Autoregressive Moving Average System Research Articles

Best stable and invertible approximations for ARMA systems

A method is proposed for finding the best stable and invertible approximations for an autoregressive moving average (ARMA) system, relative to a general quadratic metric in the coefficient space. Mathematically, the problem is equivalent to projecting the regression and moving average vectors of the system onto the set S of coefficients of monic Schur polynomials. The geometry of S is too complex to allow the problem to be approached directly in the ARMA coefficient space. A solution is obtained by constrained steepest descent in the hypercube of reflection coefficients, which is homomorphic to S. >

Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications

A compendium of recent theoretical results associated with using higher-order statistics in signal processing and system theory is provided, and the utility of applying higher-order statistics to practical problems is demonstrated. Most of the results are given for one-dimensional processes, but some extensions to vector processes and multichannel systems are discussed. The topics covered include cumulant-polyspectra formulas; impulse response formulas; autoregressive (AR) coefficients; relationships between second-order and higher-order statistics for linear systems; double C(q,k) formulas for extracting autoregressive moving average (ARMA) coefficients; bicepstral formulas; multichannel formulas; harmonic processes; estimates of cumulants; and applications to identification of various systems, including the identification of systems from just output measurements, identification of AR systems, identification of moving-average systems, and identification of ARMA systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A note on estimating autoregressive-moving average order

SUMMARY A method for the estimation of parameters and model orders in an autoregressive-moving average system was introduced by Hannan & Rissanen (1982). That method may overestimate the model orders. Here an information theoretic order estimation procedure is introduced and shown to be consistent. Simulations show that the method performs well for moderate sample sizes.

Recursive Modeling of Dynamic Systems

The dynamic response analysis of structural systems to a variety of random excitations using recursive models is presented. The methodology permits analysis of stationary, nonstationary, or transient response of structures under stochastic loads, e.g., correlated multi‐input loading due to fluctuations in wind or wave surface profile, or seismic excitation. For the nonstationary, or transient, excitation, the analysis involves a direct computation of the output covariance from a given nonstationary correlation structure of the input. The accuracy and stability of the simulated time histories is assessed. A detailed example is presented to illustrate the proposed recursive model. The concept of an ARMA (auto‐regressive moving average) system is presented in which the ARMA representation of the response is obtained in terms of the ARMA description of the stationary excitation. The usefulness of the recursive approach for nonlinear systems is demonstrated by means of an example involving an elasto‐plastic beam subjected to a suddenly applied load.

Arma Monte Carlo Simulation in Probabilistic Structural Analysis

Autoregressive moving average (ARMA) systems for sysnthesizing realizations of stochastic processes are discussed in context with the technique of Monte Carlo simulation. Strictly autoregressive (AR) or strictly moving average (MA) systems are considered as special cases of the ARMA systems. Their applicability in wind, ocean, and earthquake engineering is briefly reviewed

A fast prediction-error detector for estimating sparse-spike sequences

Based on the maximum-likelihood principle, the authors develop a locally optimal method for detecting the location and estimating the amplitude of spikes in a sequence, which is considered as the random input of a known ARMA (autoregressive moving-average) system. A Bernoulli-Gaussian product model is adopted for the sparse-spike sequence, and the available data consist of a single, noisy, output record. By using a prediction-error formulation, the authors' iterative algorithm guarantees the increase of a unique likelihood function used for the combined estimation/detection problem. Amplitude estimation is carried out with Kalman smoothing techniques, and event detection is performed in two ways, as an event adder and as an event remover. Under certain assumptions, event and amplitude estimators converge to their true values as the signal-to-noise ratio tends to infinity. Synthetic examples verify that the algorithm is self-initialized, consistent, and fast. >

ARMA systems excited by non-Gaussian processes are not always identifiable

It is a popular myth that the zeros of an ARMA (autoregressive moving-average) process excited by an unobservable non-Gaussian process can be resolved from noisy output measurements. A counterexample is given. >

Recursive covariance ladder algorithms for ARMA system identification

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. >

The iterative NCDE algorithm for ARMA system identification and spectral estimation

The problem of identifying autoregressive moving average (ARMA) models with observational output data is addressed within this report. In the absence of actual input data, the ARMA identification problem is nonlinear in the parameters. The new general ARMA algorithm derived within, entitled NCDE, makes use of the Yule-Walker equations for input estimation and a least squares input-output ARMA algorithm for initial parameter estimation. The NCDE algorithm has been tested and results show that it is both effective and efficient for autoregressive (AR), moving average (MA) and ARMA system identification via the application of an ARMA model.

