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¶

MFDFA is available from PyPI, so you can use

pip install MFDFA

Then on your favourite editor just use

from MFDFA import MFDFA

Warning

To use the extension to include Empirical Mode Decomposition detrending you will also need

pip install EMD-signal

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.unique(np.logspace(0.5, 3, 100, dtype=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
# in a double logaritmic scales, i.e., you need to
# fit the logs of the results
H_hat = np.polyfit(np.log(lag)[4:20],np.log(dfa[4:20]),1)[0]

print('Estimated H = '+'{:.3f}'.format(H_hat[0]))


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

$MFDFA of a fractional Ornstein–Uhlenbeck process$

Multifractality in one dimensional distributions¶

To show how multifractality can be studied, let us take a sample of random numbers of a symmetric Lévy distribution.

Univariate random numbers from a Lévy stable distribution¶

To obtain a sample of random numbers of Lévy stable distributions, use scipy’s levy_stable. In particular, take an $\alpha$-stable distribution, with $\alpha=1.5$

# Imports
from MFDFA import MFDFA
from scipy.stats import levy_stable

# Generate 100000 points
alpha = 1.5
X = levy_stable.rvs(alpha=alpha, beta = 0, size=10000)

For MFDFA to detect the multifractal spectrum of the data, we need to vary the parameter $q\in[-10,10]$ and exclude $0$. Let us also use a quadratic polynomial fitting by setting order=2

# Select a band of lags, which are ints
lag = np.unique(np.logspace(0.5, 3, 100).astype(int))

# Select a list of powers q
q_list = np.linspace(-10,10,41)
q_list = q_list[q_list!=0.0]

# The order of the polynomial fitting
order = 2

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

Again, we plot this in a double logarithmic scale, but now we include 6 curves, from 6 selected $q={-10,-5-2,2,5,10}$. Include as well are the theoretical curves for $q=-10$, with a slope of $1/\alpha=1/1.5$ and $q=10$, with a slope of $1/q=1/10$

Plot of the MFDFA of a Lévy stable distribution for a few q values.

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.

Warning

To use this feature, you need to first install PyEMD (EMD-signal) with

pip install EMD-signal

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]})

Extended Detrended Fluctuation Analysis¶

In the publication Detrended fluctuation analysis of cerebrovascular responses to abrupt changes in peripheral arterial pressure in rats. the authors introduce a new metric similar to the conventional Detrended Fluctuation Analysis (DFA) which they denote Extended Detrended Fluctuation Analysis (eDFA), which relies on extracting the difference of the minima and maxima for each segmentation of the data, granting a new power-law exponent to study, i.e., as in eq. (5) in the paper

\[\mathrm{d}F (n) = \mathrm{max}[F(n)] - \mathrm{min}[F(n)],\]

which in turn results in

\[\mathrm{d}F(n) \sim n^\beta.\]

Using `MFDFA`’s `eDFA` extension¶

To obtain the eDFA, simply set the extension to True and add a new output function, here denoted edfa

# 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, edfa = MFDFA(y, lag = lag, extensions = {'eDFA': True})

Moving window for segmentation¶

For short timeseries the segmentation of the data—especially for large lags—results in bad statistics, e.g. if a timeseries has 2048 datapoints and one wishes to study the flucutation analysis up to a lag of 512, only 4 segmentations of the data are possible for the lag 512. Instead one can use an moving window over the timeseries to obtain better statistics at large lags.

Using `MFDFA`’s `window` extension¶

To utilise a moving window one has to declare the moving windows step-size, i.e., the number of data points the window will move over the data. Say we wish to increase the statistics of the aforementioned example to include a moving window moving 32 steps (so one has 64 segments at a lag of 512)

# 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, edfa = MFDFA(y, lag = lag, extensions = {'window': 32})

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.

Table of Content¶

Installation¶

MFDFA is available from PyPI, so you can use

pip install MFDFA

Then on your favourite editor just use

from MFDFA import MFDFA

Warning

To use the extension to include Empirical Mode Decomposition detrending you will also need

pip install EMD-signal

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.unique(np.logspace(0.5, 3, 100, dtype=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
# in a double logaritmic scales, i.e., you need to
# fit the logs of the results
H_hat = np.polyfit(np.log(lag)[4:20],np.log(dfa[4:20]),1)[0]

print('Estimated H = '+'{:.3f}'.format(H_hat[0]))


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

$MFDFA of a fractional Ornstein–Uhlenbeck process$

Multifractality in one dimensional distributions¶

To show how multifractality can be studied, let us take a sample of random numbers of a symmetric Lévy distribution.

