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A Note on Wavelet-Based Estimator of the Hurst Parameter.

Liang Wu1

  • 1Center of Statistical Research, School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary

This study investigates empirical bias in wavelet-based Hurst parameter estimation for Fractal Brownian Motion (FBM). Findings indicate bias stems from discrete sampling errors, not simulation methods, offering insights for signal analysis.

Keywords:
Hurst parameterfractional Brownian motionlong-range dependencewavelet analysis

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Area of Science:

  • Signal Processing
  • Time Series Analysis
  • Statistical Modeling

Background:

  • Signals often exhibit scaling behaviors like long-range dependence and self-similarity, quantified by the Hurst parameter (H).
  • Fractal Brownian Motion (FBM) is crucial for modeling such self-similar signals.
  • Wavelet analysis is a standard signal processing technique, frequently employed for Hurst parameter estimation.

Purpose of the Study:

  • To conduct a detailed numerical simulation study on parameter selection for wavelet-based Hurst parameter estimation in FBM.
  • To comprehensively investigate the empirical bias of wavelet-based estimators, a factor often overlooked.
  • To identify the sources of empirical bias and their relationship with simulation methods and parameter choices.

Main Methods:

  • Detailed numerical simulations of Fractal Brownian Motion (FBM).
  • Application of wavelet analysis for Hurst parameter estimation.
  • Comparative analysis of different estimators and simulation methodologies.
  • Investigation of initialization errors due to discrete sampling.

Main Results:

  • Empirical bias in wavelet-based Hurst parameter estimation is primarily caused by initialization errors from discrete sampling.
  • The choice of simulation method does not significantly influence the observed empirical bias.
  • Selecting an appropriate orthogonal compact supported wavelet minimizes bias related to wavelet coefficient correlations.

Conclusions:

  • Discrete sampling is the main source of empirical bias in wavelet-based Hurst parameter estimation for FBM.
  • Simulation methods are not the cause of this bias.
  • Careful wavelet selection can mitigate bias arising from coefficient correlations, providing a reliable reference for signal scaling behavior analysis.