Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

TLDR

Random processes can be expressed as series expansions of deterministic functions weighted by random coefficients, with the Karhunen–Loeve expansion derived from the eigen‑decomposition of the covariance function. The paper numerically investigates the applicability of the K–L expansion as a simulation tool for both stationary and non‑stationary Gaussian random processes. Using five common covariance models, the authors assess convergence and accuracy of the K–L expansion by comparing its second‑order statistics to those of the target process and by benchmarking against the spectral representation method. Convergence depends on the process length to correlation ratio, covariance function form, and eigen‑solution method; the K–L expansion outperforms the spectral method for highly correlated processes, while the spectral method is more efficient for long stationary processes, and the K–L method’s main advantage is its ease of generalization to non‑stationary processes. © 2001 John Wiley & Sons, Ltd.

Abstract

Abstract A random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen‐decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non‐stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second‐order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen‐solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non‐stationary processes with little additional effort. Copyright © 2001 John Wiley & Sons, Ltd.