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

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

where ${\displaystyle K}$ is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have ${\displaystyle K_{\nu }=K_{-\nu }}$. 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 ${\displaystyle \alpha }$, the second having a gamma distribution with mean ${\displaystyle \mu }$ and shape parameter ${\displaystyle \beta }$.

A simpler two parameter formalization of the K-distribution can be obtained by setting ${\displaystyle \beta =1}$ as^[2]

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

where ${\displaystyle v=\alpha }$ is the shape factor, ${\displaystyle b=\alpha \beta /\mu }$ is the scale factor, and ${\displaystyle K}$ 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]

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

where ${\displaystyle \gamma =\beta -\alpha ,}$ ${\displaystyle \delta =\alpha +\beta -1,}$ ${\displaystyle \xi =\alpha \beta /\mu ,}$ and ${\displaystyle W_{-\delta /2,\gamma /2}(\cdot )}$ is the Whittaker function.

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

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

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

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

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

Other properties[]

All the properties of the distribution are symmetric in ${\displaystyle \alpha }$ and ${\displaystyle \beta .}$^[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[]

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," 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