Generalized Gaussian distribution
Generalized Gaussian Distribution (GGD)
A random variable X has generalized Gaussian distribution if its probability density function (pdf) is given by
<math>f(x;m,\sigma,\alpha)=a\ \exp(-|(x-m)/b|^\alpha)</math> ,<math>x \in \R </math>
Where m is the mean of the distribution, <math>\sigma</math> is the standard deviation, <math>\alpha</math> is the shape parameter and <math>\sigma , \alpha>0</math>. a & b are computed according to :
<math>a = \frac{1}{2\Gamma(1+1/\alpha)b }</math>
<math>b = \sigma\,\sqrt{\frac{\Gamma(1/\alpha)}{\Gamma(3/\alpha)}}</math>
b is a scaling factor which allows the variance to be <math>\sigma^2</math>.
When <math>\alpha=1</math> , <math>f(x;m,\sigma,\alpha)</math> corresponds to a Laplacian or double exponential
distribution, <math>\alpha=2</math> corresponds to a Gaussian distribution, whereas in the limiting cases where <math>\alpha</math> approches <math>+\infty</math> the pdf ( <math>f(x;m,\sigma,\alpha)</math> ) converges to a uniform distribution in <math>(m-\sqrt{3}\sigma, m+\sqrt{3}\sigma)</math>.
As GGD is symmetric around its mean (m), even central moments are zero.
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Reference:
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References
- ↑ J. Armando Domínguez-Molina, Graciela González-Farías,Ramón M. Rodríguez-Dagnino,"A practical procedure to estimate the shape parameter in the generalized Gaussian distribution" .
2.-Varanasi, M.K., Aazhang, B. (1989). Parametric generalized Gaussian density estimation,J. Acoust. Soc. Am. 86 (4), October 1989, pp. 1404.