K-distribution

{{Short description|Three-parameter family of continuous probability distributions}}

{{Probability distribution

| name = K-distribution

| type = density

| pdf_image =

| cdf_image =

| notation =

| parameters = $\mu \in (0, +\infty)$ , $\alpha \in [0, +\infty)$ , $\beta \in [0, +\infty)$

| support = $x \in [0, +\infty)\;$

| pdf = $\frac{2}{\Gamma(\alpha)\Gamma(\beta)} \, \left( \frac{\alpha \beta}{\mu} \right)^{\frac{\alpha + \beta}{2}} \, x^{ \frac{\alpha + \beta}{2} - 1} K_{\alpha - \beta} \left( 2 \sqrt{\frac{\alpha \beta x}{\mu}} \right),$

| cdf =

| mean = $\mu$

| median =

| mode =

| variance = $\mu^2 \frac{\alpha+\beta+1}{\alpha \beta}$

| skewness =

| kurtosis =

| entropy =

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

| char =

}}

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 $x>0$ :{{sfn|Redding|1999}}

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

where $K$ 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:{{sfn|Redding|1999}} 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{{sfn|Long|2001}}{{sfn|Bocquet|2011}}

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

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.{{sfn|Jakeman|Pusey|1978}} Jakeman and Tough (1987) derived the distribution from a biased random walk model.{{sfn|Jakeman|Tough|1987}} 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.{{sfn|Ward|1981}}

[[Moment (mathematics)|Moments]]

The moment generating function is given by{{sfn|Bithas|Sagias|Mathiopoulos|Karagiannidis|2006}}

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

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{{sfn|Redding|1999}}

: $\mu_n = \xi^{-n} \frac{\Gamma(\alpha+n)\Gamma(\beta+n)}{\Gamma(\alpha)\Gamma(\beta)}.$

So the mean and variance are given by{{sfn|Redding|1999}}

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

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

Other properties

All the properties of the distribution are symmetric in $\alpha$ and $\beta.$ {{sfn|Redding|1999}}

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

{{Citation |last=Redding |first=Nicholas J. |title=Estimating the Parameters of the K Distribution in the Intensity Domain |publisher=DSTO Electronics and Surveillance Laboratory |location=South Australia |id=DSTO-TR-0839 |date=1999 |pages=60 |url=https://apps.dtic.mil/sti/pdfs/ADA368069.pdf }}
{{Citation |last=Bocquet |first=Stephen |title=Calculation of Radar Probability of Detection in K-Distributed Sea Clutter and Noise |publisher=Joint Operations Division, DSTO Defence Science and Technology Organisation |location=Canberra, Australia |id=DSTO-TR-0839 |date=2011 |pages=35 |url=https://apps.dtic.mil/sti/pdfs/ADA543178.pdf }}
{{cite journal |last1=Jakeman |first1=Eric |last2=Pusey |first2=Peter N. |title=Significance of K-Distributions in Scattering Experiments |journal=Physical Review Letters |publisher=American Physical Society (APS) |volume=40 |issue=9 |date=1978-02-27 |issn=0031-9007 |doi=10.1103/physrevlett.40.546 |pages=546–550 |bibcode=1978PhRvL..40..546J }}
{{cite journal |last1=Jakeman |first1=Eric |last2=Tough |first2=Robert J. A. |title=Generalized K distribution: a statistical model for weak scattering |journal=Journal of the Optical Society of America A |publisher=The Optical Society |volume=4 |issue=9 |date=1987-09-01 |issn=1084-7529 |doi=10.1364/josaa.4.001764 |page=1764-1772|bibcode=1987JOSAA...4.1764J }}
{{cite journal |last=Ward |first=Keith D. |title=Compound representation of high resolution sea clutter |journal=Electronics Letters |publisher=Institution of Engineering and Technology (IET) |volume=17 |issue=16 |year=1981 |issn=0013-5194 |doi=10.1049/el:19810394 |page=561-565|bibcode=1981ElL....17..561W }}
{{cite journal |last1=Bithas |first1=Petros S. |last2=Sagias |first2=Nikos C. |last3=Mathiopoulos |first3=P. Takis |last4=Karagiannidis |first4=George K. |author-link4=George Karagiannidis |last5=Rontogiannis |first5=Athanasios A. |title=On the performance analysis of digital communications over generalized-k fading channels |journal=IEEE Communications Letters |publisher=Institute of Electrical and Electronics Engineers (IEEE) |volume=10 |issue=5 |year=2006 |issn=1089-7798 |doi=10.1109/lcomm.2006.1633320 |pages=353–355 |s2cid=4044765 |citeseerx=10.1.1.725.7998 }}
{{cite book |last=Long |first=Maurice W. |author-link=Maurice W. Long |title=Radar Reflectivity of Land and Sea |edition=3rd |publisher=Artech House |location=Norwood, MA |year=2001 |page=560 }}