DFA-based abacuses providing the Hurst exponent estimate for short-memory processes

Abstract

The detrended fluctuation analysis (DFA) and its higher-order variant make it possible to estimate the Hurst exponent and therefore to quantify the long-range dependence of a random process. These methods are popular and used in a wide range of applications where they have been proven to be discriminative to characterize or classify processes. Nevertheless, in practice, the signal may be short-memory. In addition, depending on the number of samples available, there is no guarantee that these methods provide the true value of the Hurst exponent, leading the user to draw erroneous conclusions on the long-range dependence of the signal under study. In this paper, using a matrix formulation and making no approximation, we first propose to analyze how the DFA and its higher-order variant behave with respect to the number of samples available. Illustrations dealing with short-memory data that can be modeled by a white noise, a moving-average process and a random process whose autocorrelation function exponentially decays are given. Finally, to avoid any wrong conclusions, we propose to derive abacuses linking the value provided by the DFA or its variant with the properties of the signal and the number of samples available.