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 $$x>0$$:


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

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: 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


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

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

So the mean and variance are given by
 * $$ \operatorname{E}(X)= \mu $$
 * $$ \operatorname{var}(X)= \mu^2 \frac{\alpha+\beta+1}{\alpha \beta} .$$

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

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.