Disentangling Sources of Multifractality in Time Series

This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as...

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Main Authors: Robert Kluszczyński, Stanisław Drożdż, Jarosław Kwapień, Tomasz Stanisz, Marcin Wątorek
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Mathematics
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Online Access:https://www.mdpi.com/2227-7390/13/2/205
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author Robert Kluszczyński
Stanisław Drożdż
Jarosław Kwapień
Tomasz Stanisz
Marcin Wątorek
author_facet Robert Kluszczyński
Stanisław Drożdż
Jarosław Kwapień
Tomasz Stanisz
Marcin Wątorek
author_sort Robert Kluszczyński
collection DOAJ
description This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only when correlations are present. These issues are illustrated here using series generated by several model mathematical cascades, which by design build correlations into these series. The thickness of the tails of fluctuations in such series is then governed by an appropriate procedure of adjusting them to q-Gaussian distributions, and q is treated as a variable parameter that, while preserving correlations, allows for tuning these distributions to the desired functional form. Multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical method for quantifying multifractality, is then used to identify the influence of the thickness of the fluctuation tails in the presence of temporal correlations on the width of multifractal spectra. The obtained results point to the Gaussian distribution, so <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, as the appropriate reference distribution to evaluate the contribution of fatter tails to the width of multifractal spectra. An appropriate procedure is presented to make such estimates.
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spelling doaj-art-b767cacd522b4144a5c714309d5b88f62025-01-24T13:39:44ZengMDPI AGMathematics2227-73902025-01-0113220510.3390/math13020205Disentangling Sources of Multifractality in Time SeriesRobert Kluszczyński0Stanisław Drożdż1Jarosław Kwapień2Tomasz Stanisz3Marcin Wątorek4Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, PolandComplex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, PolandComplex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, PolandComplex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, PolandFaculty of Computer Science and Telecommunications, Cracow University of Technology, 31-155 Kraków, PolandThis contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only when correlations are present. These issues are illustrated here using series generated by several model mathematical cascades, which by design build correlations into these series. The thickness of the tails of fluctuations in such series is then governed by an appropriate procedure of adjusting them to q-Gaussian distributions, and q is treated as a variable parameter that, while preserving correlations, allows for tuning these distributions to the desired functional form. Multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical method for quantifying multifractality, is then used to identify the influence of the thickness of the fluctuation tails in the presence of temporal correlations on the width of multifractal spectra. The obtained results point to the Gaussian distribution, so <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, as the appropriate reference distribution to evaluate the contribution of fatter tails to the width of multifractal spectra. An appropriate procedure is presented to make such estimates.https://www.mdpi.com/2227-7390/13/2/205complexitytime series analysismathematical cascadesnonlinear correlationsmultifractalitysingularity spectra
spellingShingle Robert Kluszczyński
Stanisław Drożdż
Jarosław Kwapień
Tomasz Stanisz
Marcin Wątorek
Disentangling Sources of Multifractality in Time Series
Mathematics
complexity
time series analysis
mathematical cascades
nonlinear correlations
multifractality
singularity spectra
title Disentangling Sources of Multifractality in Time Series
title_full Disentangling Sources of Multifractality in Time Series
title_fullStr Disentangling Sources of Multifractality in Time Series
title_full_unstemmed Disentangling Sources of Multifractality in Time Series
title_short Disentangling Sources of Multifractality in Time Series
title_sort disentangling sources of multifractality in time series
topic complexity
time series analysis
mathematical cascades
nonlinear correlations
multifractality
singularity spectra
url https://www.mdpi.com/2227-7390/13/2/205
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AT tomaszstanisz disentanglingsourcesofmultifractalityintimeseries
AT marcinwatorek disentanglingsourcesofmultifractalityintimeseries