Fractality and Long-Range Correlations

Fractality is a characteristic of a complex system in which self-similarity at different scales can be found. Fractality quantifies dynamically fluctuating variability of systems through multi-scale analyses and provides insights into underlying structures of objects under study. Probably the most widely used methods to analyze fractality and long-range correlations are Detrended Fluctuation Analysis (DFA; Peng et al., 1994) and Multi-Fractal Detrended Fluctuation Analysis (MFDFA; Kantelhardt et al., 2002), which is an extension of DFA. For further explanation, please refer to this medium post or this Github page.

This library implement MFDFA and provides also funtions for represent of results.

Installation

You can install mfdfa-toolkit using pip:

pip install mfdfa-toolkit

or from the Github repository:

pip install git+https://github.com/mohsenim/Multifractality.git

Using mfdfa-toolkit

To apply MFDFA to a series:

from mfdfa_toolkit import MFDFA
import numpy as np

x = np.loadtxt('./examples/Henry-James_The-Golden-Bowl.txt')
result = MFDFA(x)
print(f"Degree of fractality: {result['H']:.2f}")
print(f"Degree of multifractality: {result['multifractality']:.2f}")
print(f"Goodness of fit(R2):{result['R2'].mean():.2f}")

Fluctuations, $F_q(s)$, can be visualized using visualization:

from mfdfa_toolkit import visualization
import matplotlib.pyplot as plt

fig, ax = plt.subplots(1)
visualization.plot_fluctuations(result, ax)

Similarly, the singularity spectrum can be visualized:

from mfdfa_toolkit import visualization
import matplotlib.pyplot as plt

fig, ax = plt.subplots(1)
visualization.plot_fluctuations(result, ax)

You can view MFDFA.ipynb for running the code step by step and seeing examples.

References

• Peng, C.-K., S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger (1994). “Mosaic organization of DNA nucleotides”. In: Physical Review E 49.2, pp. 1685–1689.
• Kantelhardt, JanW., Stephan A. Zschiegner, Eva Koscielny-Bunde, Shlomo Havlin, Armin Bunde, and H.Eugene Stanley (2002). “Multifractal detrended fluctuation analysis of nonstationary time series”. In: Physica A: Statistical Mechanics and Its Applications 316.1, pp. 87–114.
• Mohseni, Mahdi,Volker Gast, and ChristophRedies (2021). “Fractality andVariability in Canonical and Non-Canonical English Fiction and in Non-Fictional Texts”. In: Frontiers in Psychology 12, p. 920.