Autoregressive Moving-average Processes Research Articles

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

The asymptotic CRLB for the spectrum of ARMA processes

This paper addresses the issue of quantifying the frequency domain accuracy of autoregressive moving average (ARMA) spectral estimates as dictated by the Cramer-Rao lower bound (CRLB). Classical work in this area has led to expressions that are asymptotically exact as both data length and model order tend to infinity, although they are commonly used in finite model order and finite data length settings as approximations. More recent work has established quantifications that, for AR models, are exact for finite model order. By employing new analysis methods based on rational orthonormal parameterizations, together with the ideas of reproducing kernel Hilbert spaces, this paper develops quantifications that extend this previous work by being exact for finite model order in all of the AR, MA, and ARMA system cases. These quantifications, via their explicit dependence on poles and zeros of the underlying spectral factor, reveal certain fundamental aspects of the accuracy achievable by spectral estimates of ARMA processes.

How to handle colored observation noise in large least-squares problems

An approach to handling colored observation noise in large least-squares (LS) problems is presented. The handling of colored noise is reduced to the problem of solving a Toeplitz system of linear equations. The colored noise is represented as an auto regressive moving-average (ARMA) process. Stability and invertability of the ARMA model allows the solution of the Toeplitz system to be reduced to two successive filtering operations using the inverse transfer function of the ARMA model. The numerical complexity of the algorithm scales proportionally to the order of the ARMA model times the number of observations. This makes the algorithm particularly suited for LS problems with millions of observations. It can be used in combination with direct and iterative algorithms for the solution of the normal equations. The performance of the algorithm is demonstrated for the computation of a model of the Earth’s gravity field from simulated satellite-derived gravity gradients up to spherical harmonic degree 300.

A subband ARMA modeling approach to high-resolution NMR spectroscopy

In this paper, a low numerical complexity method for parameter estimation of damped exponential signals is proposed. It allows one to handle with free induction decay (FID) signals of “high complexity” containing hundreds of resonances and composed of more than 100,000 samples. At first, it is recalled that the model of a FID is a particular autoregressive moving-average (ARMA) process in which the AR part contains all useful spectral information. Then the AR parameters may be estimated, by solving the high-order Yule–Walker (HOYW) equations using a singular-value decomposition procedure. To deal with high complexity signals, a subband decomposition scheme is proposed. The filtering operation involved by the decomposition produces colored noise that makes the ARMA modeling approach even more essential. Using three real-world 13C NMR signals, the results achieved by the subband ARMA approach are compared with those obtained using the Fourier transform and a deconvolution algorithm.

Spectral estimation of nonstationary ARMA processes using the evolutionary cepstrum

The spectral estimation problem of nonstationary autoregressive moving-average (ARMA) processes is considered and a new method is proposed for the estimation of the time-varying spectral content of such signals. The proposed method can be viewed as an extension of the estimator proposed earlier by the authors to the time-varying case. The AR part of the model is estimated by solving the time-varying modified Yule-Walker equations using an estimated time-varying autocorrelation function. An evolutionary cepstrum estimator is proposed, which is then used in a simple recursion to obtain the MA parameters of the model.

STOCHASTIC MODELLING AND ANALYSIS OF DYNAMIC PLASMA REFLECTION

Keyhole arc welding is a cost-effective process to weld thick materials. To successfully apply this process, the state of the keyhole must be accurately detected and used in feedback control. In this paper, the dynamic behavior of the plasma reflection is described using the reflection arc angle (RAA). It is found that the RAA series can be considered an autoregressive moving-average (ARMA) process. The parameters of the ARMA are recursively estimated using the extended least squares algorithm. It is found that the recursive estimates of the model parameters change as the state of the keyhole changes. A discriminator has been proposed to determine the state of the keyhole based on the recursive estimates of the model parameters.

Multivariate stable ARMA processes with time dependent coefficients

This paper considers the class of m-variate autoregressive moving average (ARMA) processes with stable innovations and time dependent coefficients. A set of suitable AR and MA regularity conditions is given to ensure existence and uniqueness of valid solutions. A simple form of the above solution is expressed in terms of one sided Green's matrix functions associated with the AR operator. We solve the prediction problem arising in this class of models. A few examples are added to support the general theory.

