Karhunen–Loève expansion for multi-correlated stochastic processes

• The Karhunen–Loève expansion is extended to multi-correlated random processes.
• Two different series expansion methods for correlated random processes are proposed.
• The two approaches have advantages in either convergence or computational cost.

We propose two different approaches generalizing the Karhunen–Loève series expansion to model and simulate multi-correlated non-stationary stochastic processes. The first approach (muKL) is based on the spectral analysis of a suitable assembled stochastic process and yields series expansions in terms of an identical set of uncorrelated random variables. The second approach (mcKL) relies on expansions in terms of correlated sets of random variables reflecting the cross-covariance structure of the processes. The effectiveness and the computational efficiency of both muKL and mcKL is demonstrated through numerical examples involving Gaussian processes with exponential and Gaussian covariances as well as fractional Brownian motion and Brownian bridge processes. In particular, we study accuracy and convergence rates of our series expansions and compare the results against other statistical techniques such as mixtures of probabilistic principal component analysis. We found that muKL and mcKL provide an effective representation of the multi-correlated process that can be readily employed in stochastic simulation and dimension reduction data-driven problems.