Nakagami distribution

Nakagami

Probability density function File:Nakagami pdf.svg
Cumulative distribution function File:Nakagami cdf.svg
Parameters	$m\text{ or }\mu \ge 0.5$ shape (real) $\mathrm{\Omega }\text{ or }\omega >0$ scale (real)
Support	$x>0$
PDF	$\frac{2m^m}{\mathrm{\Gamma }(m){\mathrm{\Omega }}^m}{x}^{2m-1}\exp (-\frac{m}{\mathrm{\Omega }}{x}^2)$
CDF	$\frac{\gamma (m,\frac{m}{\mathrm{\Omega }}{x}^2)}{\mathrm{\Gamma }(m)}$
Mean	$\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}{\left(\frac{\mathrm{\Omega }}{m}\right)}^{1/2}$
Median	No simple closed form
Mode	${\left(\frac{(2m-1)\mathrm{\Omega }}{2m}\right)}^{1/2}$
Variance	$\mathrm{\Omega }(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2)$

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

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

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 $Y\sim \text{<mrow data-mjx-texclass="ORD">Gamma</mrow>}(k,\theta )$, it is possible to obtain a random variable $X\sim \text{<mrow data-mjx-texclass="ORD">Nakagami</mrow>}(m,\mathrm{\Omega })$, 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) 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 $m=\frac{1}{2}$, 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