIudel di sl;UlCC is prcsclIlCd to determine the di stHlux between two pos."ihlc models whclI the prediction e rror is the primary COlleem of 1l1csc models. Through a L' omparisoll hclwccn !he model di slance from a Illwer order model Lt) a higher orde r model (which is regarded to he a tn e<.l..."urc of the imrrovemcrH of the modelling accuracy due to an increase in t.he numher uf panuneLCrs) and an estimate of the mode.lling accuracy ~l%OCialed with th e higher order model, a model di s tance cri te rion is proposed. Il is shown that the models selecte.d by th is crite rion is either optimal or nearly l)ptima] for pu rpose of predi c tion. Simulations (Ire perfonned m verify the effectiveness of the modc1 distaTl ce criterion.

Exact Maximum Likelihood Estimation of Stationary Vector ARMA Models

Journal of the American Statistical Association, 1995

The problems of evaluating and maximizing the exact likelihood function of vector ARMA models are considered separately. A new and efficient procedure far evaluating the exact likelihood function is presented. This method puts together a set of useful features which can only be found separately in currently available algoritluns. A procedure for maximizing the exact likeliliood function, which takes full advantage of the properties offered by the evaluation algorithm, is also considered, Combining these two procedures, a * This paper summarizes the author's doctoral

ARMA model parameter estimation based on the equivalent MA approach

Digital Signal Processing, 2006

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.

ARMA parameter estimation using a novel recursive estimation algorithm with selective updating

IEEE Transactions on Acoustics, Speech, and Signal Processing

Ahstract-This paper investigates an extension of a recursive estimation algorithm (the so-called OBE algorithm) [9]-[ll], which features a discerning update strategy. In particular, an extension of the algorithm to ARMA parameter estimation is presented here along with convergence analysis. The extension is similar to the extended leastsquares algorithm. However, the convergence analysis is complicated due to the discerning update strategy which incorporates an information-dependent updating factor. The virtues of such an update strategy are: I) more efficient use of the input data in terms of information processing, and 2) a modular adaptive filter structure which would facilitate the development of a parallel-pipelined signal processing architecture. It is shown in this paper that if the input noise is bounded and the moving average parameters satisfy a certain magnitude bound, then the a posteriori prediction errors are uniformly bounded. With an additional persistence of excitation condition, the parameter estimates are shown to converge to a neighborhood of the true parameters, and the a priori prediction errors are shown to he asymptotically bounded. Simulation results show that the parameter estimation error for the EOBE algorithm is comparable to that for the ELS algorithm.

A Unified Approach to Arma Model Identification and Preliminary Estimation

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.

Multivariate Arma Order Estimation via Multi-Model Partition Theory

summarizes the parametric model uncertainty into an unknown, finite dimensional parameter vector whose values are assumed to lie within a known set of finite cardinality. It is not restricted to the Gaussian case and it is also applicable to on line/adaptive operation. By applying this method a new computationally efficient order selection criterion for Multivariate ARMA models will be proposed, developed and justified as an extension to the model order selection criterion for MV AR (AutoRegressive) models . Finally it will be shown that the proposed method is also successful in tracking model order changes in real time.

A Robust ARX and ARMA Model order Estimation via Pivot-Neighbors Comparisons

Recent Patents on Computer Science, 2010

Model order selection of an Autoregressive Moving Average (ARMA) process is an important problem. This paper presents a new algorithm for the estimation of an ARMA and autoregressive with exogenous input (ARX) model orders based on a rounding approach which uses the floor and the ceiling functions. The rounding approach is implemented to deal with the precision of binary words. The proposed algorithm is based on selecting a sequence of pivot cells from an MEV matrix which is based on the minimum eigenvalue of a covariance matrix computed from the observed data. It searches for the corner that contains the estimates of the true orders using the floor and the ceiling functions of the pivot cell values and the values of its neighbors. The proposed algorithm is an expansion of the algorithm proposed by Liang et al. (IEEE Transaction on Signal Processing, 1993; 41(10): 3003-3009). Recent patents and research advances aim to apply eigenvalue decomposition in estimation and prediction. Among the patents discussed is a method that describes estimation of uncertainty of a measuring machine where covariance matrix is subjected to eigenvalue decomposition.