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 ${\displaystyle m\geq 1/2}$ and a scale parameter ${\displaystyle \Omega >0}$. 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	${\displaystyle m{\text{ or }}\mu \geq 0.5}$ shape (real) ${\displaystyle \Omega {\text{ or }}\omega >0}$ scale (real)
Support	${\displaystyle x>0\!}$
PDF	${\displaystyle {\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right)}$
CDF	${\displaystyle {\frac {\gamma \left(m,{\frac {m}{\Omega }}x^{2}\right)}{\Gamma (m)}}}$
Mean	${\displaystyle {\frac {\Gamma (m+{\frac {1}{2}})}{\Gamma (m)}}\left({\frac {\Omega }{m}}\right)^{1/2}}$
Median	No simple closed form
Mode	${\displaystyle \left({\frac {(2m-1)\Omega }{2m}}\right)^{1/2}}$
Variance	${\displaystyle \Omega \left(1-{\frac {1}{m}}\left({\frac {\Gamma (m+{\frac {1}{2}})}{\Gamma (m)}}\right)^{2}\right)}$

Characterization

Its probability density function (pdf) is^[1]

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

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

Its cumulative distribution function (CDF) is^[1]

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

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

Parameterization

The parameters ${\displaystyle m}$ and ${\displaystyle \Omega }$ are^[2]

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

and

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

No closed form solution exists for the median of this distribution, although special cases do exist, such as ${\displaystyle {\sqrt {\Omega \ln(2)}}}$ 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 ${\displaystyle \Omega }$ as σ = Ω/m.^[3]

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

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

Its logarithm is

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

Therefore

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

These derivatives vanish only when

${\displaystyle \sigma ={\frac {\sum _{i=1}^{n}x_{i}^{2}}{nm}}}$

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

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

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

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

For a chi-distribution, the degrees of freedom ${\displaystyle 2m}$ must be an integer, but for Nakagami the ${\displaystyle m}$ 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.^[4] It has been used to model attenuation of wireless signals traversing multiple paths^[5] and to study the impact of fading channels on wireless communications.^[6]

Related distributions

• Restricting m to the unit interval (q = m; 0 < q < 1)^{[dubious – discuss]} defines the Nakagami-q distribution, also known as Hoyt distribution, first studied by R.S. Hoyt in the 1940s.^[7][8][9] 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 ${\displaystyle m={\tfrac {1}{2}}}$, 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

• Mathematics portal

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