Time-singularity multifractal spectrum distribution based on detrended fluctuation analysis

• We propose the multifractal spectrum distribution based on DFA (DFA-MFSD).
• We prove the equivalent of proposed DFA-MFSD and to the standard time-singularity MFSD.
• DFA-MFSD indicates less computational cost and more adaptable than WTMM-MFSD.
• Simulation on real data shows excellent theoretical and practical performances of DFA-MFSD.

The time-singularity multifractal spectrum distribution (TS-MFSD) generalizes the singularity spectrum in a time-varying framework. In this paper, a new method to compute MFSD based on detrended fluctuation analysis (DFA-MFSD) is introduced. We relate DFA-MFSD method to the standard partition function based multifractal spectrum distribution formalism, and prove that both approaches are equivalent for fractal time series with compact support. Furthermore, we find that DFA-MFSD has equivalent results, better mathematic foundation, less computational cost and is more adapted for fractal time series with arbitrary length, compared with MFSD based on wavelet transform modulus maxima (WTMM-MFSD). By analyzing several examples, this paper shows that DFAm-MFSD with different polynomial fitting orders can reliably determine the time-varying multifractal scaling behavior of time series, including processes embodying chirp-type or oscillating singularities. To illustrate these results, simulations are executed using binomial multiplicative cascades, wavelet series and real sea clutter, and simulations indicate that DFAmm-MFSD benefits from excellent theoretical and practical performances.