Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA), which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression tau(q) = qh(q) - 1 stipulating the relationship between the multifractal exponent tau(q) and the generalized Hurst exponent h(q). As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as tau(q) = qh(q) - qH' - 1, where H' is the nonconservation parameter in the universal multifractal formalism. The singular spectra, alpha and f(alpha), are also derived according to this new relationship.