whittlehurst 0.5

Hurst exponent estimation usding Whittle's method

README

Overview

This module provides an implementation of Whittle's likelihood method to estimate the Hurst exponent of a time series. The method fits a theoretical spectral density model to the periodogram of a time series. This implementation supports multiple spectral density approximations for fractional Gaussian noise (increments of fractional Brownian motion) and ARFIMA processes.

Features

• Estimate the Hurst exponent (H) by minimizing the Whittle likelihood function.
• Spectral density options:
  • fGn
  • arfima
  • fGn_paxson
  • fGn_truncation
  • fGn_taylor
• Flexible interface with an option for a custom spectral density callback.
• Included generators for fBm, and ARFIMA:

Installation

pip install whittlehurst

Usage

fBM or fGn

import numpy as np
from whittlehurst import whittle, fbm

# Original Hurst value to test with
H=0.42

# Generate an fBm realization
fBm_seq = fbm(H=H, n=10000)

# Calculate the increments (the estimator works with the fGn spectrum)
fGn_seq = np.diff(fBm_seq)

# Estimate the Hurst exponent
H_est = whittle(fGn_seq)

print(f"Original H: {H:0.04f}, estimated H: {H_est:0.04f}")

ARFIMA

import numpy as np
from whittlehurst import whittle, arfima

# Original Hurst value to test with
H=0.42

# Generate an ARFIMA(0, H - 0.5, 0) realization
arfima_seq = arfima(H=H, n=10000)

# No need to take the increments here
# Estimate the "Hurst exponent"
H_est = whittle(arfima_seq, spectrum="arfima")

print(f"Original H: {H:0.04f}, estimated H: {H_est:0.04f}")

Notes

• The default recommended spectral model is fGn which relies on Hurwitz's zeta function.
• fGn_paxson, fGn_truncation, fGn_taylor are experimental approximations of the fGn spectrum.
• For models fGn_paxson and fGn_truncation, the parameter K is configurable (defaults: 50 and 200 respectively).
• A custom spectral density function may be provided via the spectrum_callback parameter.

References

The initial implementation of Whittle's method was based on:
https://github.com/JFBazille/ICode/blob/master/ICode/estimators/whittle.py

For details on spectral density models for fractional Gaussian noise, refer to:
https://onlinelibrary.wiley.com/doi/full/10.1111/jtsa.12750

License

This project is licensed under the MIT License (c) 2025 Bálint Csanády, aielte-research. See the LICENSE file for details.