Multifractality and heteroscedastic dynamics: An application to time series analysis

Abstract

An increasingly important problem in physics concerns scale invariance symmetry in diverse complex systems, often characterized by heteroscedastic dynamics. We investigate the nature of the relationship between the heteroscedastic and fractal aspects of the dynamics of complex systems, by analyzing the sensitivity to heteroscedasticity of the scaling properties of weakly nonstationary time series. By using multifractal detrended fluctuation analysis, we study the singularity spectra of currency exchange rate fluctuations, after partially or completely eliminating n-point correlations via data shuffling techniques. We conclude that heteroscedasticity can significantly increase multifractality and interpret these findings in the context of selforganizing and adaptive complex systems.

Fat tails, long-range correlations and multifractality as emergent properties in nonstationary time series

“…This result shows the direct relation between multifractality and heteroscedasticity [15]. This is relevant in the context of self-organizing, adaptive and evolutionary phenomena (see ref.…”

“…This is relevant in the context of self-organizing, adaptive and evolutionary phenomena (see ref. [15]). It is possible that heteroscedastic behavior in the dynamics of a system might bring advantage to it.…”

“…Volatility clustering is a stylized fact observed in finance (see ref. [15]) and can be direct related to the heteroscedastic dynamics of the return. Here we restrict the heteroscedastic analysis to the first and the second moment fluctuations of the signal…”

Turing bifurcation in a reaction–diffusion system with density-dependent dispersal

“…In Ref. [21] the dynamics of interacting particle systems governed by such density-dependent diffusion was studied using the Hurst exponent formalism [33][34][35][36][37]. The Hurst exponent H(q) is defined as…”

Lévy sections vs. partial sums of heteroscedastic time series

“…Fluctuation phenomena in such systems often do not follow Gaussian, Poisson or similar statistics, e.g., the dynamics of financial markets [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Some open questions: i) the origin of fattailed distributions (see [11] and references therein), ii) the multifractal properties of heteroscedastic signals [19] and iii) non-convergence or ultra-slow convergence to the Gaussian regime [19,20]. The latter led to the idea of Lévy flights by Mandelbrot and later to the idea of truncated Lévy flights [21] by Mantegna and Stanley.…”