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 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 $x>0$:

:$f_X(x; \mu, \alpha, \beta)= \frac \, \left( \frac \right)^ \, x^ K_ \left( 2 \sqrt \right),$

where $K$ is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have $K_ = K_$. 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

:$f_X(x; b, v)= \frac \left( \sqrt \right)^ K_ (2 \sqrt ),$

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
:$M_X(s) = \left(\frac\right)^ \exp \left( \frac \right) W_ \left(\frac\right),$

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

The n-th moments of K-distribution is given by
:$\mu_n = \xi^ \frac.$

So the mean and variance are given by
:$\operatorname(X)= \mu$
:$\operatorname(X)= \mu^2 \frac .$

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.

Notes

Sources

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Further reading

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* Ward, Keith D.; Tough, Robert J. A; Watts, Simon (2006) ''Sea Clutter: Scattering, the K Distribution and Radar Performance'', Institution of Engineering and Technology. .

{{DEFAULTSORT:K-Distribution
Radar signal processing
Continuous distributions
Compound probability distributions
Synthetic aperture radar