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 ${\displaystyle m\geq 1/2}${\displaystyle m\geq 1/2} and a scale parameter ${\displaystyle \Omega >0}${\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.

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

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

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

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

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

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

and

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

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

An alternative way of fitting the distribution is to re-parametrize ${\displaystyle \Omega }${\displaystyle \Omega } as σ = Ω/m.^[3]

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

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

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

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

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

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

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

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

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

For a chi-distribution, the degrees of freedom ${\displaystyle 2m}${\displaystyle 2m} must be an integer, but for Nakagami the ${\displaystyle m}${\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.

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]

• 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 ${\displaystyle m={\tfrac {1}{2}}}${\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.

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