Wavelet versus detrended fluctuation analysis of multifractal structures

Abstract

We perform a comparative study of applicability of the Multifractal Detrended Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM) method in proper detecting of mono-and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Lévy flights, and different sorts of multifractal binomial cascades. Our results show that in majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f (α), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

“…Among the reasons behind the popularity of DFA was its ability of detecting fractal character of signals, which was subsequently extended to the multifractal case (the MFDFA method [17]), which also proved very useful if applied to empirical data [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], especially owing to its superior reliability if compared to other methods [37]. DCCA was also generalized in order to be applicable to signals with multifractal cross-correlations and the resulting MFDCCA/MFDXA algorithm [38] also attracted some attention [39][40][41][42].…”

Comparison of detrending methods for fluctuation analysis

“…For studies comparing DFA and BMA, see [46,47]; note that [47] also discusses CMA. For studies comparing methods for detrending multifractal analysis (multifractal DFA (MF-DFA) and wavelet transform modulus maxima (WTMM) method), see [5,24,48].…”

Chaos Based Quantitative Electro-Diagnostic Marker for Diagnosis of Myopathy, Neuropathy and Motor Neuron Disease

“…Many researchers in this domain have recommended MFDFA due to its better performance than other conventional methods in the analysis of multifractality in both stationary as well as non-stationary time series [63,81,82]. Oswiecimka et al have established the superiority of MFDFA over other techniques, especially over the most popular one, the Wavelet Transform Modulus Maxima (WTMM) in terms of reliable applications [83].…”