K-distribution

In probability and statistics, the 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, and
the usual shape parameter.

Density

The model is that random variable $X$ has a gamma distribution with mean $\sigma$ and shape parameter $L$ , with $\sigma$ being treated as a random variable having another gamma distribution, this time with mean $\mu$ and shape parameter $\nu$ . The result is that $X$ has the following probability density function (pdf) for $x>0$ :^[1]

f_X(x;\mu,\nu,L)= \frac{2 \, \xi^{(\beta + 1)/2} x^{(\beta - 1)/2}}{\Gamma(L)\Gamma(\nu)} K_\alpha ( 2 \sqrt{\xi x}),

where $\alpha = \nu-L,$ $\beta = L + \nu - 1,$ $\xi = L\nu/\mu ,$ and $K$ is a modified Bessel function of the second kind. 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 $L$ , the second having a gamma distribution with mean $\mu$ and shape parameter $\nu$ .

This distribution derives from a paper by Jakeman and Pusey (1978) who used it to model microwave sea echo. Jakeman and Tough (1987) derived the distribution from a biased random walk model. 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^[2]

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

where $W_{-\beta/2,\alpha/2}(\cdot)$ is the Whittaker function.

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

\mu_n = \xi^{-n} \frac{\Gamma(L+n)\Gamma(\nu+n)}{\Gamma(L)\Gamma(\nu)}.

So the mean and variance are given^[1] by

\operatorname {E}(X)=\mu

\operatorname {var}(X)=\mu ^{2}{\frac {\nu +L+1}{L\nu }}.

Other properties

All the properties of the distribution are symmetric in $L$ and $\nu$ .^[1]

Differential equation

The pdf of the K-distribution is a solution of the following differential equation:

\left\{{\begin{array}{l}\mu x^{2}f''(x)-\mu x(L+\nu -3)f'(x)+f(x)(\mu (L-1)(\nu -1)-L\nu x)=0,\\f(1)={\frac {2\left({\frac {L\nu }{\mu }}\right)^{{{\frac {L}{2}}+{\frac {\nu }{2}}}}K_{{\nu -L}}\left(2{\sqrt {{\frac {L\nu }{\mu }}}}\right)}{\Gamma (L)\Gamma (\nu )}},\\f'(1)={\frac {2\left({\frac {L\nu }{\mu }}\right)^{{{\frac {L+\nu }{2}}}}\left((L-1)K_{{L-\nu }}\left(2{\sqrt {{\frac {L\nu }{\mu }}}}\right)-{\sqrt {{\frac {L\nu }{\mu }}}}K_{{L-\nu +1}}\left(2{\sqrt {{\frac {L\nu }{\mu }}}}\right)\right)}{\Gamma (L)\Gamma (\nu )}}\end{array}}\right\}

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

1 2 3 4 5 Redding (1999)
↑ Bithas (2006)

Sources

Redding, Nicholas J. (1999) Estimating the Parameters of the K Distribution in the Intensity Domain . Report DSTO-TR-0839, DSTO Electronics and Surveillance Laboratory, South Australia. p. 60
Jakeman, E. and Pusey, P. N. (1978) "Significance of K-Distributions in Scattering Experiments", Physical Review Letters, 40, 546–550 doi:10.1103/PhysRevLett.40.546
Jakeman, E. and Tough, R. J. A. (1987) "Generalized K distribution: a statistical model for weak scattering," J. Opt. Soc. Am., 4, (9), pp. 1764 - 1772.
Ward, K. D. (1981) "Compound representation of high resolution sea clutter," Electron. Lett., 17, pp. 561 - 565.
Bithas, P.S.; Sagias, N.C.; Mathiopoulos, P.T.; Karagiannidis, G.K.; Rontogiannis, A.A. (2006) "On the performance analysis of digital communications over generalized-K fading channels," IEEE Communications Letters, 10 (5), pp. 353 - 355.

Jakeman, E. (1980) "On the statistics of K-distributed noise", Journal of Physics A: Mathematics and General, 13, 31–48
Ward, K.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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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