[Paper Review] On Estimation of Hurst Scaling Exponent through Discrete Wavelets

P. Manimaran, Prasanta K. Panigrahi+1 more|ArXiv.org|Apr 1, 2006

Complex Systems and Time Series Analysis1 references3 citations

TL;DR

This paper proposes a discrete wavelet-based method to estimate the Hurst scaling exponent in non-stationary time series, using high-pass wavelet coefficients from the Daubechies family to extract scale-dependent fluctuations. It demonstrates that discrete wavelets provide more accurate and independent fluctuation estimates than continuous wavelets, yielding reliable Hurst exponent values without overestimation of power in fluctuations.

ABSTRACT

We study the scaling behavior of the fluctuations, as extracted through wavelet coefficients based on discrete wavelets. The analysis is carried out on a variety of physical data sets, as well as Gaussian white noise and binomial multi-fractal model time series and the results are compared with continuous wavelet based average wavelet coefficient method. It is found that high-pass coefficients of wavelets, belonging to the Daubechies family are quite good in estimating the true power in the fluctuations in a non-stationary time series. Hence, the fluctuation functions based on discrete wavelet coefficients find the Hurst scaling exponents accurately.

Motivation & Objective

• To evaluate the efficacy of discrete wavelet transforms in estimating the Hurst scaling exponent for self-similar, non-stationary time series.
• To compare the performance of discrete wavelet-based fluctuation functions with the continuous wavelet average wavelet coefficient method.
• To address the overestimation of fluctuation power inherent in continuous wavelets due to their overcomplete basis.
• To validate the method on diverse data sets, including physical plasma data, financial indices, and synthetic models like Gaussian white noise and binomial multifractals.
• To establish discrete wavelets as a superior alternative for accurate Hurst exponent estimation in fluctuation analysis.

Proposed method

• The method uses the discrete wavelet transform (DWT) to decompose time series into approximation and detail coefficients at multiple scales.
• The profile of the time series is computed as the cumulative sum after subtracting the mean, following standard multifractal analysis procedures.
• Fluctuation functions F(s) are computed from high-pass wavelet coefficients (detail coefficients) at each scale s, capturing local variations.
• The Hurst exponent H is estimated from the slope of the log-log plot of F(s) versus scale s, using F(s) ∼ s^H.
• The approach leverages the orthonormality of discrete wavelets (e.g., Daubechies) to ensure statistical independence of fluctuations across scales.
• The method is validated against continuous wavelet-based average wavelet coefficient method and analytical values for synthetic models.

Experimental results

Research questions

• RQ1 Does the discrete wavelet-based fluctuation function provide a more accurate estimate of the Hurst exponent than the continuous wavelet-based average wavelet coefficient method?
• RQ2 How do discrete wavelet coefficients compare to continuous wavelet coefficients in estimating the true power of fluctuations in non-stationary time series?
• RQ3 To what extent does the overcomplete basis of continuous wavelets lead to overestimation of fluctuation power and biased Hurst exponent estimates?
• RQ4 Can the discrete wavelet method reliably estimate H for complex real-world data such as tokamak plasma fluctuations and financial time series?
• RQ5 How do the results from discrete wavelets compare to analytical values for Gaussian white noise and binomial multifractal models?

Key findings

• The discrete wavelet method yields more accurate Hurst exponent estimates than the continuous wavelet average wavelet coefficient method, which tends to overestimate fluctuation power due to basis overcompleteness.
• For Gaussian white noise, the discrete wavelet method estimates H = 0.5077, close to the theoretical value of 0.5, while the continuous method yields H = 0.5163.
• For the binomial multifractal model, the discrete method gives H_d = 0.8421, closely matching the analytical value H_a = 0.8390.
• The method successfully detects long-range correlations (H > 0.5) in tokamak plasma data (IC: H = 0.585, ISC: H = 0.554, FP: H = 0.549) and financial indices (NASDAQ: H = 0.553, BSE: H = 0.548).
• Random matrix ensembles show anti-persistent behavior: GOE (H = 0.095), GSE (H = 0.143), GUE (H = 0.107), and uncorrelated behavior in GDE (H = 0.495), consistent with theoretical expectations.
• The discrete wavelet approach ensures independent fluctuations across scales due to its orthonormal basis, avoiding the bias introduced by redundant continuous wavelet representations.