Multifractality of Nonlinear Transformations with Application in Finances

Abstract

We study the multifractal eects of nonlinear transformations of monofractal, stationary time series and apply the found results to measure the true unbiased multifractality generated only by multiscaling properties of initial (primary) data before transformations. A dierence is stressed between naive observed multifractal eects calculated directly within detrended multifractal analysis as the spread ∆h of the generalized Hurst exponents h(q) and the more reliable unbiased multifractality received after subtraction of residual bias eects generated by nonlinear transformations of initial data and coupled with nite size eects in time series. This property is investigated for volatile series of the real main world nancial indices. A dierence between multifractal properties of intraday and interday quotes is also pointed out in this context for the Warsaw Stock Exchange WIG index.Finally, based on the observed feature of real nonstationary data, a new measure of unbiased multifractality in signals is introduced. This measure comes from an analysis of the whole generalized Hurst exponent prole instead of looking just at its edge behavior h ± ≡ h(q → ±∞). Such an approach seems to be particularly useful when h(q) is not a monotonic function of the moment order q. Interesting examples with extreme events from nance are presented. They convince that an analysis directed only on investigation of the edges h ± in multifractal spectrum may be misleading.

On the multifractal effects generated by monofractal signals

“…We discuss in this section the problem of integration and differentiation of data, both meant in a discrete manner. The case of basic nonlinear transformations is considered in the following section, while an extension to more complex nonlinear transformations is treated numerically in our separate publication [65].…”

Multifractal analysis of financial markets

“…where the linear correlation component ∆α LM can be determined by IAAFT surrogate time series of the raw time series. Note that the concepts of efficient multifractality ∆α eff and linear correlation component ∆α LM are essentially equivalent to observed multifractality and multifractal bias studied by Grech and Pamuła [942].…”

MFDFA: Efficient Multifractal Detrended Fluctuation Analysis in Python

“…[18]. MFDFA has found application in various fields, such as the analysis of heartbeat rate [19], arterial pressure [10], EEG sleep data [11,13], physiology [20], keystroke time series from Parkinson's disease patients [21], cosmic microwave radiation [22,23], seismic activity [24,25], sunspot activity [26], atmospheric scintillation [27], temperature variability [28], meteorology [29], precipitation levels [30], streamflow and sediment movement [7,[31][32][33][34][35][36], protein folding [37], finance and econophysics [38][39][40][41][42], electricity prices [43,44], power-grid frequency [45,46], epidemiology [47], music [48][49][50], ethology [51,52], multifractal harmonic signals [53], and microrheology [54].…”