Notes on poles of autoregressive type model part I: Non-robust singular pole

AbstractAn autoregressive type approximation is determined from an AR.MA model of physical process by truncating the Taylor expansion of MA part, which is called the T. AR model. The poles of the T.AR model are studied by the aid of the Rouche's theorem of the theory of complex functions. Though the power spectral density of T.AR model converges uniformly to that of AR.MA model, the pole location of T.AR model is quite different from the pole-zero location of AR.MA model. T.AR models have some of original poles of AR.MA model, a “non-robust singular” pole, and poles distributing in a circle in the complex plane which are the statistically equivalent expression of the zero of the AR.MA model closest to the unit circle in the complex plane. The non-robust singular pole has no direct relation to poles or zeros of the original AR.MA model. The zero of the AR.MA model closest to the unit circle in the complex plane has an important role in the convergence of power spectral density of T.AR model. The comparison of pole locations between T.AR model and AR model is also studied in a simple example