Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Abstract

We describe and compare how methods based on the classical Rice's formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately compute the statistical distribution of the maximum over a xed interval, the length of excursions above a level, and the geometry of oscillations, and other characteristics of the sample paths.We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a nite interval and the waiting time to rst crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.We also treat the notoriously dicult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice's original formulation and arguments and the exact numerical methods.MSC 2010 subject classications: Primary 60G17; secondary 60-08, 60G10, 60G70, 65C50.

Counting of level crossings for inertial random processes: Generalization of the Rice formula

“…The main result was the classical Rice formula for the average number of occasions, per unit time, that these processes cross a given level. While Rice was primarily concerned with applications to electrical and radio engineering, the matter has deep and far-reaching effects on other fields of knowledge such as ocean and mechanical engineering, chemical physics, material sciences, laser physics and optics, and many more (see the review [10]). After Rice the problem was first put on firmer mathematical basis by Itô [11], Ylvisaker [12], and particularly by the Scandinavian school of statistics led by Harald Cramer and collaborators [3,10,[13][14][15][16], among others (see [17][18][19] for a small sample).…”

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

“…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. We group these examples in two groups: spectra with near independent halfperiods, i.e.…”

Heavy and light inertial particle aggregates in homogeneous isotropic turbulence: A study on breakup and stress statistics

“…For a stationary stochastic process, the above can be calculated by the frequency of upward-crossing of the the threshold 𝜎 𝑐𝑟 divided by the fraction of time the trajectory stays below the threshold. The former can be expressed by means of the Rice theorem, whereas the latter follows from the probability density function (PDF) of 𝜎 𝑡𝑜𝑡 (𝑡) (Lindgren, 2019). By doing so, the proxy-breakup rate can be computed as:…”