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

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.