Effective computations of joint excursion times for stationary Gaussian processes

Abstract

This work is to popularize the method of computing the distribution of the excursion times for a Gaussian process that involves extended and multivariate Rice's formula. The approach was used in numerical implementations of the high-dimensional integration routine and in earlier work it was shown that the computations are more effective and thus more precise than those based on Rice expansions.The joint distribution of successive excursion times is clearly related to the distribution of the number of level crossings, a problem that can be attacked via the Rice series expansion, based on the moments of the number of crossings. Another point of attack is the "Independent Interval Approximation" (IIA) intensively studied for the persistency of physical systems. It treats the lengths of successive crossing intervals as statistically independent. Under IIA, a renewal type argument leads to an expression that provides the approximate interval distribution via its Laplace transform.However, the independence is not valid in most typical situations. Even if it leads to acceptable results for the persistency exponent of the long excursion time distribution or some classes of processes, rigorous assessment of the approximation error is not readily available. Moreover, we show that the IIA approach cannot deliver properly defined probability distributions and thus the method is limited only to persistence studies.The ocean science community favours a third approach, in which a class of parametric marginal distributions, either fitted to excursion data or derived from a narrow band approximation, is extended by a copula technique to bivariate and higher order distributions.This paper presents an alternative approach that is both more general, more accurate and relatively unknown. It is based on exact expressions for the probability density for one and for two successive excursion lengths. The numerical routine RIND

Effective persistency evaluation via exact excursion distributions for random processes and fields

“…Here, we give an account of numerical evaluation of the degree of the dependence for Gaussian processes. The exact approach through the generalized Rice formula is explained and evaluated using the RIND routine, the technical details of the computations are given in [45]. We compare the exact RIND pdf with the IIA-based pdf for the spectra in table 1, extending the cases extensively studied in [50], and illustrate the results.…”

“…A WAFO-tutorial [45] guides the user through the technicalities of using the computational tools. It also gives information about the main computational engine, the routine RIND, and presents examples of the code.…”