Detrended Fluctuation Analysis for Continuous Real Functions

Abstract

Detrended Fluctuation Analysis (DFA) is a widely used method applied to determine the long-term autocorrelation of a time series. It consists of an algorithm that first transforms the original time series into a self similar process and then removes its trend. As a result, an approximation to a power law is obtained. The autocorrelation of the original time series will be established depending on the value of the scaling exponent of the resulting power law. In this work, we propose a DFA that applies to continuous real functions. It also consists of an algorithm that we prove transforms a continuous function into the sum of two self similar fractal functions. After removing the trend, an approximation of a power law is obtained. However, unlike the discrete case, numerical simulations show that in a log-log graph, the line of best fit always has slope approximately equal to one, thus establishing that the DFA for continuous functions is approximately a power law with scaling exponent one. This agrees with the fact that a function, as a continuous-time time series, may be regarded as a deterministic object and hence is highly autocorrelated.