Simplified models for perturbed parameter Markov equations with application to ARMA systems

A general Markov vector equation y(n+l) = Φ(ξ(n), y(n), n), in which ξ(n) is a stochastic parameter vector, is well approximated by a model of form y(n+l) = ϕ([ytilde](n), n), where ϕ(y(n), n) ≜E[Φ(ξ(n), y(n), n)|y(n), n], if the perturbation effects of the process ξ(n) are small, and if the approximation operator satisfies a Lipschitz Condition ‖ϕ(y′(n), n) − ϕ(y″(n), n)‖ < K ‖y″(n) − y″(n)‖. A bound on the error for such an approximation is developed, and it is shown that the condition K<1 is sufficient for the error to converge asymptotically. The results are applied to the perturbed parameter auto regressive-moving average (ARMA) model as an example, and the Lipschitz constant is seen to have an interesting interpretation in this case.

ARMA system identification via the Cholesky least squares method

A computationally efficient method for the identification of scalar autoregressive moving average (ARMA) models of the form a(z)y = b(z)u+ c(z)\epsilon is introduced; it is called the Cholesky Least Squares (CLS) method. This technique iteratively estimates the coefficients of the polynomials a(z) and b(z) by using the least squares method on data which, at each iteration step, are a filtered version of the original observations \{y_{1},...,y_{N}; u_{1},...,u_{N}\} . The filter employed at each stage is the inverse of the current estimate of c(z) and this estimate is generated by factoring the sample covariance matrix of the residual sequence by using a fast Cholesky factorization algorithm [6], [7]. We describe a natural extension of the CLS method for the identification of multivariable ARMA systems and present computational experiments demonstrating the operation of this extended version of the algorithm. Our method is a variant of the Generalized Least Squares method [1]-[3], and computational experiments comparing a particular version of this method with the CLS algorithm are presented. Finally, some evidence is presented to support the view that the CLS algorithm, like many other identification methods, computes approximations to the true system's impulse response when it is provided with a (possibly incorrect) set of orders for the polynomials a(z), b(z), c(z) .

Iterative identification of non-invertible autoregressive moving-average systems with seismic applications

Let A be an invertible (i.e. minimum-delay) polynomial of degree m and let E be any polynomial of degree n. Then S = E A represents the transfer function of an autoregressive moving-average (ARMA) system. The numerator E may also be invertible, in which case the ARMA system S is invertible. In seismic exploration the sequence with z-transform given by S represents a model of an observable seismogram, where E is the z-transform of the reflection coefficient sequence of the unknown (and desired) deep reflecting interfaces and where A represents the (invertible) z-transform of the unknown inverse reverberation filter due to the surface layers. Because there is no physical reason for the reflection coefficient sequence to be invertible, the seismic model S = E A will in general be a non-invertible ARMA system. Given such a non-invertible sequence with z-transform S, the problem is to estimate E and A under the restriction that A is invertible. We give an iterative identification scheme based on the following algorithm. Initially we let the initial estimate E ( o) be unity. We then apply an iterative algorithm which for the k th step has inputs S and E ( k−1) and outputs A ( k) and E ( k) , where A ( k) is required to be invertible. Iteration is terminated when S and E (k) A (k) are approximately the same, and then E ( k) and A ( k) are the required estimates of E and A, respectively.

Linear Estimation of ARMA Systems

This paper presents a new method for the estimation of the parameters of a mixed autoregressive-moving average (ARMA) system of known order. Since this is a nonlinear estimation problem, existing algorithms, unless they sacrifice efficiency as in instrumental variable methods, require an infinite number of iterations with consequent high computation cost and possible convergence to a local maximum of the likelihood function. The method presented in this paper requires the solution of a few (e.g. three) linear estimation problems: the first yields an autoregressive model, the second yields an ARMA model which is used to prefilter the data, and the third and (possible) subsequent iterations yield improved estimates of the ARMA system. The method is closer in spirit to the classical generalized least squares method (which yield a sequence of unbiassed estimates) than Clarke’s algorithm. The identification procedure yields an estimate which is consistent and asymptotically efficient.

The identification of vector mixed autoregressive-moving average system

The identification of vector mixed autoregressive-moving average system