K-distribution

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

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

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

1. Redding 1999.
2. Long 2001.
3. Bocquet 2011.
4. Jakeman & Pusey 1978.
5. Jakeman & Tough 1987.
6. Ward 1981.
7. Bithas et al. 2006.

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. Bibcode:1978PhRvL..40..546J. 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. Bibcode:1987JOSAA...4.1764J. 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. Bibcode:1981ElL....17..561W. 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. Bibcode:1980JPhA...13...31J. 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.