A numerical recipe for the computation of stationary stochastic processes’ autocorrelation function

Abstract

Many natural phenomena exhibit a stochastic nature that one attempts at modelling by using stochastic processes of different types. In this context, often one is interested in investigating the memory properties of the natural phenomenon at hand. This is usually accomplished by computing the autocorrelation function of the numerical series describing the considered phenomenon. Often, especially when considering real world data, the autocorrelation function must be computed starting from a single numerical series: i.e. with a time-average approach. Hereafter, we will propose a novel way of evaluating the time-average autocorrelation function, based on the preliminary evaluation of the quantity N(\tau, g_\mu., g_\nu), that, apart from normalization factors, represents a numerical estimate, based on a single realization of the process, of the 2-point joint probability density function P(x_2, \tau, x_1.0). This approach is not necessarily faster or more precise than the usual methods already known in the literature. Rather, our contribution is mainly of methodological nature as it allows to quantitatively assess the finite size effects that one encounters when numerically evaluating the autocorrelation function due to the fact that any simulated time series is necessarily bounded. In fact, we show that, for a wide class of stochastic processes admitting a nonlinear Langevin equation with white noise and that can be described by using a Fokker–Planck equation, the way the numerical estimate of the autocorrelation function converges to its theoretical prediction depends on the pdf tails. Moreover, the knowledge of N(\tau, g_\mu., g_\nu) allows to easily compute the process histogram and to characterize processes with multiple timescales. We will show the effectiveness of our new methodology by considering three stochastic processes whose autocorrelation function and two-point probability density function are both known in an analytical or numerical form, thus allowing direct comparisons.