Estimation of 2-D ARMA model parameters by using EAR model approach

Bu çalışmada, çeyrek-düzlem  destek bölgesine sahip doğrusal zamanla değişmeyen durağan iki-boyutlu özbağlanımlı kayan ortalamalı (2-B ARMA) modelin parametrelerinin kestirim problemi ele alınmakta ve bu problemin çözümü için, 2-B ARMA model parametreleri ile bu modele eşdeğer sonsuz mertebeden iki-boyutlu özbağlanımlı (2-B EAR) modelin parametreleri arasındaki ilişki incelenmektedir. Bu ilişkiyi esas alarak sonlu mertebeden EAR modelin katsayılarından (p1, p2, q1, q2). mertebeden 2-B ARMA modelin parametrelerini kestirmek amacıyla; doğrusal denklem takımlarının çözümüyle parametre kestirimlerini gerçekleştiren, yakınsama sorunu olmayan, hesapsal karmaşıklığı düşük yeni bir yöntem önerilmektedir. Önerilen bu yöntem, üç aşamalı olup; birinci aşamada, (p1, p2, q1, q2). mertebeden 2-B ARMA modeli yaklaşık olarak temsil eden (L1, L2).mertebeden 2-B EAR modelin parametreleri değiştirilmiş Yule-Walker denklemleri olarak adlandırılan doğrusal denklem takımlarının çözümüyle elde edilmektedir. İkinci aşamada ise, birinci aşamada elde edilen EAR model katsayılarını önerilen yöntem ile türetilen eşitliklerde kullanarak 2-B ARMA modelin kayan ortalamalı (MA) parametrelerinin kestirimi gerçekleştirilmektedir. Son olarak, birinci ve ikinci aşamalarda hesaplanan EAR ve MA parametre kestirimlerini türetilen doğrusal denklem ifadesinde yerine koyarak 2-B ARMA modelin özbağlanımlı (AR) kısmını tanımlayan katsayıların hesabı yapılmaktadır. Önerilen yöntemin başarımı, bilgisayar benzetimleri sınanmıştır. Bu amaçla, önerilen yöntemin literatürdeki yöntem ile eşzamanlı çalıştırılması sonucunda üretilen parametre kestirimleri ve bu parametrelere karşı düşen güç izge yoğunluk kestirimleri çeşitli başarım ölçütlerine göre karşılaştırılmıştır. Sonuç olarak, önerilen yöntemle oldukça iyi ve tatmin edici sonuçlara ulaşıldığı gözlenmiştir. Anahtar Kelimeler: 2-B ARMA model, 2-B EAR model, parametre kestirimi, çeyrek-düzlem destek bölgesi.     This paper considers the parameter estimation problem of a quarter-plane (QP) linear time-invariant (LTI) two-dimensional autoregressive moving average (2-D ARMA) model excited by an unknown zero-mean white Gaussian noise with variance w2. Since the use of nonparametric methods such as Fast Fourier Transform (FFT) yield low-resolution results, 2-D system identification and parametric representations of 2-D stationary random fields based on parametric 2-D autoregressive (AR), moving average (MA), and ARMA models have received great attention in a wide range of image and signal processing applications. These applications include image restoration, image compression, stochastic texture analysis and synthesis, modeling, and high-resolution spectrum estimation of 2-D data, etc. Note that AR and MA models correspond to the special case of ARMA models. The most general models used in modeling the random fields are the ARMA models. In modeling of 2-D random fields, AR models have been used extensively since their parameters are estimated easily by solving the set of linear equations called as Modified Yule-Walker (MYW) equations. However, as in the one-dimensional case, the parameter estimation procedures for the MA and ARMA models are much more difficult than the AR models since these procedures require a heavy computational burden and there are convergence problems. All of these reasons and intrinsic nonlinearity of estimating the MA parameters cause restriction on making studies based upon MA and ARMA models. In spite of these difficulties, ARMA models are preferred frequently because of their relations with the linear filters having rational transfer function and their abilities on simulating the behavior similar to the noise correctly. From the parameter parsimony point of view, 2-D ARMA models usually provide the most effective linear models of the 2-D homogeneous random fields and are therefore preferable over its AR or MA counterparts: as compared to the AR and MA models, ARMA models can perform more accurate modeling with a few number of parameters. From the spectral estimation viewpoint, while the ARMA models can characterize both the peaks and the valleys, the AR and MA models can characterize only the peaks and the valleys, respectively. In spite of its advantages, there are a few methods in the literature related to the parameter and spectral estimation of 2-D ARMA models. For the aim of modeling 2-D random fields, the existing 2-D ARMA model-based estimation methods can be classified into two main groups. In the first group of methods, AR and MA parameters of the ARMA model are estimated explicitly from the given data set or its second-order statistics. Thus, the given data set is characterized by either the transfer function of the ARMA model or its power spectral density (PSD) function obtained using the estimated AR and MA parameters. In the second group of methods, the estimation processes are realized on the basis of the PSD function of the ARMA model. AR parameters are estimated explicitly from the given data record or its statistics, and then the MA spectrum parameters are calculated using the estimated AR parameters and the second-order statistics of the data set. Hence, the observation data are characterized by the ARMA model PSD function formed by the estimated AR parameters and MA spectrum parameters. Note that while the MA parameters are acquired explicitly in the first group of methods, the methods involving to the second group obtain the MA spectrum parameters rather than estimating the MA parameters explicitly. In this paper, we have introduced a simple and computationally efficient method for estimating the parameters of a LTI 2-D ARMA model having QP support region. The suggested method is based on the relation between the parameters of the 2-D ARMA model and those of the equivalent autoregressive (2-D EAR) model. On the basis of this relation, linear equations performing the ARMA model parameter estimation process from the coefficients of the EAR model are derived. The method proposed for this purpose is a three-step approach: firstly, the 2-D EAR model parameters are obtained solving the set of linear equations called as MYW equations; then, the MA parameters are estimated benefiting from the EAR model coefficients; finally, the AR parameters are calculated exploiting the estimated EAR and MA parameters in the derived formula. Performance of the proposed method is analyzed via computer simulations. We demonstrate with simulations that satisfactory results are obtained by the proposed method. Keywords: 2-D ARMA model, 2-D EAR model, parameter estimation, quarter-plane support region

