Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

A new representation of spatio-temporal random processes is proposed in this work.In practical applications,such processes are used to model velocity fields,temperature distributions,response of vibrating systems,to name a few.Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations,for instance,in a computational mechanics problem.For a single-parameter process such as spatial or temporal process,the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Lo`eve(KL)decomposition.However,for multiparameter processes such as a spatio-temporal process,the covariance function itself can be defined in multiple ways.Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants.Then the spatial covariance matrix at different time instants are considered to define the covariance of the process.This set of square,symmetric,positive semi-definite matrices is then represented as a thirdorder tensor.A suitable decomposition of this tensor can identify the dominant components of the process,and these components are then used to define a closed-form representation of the process.The procedure is analogous to the KL decomposition for a single-parameter process,however,the decompositions and interpretations vary significantly.The tensor decompositions are successfully applied on(i)a heat conduction problem,(ii)a vibration problem,and(iii)a covariance function taken from the literature that was fitted to model a measured wind velocity data.It is observed that the proposed representation provides an efficient approximation to some processes.Furthermore,a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL,both in terms of computer memory and execution time.