Nakagami 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 $m\geq 1/2$ and a scale parameter $\Omega >0$. It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.

Nakagami

Probability density function
Cumulative distribution function
Parameters	$m{\text{ or }}\mu \geq 0.5$ shape (real) $\Omega {\text{ or }}\omega >0$ scale (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	No simple closed form
Mode	$\left({\frac {(2m-1)\Omega }{2m}}\right)^{1/2}$
Variance	$\Omega \left(1-{\frac {1}{m}}\left({\frac {\Gamma (m+{\frac {1}{2}})}{\Gamma (m)}}\right)^{2}\right)$

Characterization

Its probability density function (pdf) is

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

where $m\geq 1/2$ and $\Omega >0$.

Its cumulative distribution function (CDF) is

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

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

Parameterization

The parameters $m$ and $\Omega$ are

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

and

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

No closed form solution exists for the median of this distribution, although special cases do exist, such as ${\sqrt {\Omega \ln(2)}}$ 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 $\Omega$ as σ = Ω/m.

Given independent observations ${\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}}$ from the Nakagami distribution, the likelihood function is

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

Its logarithm is

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

Therefore

${\begin{aligned}{\frac {\partial \ell }{\partial \sigma }}={\frac {-nm\sigma +\sum _{i=1}^{n}x_{i}^{2}}{\sigma ^{2}}}\quad {\text{and}}\quad {\frac {\partial \ell }{\partial m}}=-n{\frac {\Gamma '(m)}{\Gamma (m)}}-n\log \sigma +2\sum _{i=1}^{n}\log x_{i}.\end{aligned}}$

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 $Y\,\sim {\textrm {Gamma}}(k,\theta )$, 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 $Y$:

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

Alternatively, 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.

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

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) defines the Nakagami-q distribution, also known as Hoyt 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 $m={\tfrac {1}{2}}$, 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