Nakagami distribution

Template:Short description Template:Multiple issues Template:Probability distribution The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter $m\ge 1/2$ and a scale parameter $\mathrm{\Omega }>0$. It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.

Characterization

Its probability density function (pdf) is

$f(x;m,\mathrm{\Omega })=\frac{2m^m}{\mathrm{\Gamma }(m){\mathrm{\Omega }}^m}{x}^{2m-1}\exp (-\frac{m}{\mathrm{\Omega }}{x}^2)\text{ for }x\ge 0.$

where $m\ge 1/2$ and $\mathrm{\Omega }>0$.

Its cumulative distribution function (CDF) is

$F(x;m,\mathrm{\Omega })=\frac{\gamma (m,\frac{m}{\mathrm{\Omega }}{x}^2)}{\mathrm{\Gamma }(m)}=P(m,\frac{m}{\mathrm{\Omega }}{x}^2)$

where P is the regularized (lower) incomplete gamma function.

Parameterization

The parameters $m$ and $\mathrm{\Omega }$ are

$m=\frac{{(\mathrm{E}[X^2])}^2}{\mathrm{Var}[X^2]},$

and

$\mathrm{\Omega }=\mathrm{E}[X^2].$

No closed form solution exists for the median of this distribution, although special cases do exist, such as $\sqrt{\mathrm{\Omega }\ln (2)}$ when m = 1. For practical purposes the median would have to be calculated as the 50th-percentile of the observations.

Parameter estimation

An alternative way of fitting the distribution is to re-parametrize $\mathrm{\Omega }$ as σ = Ω/m.

Given independent observations $X_1=x_1,\dots ,X_n=x_n$ from the Nakagami distribution, the likelihood function is

$L(\sigma ,m)={\left(\frac{2}{\mathrm{\Gamma }(m){\sigma }^m}\right)}^n{\left(\prod _{i=1}^n{x}_i\right)}^{2m-1}\exp (-\frac{\sum _{i=1}^n{x}_i^2}{\sigma }).$

Its logarithm is

$\ell (\sigma ,m)=\log L(\sigma ,m)=-n\log \mathrm{\Gamma }(m)-nm\log \sigma +(2m-1)\sum _{i=1}^n\log x_i-\frac{\sum _{i=1}^n{x}_i^2}{\sigma }.$

Therefore

$\begin{array}{c}\frac{\partial \ell }{\partial \sigma }=\frac{-nm\sigma +\sum _{i=1}^n{x}_i^2}{{\sigma }^2}\text{and}\frac{\partial \ell }{\partial m}=-n\frac{{\mathrm{\Gamma }}^{\prime }(m)}{\mathrm{\Gamma }(m)}-n\log \sigma +2\sum _{i=1}^n\log x_i.\end{array}$

These derivatives vanish only when

$\sigma =\frac{\sum _{i=1}^n{x}_i^2}{nm}$

and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the Newton–Raphson method.

It can be shown that at the critical point a global maximum is attained, so the critical point is the maximum-likelihood estimate of (m,σ). Because of the equivariance of maximum-likelihood estimation, a maximum likelihood estimate for Ω is obtained as well.

Random variate generation

The Nakagami distribution is related to the gamma distribution. In particular, given a random variable $Y\sim \text{Gamma}(k,\theta )$, it is possible to obtain a random variable $X\sim \text{Nakagami}(m,\mathrm{\Omega })$, by setting $k=m$, $\theta =\mathrm{\Omega }/m$, and taking the square root of $Y$:

$X=\sqrt{Y}.$

Alternatively, the Nakagami distribution $f(y;m,\mathrm{\Omega })$ can be generated from the chi distribution with parameter $k$ set to $2m$ and then following it by a scaling transformation of random variables. That is, a Nakagami random variable $X$ is generated by a simple scaling transformation on a chi-distributed random variable $Y\sim \chi (2m)$ as below.

$X=\sqrt{(\mathrm{\Omega }/2m)}Y.$

For a chi-distribution, the degrees of freedom $2m$ must be an integer, but for Nakagami the $m$ can be any real number greater than 1/2. This is the critical difference and accordingly, Nakagami-m is viewed as a generalization of chi-distribution, similar to a gamma distribution being considered as a generalization of chi-squared distributions.

History and applications

The Nakagami distribution is relatively new, being first proposed in 1960 by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation. It has been used to model attenuation of wireless signals traversing multiple paths and to study the impact of fading channels on wireless communications.

Related distributions

• Restricting m to the unit interval (q = m; 0 < q < 1)Template:Dubious defines the Nakagami-q distribution, also known as Template:Vanchor distribution, first studied by R.S. Hoyt in the 1940s. In particular, the radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable also does.
• With 2m = k, the Nakagami distribution gives a scaled chi distribution.
• With $m=\frac{1}{2}$, the Nakagami distribution gives a scaled half-normal distribution.
• A Nakagami distribution is a particular form of generalized gamma distribution, with p = 2 and d = 2m.

See also

• Gamma distribution
• Modified half-normal distribution
• Sub-Gaussian distribution