Combining the global trends of DFA or CDFA of different orders

Abstract

When dealing with the detrended fluctuation analysis (DFA) and its variants such as the higher-order DFA and the continuous DFA (CDFA), removing the mean, integrating the signal and detrending it amount to applying an equivalent linear filter characterized by its frequency response when processing a wide-sense stationary random process. By studying the frequency response of each variant, one can better understand its behavior. Thus, in this paper, we first compare the frequency responses of two DFAs whose orders are different. Then, by deriving the higher-order CDFA using Lagrange multipliers, we can show how the frequency responses of the equivalent filters of the CDFA and the DFA are related to each other. Finally, we propose to combine the global trends obtained with the DFA or the CDFA of different orders to derive a new variant. Once again, the behavior is studied using the frequency response of the equivalent filter. Illustrations and comments on ARFIMA processes and Weierstrass functions are also given to evaluate their long range dependence.