Extended self-similarity based multi-fractal detrended fluctuation analysis: A novel multi-fractal quantifying method

Abstract

Based on multi-fractal detrended fluctuation analysis (MF-DFA) method, the extended self-similarity (ESS) is incorporated to develop a novel method to quantify multi-fractal characteristics. Detailed results show that compared with MF-DFA this new extended self-similarity based MF-DFA (ESS-MF-DFA) method can significantly extend scaling range and reduce uncertainties in estimating the exponents. Moreover, although ESS-MF-DFA method is developed from the DFA method with a fundamental assumption of a definite scaling range between fluctuation function and scale, ESS-MF-DFA can still work well even when DFA fails due to no scaling range between fluctuation function and scale. Furthermore, a criterion without estimating the generalized Hurst exponents is developed based on ESS-MF-DFA to distinguish multi-fractal from mono-fractal behavior and to quantify multi-fractal strength. All these results indicate that ESS-MF-DFA outperforms MF-DFA in reliably handling multi-fractal quantifications for much wider fields.