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

Abstract

Detrended fluctuation analysis (DFA) is a method widely used for the study of long-rangecorrelation and fractal scaling properties of time series. Based on DFA, multifractaldetrended fluctuation analysis (MF-DFA) was proposed to give a full description ofmore complicated time series. However, the removal of local trends in DFA isbased on discontinuous polynomial fitting. It has been shown that oscillationsin the fluctuation function and significant errors in crossover locations can beintroduced in actual implementations. In terms of time series, it is generally naturalthat points near in time are more related than points some distance apart. Suchprinciples can help us circumvent the above problems in the detrending step ofMF-DFA. Based on this rationale, the ideas of moving windows and weighted movingwindows are proposed for smoothing the log–log plot of the fluctuation functionF(s) versusthe scale s so that local effects can be taken into consideration and crossover timescales, particularlylarge timescales, can be effectively detected. The multifractal moving-window detrendedfluctuation analysis (MF-MWDFA) and the more general multifractal temporally weighteddetrended fluctuation analysis (MF-TWDFA) are proposed in this paper. Numericalsimulations and the analysis of a real-life daily temperature time series are performed inorder to substantiate the arguments and evaluate the performance. With the help ofMF-TWDFA, two more crossover points, which cannot be found by the conventionalMF-DFA, have been found in the annual-detrended temperature series by theproposed model. The crossover timescales appear to correspond rather closely withthe actual variation of temperature over time under different climate regimes.