Consistency of detrended fluctuation analysis.

The scaling function F(s) in detrended fluctuation analysis (DFA) scales as F(s)∼s^{H} for stochastic processes with Hurst exponent H. This scaling law is proven for stationary stochastic processes with 0<H<1 and nonstationary stochastic processes with 1<H<2. For H<0.5, it is observed that the asymptotic (power-law) autocorrelation function (ACF) scales as ∼s^{1/2}. It is also demonstrated that the fluctuation function in DFA is equal in expectation to (i) a weighted sum of the ACF and (ii) a weighted sum of the second-order structure function. These results enable us to compute the exact finite-size bias for signals that are scaling and to employ DFA in a meaningful sense for signals that do not exhibit power-law statistics. The usefulness is illustrated by examples where it is demonstrated that a previous suggested modified DFA will increase the bias for signals with Hurst exponents 1<H≤1.5. As a final application of these developments, an estimator F[over ̂](s) is proposed. This estimator can handle missing data in regularly sampled time series without the need of interpolation schemes. Under mild regularity conditions, F[over ̂](s) is equal in expectation to the fluctuation function F(s) in the gap-free case.