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$L$ and the correlation coefficient of the linear regression R²$R^2$ 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$\lambda$ is found to change linearly—both with L$L$ and R²$R^2$. We show how this dependence can be generalized to a simple unified model describing the relation λ=λ(L,R²,H)$\lambda =\lambda (L,R^2,H)$ where H$H$ (1/2≤H≤1$1/2\le H\le 1$) 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.