Robust estimation for ARMA models

This paper introduces a new class of robust estimates for ARMA models. They are M-estimates, but the residuals are computed so the effect of one outlier is limited to the period where it occurs. These estimates are closely related to those based on a robust filter, but they have two important advantages: they are consistent and the asymptotic theory is tractable. We perform a Monte Carlo where we show that these estimates compare favorably with respect to standard M-estimates and to estimates based on a diagnostic procedure.Comment: Published in at http://dx.doi.org/10.1214/07-AOS570 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some Computational Aspects of Gaussian CARMA Modelling

Representation of continuous-time ARMA, CARMA, models is reviewed. Computational aspects of simulating and calculating the likelihood-function of CARMA are summarized. Some numerical properties are illustrated by simulations. Some real data applications are shown.CARMA, maximum-likelihood, spectrum, Kalman filter, computation

A Likelihood Ratio Test of Stationarity Based on a Correlated Unobserved Components Model

We propose a likelihood ratio (LR) test of stationarity based on a widely-used correlated unobserved components model. We verify the asymptotic distribution and consistency of the LR test, while a bootstrap version of the test is at least first-order accurate. Given empirically-relevant processes estimated from macroeconomic data, Monte Carlo analysis reveals that the bootstrap version of the LR test has better small-sample size control and higher power than commonly used bootstrap Lagrange multiplier (LM) tests, even when the correct parametric structure is specified for the LM test. A key feature of our proposed LR test is its allowance for correlation between permanent and transitory movements in the time series under consideration, which increases the power of the test given the apparent presence of non-zero correlations for many macroeconomic variables. Based on the bootstrap LR test, and in some cases contrary to the bootstrap LM tests, we can reject trend stationarity for U.S. real GDP, the unemployment rate, consumer prices, and payroll employment in favor of nonstationary processes with volatile stochastic trends.Stationarity Test, Likelihood Ratio, Unobserved Components, Parametric Bootstrap, Monte Carlo Simulation, Small-Sample Inference

The Relationship between the Beveridge-Nelson Decomposition andUnobserved Component Models with Correlated Shocks

Many researchers believe that the Beveridge-Nelson decomposition leads to permanent and transitory components whose shocks are perfectly negatively correlated. Indeed, some even consider it to be a property of the decomposition. We demonstrate that the Beveridge-Nelson decomposition does not provide definitive information about the correlation between permanent and transitory shocks in an unobserved components model. Given an ARIMA model describing the evolution of U.S. real GDP, we show that there are many state space representations that generate the Beveridge-Nelson decomposition. These include unobserved components models with perfectly correlated shocks and partially correlated shocks. In our applications, the only knowledge we have about the correlation is that it lies in a restricted interval that does not include zero. Although the filtered estimates of the trend and cycle are identical for models with different correlations, the observationally equivalent unobserved components models produce different smoothed estimates.

Forecasting VARMA processes using VAR models and subspace-based state space models

VAR modelling is a frequent technique in econometrics for linear processes. VAR modelling offers some desirable features such as relatively simple procedures for model specification (order selection) and the possibility of obtaining quick non-iterative maximum likelihood estimates of the system parameters. However, if the process under study follows a finite-order VARMA structure, it cannot be equivalently represented by any finite-order VAR model. On the other hand, a finite-order state space model can represent a finite-order VARMA process exactly, and, for state-space modelling, subspace algorithms allow for quick and non-iterative estimates of the system parameters, as well as for simple specification procedures. Given the previous facts, we check in this paper whether subspace-based state space models provide better forecasts than VAR models when working with VARMA data generating processes. In a simulation study we generate samples from different VARMA data generating processes, obtain VAR-based and state-space-based models for each generating process and compare the predictive power of the obtained models. Different specification and estimation algorithms are considered; in particular, within the subspace family, the CCA (Canonical Correlation Analysis) algorithm is the selected option to obtain state-space models. Our results indicate that when the MA parameter of an ARMA process is close to 1, the CCA state space models are likely to provide better forecasts than the AR models. We also conduct a practical comparison (for two cointegrated economic time series) of the predictive power of Johansen restricted-VAR (VEC) models with the predictive power of state space models obtained by the CCA subspace algorithm, including a density forecasting analysis.subspace algorithms; VAR; forecasting; cointegration; Johansen; CCA