Modified Yule-Walker Equations Research Articles

Alternative dependency measures-based approach for estimation of the α–stable periodic autoregressive model

The paper deals with the estimation methodology for the α-stable periodic autoregressive (PAR) models. For the classical (Gaussian) PAR time series is cyclostationary systems the periodicity is manifested in the model characteristics. One of the most common characteristics used in this context is the autocovariance function. One of the methods of the PAR model estimation utilizes the Yule-Walker equations that contain the autocovariance function of the given process. However, for the infinite variance version of the PAR model (i.e., α-stable-distributed) it is expected that the autocovariance function is not properly defined. Thus, alternative measures need to be used. In this paper, we present the general idea of the modified Yule-Walker equations for the considered model. It is proposed to replace the classical dependency measure by the dependency measures properly defined for infinite variance models. We demonstrate two possible modifications based on the covariation and fractional lower order covariance. The first approach was recently proposed in the literature while the latter is a novel algorithm. We compare the effectiveness of both estimation methods for the considered model. Finally, the results are compared with the classical Yule-Walker approach. We demonstrate the limitations of the classical method for the considered model. The possible application for real data analysis is demonstrated.

Design and implementation of microwave multi-band/multi-level filters using equal-length transmission lines and Yule–Walker scheme

A new method to design microwave filter with multiple-band/multiple-level response is proposed. The modified Yule–Walker equation is used to define the frequency-domain response of filter in the Z domain. To implement a filter having transfer function in the Z domain, an equal-length transmission-line configuration is employed to emulate the filter. A band-pass filter a dual-band band-pass filter and a dual-level band-pass filter are implemented and their frequency responses are measured to illustrate the validity of this design method.

Evaluating some Yule-Walker Methods with the Maximum-Likelihood Estimator for the Spectral ARMA Model

The aim of this work is to compare some ARMA spectral separatede stimation methods based on the modified Yule-Walker equation and least squares method with the Maximum-Likelihood estimator, using the convergence curve of the relative mean error (RME), generated by Monte Carlo simulation.

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.

A delta MYWE algorithm for parameter estimation of noisy AR processes

We develop a delta-operator-based modified Yule-Walker equation algorithm (MYWE) for parameter estimation of a noisy autoregressive (AR) process. The methodology in developing this new algorithm is similar to the previous works on pure AR processes. Computer simulation results are given to show the improvement of performance in estimating AR parameters in white noise over the q-operator MYWE algorithm.

Asymptotic properties of estimators for autoregressive models with errors in variables

Let ${X_t, t \epsilon \mathbb{Z}}$ be an observable strictly stationary sequence of random variables and let $X_t = U_t + \varepsilon_t$, where ${U_t}$ is an AR (p) and ${\varepsilon_t}$ is a strictly stationary sequence representing errors of measurement in ${X_t}$, with $E{\varepsilon_1} = 0$. Under some broad assumptions on ${\varepsilon_t}$ we establish the consistency properties as well as the rates of convergence for the standard estimators for the autoregressive parameters computed from a set of modified Yule-Walker equations.

Reduced order ARMA spectral estimation of ocean waves

Several system identification techniques are available for determining parametric models of dynamic systems based on the input and output of stochastic processes such as ocean waves. Here we establish a reduced order Autoregressive Moving Average (ARMA) algorithm which is based on the calculation of modal energies. We apply this system identification technique to reduce time series data or target spectra into a few parameters which are the coefficients of the ARMA model. After selecting the initial model order based on the Akaike Information Criterion method, a novel model order reduction technique is applied to obtain the final reduced order ARMA model. First estimates of the higher order autoregressive coefficients are determined using the modified Yule-Walker equations and then first and second order real modes are obtained from the autoregressive polynomial. The energy in each mode is then determined. Considering only the higher energy modes, the autoregressive part of the reduced order ARMA model is obtained. The moving average part is determined based on partial fraction and recursive methods. The above system identification models and model order reduction technique are shown here to be successfully applied to the estimation of ocean wave spectra.

High resolution two-dimensional ARMA spectral estimation

The authors present a practical algorithm for estimating the power spectrum of a 2-D homogeneous random field based on 2-D autoregressive moving average (ARMA) modeling. This algorithm is a two-step approach: first, the AR parameters are estimated by solving a version of the 2-D modified Yule-Walker equation, for which some existing efficient algorithms are available; then the MA spectrum parameters are obtained by simple computations. The potential capability and the high-resolution performance of the algorithm are demonstrated by using some numerical examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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).

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.

Instrumental Variable Methods for Arma Parameter Estimation

The paper reviews the application of instrumental variable (IV) methods to the estimation of the autoregresslve (AR) parameters of time series. The relationship of the modified Yule-Walker equations to IV methods is discussed. The asymptotic properties of the IV estimates are summarized in the context of optimizing estimation accuracy. Two multistep algorithms for optimal estimation of AR parameters are presented. Finally, the recursive IV algorithm and the overdetermined recursive IV algorithm are briefly discussed

A Parametric Technique for Time Delay Estimation

The problem of estimating the time difference of arrival of a signal with unknown spectrum to two receivers is treated. A signal model containing both spectral and delay parameters is derived. The model parameters are computed by a two-step procedure: (1) the modified Yule-Walker equations are used to estimate the autoregressive spectrum of the source signal, and (2) a frequency domain squared error criterion is minimized to provide the delay estimate. The performance of the proposed technique is illustrated by simulation results.

The Modified Yule-Walker Method of ARMA Spectral Estimation

An overview of ARMA spectral estimation techniques based on the modified Yule-Walker equations is presented. The importance of using order overestimation, as well as of using an overdetermined set of equations, is emphasized. The Akaike information criterion is proposed for determining the equation order. A procedure for removing spurious noise modes based on modal decomposition of the sample covariance matrix is derived. The role of the singular value decomposition method in solving the modified Yule-Walker equations is discussed. A number of techniques for estimating MA spectral parameters are presented.

Instrumental variable methods for ARMA spectral estimation

The modified Yule-Walker technique of ARMA spectral estimation is shown to be a special case of the instrumental variable method of system identification. Several recursive instrumental variable algorithms are proposed for adaptive spectral estimation, including a new algorithm for the overdetermined case. An efficient lattice algorithm is presented for solving the modified Yule-Walker equations, given the sample correlation coefficients. The performance of some of the algorithms is illustrated by simulation results.

Efficient algorithm for ARMA spectral estimation

An ARMA spectral estimation technique based on the modified Yule-Walker equations is presented. Several recursive lattice algorithms are proposed for estimating the AR and MA spectral parameters. These computationally efficient algorithms provide spectral estimates of different orders. The choice of the AR model order and its effect on the quality of the spectral estimates are briefly discussed.