AR Parameters Research Articles

Prony estimation of AR parameters of an ARMA time series

The auto-covariance function of a white noise excited time series can be decomposed into the contributions of different modes, therefore having the same structure as that of the impulse response of a deterministic system. By matching the auto-covariance of the data with that of the ARMA model, the estimation of the characteristic roots of the system and the dispersion coefficients can be implemented using the Prony method, therefore the estimation of the AR parameters becomes a linear least squares problem. It is found that this estimate for the AR parameters of an ARMA model is identical to the asymptotically unbiased estimate using modified Yule-Walker equation for ARMA ( n, n−1).

ARMA bispectrum approach to nonminimum phase system identification

An identification procedure is proposed for a nonGaussian white-noise-driven, linear, time-invariant, nonminimum-phase FIR (finite-impulse response) system. The method is based on parametric modeling of the third moments of the output sequence and uses causal and anticausal autoregressive moving-average (ARMA) models. The magnitude and phase response of the system are expressed in terms of the AR parameters of the ARMA models. In particular, the AR part of the causal ARMA model captures the minimum-phase component of the system, and the AR part of the anticausal ARMA captures the maximum-phase component. Both sets of parameters are obtained by solving overdetermined linear systems of equations. A model-order-selection criterion based on third-order moments is proposed. The ARMA bispectrum approach is compared to more conventional approaches for magnitude and phase reconstruction. It is demonstrated that the proposed identification procedure exhibits improved modeling performance. The method does not require knowledge of the non-Gaussian noise distribution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Application of the lattice filter to robust estimation of AR and ARMA models

A method for estimating the parameters of an autoregressive moving-average (ARMA) model is proposed. A (p+q)-stage lattice whitening filter is used to obtained consistent estimates of the AR parameters of an ARMA (p,p) model. It is shown that a set of MA parameter estimates can also be obtained using this approach with little added computation. The method is used for estimation of the AR parameters of an ARMA process in additive white noise. It is shown that the lattice filter is also useful in robust estimation of the AR parameter on an ARMA process with additive outliers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A method to estimate the parameters of an ARMA model

A method of estimating the parameters of an ARMA model is proposed. In the transient case, we show that the formulas recently given by Li and Dickinson [7] for the AR parameters are easily obtained by considering the formulation of the one-step predictor. In the steady-state case, we can get separately the AR parameters and the MA ones.

Further Results on Testing AR (1) Against MA (1) Disturbances in the Linear Regression Model

This paper examines testing for AR(1) disturbances against MA(1) disturbances in the linear regression model. A Monte Carlo experiment compares the small-sample properties of the Cox test, some linearized Cox tests, and an approximate point optimal test, as well as a Lagrange multiplier test of AR (1) disturbances against ARM A (1,1) disturbances. The main findings are that the true sizes of the asymptotic non-nested tests can differ considerably from their nominal sizes, the Lagrange multiplier test's sizes are reasonably accurate and the point optimal test is generally more powerful than the other tests when appropriate critical values are used. When sizes are controlled at an arbitrary value of the AR (1) parameter, the relative power of the Cox test is increased substantially.

High-order ar estimation method of ARMA power spectrum

In this paper, the high-order AR estimation method of ARMA power spectrum and the whitened-noise decision order criterion of AR order are presented. It is indicated that the quality of high-order AR estimation is related to the ill-condition problem and the algorithmic stability of numerical calculation in the paper. The latter can be solved well by using the recurrence algorithm of Householder transform in the solution of high-order AR parameter estimation.

An approach to time series analysis and ARMA spectral estimation

This paper presents an approach to time series analysis and ARMA spectral estimation from only the output data corrupted by noise. It is shown that the generalized (not well-known) modified Yule-Walker (MYW) equations hold when the residual is some correlated noise. To solve such equations, a new version of the generalized least squares (GLS) method is proposed, yielding AR parameter estimates with higher accuracy. This GLS method can also be used to enhance the estimation accuracy of AR parameters for short and noisy data in case of the MYW equation holding theoretically. Furthermore, a simple procedure for improving MA parameter estimates is studied. Our approach, as Cadzow's singular value decomposition (SVD) method, has provided significantly higher performance spectral estimates for a low-order ARMA model than those obtained via usual techniques which are based upon direct solution of the MYW equations and which use a high-order model without considering an additive noise.

Order-recursive methods for instrumental variable estimates and instrumental variable inverses

A new pseudoinverse called the instrumental variable (IV) inverse is introduced. The inverse and the least-squares (LS) inverse are two special examples of the instrumental variable inverse. This paper presents order-recursive methods for the instrumental variable inverse and IV estimates of the AR and MA parameters of an ARM A process. In simple cases, this paper's methods provide directly the order-recursive algorithms for the least-squares inverse and LS estimates of parameters. Since the matrix form of parameter estimates for any basic or optimal IV method is the same, the order-recursive algorithm for the IV estimates is available for the offline forms of any basic or optimal IV methods. As regards programming the recursive method for IV estimates, such a subroutine is available for several purposes, for example, it can be used in the LS, generalized least-squares (GLS), IV and optimal IV estimations.

On instrumental variable estimation of sinusoid frequencies and the parsimony principle

Multiple sinusoids in noise can be modeled as an ARMA process with the AR parameters satisfying certain symmetry relations. According to the "parsimony principle" the constraints on the AR parameters should he taken into account to get improved estimation accuracy. It is shown in this note that when estimating the AR parameters by a general instrumental-variable method, such a parsimony does not necessarily apply. However, the parsimony principle does hold when an optimal instrumental-variable method is used.

