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  1. 2 Acta Physica Polonica A

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  1. 1 2013

  2. 1 2012

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  1. 2 Grech D.

  2. 2 Pamuła G.

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Multifractality of Nonlinear Transformations with Application in Finances

100%
Grech D., Pamuła G.
Acta Physica Polonica A
|
2013
|
vol. 123
|
issue 3
529-537
EN
We study the multifractal effects 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 difference is stressed between "naive" observed multifractal effects calculated directly within detrended multifractal analysis as the spread Δh of the generalized Hurst exponents h(q) and the more reliable unbiased multifractality received after subtraction of residual bias effects generated by nonlinear transformations of initial data and coupled with finite size effects in time series. This property is investigated for volatile series of the real main world financial indices. A difference 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 profile 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 finance are presented. They convince that an analysis directed only on investigation of the edges h^{±} in multifractal spectrum may be misleading.
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Multifractal Background Noise of Monofractal Signals

100%
Grech D., Pamuła G.
Acta Physica Polonica A
|
2012
|
vol. 121
|
issue 2B
B-34-B-39
EN
We investigate the presence of multifractal residual background effect for monofractal signals which appears due to the finite length of the signals and (or) due to the constant long memory the signals reveal. This phenomenon is investigated numerically within the multifractal detrended fluctuation analysis (MF-DFA) for artificially generated time series. Next, the analytical formulas enabling to describe the multifractal content in such signals are provided. Final results are shown in the frequently used generalized Hurst exponent h(q) multifractal scenario as a function of time series length L and the autocorrelation scaling exponent value γ. The obtained results may be significant in any practical application of multifractality, including financial data analysis, because the "true" multifractal effect should be clearly separated from the so called "multifractal noise" resulting from the finite data length. Examples from finance in this context are given. The provided formulas may help to decide whether one deals with the signal of real multifractal origin or not and make further step in analysis of the so called spurious or corrupted multifractality discussed in literature.
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