Detrended Fluctuation Analysis of Autoregressive Processes

TL;DR

This paper explores how Detrended Fluctuation Analysis (DFA) can characterize correlation structures in autoregressive processes, revealing relationships between AR model parameters and DFA exponents, aiding model identification.

Contribution

It systematically investigates AR(1) and AR(2) processes using DFA, establishing links between AR parameters and correlation exponents, and demonstrating model distinguishability.

Findings

Higher AR interaction constants lead to higher short-range correlation exponents.

Distant positive interactions increase correlation range and exponent.

Distant negative interactions decrease correlation range significantly.

Abstract

Autoregressive processes (AR) have typical short-range memory. Detrended Fluctuation Analysis (DFA) was basically designed to reveal long range correlation in non stationary processes. However DFA can also be regarded as a suitable method to investigate both long-range and short range correlation in non-stationary and stationary systems. Applying DFA to AR processes can help understanding the non uniform correlation structure of such processes. We systematically investigated a first order autoregressive model AR(1) by DFA and established the relationship between the interaction constant of AR(1) and the DFA correlation exponent. The higher the interaction constant the higher is the short range correlation exponent. They are exponentially related. The investigation was extended to AR(2) processes. The presence of a distant positive interaction in addition to a near by interaction will increase the correlation exponent and the range of correlation while the effect of a distant negative interaction will decrease significantly only the range of interaction. This analysis demonstrate the possibility to identify and AR(1) model in an unknown DFA plot or to distinguish among AR(1) and AR(2) models. The analysis was performed on medium long series of 1000 terms.