Univariate random numbers from a Lévy stable distribution¶

To obtain a sample of random numbers of Lévy stable distributions, use scipy’s levy_stable. In particular, take an $\alpha$-stable distribution, with $\alpha=1.5$

# Imports
from MFDFA import MFDFA
from scipy.stats import levy_stable

# Generate 100000 points
alpha = 1.5
X = levy_stable.rvs(alpha=alpha, beta = 0, size=10000)

For MFDFA to detect the multifractal spectrum of the data, we need to vary the parameter $q\in[-10,10]$ and exclude $0$. Let us also use a quadratic polynomial fitting by setting order=2

# Select a band of lags, which are ints
lag = np.unique(np.logspace(0.5, 3, 100).astype(int))

# Select a list of powers q
q_list = np.linspace(-10,10,41)
q_list = q_list[q_list!=0.0]

# The order of the polynomial fitting
order = 2

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

Again, we plot this in a double logarithmic scale, but now we include 6 curves, from 6 selected $q={-10,-5-2,2,5,10}$. Include as well are the theoretical curves for $q=-10$, with a slope of $1/\alpha=1/1.5$ and $q=10$, with a slope of $1/q=1/10$

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.

Warning

To use this feature, you need to first install PyEMD (EMD-signal) with

pip install EMD-signal

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]})

Extended Detrended Fluctuation Analysis¶

In the publication Detrended fluctuation analysis of cerebrovascular responses to abrupt changes in peripheral arterial pressure in rats. the authors introduce a new metric similar to the conventional Detrended Fluctuation Analysis (DFA) which they denote Extended Detrended Fluctuation Analysis (eDFA), which relies on extracting the difference of the minima and maxima for each segmentation of the data, granting a new power-law exponent to study, i.e., as in eq. (5) in the paper

\[\mathrm{d}F (n) = \mathrm{max}[F(n)] - \mathrm{min}[F(n)],\]

which in turn results in

\[\mathrm{d}F(n) \sim n^\beta.\]

Using `MFDFA`’s `eDFA` extension¶

To obtain the eDFA, simply set the extension to True and add a new output function, here denoted edfa

# 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, edfa = MFDFA(y, lag = lag, extensions = {'eDFA': True})

Moving window for segmentation¶

For short timeseries the segmentation of the data—especially for large lags—results in bad statistics, e.g. if a timeseries has 2048 datapoints and one wishes to study the flucutation analysis up to a lag of 512, only 4 segmentations of the data are possible for the lag 512. Instead one can use an moving window over the timeseries to obtain better statistics at large lags.

Using `MFDFA`’s `window` extension¶

To utilise a moving window one has to declare the moving windows step-size, i.e., the number of data points the window will move over the data. Say we wish to increase the statistics of the aforementioned example to include a moving window moving 32 steps (so one has 64 segments at a lag of 512)

# 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, edfa = MFDFA(y, lag = lag, extensions = {'window': 32})

Functions¶

MFDFA¶

MFDFA.MFDFA.MFDFA(timeseries: numpy.ndarray, lag: numpy.ndarray, order: int = 1, q: numpy.ndarray = 2, stat: bool = False, modified: bool = False, extensions: dict = {'EMD': False, 'eDFA': False, 'window': False}) → Tuple[numpy.array, numpy.ndarray]¶

Multifractal Detrended Fluctuation Analysis of timeseries. MFDFA generates a fluctuation function F²(q,s), with s the segment size and q the q-powers, Take a timeseries Xₜ, find the integral Yₜ = cumsum(Xₜ), and segment the timeseries into Nₛ segments of size s.

\[F^2(v,s) = \dfrac{1}{s} \sum_{i=1}^s [Y_{(v-1)s + i} - y_{v,i}]^2, ~\mathrm{for}~v=1,2, \dots, N_s,\]

with $y_{v,i}$ the polynomial fittings of order m. Having obtained the variances of each (detrended) segment, average over s and increase s, to obtain the fluctuation function $F_q^2(s)$ depending on the segment length.

\[F_q^2(s) = \Bigg\{\dfrac{1}{N_s} \sum_{v=1}^{N_s} [F^2(v,s)]^{q/2}\Bigg\}^{1/q}\]

The fluctuation function $F_q^2(s)$ can now be plotted in a log-log scale, the slope of the fluctuation function $F_q^2(s)$ vs the s-segment size is the self-similarity scaling $h(q)$

\[F_q^2(s) \sim s^{h(q)}.\]

If $H ≈ 0$ in a monofractal series, use a second integration step by setting modified = True.

