Crossover detection based on variances of slope differences for multi-fractal detrended fluctuation analysis (MF-DFA)

Abstract Multi-fractal detrended fluctuation analysis (MF-DFA) describes long-range correlations across scales in time series in terms of the linear fittings of the fluctuation functions $$F_q(s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for every moment q considered. It is relatively common to find time scales, also called crossovers, where different linearities are found at each side of the $$F_q(s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , leading to different (multi-)fractal properties. Generally, crossovers are manually identified by experts, subject then to their skills and possible bias. In this work, a novel methodology based on the variances of the slope differences (CDV-A) is proposed to automatically detect crossovers. A set of synthetic fluctuation functions with different mono-fractal and multi-fractal features, different noise levels and known crossovers is generated to assess the CDV-A. Moreover, the proposed methodology is also applied to state of the art fluctuation functions in order to compare the proposed manual and automated locations of the crossovers. Results show the proposed methodology is highly accurate, specially in low noise synthetic fluctuation functions and in most of the real cases.