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 excursion times is 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" intensively studied for the persistence of physical systems. It treats the lengths of successive crossing intervals as statistically independent. A renewal type argument leads to an expression that provides the approximate interval distribution via its Laplace transform. However, independence is not valid in typical situations. Even if it leads to acceptable results for the persistency exponent, rigorous assessment of the approximation error is not available. Moreover, we show that the IIA approach cannot deliver properly defined probability distributions and thus the method is limited to persistence studies. 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 computes the densities using recent advances in scientific computing and is easily accessible via a simple Matlab interface. The result solves the problem of two-step excursion dependence for a general stationary differentiable Gaussian process. The work offers also some analytical results that explain the effectiveness of the implemented method.