Consistency of detrended fluctuation analysis

TL;DR

This paper provides a rigorous theoretical foundation for detrended fluctuation analysis (DFA), proving its scaling law for various stochastic processes, analyzing biases, and introducing an estimator that handles missing data effectively.

Contribution

It offers the first comprehensive proof of DFA's scaling law for different Hurst exponents, analyzes bias in existing methods, and proposes a new estimator for incomplete data.

Findings

DFA's fluctuation function scales as $s^{H}$ for $0<H<2$.

Asymptotic ACF yields $F(s) o s^{1/2}$ for $H<0.5$.

New estimator effectively handles missing data in time series.

Abstract

The scaling function $F(s)$ in detrended fluctuation analysis (DFA) scales as $F(s)\sim s^{H}$ for stochastic processes with Hurst exponents $H$. We prove this scaling law for both stationary stochastic processes with $0<H<1$, and non-stationary stochastic processes with $1<H<2$. For $H<0.5$ we observe that using the asymptotic (power-law) auto-correlation function (ACF) yield $F(s)\sim s^{1/2}$. We also show that the fluctuation function in DFA is equal in expectation to: i) A weighted sum of the ACF ii) A weighted sum of the second order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling, as well as studying DFA for signals that do not have power-law statistics. We illustrate this with examples, where we find that a previous suggested modified DFA will increase the bias for signals with Hurst exponents $H>1$. As a final application of the new theory, we present an estimator $\hat F(s)$ that can handle missing data in regularly sampled time series without the need for interpolation schemes. Under mild regularity conditions, $\hat F(s)$ is equal in expectation to the fluctuation function $F(s)$ in the gap-free case.