ARMA SIMULATION OF MULTIVARIATE AND MULTIDIMENSIONAL RANDOM PROCESSES
MIGNOLET, MARC PAUL
Doctor of Philosophy
Autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) systems for the simulation of multivariate multidimensional random processes are investigated. The AR system is reviewed first. A meaningful measure of the matching between the target and AR spectral matrices is introduced, and the properties of the corresponding AR approximations are discussed. Next, AR to ARMA modelling approaches are developed on the basis of the minimization of frequency domain errors. The final values of these errors, which quantify the quality of the matching between AR and ARMA spectra, are used to compare the various design approaches. A sufficient condition for the stability and/or invertibility of the corresponding AR and ARMA systems is derived. Further, matching properties between the AR and ARMA auto- and cross-correlations are proved. The computation of the parameters of MA models is approached from the perspective of Fourier approximation of any of the decompositions of the target spectral matrix. The selection of the "most suitable" one is formulated as an optimization problem whose solution involves an eigenvalue problem. Three approaches to derive an ARMA approximation of a target spectral matrix from a prior MA model are also presented. The first one involves an approximation of the MA transfer function in the form of a ratio of two-sided matrix polynomials. The two other approaches rely on separate rational approximations of the causal and anticausal parts of the autocorrelation sequence and MA transfer function, respectively. Examples of application are given from the field of structural engineering.