Efficient procedure to generate generalized Gaussian noise using linear spline tools

In this paper, we propose a simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution. This inverse is expressible by means of the inverse incomplete Gamma function. Since the implementation of Newton’s method is rather simple, for approximating inverse incomplete Gamma function, we propose a better and new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant. The numerical results highlight that the proposed method simulates well the univariate and bivariate generalized Gaussian noises.

[1]  Nadia Solaro.  Random variate generation from Multivariate Exponential Power distribution , 2004 .

[2]  Mark E. Johnson.  Multivariate Statistical Simulation: Johnson/Multivariate , 1987 .

[3]  Abdelilah Monir.  Contribution à la modélisation et à la synthèse des signaux aléatoires : signaux non gaussiens, signaux à corrélation non exponentielle , 2003 .

[4]  Mark E. Johnson,et al.  Multivariate Statistical Simulation , 1989, International Encyclopedia of Statistical Science.

[5]  Thomas Luu.  Efficient and Accurate Parallel Inversion of the Gamma Distribution , 2015, SIAM J. Sci. Comput..

[6]  Abdelilah Monir,et al.  Spline approximations of the Lambert W function and application to simulate generalized Gaussian noise with exponent α = 1/2 , 2014, Digit. Signal Process..

[7]  Mischa Dohler,et al.  Inverse incomplete gamma function and its application , 2006 .

[8]  Luc Devroye,et al.  Random variate generation for multivariate unimodal densities , 1997, TOMC.

[9]  Tien D. Bui,et al.  Multivariate statistical modeling for image denoising using wavelet transforms , 2005, Signal Process. Image Commun..

[10]  M. A. Gómez–Villegas,et al.  A MATRIX VARIATE GENERALIZATION OF THE POWER EXPONENTIAL FAMILY OF DISTRIBUTIONS , 2002 .

[11]  Nico M. Temme,et al.  Numerical methods for special functions , 2007 .

[12]  Emiliano A. Valdez,et al.  Wang’s capital allocation formula for elliptically contoured distributions , 2003 .

[13]  Jean-Yves Tourneret,et al.  Parameter Estimation For Multivariate Generalized Gaussian Distributions , 2013, IEEE Transactions on Signal Processing.

[14]  Eugene Isaacson,et al.  Numerical Recipes in C: The Art of Scientific Computing (William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling); Numerical Recipes: Example Book (C) (William T. Vetterling, Saul A. Teukolsky, William H. Press, and Brian P. Flannery) , 1989, SIAM Rev..

[15]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[16]  Enrique Castillo,et al.  Multivariate distributions defined in terms of contours , 2008 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Nico M. Temme,et al.  Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios , 2012, SIAM J. Sci. Comput..

[19]  Jonathan M. Borwein,et al.  Crandall's computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series , 2015, Appl. Math. Comput..

[20]  François Chapeau-Blondeau,et al.  Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2 , 2002, IEEE Trans. Signal Process..

[21]  Hideaki Kaneko,et al.  Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind , 1994 .

[22]  Armido R. Didonato,et al.  Computation of the incomplete gamma function ratios and their inverse , 1986, TOMS.

[23]  Douglas Kelker,et al.  DISTRIBUTION THEORY OF SPHERICAL DISTRIBUTIONS AND A LOCATION-SCALE PARAMETER GENERALIZATION , 2016 .

[24]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .