Series Multifractal Spectrum Analysis

Application Scenarios

The CFASeriesMFS class is used to calculate the multifractal spectrum of 1D time series, serving as an important tool for analyzing sequence complexity and long-range correlations. Main application scenarios include:

Financial Time Series: Analyze multifractal properties of stock prices and returns
Physiological Signal Analysis: Complexity analysis of ECG and EEG
Climate Data: Multi-scale features of temperature and precipitation sequences
Network Traffic: Self-similarity analysis of internet traffic
Seismic Data: Multifractal features of seismic waveforms

Usage Examples

Basic Usage

import numpy as np
from FreeAeonFractal.FASeriesMFS import CFASeriesMFS

# Generate random walk sequence
x = np.cumsum(np.random.randn(5000))

# Create multifractal spectrum analysis object
q_list = np.linspace(-5, 5, 21)
mfs = CFASeriesMFS(x, q_list=q_list)

# Calculate multifractal spectrum
df_mfs = mfs.get_mfs()

# View results
print(df_mfs)

# Visualize results
mfs.plot(df_mfs)

Custom Scale Parameters

from FreeAeonFractal.FASeriesMFS import CFASeriesMFS, recommended_lag

# Use recommended scale function
lag_list = recommended_lag(len(x), order=2, num_scales=40)

# Calculate multifractal spectrum
df_mfs = mfs.get_mfs(lag_list=lag_list, order=2)

Installation

pip install FreeAeon-Fractal

Class Description

CFASeriesMFS

Description: Class for calculating 1D time series multifractal spectrum based on MFDFA (Multifractal Detrended Fluctuation Analysis) method.

Initialization Parameters

Parameter	Type	Default	Description
`data`	numpy.ndarray	Required	Input time series (1D array)
`q_list`	array-like	linspace(-5,5,51)	q value list
`with_progress`	bool	True	Whether to show progress bar

Module-level Helper

recommended_lag(x_len, order=2, num_scales=40, s_min=None, s_max_ratio=0.25)

Description: Generate a recommended scale (lag) list for MFDFA.

Parameters:

x_len (int): Length of the input series
order (int): DFA polynomial order
num_scales (int): Number of scale points
s_min (int or None): Minimum scale (auto if None)
s_max_ratio (float): Maximum scale as fraction of series length (default 0.25)

Return Value: numpy array of integer lag values

Main Methods

1. get_mfs(lag_list=None, order=2)

Description: Calculate multifractal spectrum, including generalized Hurst exponent, mass exponent, singularity strength, and multifractal spectrum.

Parameters:

lag_list (array-like): Custom scale window list. If None, automatically generates recommended scales using recommended_lag
order (int): DFA polynomial fitting order, default is 2

Return Value (pandas.DataFrame):

Column	Description
`q`	q value
`h(q)`	Generalized Hurst exponent
`t(q)`	τ(q) mass exponent
`a(q)`	α(q) singularity strength
`f(a)`	f(α) multifractal spectrum
`d(q)`	D(q) generalized dimension

Example:

df_mfs = mfs.get_mfs()
print(df_mfs.head())
#    q      h(q)     t(q)     a(q)     f(a)     d(q)
# 0 -5.0   0.8234  -5.9170   0.9523   0.7445   1.9835
# 1 -4.0   0.7891  -4.1564   0.9012   0.7484   1.7854
# ...

2. plot(df_mfs)

Description: Visualize multifractal spectrum analysis results with 6 subplots:

H(q) vs q: Generalized Hurst exponent
τ(q) vs q: Mass exponent
D(q) vs q: Generalized dimension
α(q) vs q: Singularity strength
f(α) vs α: Multifractal spectrum
Overview: Normalized comparison of all indices

Parameters:

df_mfs (DataFrame): Results returned by get_mfs()

Theoretical Background

MFDFA Method

Multifractal Detrended Fluctuation Analysis (MFDFA) is a generalization of the DFA method with the following steps:

1. Cumulative Deviation

Y(i) = Σ[x(k) - x̄] (i = 1, 2, ..., N)

2. Segment Detrending

Divide Y(i) into non-overlapping windows of size s, fit polynomials in each window, and calculate variance.

3. Fluctuation Function

F²(s,v) = (1/s) Σ{Y[(v-1)s+i] - yᵥ(i)}²

4. q-order Fluctuation Function

Fq(s) = {(1/Ns) Σ[F²(s,v)]^(q/2)}^(1/q)

5. Scaling Law

Fq(s) ~ s^h(q)

Where h(q) is the generalized Hurst exponent.

Multifractal Parameters

1. Mass Exponent τ(q)

τ(q) = q·h(q) - 1

2. Singularity Strength α(q)

α(q) = dτ(q)/dq = h(q) + q·dh(q)/dq

3. Multifractal Spectrum f(α)

f(α) = q·α(q) - τ(q)

4. Generalized Dimension D(q)

D(q) = τ(q) / (q - 1), q ≠ 1

Important Notes

Sequence Length:
- Minimum length recommended ≥ 1000
- Longer sequences produce more stable results
- For short sequences, reduce number of scales
q Value Selection:
- Negative q: Sensitive to small fluctuations
- Positive q: Sensitive to large fluctuations
- Recommended range: -5 to 5
- Recommended points: 20-50
Scale Parameters:
- order=1: Suitable for trendless sequences
- order=2: Suitable for linear trends (recommended)
- order=3: Suitable for quadratic trends
- Use recommended_lag() for optimal scale selection
Result Interpretation:
- h(q) monotonically decreasing: Multifractal
- h(q) constant: Monofractal
- Δh > 0.1: Significant multifractality
- Wider f(α): Stronger heterogeneity
Performance Considerations:
- Long sequences compute slowly
- Reduce number of q values to speed up
- Use recommended_lag function for optimized scale selection

References

Kantelhardt, J. W., et al. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica A.
Peng, C. K., et al. (1994). Mosaic organization of DNA nucleotides. Physical Review E.