Multifractal temporally weighted detrended fluctuation analysis and its application in the analysis of scaling behavior in temperature series

Detrended fluctuation analysis (DFA) is a method widely used for the study of long-range correlation and fractal scaling properties of time series. Based on DFA, multifractal detrended fluctuation analysis (MF-DFA) was proposed to give a full description of more complicated time series. However, the removal of local trends in DFA is based on discontinuous polynomial fitting. It has been shown that oscillations in the fluctuation function and significant errors in crossover locations can be introduced in actual implementations. In terms of time series, it is generally natural that points near in time are more related than points some distance apart. Such principles can help us circumvent the above problems in the detrending step of MF-DFA. Based on this rationale, the ideas of moving windows and weighted moving windows are proposed for smoothing the log-log plot of the fluctuation function F(s) versus the scale s so that local effects can be taken into consideration and crossover timescales, particularly large timescales, can be effectively detected. The multifractal moving-window detrended fluctuation analysis (MF-MWDFA) and the more general multifractal temporally weighted detrended fluctuation analysis (MF-TWDFA) are proposed in this paper. Numerical simulations and the analysis of a real-life daily temperature time series are performed in order to substantiate the arguments and evaluate the performance. With the help of MF-TWDFA, two more crossover points, which cannot be found by the conventional MF-DFA, have been found in the annual-detrended temperature series by the proposed model. The crossover timescales appear to correspond rather closely with the actual variation of temperature over time under different climate regimes.