An efficient method to compute consistent estimates of the AR parameters of an ARMA model

A computationally efficient method for consistent estimation of the AR parameters of an ARMA model is obtained, based on an iterated least-squares approach proposed by Tsay and Tiao. The method is derived from the lattice filter structure which generates an orthogonal basis for the Hilbert space of interest. It is shown that for the ARMA ( p, q ) model, we can get consistent estimates of the AR parameters by using a least-squares lattice estimation algorithm and solving a system of q linear equations.

Detecting changes in the ar parameters of a nonstationary arma process

We present a method for detecting changes in the AR parameters of an ARMA process with arbitrarily time varying MA parameters. Assuming that a collection of observations and a set of nominal time invariant AR parameters are given, we test if the observations are generated by the nominal AR parameters or by a different set of time invariant AR parameters. The detection method is derived by using a local asymptotic approach and it is based on an estimation procedure which was shown to be consistent under nonstationarities.

Detection of changes in the AR parameters of a vector arma process: Application to vibration monitoring

Detection of changes in the AR parameters of a vector arma process: Application to vibration monitoring

Optimal instrumental variable estimates of the AR parameters of an ARMA process

The modified Yule-Walker (MYW) equations for estimating the AR parameters of an ARMA process are presented as a special case of an instrumental variable (IV) method. The consistency and accuracy of the AR parameter estimates are studied. It is shown that estimation accuracy increases monotonically with the number of MYW equations for an optimal choice of the weighting matrix used in the least-squares solution of these equations. The asymptotic error covariance of the optimal IV method equals that of the prediction error method. The results of this paper verify experimental results reported in the literature regarding the performance of the MYW method, and provide the necessary accuracy analysis. Furthermore, they suggest several simple, asymptotically efficient, multistep algorithms for estimating the AR parameters, which are presented in a companion paper.

Large-sample estimation of the AR parameters of an ARMA model

The purpose of this note is to show that the optimal instrumental-variable estimate of the AR parameters of an ARMA model, recently proposed in [1] can also be interpreted as a large-sample approximation of a maximum-likelihood estimate.

Predictive spectral analysis of short records using maximum entropy

A method of spectral analysis based on the prediction of the signal by means of AR parameters is proposed. The essence of this method ranks it between the classical methods of spectral analysis and the method of maximum entropy. For sufficiently high SNR its resolution is higher than that of the classical methods. The new method enables power and phase spectra of the signal to be determined, and provides a better determination of the power spectrum amplitude. than the method of maximum entropy. A regularization procedure is presented which abolished the instability of the prediction filter, obtained by the least-squares method.

Spectral Estimation of Speech Corrupted by Colored Noise

A modified SEARMA method is proposed for estimating the speech spectrum in the presence of colored background noise. The following assumptions are used in developing the analysis. The speech production process is represented by an autoregressive moving-average (ARMA) model. The background noise is represented by an AR process, and is always present regardless of presence and absence of speech activity. The noise process is locally stationary during speech activity. Following these assumptions, the process during speech activity can be represented by an extended ARMA model. In this formulation, unique estimation of AR parameters of the vocal tract transfer function is always possible if the AR parameters of the noise process can be estimated separately, but the estimation of MA parameters requires further assumption of a high SNR. The validity of| the proposed method is demonstrated by spectral estimation of synthetic speech sounds in the presence of additive colored noise, and by comparing the results with those obtained by a method using an AR model.

A Lattice Structured Algorithm for Recursive Solution of the Extended Yule-Walker Equations

Abstract Estimation of LPC (AR) parameters of speech signals in additive noise can be approached as the estimation of the AR parameters of an ARMA process. A recursive algorithm is presented that estimates the AR parameters without noise bias. This is based on the Toeplitz extended Yule-Walker equations. Using the Trench algorithm, a lattice structure is developed, leading to a gradient-type algorithm which may be steepest ascent or steepest descent.

Global versus local minimization in least-squares AR spectral estimation

A general framework for power spectral estimation is presented. The equivalence among MEM, AR modeling, and predictive estimation is established. The practical determination of the AR parameters is reviewed. A new classification method for the least-squares AR spectral estimation methods is introduced: the global and local minimization methods. The bordering algorithm for the solution of a system of linear equations is used to deduce Levinson algorithm and to show that Burg Method and a Modified Burg Method are local methods and approximations to a global minimization method. A brief comparative discussion of global versus local least-squares estimation is done.

The exact likelihood for a multivariate ARMA model

A number of algorithms are presented for calculating the exact likelihood of a multivariate ARMA model. There are two aspects to the algorithms. Firstly, the parameterization is in terms of AR parameters and autocovariances. This obviates difficulties with initial MA estimates. Secondly, the algorithms explicitly account for specification of the lag structure of the multivariate time series. Additionally, an algorithm is presented to deal with missing data. The algorithms are, of themselves, not new but they have not been applied to likelihood construction in the manner discussed here.

The small-sample properties of some preliminary test estimators in a linear model with autocorrelated errors

A Monte Carlo study is used to examine the finite-sample relative efficiency of a number of estimators (including pre-test estimators), and the accuracy of asymptotic confidence intervals derived from them, for a linear model with AR(1) or MA(1) errors. The bias of the ML estimator of the AR(1) parameter, and the frequency of ML estimates equal to ±1 for the MA(1) parameter, have a large bearing on the properties of the estimators of the regression coefficient. The traditional OLS-AR pre-test estimator compares relatively favorably, even under an MA error specification. However, the confidence intervals associated with positive autoregressive errors and a trended explanatory variable can be quite inaccurate.