Karhunen-Loeve Representation of Periodic Second-Order Autoregressive Processes

In dynamic data driven applications modeling accurately the uncertainty of various inputs is a key step of the process. In this paper, we first review the basics of the Karhunen-Loève decomposition as a means for representing stochastic inputs. Then, we derive explicit expressions of one-dimensional covariance kernels associated with periodic spatial second-order autoregressive processes. We also construct numerically those kernels by employing the Karhunen-Loève expansion and making use of Fourier representation in order to solve efficiently the associated eigenvalue problem. Convergence and accuracy of the numerical procedure are checked by comparing the covariance kernels obtained from the Karhunen-Loève expansions against theoretical solutions.