Simulation of Ergodic Multivariate Stochastic Processes

TLDR

The process is nonhomogeneous in space when its components correspond to different spatial locations. The study proposes a simulation algorithm to generate sample functions of a stationary multivariate stochastic process based on its cross‑spectral density matrix. The algorithm, an extension of the spectral representation method that exploits the fast Fourier transform, produces ergodic, periodic sample functions whose temporal cross‑correlation matches the target and is demonstrated with turbulent wind velocity simulations. The generated sample functions reproduce the target ensemble cross‑correlation matrix exactly, and become Gaussian as the frequency discretization is refined.

Abstract

A simulation algorithm is proposed to generate sample functions of a stationary, multivariate stochastic process according to its prescribed cross-spectral density matrix. If the components of the vector process correspond to different locations in space, then the process is nonhomogeneous in space. The ensemble cross-correlation matrix of the generated sample functions is identical to the corresponding target. The simulation algorithm generates ergodic sample functions in the sense that the temporal cross-correlation matrix of each and every generated sample function is identical to the corresponding target, when the length of the generated sample function is equal to one period (the generated sample functions are periodic). The proposed algorithm is based on an extension of the spectral representation method and is very efficient computationally since it takes advantage of the fast Fourier transform technique. The generated sample functions are Gaussian in the limit as the number of terms in the frequency discretization of the cross-spectral density matrix approaches infinity. An example involving simulation of turbulent wind velocity fluctuations is presented in order to demonstrate the capabilities and efficiency of the proposed algorithm.