K-distribution

K-distribution

Parameters	${\displaystyle \mu \in (0,+\infty )}$, ${\displaystyle \alpha \in [0,+\infty )}$, ${\displaystyle \beta \in [0,+\infty )}$
Support	${\displaystyle x\in [0,+\infty )\;}$
PDF	${\displaystyle {\frac {2}{\Gamma (\alpha )\Gamma (\beta )}}\,\left({\frac {\alpha \beta }{\mu }}\right)^{\frac {\alpha +\beta }{2}}\,x^{{\frac {\alpha +\beta }{2}}-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu }}}\right),}$
Mean	${\displaystyle \mu }$
Variance	${\displaystyle \mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta }}}$
MGF	${\displaystyle \left({\frac {\xi }{s}}\right)^{\beta /2}\exp \left({\frac {\xi }{2s}}\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s}}\right)}$

In probability and statistics, the generalized 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,
• 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. A simpler special case of the generalized K-distribution is often referred as the K-distribution.

Density

Suppose that a random variable ${\displaystyle X}$ has 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][3]

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

where ${\displaystyle v=\alpha }$ is the shape factor, ${\displaystyle b=\alpha /\mu }$ is the scale factor, and ${\displaystyle K}$ is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting ${\displaystyle \alpha =1}$, ${\displaystyle v=\beta }$, and ${\displaystyle b=\beta /\mu }$, albeit with different physical interpretation of ${\displaystyle b}$ and ${\displaystyle v}$ parameters. This two parameter formalization is often referred to as the K-distribution, while the three parameter formalization is referred to as the generalized K-distribution.

This distribution derives from a paper by Eric Jakeman and Peter Pusey (1978) who used it to model microwave sea echo.^[4] Jakeman and Tough (1987) derived the distribution from a biased random walk model.^[5] Keith D. 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.^[6]

Moments

The moment generating function is given by^[7]

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

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 by^[1]

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

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

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 (PDF), South Australia: DSTO Electronics and Surveillance Laboratory, p. 60, DSTO-TR-0839
• Bocquet, Stephen (2011), Calculation of Radar Probability of Detection in K-Distributed Sea Clutter and Noise (PDF), Canberra, Australia: Joint Operations Division, DSTO Defence Science and Technology Organisation, p. 35, DSTO-TR-0839
• Jakeman, Eric; Pusey, Peter N. (1978-02-27). "Significance of K-Distributions in Scattering Experiments". Physical Review Letters. 40 (9). American Physical Society (APS): 546–550. doi:10.1103/physrevlett.40.546. ISSN 0031-9007.
• Jakeman, Eric; Tough, Robert J. A. (1987-09-01). "Generalized K distribution: a statistical model for weak scattering". Journal of the Optical Society of America A. 4 (9). The Optical Society: 1764-1772. doi:10.1364/josaa.4.001764. ISSN 1084-7529.
• Ward, Keith D. (1981). "Compound representation of high resolution sea clutter". Electronics Letters. 17 (16). Institution of Engineering and Technology (IET): 561-565. doi:10.1049/el:19810394. ISSN 0013-5194.
• Bithas, Petros S.; Sagias, Nikos C.; Mathiopoulos, P. Takis; Karagiannidis, George K.; Rontogiannis, Athanasios A. (2006). "On the performance analysis of digital communications over generalized-k fading channels". IEEE Communications Letters. 10 (5). Institute of Electrical and Electronics Engineers (IEEE): 353–355. CiteSeerX 10.1.1.725.7998. doi:10.1109/lcomm.2006.1633320. ISSN 1089-7798. S2CID 4044765.
• Long, Maurice W. (2001). Radar Reflectivity of Land and Sea (3rd ed.). Norwood, MA: Artech House. p. 560.

Further reading

• Jakeman, Eric (1980-01-01). "On the statistics of K-distributed noise". Journal of Physics A: Mathematical and General. 13 (1). IOP Publishing: 31–48. doi:10.1088/0305-4470/13/1/006. ISSN 0305-4470.
• Ward, Keith 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.