Parameters:

timeseries (np.ndarray) – A 1-dimensional timeseries (N, 1). The timeseries of length N.
lag (np.ndarray of ints) – An array with the window sizes to calculate (ints). Notice min(lag) > order + 1 given a polynomial fit of order m needs at least m points. The results are meaningless for ‘order = m’ and for lag > size of data / 4 since there is low statistics with < 4 windows to divide the timeseries.
order (int (default 1)) – The order of the polynomials to approximate. order = 1 is the DFA1, which is a least-square fit of the data with a first order polynomial (a line), order = 2 is a second-order polynomial, etc.. order = 0 skips the detrending process and hence gives the non-detrended fluctuation functions, i.e., simply Fluctuation Analysis.
q (np.ndarray (default 2)) – Fractal exponent to calculate. Array in [-10,10]. The values = 0 will be removed, since the code does not converge there. q = 2 is the standard Detrended Fluctuation Analysis as is set a default.
stat (bool (default False)) – Calculates the standard deviation associated with each segment’s averaging.
modified (bool (default False)) – For data with the Hurst index ≈ 0, i.e., strongly anticorrelated, a standard MFDFA will result in inacurate results, thus a further integration of the timeseries yields a modified scaling coefficient.
extensions (dict) –
- EMD: list (default False)
  
  If not None, requires a list of indices of the user-chosen IMFs obtained from an (externally performed) EMD analysis. The indexing starts from 0. Will enforce order = 0 since there is no need for a polynomial detrending.
- eDFA: bool (default False)
  
  A method to evaluate the strength of multifractality. Calls function eDFA().
- window: bool (default False)
  
  A moving window for smaller timeseries. Set window as int > 0 with the number of steps the window shoud move over the data. window = 1 will move window by 1 step. Since the timeseries is segmented at each lag lenght, any window choise > lag is only segmented once.

Returns:

lag (np.ndarray of ints) – Array of lags, realigned, preserving only different lags and with entries > order + 1
f (np.ndarray) – A array of shape (size(lag),size(q)) of variances over the indicated lag windows and the indicated q-fractal powers.

References

[Peng1994]

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger. “Mosaic organization of DNA nucleotides.” Phys. Rev. E, 49(2), 1685–1689, 1994.

[Kantelhardt2002]

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H. E. Stanley. “Multifractal detrended fluctuation analysis of nonstationary time series.” Physica A, 316(1-4), 87–114, 2002.

MFDFA.MFDFA.eDFA(F: numpy.ndarray) → numpy.ndarray¶

In the reference indicated below a measure of nonstationarity was added by including a subsequent calculation of the extrema of the DFA. Denoted $dF_q^2(s)$ the difference of the extrema at each segment, i.e.,

\[dF_q^2(s) = \max[F_q^2(s)] - \min[F_q^2(s)]\]

Parameters:	F (np.ndarray) – Fluctuation function given by the MFDFA().
Returns:	res – Difference of max and min.
Return type:	np.ndarray

Notes

New in version 0.3.

References

[Pavlov2020]

A. N. Pavlov, A. S. Abdurashitov, A. A. Koronovskii Jr., O. N. Pavlova, O. V. Semyachkina-Glushkovskaya, and J. Kurths. “Detrended fluctuation analysis of cerebrovascular responses to abrupt changes in peripheral arterial pressure in rats.” CNSNS 85, 105232, 2020

Empirical Mode Decomposition for detrending timeseries¶

MFDFA.emddetrender.detrendedtimeseries(timeseries: numpy.ndarray, modes: list) → numpy.ndarray¶

