Wavelet versus Detrended Fluctuation Analysis of multifractal structures

TL;DR

This study compares the effectiveness of MFDFA and WTMM methods in detecting multifractal structures in artificial and real financial data, recommending MFDFA for general use due to its reliability and automation capabilities.

Contribution

The paper provides a comprehensive comparison of MFDFA and WTMM, highlighting the biases of WTMM and recommending MFDFA for more accurate and automatic multifractal analysis.

Findings

MFDFA is generally more reliable for unknown data properties.

WTMM can produce biased results, especially for fractional Brownian motion.

Both methods detect multifractality in stock market data, with WTMM indicating stronger multifractality.

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 Levy 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(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.