Increment definitions for scale dependent analysis of stochastic data

Abstract

It is common for scale-dependent analysis of stochastic data to use the increment $Δ(t,r) = ξ(t+r) - ξ(t)$ of a data set $ξ(t)$ as a stochastic measure, where $r$ denotes the scale. For joint statistics of $Δ(t,r)$ and $Δ(t,r')$ the question how to nest the increments on different scales $r,r'$ is investigated. Here we show that in some cases spurious correlations between scales can be introduced by the common left-justified definition. The consequences for a Markov process are discussed. These spurious correlations can be avoided by an appropriate nesting of increments. We demonstrate this effect for different data sets and show how it can be detected and quantified. The problem allows to propose a unique method to distinguish between experimental data generated by a noiselike or a Langevin-like random-walk process, respectively.