Nakagami distribution

Nakagami

Probability density function 325px
Cumulative distribution function 325px
Parameters	$m\ or\ \mu >= 0.5$ shape (real) $\Omega\ or\ \omega > 0$ spread (real)
Support	$x > 0\!$
PDF	$\frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2 \right)$
CDF	$\frac{\gamma \left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}$
Mean	$\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}$
Median	$\sqrt{\Omega}\!$
Mode	$\frac{\sqrt{2}}{2} \left(\frac{(2m-1)\Omega}{m}\right)^{1/2}$
Variance	$\Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)$

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter $m$ and a second parameter controlling spread, $\Omega$.

Characterization

Its probability density function (pdf) is

$f(x;\,m,\Omega) = \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1}\exp\left(-\frac{m}{\Omega}x^2\right).$

Its cumulative distribution function is

$F(x;\,m,\Omega) = P\left(m, \frac{m}{\Omega}x^2\right)$

where P is the incomplete gamma function (regularized).

Differential equation

$\left\{x \Omega f'(x)+f(x) \left(2 m x^2-2 m \Omega +\Omega \right)=0,f(1)=\frac{2 m^m e^{-\frac{m}{\Omega }} \Omega ^{-m}}{\Gamma (m)}\right\}$

Parameter estimation

The parameters $m$ and $\Omega$ are

$m = \frac{\operatorname{E}^2 \left[X^2 \right]} {\operatorname{Var} \left[X^2 \right]},$

and

$\Omega = \operatorname{E} \left[X^2 \right].$

An alternative way of fitting the distribution is to re-parametrize $\Omega$ and m as σ = Ω/m and m. Then, by taking the derivative of log likelihood with respect to each of the new parameters, the following equations are obtained and these can be solved using the Newton-Raphson method:

$\Gamma(m)= \frac{x^{2m}}{\sigma^m},$

and

$\sigma= \frac{x^2}{m}$

It is reported by authors that modelling data with Nakagami distribution and estimating parameters by above mention method results in better performance for low data regime compared to moments based methods.

Generation

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

$X = \sqrt{Y} \,$.

The Nakagami distribution $f(y; \,m,\Omega)$ 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:

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

Currently, the more efficient generation method is provided in.

History and applications

The Nakagami distribution is relatively new, being first proposed in 1960. It has been used to model attenuation of wireless signals traversing multiple paths.

Related distributions

• Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as Hoyt distribution.

"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 does."