Estimation of self-similar Gaussian fields using wavelet transform

We propose an estimator (H) over cap for the Hurst parameters H = (H-1,..., H-d) of fractional Brownian sheet (FBS), which plays an important role in modeling anisotropic self-similar Gaussian fields. The proposed estimator is based on the stationarity and scale invariance of hyperbolic wavelet coefficients of FBS and the weighted least squares. We use simulated fields to validate the proposed estimator. The estimator performs well in both the standard deviation and the root mean square error.

[1]  J. L. Véhel,et al.  Stochastic fractal models for image processing , 2002, IEEE Signal Process. Mag..

[2]  Patrice Abry,et al.  Wavelet Analysis of Long-Range-Dependent Traffic , 1998, IEEE Trans. Inf. Theory.

[3]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[4]  B. Øksendal,et al.  Stochastic Calculus for Fractional Brownian Motion and Applications , 2008 .

[5]  Patrice Abry,et al.  Self-Similar Anisotropic Texture Analysis: The Hyperbolic Wavelet Transform Contribution , 2013, IEEE Transactions on Image Processing.

[6]  Brani Vidakovic,et al.  2D wavelet-based spectra with applications , 2011, Comput. Stat. Data Anal..

[7]  Mark M. Meerschaert,et al.  Operator scaling stable random fields , 2006 .

[8]  Murad S. Taqqu,et al.  Theory and applications of long-range dependence , 2003 .

[9]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[10]  Brani Vidakovic,et al.  A wavelet‐based spectral method for extracting self‐similarity measures in time‐varying two‐dimensional rainfall maps , 2011 .

[11]  Dongsheng Wu,et al.  Geometric Properties of Fractional Brownian Sheets , 2007 .

[12]  C. R. Dietrich,et al.  Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix , 1997, SIAM J. Sci. Comput..

[13]  Patrice Abry,et al.  Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter , 2011 .

[14]  Liang Wu,et al.  Wavelet-based Estimator for the Hurst Parameters of Fractional Brownian Sheet , 2015, ArXiv.

[15]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[16]  Patrice Abry,et al.  Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures , 2012 .

[17]  Anne Estrade,et al.  Anisotropic Analysis of Some Gaussian Models , 2003 .

[18]  Philippe Bolon,et al.  2-D Wavelet Packet Spectrum for Texture Analysis , 2013, IEEE Transactions on Image Processing.

[19]  M. Moseley.  Diffusion tensor imaging and aging – a review , 2002, NMR in biomedicine.

[20]  Patrice Abry,et al.  The hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic fields , 2015 .

[21]  Dirk P. Kroese,et al.  Spatial Process Generation , 2013, 1308.0399.

[22]  Murad S. Taqqu,et al.  Semi-parametric estimation of the long-range dependence parameter : A survey , 2003 .