On the scaling ranges of detrended fluctuation analysis for long-memory correlated short series of data

TL;DR

This paper investigates how the scaling range in detrended fluctuation analysis (DFA) depends on data length and correlation strength, providing a unified model to improve DFA application on short and long-memory data series.

Contribution

It introduces a simple model linking the DFA scaling range to data length, correlation, and Hurst exponent, enhancing analysis of short and long-memory time series.

Findings

Scaling range varies linearly with data length and regression fit quality.

A unified model relates scaling range to data length, R^2, and Hurst exponent.

Results improve DFA application for short and locally analyzed data series.

Abstract

We examine the scaling regime for the detrended fluctuation analysis (DFA) - the most popular method used to detect the presence of long memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of length $L$ of time series and regression line coefficient $R^2$ at various confidence levels. Next, an analysis of artificial short series with long memory is performed. In both cases the scaling range $\lambda$ is found to change linearly -- both with $L$ and $R^2$. We show how this dependence can be generalized to a simple unified model describing the relation $\lambda=\lambda(L, R^2, H)$ where $H$ ($1/2\leq H \leq 1$) stands for the Hurst exponent of long range autocorrelated data. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where enormous number of short time series has to be examined at once, without possibility for preliminary check of the scaling range of each series separately.