Detrended Fluctuation Analysis for Continuous Real Variable Functions

Abstract

Detrended Fluctuation Analysis (DFA) is a widely used technique for estimating scaling exponents and long-range correlations in time series data. In this work, we formulate DFA in a continuous framework (which we call Continuous DFA or CDFA for short) and provide a rigorous analytical description of its behavior for real-valued continuous functions. We derive an explicit integral representation of the fluctuation function and establish its regularity properties. We show that the fluctuation grows linearly at sufficiently small scales for all signals, so that the associated scaling exponent converges universally to one. Although finite scaling ranges may exhibit apparent power law fits with different exponents, these reflect only local approximations rather than the true asymptotic behavior. Our results show that CDFA primarily measures first-order regularity rather than correlation structure. Consequently, while the classical discrete DFA distinguishes rough stochastic processes, CDFA loses discriminating power for continuous deterministic signals. These findings provide a clear theoretical explanation of both the capabilities and the intrinsic limitations of DFA.