K-distribution

Short description: Three-parameter family of continuous probability distributions

K-distribution

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

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

$f_X(x;\mu ,\alpha ,\beta )=\frac{2}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}{\left(\frac{\alpha \beta }{\mu }\right)}^{\frac{\alpha +\beta }{2}}{x}^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\frac{\alpha \beta x}{\mu }}\right),$

where $K$ is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have $K_{\nu }=K_{-\nu }$. In this derivation, the K-distribution is a compound probability distribution. It is also a product distribution: it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter $\alpha$, the second having a gamma distribution with mean $\mu$ and shape parameter $\beta$.

A simpler two parameter formalization of the K-distribution can be obtained by setting $\beta =1$ as

$f_X(x;b,v)=\frac{2b}{\mathrm{\Gamma }(v)}{\left(\sqrt{bx}\right)}^{v-1}{K}_{v-1}(2\sqrt{bx}),$

where $v=\alpha$ is the shape factor, $b=\alpha /\mu$ is the scale factor, and $K$ is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting $\alpha =1$, $v=\beta$, and $b=\beta /\mu$, albeit with different physical interpretation of $b$ and $v$ 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. Jakeman and Tough (1987) derived the distribution from a biased random walk model. 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.

Moments

The moment generating function is given by

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

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

The n-th moments of K-distribution is given by

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

So the mean and variance are given by

$\mathrm{E}(X)=\mu$

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

Other properties

All the properties of the distribution are symmetric in $\alpha$ and $\beta .$

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.

Sources

• Redding, Nicholas J. (1999), Estimating the Parameters of the K Distribution in the Intensity Domain, South Australia: DSTO Electronics and Surveillance Laboratory, pp. 60, DSTO-TR-0839, https://apps.dtic.mil/sti/pdfs/ADA368069.pdf 
• Bocquet, Stephen (2011), Calculation of Radar Probability of Detection in K-Distributed Sea Clutter and Noise, Canberra, Australia: Joint Operations Division, DSTO Defence Science and Technology Organisation, pp. 35, DSTO-TR-0839, https://apps.dtic.mil/sti/pdfs/ADA543178.pdf 
• Jakeman, Eric; Pusey, Peter N. (1978-02-27). "Significance of K-Distributions in Scattering Experiments". Physical Review Letters (American Physical Society (APS)) 40 (9): 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 (The Optical Society) 4 (9): 1764-1772. doi:10.1364/josaa.4.001764. ISSN 1084-7529. 
• Ward, Keith D. (1981). "Compound representation of high resolution sea clutter". Electronics Letters (Institution of Engineering and Technology (IET)) 17 (16): 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 (Institute of Electrical and Electronics Engineers (IEEE)) 10 (5): 353–355. doi:10.1109/lcomm.2006.1633320. ISSN 1089-7798. 
• 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 (IOP Publishing) 13 (1): 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.