Elucidating appealing features of differentiable auto-correlation functions: A study on the modified exponential kernel

Abstract

Research on stochastic processes in recent decades has pointed out that, in the context of modeling spatial or temporal uncertainties, auto-correlation functions that are differentiable at the origin have advantages over functions that are not differentiable. For instance, the non-differentiability of e.g., single exponential auto-correlation functions yields non-smooth sample paths. Such sample paths might not be physically possible or may yield numerical difficulties when used as random parameters in partial differential equations (such as encountered in e.g., mechanical equilibrium problems). Further, it is known that due to the non-differentiability of certain auto-correlation functions, more terms are required in the series expansion representations of the associated stochastic processes. This makes these representations less efficient from a computational standpoint.This paper elucidates some additional appealing features of auto-correlation functions which are differentiable at the origin. Further, it focuses on enhancing the arguments in favor of these functions already available in literature. Specifically, attention is placed on single exponential, modified exponential and squared exponential auto-correlation functions, which can be shown to be all part of the Whittle–Matérn family of functions. To start, it is shown that the power spectrum of differentiable kernels converges faster to zero with increasing frequency as compared to non-differentiable ones. This property allows capturing the same percentage of the total energy of the spectrum with a smaller cut-off frequency, and hence, less stochastic terms in the harmonic representation of stochastic processes. Further, this point is examined with regards to the Karhunen–Loève series expansion and first and second order Markov processes, generated by auto-regressive representations. The need for finite differentiability is stressed and illustrated.