A tutorial on reproducing a predefined autocovariance function through AR models

Sequential methods for synthetic realisation of random processes have a number of advantages as compared with spectral methods. In this article, the determination of optimal autoregressive (AR) models for reproducing a predefined target autocovariance function of a random process is addressed. To this end, a general formulation of the problem is developed. This formulation is linear and generalises the well-known Yule-Walker (YW) equations and a recent approach based on restricted AR models (Krenk approach). Two main features characterise the introduced formulation: (i) flexibility in the choice for the autocovariance equations employed in the model determination, and (ii) flexibility in the definition of the AR model scheme. Both features were exploited by a genetic algorithm to obtain optimal AR models for the particular case of synthetic generation of homogeneous stationary isotropic turbulence time series. The obtained models improved those obtained with the YW and Krenk approaches for the same model parsimony in terms of the global fitting of the target autocovariance function. The formulation for the one-dimensional multivariate case was also presented, highlighting the causes behind some computational bottlenecks.