MFDFA’s documentation¶

MFDFA is a python implementation of Multifractal Detrended Fluctuation Analysis, first developed by by Peng et al. ¹ and later extended to study multifractality MFDFA by Kandelhardt et al. ².

Installation¶

For the moment the library is available from TestPyPI, so you can use

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

Then on your favourite editor just use

from MFDFA import MFDFA

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

$MFDFA of a fractional Ornstein–Uhlenbeck process$

Extensions¶

MFDFA as seen since its development a set of enhancements. In particular the usage of Empirical Mode Decomposition as a source of detrending, instead of polynomial fittings, which allows for a more precise removal of known trends in the timeseries.

Employing Empirical Mode Decompositions for detrending¶

Empirical Mode Decomposition (EMD), or maybe more correctly described, the Hilbert─Huang transform is a transformation analogous to a Fourier or Hilbert transform that decomposes a one-dimensional timeseries or signal into its Intrinsic Mode Functions (IMFs). For our purposes, we simply want to employ EMD to detrend a timeseries.

Understanding `MFDFA`’s `EMD` detrender¶

Take a timeseries y and extract the Intrinsic Mode Functions (IMFs)

# Import
from MFDFA import IMFs

# Extract the IMFs simply by employing
IMF = IMFs(y)

From here one obtains a (..., y.size). Best now to study the different IMFs is to plot them and the timeseries y

# Import
import matplotlib.pyplot as plt

# Plot the timeseries and the IMFs 6,7, and 8
plt.plot(X, color='black')
plt.plot(np.sum(IMF[[6,7,8],:], axis=0).T)

Using `MFDFA` with `EMD`¶

To now perform the multifractal detrended fluctuation analysis, simply insert the IMFs desired to be subtracted from the timeseries. This will also for order = 0, not to do any polynomial detrending.

# 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)

# Obtain the (MF)DFA by declaring the IMFs to subtract
# in a list in the dictionary of the extensions
lag, dfa = MFDFA(y, lag = lag, extensions = {"EMD": [6,7,8]})

Literature¶

¹ Peng, 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

² Kantelhardt, 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

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.