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 , ,
Support
PDF
Mean
Variance
MGF

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

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Suppose that a random variable   has gamma distribution with mean   and shape parameter  , with   being treated as a random variable having another gamma distribution, this time with mean   and shape parameter  . The result is that   has the following probability density function (pdf) for  :[1]

 

where   is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have  . 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  , the second having a gamma distribution with mean   and shape parameter  .

A simpler two parameter formalization of the K-distribution can be obtained by setting   as[2][3]

 

where   is the shape factor,   is the scale factor, and   is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting  ,  , and  , albeit with different physical interpretation of   and   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]

The moment generating function is given by[7]

 

where       and   is the Whittaker function.

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

 

So the mean and variance are given by[1]

 
 

Other properties

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All the properties of the distribution are symmetric in   and  [1]

Applications

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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

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Sources

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  • 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

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