On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data

We examine the scaling regime for the detrended fluctuation analysis (DFA)—the most popular method used to detect the presence of long-term memory in data and the fractal structure of time series. First, the scaling range for DFA is studied for uncorrelated data as a function of time series length L and the correlation coefficient of the linear regression R 2 at various confidence levels. Next, a similar analysis for artificial short series of data with long-term memory is performed. In both cases the scaling range λ is found to change linearly—both with L and R 2 . We show how this dependence can be generalized to a simple unified model describing the relation λ = λ ( L , R 2 , H ) where H ( 1 / 2 ≤ H ≤ 1 ) stands for the Hurst exponent of the long range autocorrelated signal. Our findings should be useful in all applications of DFA technique, particularly for instantaneous (local) DFA where a huge number of short time series has to be analyzed at the same time, without possibility of checking the scaling range in each of them separately.