[Paper Review] Wavelet-based Estimator for the Hurst Parameters of Fractional Brownian Sheet

Liang Wu, Yiming Ding|arXiv (Cornell University)|Feb 3, 2015

Complex Systems and Time Series Analysis20 references3 citations

TL;DR

This paper proposes a wavelet-based two-step generalized least squares estimator for the Hurst parameters of fractional Brownian sheets in multidimensional settings. By leveraging wavelet decomposition and regression on log-variance of coefficients, the method achieves asymptotic normality and improved efficiency, with low bias for $H_i \geq 0.5$ and moderate bias for $H_i < 0.5$, validated through simulations in 2D and 3D.

ABSTRACT

It is proposed a class of statistical estimators $\hat H =(\hat H_1, \ldots, \hat H_d)$ for the Hurst parameters $H=(H_1, \ldots, H_d)$ of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when $H_i \geq 1/2$, the estimators are efficient, and when $H_i &lt; 1/2$, there are some bias.

Motivation & Objective

• To develop a robust statistical estimator for the Hurst parameters of multidimensional fractional Brownian sheets.
• To address the challenge of estimating anisotropic long-range dependence and self-similarity in multi-dimensional signals.
• To improve estimation accuracy and reduce variance compared to ordinary least squares (OLS) estimators.
• To validate the estimator’s performance on synthetic fractional Brownian sheet data in 2D and 3D.
• To provide a computationally efficient and scalable method applicable to texture classification and diffusion tensor imaging (DTI) in NMR.

Proposed method

• The method uses multi-dimensional wavelet analysis to decompose the fractional Brownian sheet into wavelet coefficients.
• It applies regression on the log-variance of wavelet coefficients to estimate the Hurst parameters.
• A two-step generalized least squares (GLS) procedure is implemented to minimize estimator variance, improving efficiency over OLS.
• The wavelet coefficients' properties are derived under the fBs model, with assumptions on stationarity and scaling behavior.
• The asymptotic normality of the estimator is proven using a sequence of random vectors and a functional central limit theorem.
• The method accounts for discrete sampling effects by simulating fBs via circulant embedding and evaluating bias and mean squared error (MSE) across 500 realizations.

Experimental results

Research questions

• RQ1 Can wavelet-based regression on log-variance of wavelet coefficients provide a consistent and efficient estimator for the Hurst parameters of a fractional Brownian sheet in multiple dimensions?
• RQ2 How does the performance of the two-step generalized least squares (GLS) estimator compare to ordinary least squares (OLS) in terms of bias and variance for different Hurst parameter values?
• RQ3 What is the impact of $H_i < 0.5$ versus $H_i \geq 0.5$ on the accuracy and bias of the wavelet-based estimator?
• RQ4 To what extent does the discrete sampling of the fBs affect the estimator’s mean and convergence properties compared to continuous-time theory?
• RQ5 Can the proposed method effectively detect self-similarity and long-range dependence in multidimensional signals such as in texture classification or DTI of NMR?

Key findings

• The proposed two-step GLS estimator $\hat{H}_{og}$ achieves lower variance and smaller root mean squared error (RMSE) than the OLS estimator $\hat{H}_o$ across all tested Hurst parameters.
• For $H_i \geq 0.5$, the bias of the estimator is small and negligible, with mean estimates close to the true values (e.g., $H=0.5$: mean ≈ 0.489–0.491; $H=0.8$: mean ≈ 0.791–0.796).
• For $H_i < 0.5$, the bias increases, with mean estimates consistently below the true value (e.g., $H=0.3$: mean ≈ 0.271 for $\hat{H}_{og}$), and RMSE values rising to 0.032–0.042.
• The standard deviation and RMSE of $\hat{H}_{og}$ are consistently lower than those of $\hat{H}_o$, confirming the theoretical advantage of GLS in variance reduction.
• The convergence rate of the estimator improves with increasing sample size, scaling as $\sqrt{\prod_{i=1}^{d} T_i}$, indicating improved accuracy with larger data volumes.
• The method is robust to nonstationarities due to the vanishing moments of the wavelet basis, and the estimator remains efficient even in 3D simulations with $256^3$ grids.