Multifractal Detrended Fluctuation Analysis

The MFDFA library supports several enhancements to the standard Multifractal Detrended Fluctuation Analysis. Key extensions include:

• Empirical Mode Decomposition (EMD) Detrending: Uses EMD as a source of detrending instead of traditional polynomial fittings. This allows for more precise removal of known trends in a time series.
• Extended DFA: Advanced variations of the Detrended Fluctuation Analysis.
• Moving Window Analysis: Implementation of window-based analysis techniques.

When analyzing short time series, large lags can result in poor statistics because there are too few possible segments (e.g., a lag of 512 on a 2048-point series only allows 4 segments). To improve statistical reliability at large lags, you can use the window extension in the MFDFA function.

To implement this, pass a dictionary to the extensions parameter where the key 'window' is set to the desired step-size (the number of data points the window moves over the data).

# Example: Using a moving window with a step-size of 32
# to increase statistics for large lags
lag, dfa, edfa = MFDFA(y, lag=lag, extensions={'window': 32})

To perform multifractal detrended fluctuation analysis using EMD, pass the desired IMF indices to the extensions dictionary in the MFDFA function.

When you provide IMFs via the extensions dictionary (e.g., extensions = {"EMD": [6,7,8]}), the library subtracts those specific IMFs from the timeseries. This effectively sets the polynomial detrending order to 0, as the trend is handled by the EMD components instead of polynomial fitting.

import numpy as np
from MFDFA import MFDFA

# Define the lag range (e.g., from small to long segments)
lag = np.logspace(0.7, 4, 30).astype(int)

# Perform (MF)DFA by declaring which IMFs to subtract via the 'extensions' dictionary
lag, dfa = MFDFA(y, lag=lag, extensions={"EMD": [6,7,8]})

To estimate the Hurst exponent ($H$) from the MFDFA output, you must fit a line to the results on a log-log scale.

Note that because the plot is double-logarithmic, you must fit the logs of the lag and dfa values. The slope of this fit relates to the Hurst exponent as: $\text{slope} = H + 1$ (or similar depending on the specific scaling implementation, but the example uses H_hat[0] as the slope).

import numpy as np
import matplotlib.pyplot as plt

# Assuming 'lag' and 'dfa' are obtained from MFDFA()
plt.loglog(lag, dfa, 'o')

# Fit the line to the logs of the results
# Using a slice (e.g., [4:20]) to avoid edge effects at very small/large scales
H_hat = np.polyfit(np.log(lag)[4:20], np.log(dfa[4:20]), 1)[0]

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

To use the library in your Python scripts, import the MFDFA class and the fgn utility (used for generating fractional Gaussian noise):

from MFDFA import MFDFA
from MFDFA import fgn

You can install the MFDFA library directly from PyPI using pip.

If you intend to use the extension for Empirical Mode Decomposition (EMD) detrending, you must also install the EMD-signal package.

pip install MFDFA

# Required if using Empirical Mode Decomposition detrending
pip install EMD-signal

Extended Detrended Fluctuation Analysis (eDFA) is a metric that calculates the difference between the maximum and minimum values for each data segmentation: $\mathrm{d}F (n) = \mathrm{max}[F(n)] - \mathrm{min}[F(n)]$. This results in a power-law exponent $\mathrm{d}F(n) \sim n^\beta$.

To use eDFA with the MFDFA library, pass a dictionary to the extensions parameter where the key 'eDFA' is set to True. When this extension is enabled, the MFDFA function will return an additional output value representing the eDFA results.

# 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})

To use Empirical Mode Decomposition (EMD) for detrending in MFDFA, you must install the EMD-signal package via pip.

pip install EMD-signal

The Hurst exponent $H$ can be estimated by analyzing the relationship between the lag and the fluctuation function on a log-log scale.

In a double logarithmic scale, the slope of the line is equal to $H + 1$. To find this slope, fit a first-order polynomial to the logarithms of the lag and dfa values using numpy.polyfit.

Note: It is common practice to exclude the very smallest lags (e.g., using a slice like [4:20]) to avoid instabilities in the estimation.

import numpy as np
import matplotlib.pyplot as plt

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

# Fit the line to find the slope in double logarithmic scales
# The slope in log-log space is H + 1
H_hat = np.polyfit(np.log(lag)[4:20], np.log(dfa[4:20]), 1)[0]

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

Install the MFDFA library using pip:

pip install MFDFA