K-distribution

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:

the mean of the distribution, and
the usual shape parameter.

Density

The model is that random variable $X$ has a gamma distribution with mean $\sigma$ and shape parameter $L$ , with $\sigma$ being treated as a random variable having another gamma distribution, this time with mean $\mu$ and shape parameter $\nu$ . The result is that $X$ has the following probability density function (pdf) for $x>0$ :^[1]

f_{X}(x;\mu ,\nu ,L)={\frac {2\,\xi ^{(\beta +1)/2}x^{(\beta -1)/2}}{\Gamma (L)\Gamma (\nu )}}K_{\alpha }(2{\sqrt {\xi x}}),

where $\alpha =\nu -L,$ $\beta =L+\nu -1,$ $\xi =L\nu /\mu ,$ 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:^[1] 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 $\mu$ and shape parameter $\nu$ .

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^[2]

M_{X}(s)=\left({\frac {\xi }{s}}\right)^{\beta /2}\exp \left({\frac {\xi }{2s}}\right)W_{-\beta /2,\alpha /2}\left({\frac {\xi }{s}}\right),

where $W_{-\beta /2,\alpha /2}(\cdot )$ is the Whittaker function.

The n-th moments of K-distribution is given by^[1]

\mu _{n}=\xi ^{-n}{\frac {\Gamma (L+n)\Gamma (\nu +n)}{\Gamma (L)\Gamma (\nu )}}.

So the mean and variance are given^[1] by

\operatorname {E} (X)=\mu

\operatorname {var} (X)=\mu ^{2}{\frac {\nu +L+1}{L\nu }}.

Other properties

All the properties of the distribution are symmetric in $L$ and $\nu$ .^[1]

Differential equation

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

\left\{{\begin{array}{l}\mu x^{2}f''(x)-\mu x(L+\nu -3)f'(x)+f(x)(\mu (L-1)(\nu -1)-L\nu x)=0,\\f(1)={\frac {2\left({\frac {L\nu }{\mu }}\right)^{{\frac {L}{2}}+{\frac {\nu }{2}}}K_{\nu -L}\left(2{\sqrt {\frac {L\nu }{\mu }}}\right)}{\Gamma (L)\Gamma (\nu )}},\\f'(1)={\frac {2\left({\frac {L\nu }{\mu }}\right)^{\frac {L+\nu }{2}}\left((L-1)K_{L-\nu }\left(2{\sqrt {\frac {L\nu }{\mu }}}\right)-{\sqrt {\frac {L\nu }{\mu }}}K_{L-\nu +1}\left(2{\sqrt {\frac {L\nu }{\mu }}}\right)\right)}{\Gamma (L)\Gamma (\nu )}}\end{array}}\right\}

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.

Notes

^ ^a ^b ^c ^d ^e Redding (1999)
^ Bithas (2006)

Sources

Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain [1]. Report DSTO-TR-0839, DSTO Electronics and Surveillance Laboratory, South Australia. p. 60
Jakeman, E. and Pusey, P. N. (1978) "Significance of K-Distributions in Scattering Experiments", Physical Review Letters, 40, 546–550 doi:10.1103/PhysRevLett.40.546
Jakeman, E. and Tough, R. J. A. (1987) "Generalized K distribution: a statistical model for weak scattering," J. Opt. Soc. Am., 4, (9), pp. 1764 - 1772.
Ward, K. D. (1981) "Compound representation of high resolution sea clutter," Elecron. Lett., 17, pp. 561 - 565.
Bithas, P.S.; Sagias, N.C.; Mathiopoulos, P.T.; Karagiannidis, G.K.; Rontogiannis, A.A. (2006) "On the performance analysis of digital communications over generalized-K fading channels," IEEE Communications Letters, 10 (5), pp. 353 - 355.