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.

K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution.

Density

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

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

where ${\displaystyle \alpha =\nu -L,}$ ${\displaystyle \beta =L+\nu -1,}$ ${\displaystyle \xi =L\nu /\mu ,}$ and ${\displaystyle K}$ 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 ${\displaystyle L}$, the second having a gamma distribution with mean ${\displaystyle \mu }$ and shape parameter ${\displaystyle \nu }$.

A simpler two parameter formalization of the K-distribution is[2]

${\displaystyle f_{X}(x;b,v)={\frac {2b}{\Gamma (v)}}\left({\frac {bx}{2}}\right)^{v}K_{v-1}(bx),}$

where v is the shape factor, b is the scale factor, and K is the modified Bessel function of second kind.

This distribution derives from a paper by Eric Jakeman and Peter 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[3]

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

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

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

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

So the mean and variance are given[1] by

${\displaystyle \operatorname {E} (X)=\mu }$

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

Other properties

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

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

1. Redding (1999)
2. Long (2001)
3. Bithas (2006)

Sources

• Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain . 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," Electron. Lett., 17, pp. 561 - 565.
• Long, M.W. (2001) "Radar Reflectivity of Land and Sea," 3rd ed., Artech House, Norwood, MA, 2001.
• 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.

Further reading

• Jakeman, E. (1980) "On the statistics of K-distributed noise", Journal of Physics A: Mathematics and General, 13, 31–48
• Ward, K.D.; Tough, Robert J.A; Watts, Simon (2006) Sea Clutter: Scattering, the K Distribution and Radar Performance, Institution of Engineering and Technology. ISBN 0-86341-503-2