Self-similar texture modeling using FARIMA processes with applications to satellite images

A texture model for synthetic aperture radar (SAR) images is presented. Specifically, a sea surface in satellite images is modeled using the two-dimensional (2-D) fractionally integrated autoregressive-moving average (FARIMA) process with a non-Gaussian white driving sequence. The FARIMA process is an ARMA type model which is asymptotically self-similar. It captures the long-range as well as short-range spatial dependence structure of an image with a small number of parameters. To estimate these parameters, an efficient estimation procedure based on a spectral fit is presented. Real-life ocean surveillance radar images collected by the RADARSAT sensor are used to evaluate the practicality of this FARIMA approach. Using the radial power spectral density, the new model is shown to provide a more accurate description of the SAR images than the conventional moving-average (MA), autoregressive (AR), and fractionally differenced (FD) models.

On Stein's equation, Vandermonde matrices and Fisher's information matrix of time series processes. Part I: The autoregressive moving average process

This paper introduces several forms of relationships between Fisher's information matrix of an autoregressive-moving average or ARMA process and the solution of a corresponding Stein equation. Fisher's information matrix consists of blocks associated with the autoregressive and moving average parameters. An interconnection with a solution of Stein's equation is set forth for the block case as well as for Fisher's information matrix as a global matrix involving all parameter blocks. Both cases have their importance for the interpretation of the estimated parameters. The cases of distinct and multiple eigenvalues are addressed. The obtained links involve equations with left and right inverses, these can be expressed in terms of the inverse of appropriate Vandermonde matrices. A condition is set forth for establishing an equality between Fisher's information matrix and a solution to Stein's equation. Two examples are presented for illustrating some of the results obtained. The global and off-diagonal block case with distinct and multiple roots, respectively, are considered.

Spectral estimation of ARMA processes using ARMA-cepstrum recursion

In this letter, the spectral estimation problem of a stationary autoregressive moving average (ARMA) process is considered, and a new method for the estimation of the MA part is proposed. A simple recursion relating the ARMA parameters and the cepstral coefficients of an ARMA process is derived and utilized for the estimation of the MA parameters. The method requires neither any initial estimates nor fitting of a large order AR model, both of which require further a priori knowledge of the signal and increase the computational complexity. Simulation results illustrating the performance of the new method are also given.

A metric for ARMA processes

Autoregressive-moving-average (ARMA) models seek to express a system function of a discretely sampled process as a rational function in the z-domain. Treating an ARMA model as a complex rational function, we discuss a metric defined on the set of complex rational functions. We give a natural measure of the between two ARMA processes. The paper concentrates on the mathematics behind the problem and shows that the various algebraic structures endow the choice of metric with some interesting and remarkable properties, which we discuss. We suggest that the metric can be used in at least two circumstances: (i) in which we have signals arising from various models that are unknown (so we construct the distance matrix and perform cluster analysis) and (ii) where there are several possible models M/sub i/, all of which are known, and we wish to find which of these is closest to an observed data sequence modeled as M.

GPS time series modeling by autoregressive moving average method: Application to the crustal deformation in central Japan

Autoregressive moving average (ARMA) method is applied to modeling the time series of position changes of GPS sites, obtained by the Geographical Survey Institute (GSI) of Japan during the period from April 1996 to March 1998. The present application is focused on denoising of the GPS time series data where only white noise is considered, and detection of data discontinuities and outliers in order to obtain time-averaged velocity and strain fields in central Japan. The data discontinuities are detected by a typical Kalman filter algorithm. The outliers are eliminated by using robust estimation techniques during the ARMA process. The averaged strain field, calculated by the least-squares collocation method from the improved two-year time series data, distinguishes clearly between the tectonically active and inactive regions. Higher maximum shear strain rates were detected in the southern area of the Kanto district. In the areas with very high seismicities, however, the difference between the maximum shear strain rates, that were estimated from the raw time series data and the ARMA-analyzed data, amounted to about 0.2 microstrain/yr. This indicates that the existence of noise and discontinuities can lead to an over-prediction of the strain field.

EXPERIMENTAL MODAL ANALYSIS OF A STRUCTURE EXCITED BY A RANDOM FORCE

A multivariate maximum likelihood procedure for the estimation of modal parameters is presented. The vibrating system is excited by a random force and only output sensors are used to estimate the natural frequencies and damping factors of the system. The method works in time domain and a vector autoregressive moving-average (VARMA) process is used. The information about modal parameters is contained in the multivariate AR part, which is estimated using an iterative maximum likelihood algorithm. This algorithm uses a score technique and output data only. The order of the AR part is obtained via the minimum description length associated with an instrumental variable procedure. Experimental results show the effectiveness of the method for model order selection and modal parameters estimation.