The function calculates the Intrinsic Mode Functions (IMFs) of a given timeseries, subtracts the user-chosen IMFs for detrending, and returns the detrended timeseries. Based on based on Dawid Laszuk’s PyEMD found at https://github.com/laszukdawid/PyEMD

Parameters:	timeseries (np.ndarray) – A 1-dimensional timeseries of length N. modes (list) – List of integers indicating the indices of the IMFs to be subtracted/detrended from the timeseries.
Returns:	detrendedTimeseries – Detrended 1-dimensional timeseries.
Return type:	np.ndarray

Warning

To use the extension Empirical Mode Decomposition for detrending, the pyEMD library is needed.

pip install EMD-signal

Notes

New in version 0.3.

References

[Huang1998]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non stationary time series analysis”, Proc. Royal Soc. London A, Vol. 454, pp. 903-995, 1998.

[Rilling2003]

G. Rilling, P. Flandrin, and P. Goncalves, “On Empirical Mode Decomposition and its algorithms”, IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, Grado (I), June 2003.

MFDFA.emddetrender.IMFs(timeseries: numpy.ndarray) → numpy.ndarray¶

Extract the Intrinsic Mode Functions (IMFs) of a given timeseries.

Parameters:	timeseries (np.ndarray) – A 1-dimensional timeseries of length N.

Notes

New in version 0.3.

Returns:	IMFs – The Intrinsic Mode Functions (IMFs) of the Empirical Mode Decomposition. These are of shape (…, timeseries.size), with the first dimension varying depending on the data. Last entry is the residuals.
Return type:	np.ndarray

Fractional Gaussian noise¶

MFDFA.fgn.fgn(N: int, H: float) → numpy.ndarray¶

Generates fractional Gaussian noise with a Hurst index H in (0,1). If H = 1/2 this is simply Gaussian noise. The current method employed is the Davies–Harte method, which fails for H ≈ 0. A Cholesky decomposition method and the Hosking’s method will be implemented in later versions.

Parameters:	N (int) – Size of fractional Gaussian noise to generate. H (float) – Hurst exponent H in (0,1).
Returns:	f – A array of size N of fractional Gaussian noise with a Hurst index H.
Return type:	np.ndarray

License¶

MIT License

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Contact¶

If you need help with something, find a bug, issue, or typo on the repository or in the code, you can contact me here: leonardo.rydin@gmail.com or open an issue on the GitHub repository.

MFDFA’s documentation¶

Installation¶

An exemplary one-dimensional fractional Ornstein–Uhlenbeck process¶

Generating a fractional Ornstein–Uhlenbeck process¶

Using the MFDFA¶

Multifractality in one dimensional distributions¶

Univariate random numbers from a Lévy stable distribution¶

Extensions¶

Employing Empirical Mode Decompositions for detrending¶

Understanding MFDFA’s EMD detrender¶

Using MFDFA with EMD¶

Extended Detrended Fluctuation Analysis¶

Using MFDFA’s eDFA extension¶

Moving window for segmentation¶

Using MFDFA’s window extension¶

Literature¶

Funding¶

Table of Content¶

Installation¶

An exemplary one-dimensional fractional Ornstein–Uhlenbeck process¶

Generating a fractional Ornstein–Uhlenbeck process¶

Using the MFDFA¶

Multifractality in one dimensional distributions¶

Univariate random numbers from a Lévy stable distribution¶

Extensions¶

Employing Empirical Mode Decompositions for detrending¶

Understanding MFDFA’s EMD detrender¶

Using MFDFA with EMD¶

Extended Detrended Fluctuation Analysis¶

Using MFDFA’s eDFA extension¶

Moving window for segmentation¶

Using MFDFA’s window extension¶

Functions¶

MFDFA¶

Empirical Mode Decomposition for detrending timeseries¶

Fractional Gaussian noise¶

License¶

Contact¶

Using the `MFDFA`¶

Understanding `MFDFA`’s `EMD` detrender¶

Using `MFDFA` with `EMD`¶

Using `MFDFA`’s `eDFA` extension¶

Using `MFDFA`’s `window` extension¶

Using the `MFDFA`¶

Understanding `MFDFA`’s `EMD` detrender¶

Using `MFDFA` with `EMD`¶

Using `MFDFA`’s `eDFA` extension¶

Using `MFDFA`’s `window` extension¶