Kernel density approach to error estimation of MF-DFA measures on time series

MF-DFA is a widely used method for the extraction of multifractal patterns that may appear in time series. Current techniques for determining the range of scales employed in the MF-DFA procedure are mainly empirical and in some cases require the researcher to visually select such range, so multifractal measures often differ among experiments, and the estimation of errors in such measures is often disregarded. This problem also represents a serious obstacle for incorporating MF-DFA in automated processes. In this paper, we present a Kernel Density based method that provides metrics for errors on the estimations of the multifractal spectrum obtained by MF-DFA, and gives the probabilities of errors being inside a certain range for each moment considered in the spectrum. Our approach makes use of Kernel Density Estimation in combination with a variant of the Theil–Sen estimator to compute MF-DFA exponents. We tested our technique on simulated and real-world time series. Comparison between this approach and the conventional method applied to deterministic and random multiplicative processes demonstrates robustness to spurious estimations of generalized fractal dimensions in MF-DFA.