K-distribution

Short description: Three-parameter family of continuous probability distributions

K-distribution

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

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

[math]\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), }[/math]

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

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

[math]\displaystyle{ f_X(x; b, v)= \frac{2b}{\Gamma(v)} \left( \sqrt{bx} \right)^{v-1} K_{v-1} (2 \sqrt{bx} ), }[/math]

where [math]\displaystyle{ v = \alpha }[/math] is the shape factor, [math]\displaystyle{ b = \alpha/\mu }[/math] is the scale factor, and [math]\displaystyle{ K }[/math] is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting [math]\displaystyle{ \alpha = 1 }[/math], [math]\displaystyle{ v = \beta }[/math], and [math]\displaystyle{ b = \beta/\mu }[/math], albeit with different physical interpretation of [math]\displaystyle{ b }[/math] and [math]\displaystyle{ v }[/math] 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

[math]\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), }[/math]

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

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

[math]\displaystyle{ \mu_n = \xi^{-n} \frac{\Gamma(\alpha+n)\Gamma(\beta+n)}{\Gamma(\alpha)\Gamma(\beta)}. }[/math]

So the mean and variance are given by

[math]\displaystyle{ \operatorname{E}(X)= \mu }[/math]

[math]\displaystyle{ \operatorname{var}(X)= \mu^2 \frac{\alpha+\beta+1}{\alpha \beta} . }[/math]

Other properties

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

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.