Published

2022-01-01

Asymmetric Prior in Wavelet Shrinkage

Priori asimétrico en contracción de ondículas

Keywords:

Wavelet shrinkage, Nonparametric regression, Asymmetric beta distribution (en)
Contracción de las ondículas, Regresión no paramétrico, Distribución beta asimétrica (es)

Downloads

Authors

  • Alex Rodrigo dos Santos Sousa University of São Paulo

In bayesian wavelet shrinkage, the already proposed priors to wavelet coefficients are assumed to be symmetric around zero. Although this assumption is reasonable in many applications, it is not general. The present paper proposes the use of an asymmetric shrinkage rule based on the discrete mixture of a point mass function at zero and an asymmetric beta distribution as prior to the wavelet coefficients in a non-parametric regression model. Statistical properties such as bias, variance, classical and bayesian risks of the associated asymmetric rule are provided and performances of the proposed rule are obtained in simulation studies involving artificial asymmetric distributed coefficients and the Donoho-Johnstone test functions. Application in a seismic real dataset is also analyzed.

En la contracción de las ondículas bayesianas, se supone que los coeficientes a priori ya propuestos de las ondículas son simétricos alrededor de cero. Aunque esta suposición es razonable en muchas aplicaciones, no
es general. El presente artículo propone el uso de una regla de contracción asimétrica basada en la mezcla discreta de una función de masa puntual en cero y una distribución beta asimétrica como priori de los coeficientes de ondícula en un modelo de regresión no paramétrico. Se proporcionan propiedades estadísticas tales como sesgo, varianza, riesgos clásicos y bayesianos de la regla asimétrica asociada y se obtienen los rendimientos de la regla propuesta en estudios de simulación que involucran coeficientes distribuidos asimétricos artificiales y las funciones de prueba de Donoho-Johnstone. También se analiza la aplicación en un conjunto de datos sísmicos reales.

Downloads

Download data is not yet available.

References

Abramovich, F. & Benjamini, Y. (1996), ‘Adaptive thresholding of wavelet coefficients’, Computational Statistics and Data Analysis 22(4), 351–361.

Abramovich, F., Sapatinas, T. & Silverman, B. (1998), ‘Wavelet thresholding via a bayesian approach’, Royal Statistical Society pp. 725–749.

Angelini, C. & Vidakovic, B. (2004), ‘Gama-minimax wavelet shrinkage: a robust incorporation of information about energy of a signal in denoising applications’, Statistica Sinica (14), 103–125.

Antoniadis, A., Bigot, J. & Sapatinas, T. (2001), ‘Wavelet estimators in nonparametric regression: a comparative simulation study’, Journal of Statistical Software (6), 1–83.

Beenamol, M., Prabavathy, S. & Mohanalin, J. (2012), ‘Wavelet based seismic signal de-noising using shannon and tsallis entropy’, Computers and Mathematics with Applications (64), 3580–3593.

Bhattacharya, A., Pati, D., Pillai, N. & Dunson, D. (2015), ‘Dirichlet-laplace priors for optimal shrinkage’, Journal of the American Statistical Association (110), 1479–1490.

Chipman, H., Kolaczyk, E. & McCulloch, R. (1997), ‘Adaptive bayesian wavelet shrinkage’, Journal of the American Statistical Association (92), 14131421.

Cutillo, L., Jung, Y., Ruggeri, F. & Vidakovic, B. (2008), ‘Larger posterior mode wavelet thresholding and applications’, Journal of Statistical Planning and Inference (138), 3758–3773.

Donoho, D. L. (1995a), ‘De-noising by soft-thresholding’, IEEE Transactions on Information Theory (41), 613627.

Donoho, D. L. (1995b), ‘Nonlinear solution of linear inverse problems by waveletvaguelette decomposition’, Applied and Computational Harmonic Analysis (2), 10126.

Donoho, D. L. & Johnstone, I. M. (1994a), ‘Ideal denoising in an orthonormal basis chosen from a library of bases’, Comptes Rendus de l Académie des Sciences (319), 13171322.

Donoho, D. L. & Johnstone, I. M. (1994b), ‘Ideal spatial adaptation by wavelet shrinkage’, Biometrika (81), 425455.

Donoho, D. L. & Johnstone, I. M. (1995), ‘Adapting to unknown smoothness via wavelet shrinkage’, Journal of the American Statistical Association (90), 12001224.

Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. & Picard, D. (1995), ‘Wavelet shrinkage: Asymptopia? (with discussion)’, Royal Statistical Society B (57), 301369.

Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. & Picard, D. (1996), ‘Density estimation by wavelet thresholding’, Annals of Statistics 24, 508539.

Griffin, J. & Brown, P. (2017), ‘Hierarquical shrinkage priors for regression models’, Bayesian Analysis 12(1), 135–159.

Jansen, M. (2001), Noise reduction by wavelet thresholding, Springer, New York.

Johnstone, L. & Silverman, B. (2005), ‘Empirical bayes selection of wavelet thresholds’, The Annals of Statistics 33, 1700–1752.

Karagiannis, G., Konomi, B. & Lin, G. (2015), ‘A bayesian mixed shrinkage prior procedure for spatial-stochastic basis selection and evaluation of gpc expansion: applications to elliptic spdes’, Journal of Computational Physics 284, 528–546.

Lian, H. (2011), ‘On posterior distribution of bayesian wavelet thresholding’, Journal of Statistical Planning and Inference 141, 318–324.

Mallat, S. G. (1998), A Wavelet Tour of Signal Processing, Academic Press, San Diego.

Nason, G. P. (1996), ‘Wavelet shrinkage using cross-validation’, Journal of the Royal Statistical Society B 58, 463479.

Reményi, N. & Vidakovic, B. (2015), ‘Wavelet shrinkage with double weibull prior’, Communications in Statistics: Simulation and Computation 44(1), 88–104.

Sousa, A. (2020), ‘Bayesian wavelet shrinkage with logistic prior’, Communications in Statistics: Simulation and Computation.

Sousa, A., Garcia, N. & Vidakovic, B. (2021), ‘Bayesian wavelet shrinkage with beta prior’, Computational Statistics 36(2), 1341–1363.

Torkamani, R. & Sadeghzadeh, R. (2017), ‘Bayesian compressive sensing using wavelet based markov random fields’, Signal Processing: Image Communication 58, 65–72.

Vidakovic, B. (1998), ‘Nonlinear wavelet shrinkage with bayes rules and bayes factors’, Journal of the American Statistical Association 93, 173–179.

Vidakovic, B. (1999), Statistical Modeling by Wavelets, Wiley, New York.

Vidakovic, B. & Ruggeri, F. (2001), ‘Bams method: Theory and simulations’, Sankhya: The Indian Journal of Statistics 63, 234–249.

License

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).