A tutorial on reproducing a predefined autocovariance function through AR models: Application to stationary homogeneous isotropic turbulence

TL;DR

This paper introduces a flexible, linear formulation for designing optimal autoregressive models to reproduce a predefined autocovariance function, demonstrated on stationary isotropic turbulence, outperforming traditional methods.

Contribution

A novel, flexible linear formulation for AR model determination is developed, generalizing Yule-Walker and Krenk-Moller approaches, and optimized using genetic algorithms.

Findings

Improved AR models for turbulence autocovariance reproduction.

Enhanced spectral reproduction compared to traditional methods.

Multivariate formulation with identified computational challenges.

Abstract

Sequential methods for synthetic realisation of random processes have a number of advantages 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 novel formulation of the problem is developed. This formulation is linear and generalises the well-known Yule-Walker (Y-W) equations and a recent approach based on restricted AR models (Krenk-Moller approach, K-M). 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 Y-W and K-M approaches for the same model parsimony in terms of the global fitting of the target autocovariance function. Implications for the reproduced spectra are also discussed. The formulation for the multivariate case is also presented, highlighting the causes behind some computational bottlenecks.