EAR model yaklaşımını kullanarak 2-B ARMA model parametrelerinin kestirimi

Özet

Bu çalışmada, çeyrek-düzlem destek bölgesine sahip doğrusal zamanla değişmeyen durağan ikiboyutlu 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.

Abstract

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 zeromean white Gaussian noise with variance $sigma_w^2$. 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 highresolution 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.