Models for space–time random functions

Models are developed for random functions Q ( x , t ) of space x ∈ D and time t ∈ [ 0 , τ ] from samples of these functions and any other information when available. Most of the models in the paper can be viewed as extensions of Karhunen–Loeve (KL) representations for random fields. Their samples are linear forms of basis functions with random coefficients which are extracted from samples of Q ( x , t ) by singular value decomposition. The coefficients of these forms are stochastic processes rather than random variables. The proposed models can be used to generate large sets of samples whose statistics are similar to those of target random functions. Theoretical arguments and numerical examples are presented to establish properties of the proposed models, assess their accuracy, and illustrate their implementation.