K distribution - Alchetron, The Free Social Encyclopedia

In probability and statistics, the K-distribution is a three-parameter family of continuous probability distributions. The distribution arises by compounding two gamma distributions. In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:

Density

The model is that random variable X has a gamma distribution with mean σ and shape parameter L , with σ being treated as a random variable having another gamma distribution, this time with mean μ and shape parameter ν . The result is that X has the following probability density function (pdf) for x > 0 :

f X ( x ; μ , ν , L ) = 2 ξ ( β + 1 ) / 2 x ( β − 1 ) / 2 Γ ( L ) Γ ( ν ) K α ( 2 ξ x ) ,

where α = ν − L , β = L + ν − 1 , ξ = L ν / μ , and K is a modified Bessel function of the second kind. In this derivation, the K-distribution is a compound probability distribution. It is also a product distribution: it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter L , the second having a gamma distribution with mean μ and shape parameter ν .

This distribution derives from a paper by Jakeman and Pusey (1978) who used it to model microwave sea echo. Jakeman and Tough (1987) derived the distribution from a biased random walk model. Ward (1981) derived the distribution from the product for two random variables, z = a y, where a has a chi distribution and y a complex Gaussian distribution. The modulus of z, |z|, then has K distribution.

Moments

The moment generating function is given by

M X ( s ) = ( ξ s ) β / 2 exp ⁡ ( ξ 2 s ) W − β / 2 , α / 2 ( ξ s ) ,

where W − β / 2 , α / 2 ( ⋅ ) is the Whittaker function.

The n-th moments of K-distribution is given by

μ n = ξ − n Γ ( L + n ) Γ ( ν + n ) Γ ( L ) Γ ( ν ) .

So the mean and variance are given by

E ⁡ ( X ) = μ var ⁡ ( X ) = μ 2 ν + L + 1 L ν .

Other properties

All the properties of the distribution are symmetric in L and ν .

Differential equation

The pdf of the K-distribution is a solution of the following differential equation:

{ μ x 2 f ″ ( x ) − μ x ( L + ν − 3 ) f ′ ( x ) + f ( x ) ( μ ( L − 1 ) ( ν − 1 ) − L ν x ) = 0 , f ( 1 ) = 2 ( L ν μ ) L 2 + ν 2 K ν − L ( 2 L ν μ ) Γ ( L ) Γ ( ν ) , f ′ ( 1 ) = 2 ( L ν μ ) L + ν 2 ( ( L − 1 ) K L − ν ( 2 L ν μ ) − L ν μ K L − ν + 1 ( 2 L ν μ ) ) Γ ( L ) Γ ( ν ) }

Applications

K-distribution arises as the consequence of a statistical or probabilistic model used in Synthetic Aperture Radar (SAR) imagery. The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging. It is also used in wireless communication to model composite fast fading and shadowing effects.

References

K-distribution Wikipedia

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