Quantitative performance evaluation of the lossless compression approach using pole-zero modeling

A technique for lossless compression of seismic signals is proposed. The algorithm employed is based on the equation-error structure, which approximates the signal by minimizing the error in the least square sense and estimates the transfer characteristic as a rational function or equivalently, as an autoregressive moving-average process. The algorithm is implemented in the frequency domain. The performance of the proposed technique is compared with the lossless linear predictor and the differentiator approaches for compressing seismic signals. The residual sequence of these schemes is coded using arithmetic coding. The suggested approach yields compression measures (in terms of bits per sample) lower than the lossless linear predictor and the differentiator for compressing different classes of seismic signals.

A Class of Non‐Embeddable ARMA Processes

We show that a stationary ARMA(p, q) process {Xn = 0, 1, 2, ...} whose moving‐average polynomial has a root on the unit circle cannot be embedded in any continuous‐time autoregressive moving‐average (ARMA) process {Y}(t), t≥ 0}, i.e. we show that it is impossible to find a continuous‐time ARMA process {Y}(t)} whose autocovariance function at integer lags coincides with that of {Xn}. This provides an answer to the previously unresolved question raised in the papers of Chan and Tong (J. Time Ser. Anal. 8 (1987), 277–81), He and Wang (J. Time Ser. Anal. 10 (1989), 315–23) and Brockwell (J. Time Ser. Anal. 16 (1995), 451–60).

Robust Estimation in Vector Autoregressive Moving‐Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving‐average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness‐of‐fit test based on the RA estimates is also obtained.

An improved method for uniform simulation of stable minimum phase real ARMA (p,q) processes

An improvement over the Beadle-Djuric (see ibid., vol.4, p.259-61, 1997) method to simulate the parameters of stable invertible autoregressive moving average (ARMA) (p,q) processes is presented. It is computationally efficient and is especially suitable for simulation of high-order models.

The local linearization scheme for nonlinear diffusion models with discontinuous coefficients

Abstract The local linearization scheme, defined in Shoji and Ozaki [1997, J. Time Ser. Anal. 18, 485–506] for diffusions with smooth coefficients, is studied under some regularity conditions when the coefficients are not necessarily continuous. It is shown that the local scheme converges weakly to the diffusion itself. These results are applied to continuous-time threshold autoregressive moving-average processes and multi-dimensional continuous-time threshold AR models.

Variance and Covariance Computations for 2-D ARMA Processes

An algorithm is presented to compute the variance of the output of a two-dimensional (2-D) stable auto-regressive moving-average (ARMA) process driven by a white noise bi-sequence with unity variance. Actually, the algorithm is dedicated to the evaluation of a complex integral of the form I= \frac{1}{(2 \pi i)^2} \oint_{|z_1| =1} \oint_{|z_2| =1} G(z_1, z_2)\linebreak[4] G(z^{-1}_1, z_2^{-1}) \frac{dz_2 dz_1}{z_2 z_1}, where i = \sqrt{-1} and G(z_1, z_2) = B(z_1, z_2) / A(z_1, z_2) is stable (z_1,z_2)-transfer function. Like other existing methods, the proposed algorithm is based on the partial-fraction decomposition G(z_1,z_2) G(z_1^{-1}, z_2^{-1}) \linebreak[4] = X(z_1, z_1) / A(z_1,z_2) + X(z^{-1}_1 , z^{-1}_2) / A(z^{-1}_1, z^{-1}_2). However, the general and systematic partial-fraction decomposition scheme of Gorecki and Popek [1] is extended to determine X(z_1,z_2). The key to the extension is that of bilinearly transforming the discrete (z_1, z_2)-transfer function G(z_1,z_2) into a mixed continuous-discrete (s_1, z_2)-transfer function \hat{G} (s_1, z_2). As a result, the partial-fraction decomposition involves only efficient DFT computations for the inversion of a matrix polynomial, and the value of I is finally determined by the residue method with finding the roots of a 1-D polynomial. The algorithm is very easy to implement and it can be extended to the covariance computation for two 2-D ARMA processes.