MFDFA

Multifractal Detrended Fluctuation Analysis MFDFA is a model-independent method to uncover the self-similarity of a stochastic process or auto-regressive model. DFA was first developed by Peng et al.¹ and later extended to study multifractality MFDFA by Kandelhardt et al.².

Installation

For now MFDFA is on the test PyPI system, so you can install it via

pip install -i https://test.pypi.org/simple/ MFDFA

On your favourite editor simply import MFDFA as

from MFDFA import MFDFA

There is an added library fgn to generate fractional Gaussian noise.

The MFDFA library

The MFDFA library is a standalone numpy-dependent only library to study self-similarity and factality in stochastic processes. It is designed take advantage of numpy's polynomial package, thus scales well with several CPUs.

Employing the MFDFA library

An exemplary one-dimensional fractional Ornstein–Uhlenbeck process

For a more detailed explanation on how to integrate an Ornstein–Uhlenbeck process, see the kramersmoyal's package. You can also follow the fOU.ipynb

Generating a fractional Ornstein–Uhlenbeck process

This is one method of generating a (fractional) Ornstein–Uhlenbeck process with H=0.7, employing a simple Euler–Maruyama integration method

# Imports
from MFDFA import MFDFA
from MFDFA import fgn
# where this second library is to generate fractional Gaussian noises

# integration time and time sampling
t_final = 500
delta_t = 0.001

# Some drift theta and diffusion sigma parameters
theta = 0.3
sigma = 0.1

# The time array of the trajectory
time = np.arange(0, t_final, delta_t)

# The fractional Gaussian noise
H = 0.7
dB = (t_final ** H) * fgn(N = time.size, H = H)

# Initialise the array y
y = np.zeros([time.size])

# Integrate the process
for i in range(1, time.size):
    y[i] = y[i-1] - theta * y[i-1] * delta_t + sigma * dB[i]

And now you have a fractional process with a self-similarity exponent H=0.7

Using the MFDFA

To now utilise the MFDFA, we take this exemplary process and run the (multifractal) detrended fluctuation analysis. For now lets consider only the monofractal case, so we need only q=2.

# Select a band of lags, which usually ranges from
# very small segments of data, to very long ones, as
lag = np.logspace(0.7, 4, 30).astype(int)
# Notice these must be ints, since these will segment
# the data into chucks of lag size

# Select the power q
q = 2

# The order of the polynomial fitting
order = 1

# Obtain the (MF)DFA as
lag, dfa = MFDFA(y, lag = lag, q = q, order = order)

Now we need to visualise the results, which can be understood in a log-log scale. To find H we need to fit a line to the results in the log-log plot

# To uncover the Hurst index, lets get some log-log plots
plt.loglog(lag, dfa, 'o', label='fOU: MFDFA q=2')

# And now we need to fit the line to find the slope. We will
# fit the first points, since the results are more accurate
# there. Don't forget that if you are seeing in log-log
# scales, you need to fit the logs of the results
np.polyfit(np.log(lag[:15]), np.log(dfa[:15]),1)[0]

# Now what you should obtain is: slope = H + 1

Uncovering multifractality in stochastic processes

MFDFA, as an extension to DFA, was developed to uncover if a given process is mono or multi fractal. Let Xₜ be a multi fractal stochastic process. This mean Xₜ scales with some function alpha(t) as Xcₜ = |c|alpha(t) Xₜ. With the help of taking different powers variations of the DFA, one we can distinguish monofractal and multifractal processes.

An example of multifractal behaviour

To be continued

Changelog

• Version 0.2 - Removed experimental features. Added documentation
• Version 0.1 - Uploaded initial working code

Contributions

I welcome reviews and ideas from everyone. If you want to share your ideas or report a bug, open an issue here on GitHub, or contact me directly. If you need help with the code, the theory, or the implementation, do not hesitate to reach out, I am here to help. This package abides to a Conduct of Fairness.

Literature and Support

Submission history

This library has been submitted for publication at The Journal of Open Source Software in December 2019. It was rejected. The review process can be found here on GitHub. The plan is to extend the library and find another publisher.

History

This project was started in 2019 at the Department of Mathematics, University of Oslo in the Risk and Stochastics section by Leonardo Rydin Gorjão and is supported by Dirk Witthaut and the Institute of Energy and Climate Research Systems Analysis and Technology Evaluation. I'm very thankful to all the folk in Section 3 in the Department of Mathematics, University of Oslo, for helping me getting around the world of stochastic processes: Dennis, Anton, Michele, Fabian, Marc, Prof. Benth and Prof. di Nunno.

Funding

Helmholtz Association Initiative Energy System 2050 - A Contribution of the Research Field Energy and the grant No. VH-NG-1025, STORM - Stochastics for Time-Space Risk Models project of the Research Council of Norway (RCN) No. 274410, and the E-ON Stipendienfonds.

References

¹Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., & Goldberger, A. L. (1994). Mosaic organization of DNA nucleotides. Physical Review E, 49(2), 1685–1689
²Kantelhardt, J. W., Zschiegner, S. A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H. E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and Its Applications, 316(1-4), 87–114