Jump to content
Wikipedia The Free Encyclopedia

K-distribution

From Wikipedia, the free encyclopedia
Three-parameter family of continuous probability distributions
K-distribution
Parameters μ ∈ ( 0 , + ∞ ) {\displaystyle \mu \in (0,+\infty )} {\displaystyle \mu \in (0,+\infty )}, α ∈ [ 0 , + ∞ ) {\displaystyle \alpha \in [0,+\infty )} {\displaystyle \alpha \in [0,+\infty )}, β ∈ [ 0 , + ∞ ) {\displaystyle \beta \in [0,+\infty )} {\displaystyle \beta \in [0,+\infty )}
Support x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )\;} {\displaystyle x\in [0,+\infty )\;}
PDF 2 Γ ( α ) Γ ( β ) ( α β μ ) α + β 2 x α + β 2 − 1 K α − β ( 2 α β x μ ) , {\displaystyle {\frac {2}{\Gamma (\alpha )\Gamma (\beta )}},円\left({\frac {\alpha \beta }{\mu }}\right)^{\frac {\alpha +\beta }{2}},円x^{{\frac {\alpha +\beta }{2}}-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu }}}\right),} {\displaystyle {\frac {2}{\Gamma (\alpha )\Gamma (\beta )}},円\left({\frac {\alpha \beta }{\mu }}\right)^{\frac {\alpha +\beta }{2}},円x^{{\frac {\alpha +\beta }{2}}-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu }}}\right),}
Mean μ {\displaystyle \mu } {\displaystyle \mu }
Variance μ 2 α + β + 1 α β {\displaystyle \mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta }}} {\displaystyle \mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta }}}
MGF ( ξ s ) β / 2 exp ⁡ ( ξ 2 s ) W − δ / 2 , γ / 2 ( ξ s ) {\displaystyle \left({\frac {\xi }{s}}\right)^{\beta /2}\exp \left({\frac {\xi }{2s}}\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s}}\right)} {\displaystyle \left({\frac {\xi }{s}}\right)^{\beta /2}\exp \left({\frac {\xi }{2s}}\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s}}\right)}

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

[edit ]

Suppose that a random variable X {\displaystyle X} {\displaystyle X} has gamma distribution with mean σ {\displaystyle \sigma } {\displaystyle \sigma } and shape parameter α {\displaystyle \alpha } {\displaystyle \alpha }, with σ {\displaystyle \sigma } {\displaystyle \sigma } being treated as a random variable having another gamma distribution, this time with mean μ {\displaystyle \mu } {\displaystyle \mu } and shape parameter β {\displaystyle \beta } {\displaystyle \beta }. The result is that X {\displaystyle X} {\displaystyle X} has the following probability density function (pdf) for x > 0 {\displaystyle x>0} {\displaystyle x>0}:[1]

f X ( x ; μ , α , β ) = 2 Γ ( α ) Γ ( β ) ( α β μ ) α + β 2 x α + β 2 − 1 K α − β ( 2 α β x μ ) , {\displaystyle f_{X}(x;\mu ,\alpha ,\beta )={\frac {2}{\Gamma (\alpha )\Gamma (\beta )}},円\left({\frac {\alpha \beta }{\mu }}\right)^{\frac {\alpha +\beta }{2}},円x^{{\frac {\alpha +\beta }{2}}-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu }}}\right),} {\displaystyle f_{X}(x;\mu ,\alpha ,\beta )={\frac {2}{\Gamma (\alpha )\Gamma (\beta )}},円\left({\frac {\alpha \beta }{\mu }}\right)^{\frac {\alpha +\beta }{2}},円x^{{\frac {\alpha +\beta }{2}}-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu }}}\right),}

where K {\displaystyle K} {\displaystyle K} is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have K ν = K − ν {\displaystyle K_{\nu }=K_{-\nu }} {\displaystyle K_{\nu }=K_{-\nu }}. 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 α {\displaystyle \alpha } {\displaystyle \alpha }, the second having a gamma distribution with mean μ {\displaystyle \mu } {\displaystyle \mu } and shape parameter β {\displaystyle \beta } {\displaystyle \beta }.

