Extreme statistics for time series: Distribution of the maximum relative to the initial value

TL;DR

This paper analyzes the distribution of the maximum relative to the initial value in time series, deriving exact results for certain cases and exploring the effects of correlations and boundary conditions on the distribution.

Contribution

It provides exact distributions for the maximum relative height in correlated Gaussian signals and investigates the impact of correlations and boundary conditions.

Findings

Exact MRH_I distribution for alpha=0, 2, 4, infinity

MRH_I distribution differs significantly from maximum relative to average

Boundary conditions influence the shape of the distribution

Abstract

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/f^alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRH_I). The exact MRH_I distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha, the distribution is determined from simulations. We find that the MRH_I distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRH_I distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some non-periodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRH_I distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.