A simpler two parameter formalization of the K-distribution can be obtained by setting β = 1 {\displaystyle \beta =1} {\displaystyle \beta =1} as[2] [3]

f X ( x ; b , v ) = 2 b Γ ( v ) ( b x ) v − 1 K v − 1 ( 2 b x ) , {\displaystyle f_{X}(x;b,v)={\frac {2b}{\Gamma (v)}}\left({\sqrt {bx}}\right)^{v-1}K_{v-1}(2{\sqrt {bx}}),} {\displaystyle f_{X}(x;b,v)={\frac {2b}{\Gamma (v)}}\left({\sqrt {bx}}\right)^{v-1}K_{v-1}(2{\sqrt {bx}}),}

where v = α {\displaystyle v=\alpha } {\displaystyle v=\alpha } is the shape factor, b = α / μ {\displaystyle b=\alpha /\mu } {\displaystyle b=\alpha /\mu } is the scale factor, and K {\displaystyle K} {\displaystyle K} is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting α = 1 {\displaystyle \alpha =1} {\displaystyle \alpha =1}, v = β {\displaystyle v=\beta } {\displaystyle v=\beta }, and b = β / μ {\displaystyle b=\beta /\mu } {\displaystyle b=\beta /\mu }, albeit with different physical interpretation of b {\displaystyle b} {\displaystyle b} and v {\displaystyle v} {\displaystyle 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.[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]

M X ( s ) = ( ξ s ) β / 2 exp ⁡ ( ξ 2 s ) W − δ / 2 , γ / 2 ( ξ s ) , {\displaystyle 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),} {\displaystyle 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 γ = β − α , {\displaystyle \gamma =\beta -\alpha ,} {\displaystyle \gamma =\beta -\alpha ,} δ = α + β − 1 , {\displaystyle \delta =\alpha +\beta -1,} {\displaystyle \delta =\alpha +\beta -1,} ξ = α β / μ , {\displaystyle \xi =\alpha \beta /\mu ,} {\displaystyle \xi =\alpha \beta /\mu ,} and W − δ / 2 , γ / 2 ( ⋅ ) {\displaystyle W_{-\delta /2,\gamma /2}(\cdot )} {\displaystyle W_{-\delta /2,\gamma /2}(\cdot )} is the Whittaker function.

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

μ n = ξ − n Γ ( α + n ) Γ ( β + n ) Γ ( α ) Γ ( β ) . {\displaystyle \mu _{n}=\xi ^{-n}{\frac {\Gamma (\alpha +n)\Gamma (\beta +n)}{\Gamma (\alpha )\Gamma (\beta )}}.} {\displaystyle \mu _{n}=\xi ^{-n}{\frac {\Gamma (\alpha +n)\Gamma (\beta +n)}{\Gamma (\alpha )\Gamma (\beta )}}.}

So the mean and variance are given by[1]

E ⁡ ( X ) = μ {\displaystyle \operatorname {E} (X)=\mu } {\displaystyle \operatorname {E} (X)=\mu }
var ⁡ ( X ) = μ 2 α + β + 1 α β . {\displaystyle \operatorname {var} (X)=\mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta }}.} {\displaystyle \operatorname {var} (X)=\mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta }}.}

Other properties

[edit ]

All the properties of the distribution are symmetric in α {\displaystyle \alpha } {\displaystyle \alpha } and β . {\displaystyle \beta .} {\displaystyle \beta .}[1]

Applications

[edit ]

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

[edit ]

Sources

[edit ]

Further reading

[edit ]
Discrete
univariate
with finite
support
with infinite
support
Continuous
univariate
supported on a
bounded interval
supported on a
semi-infinite
interval
supported
on the whole
real line
with support
whose type varies
Mixed
univariate
continuous-
discrete
Multivariate
(joint)
Directional
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families

AltStyle によって変換されたページ (->オリジナル) /