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Parameters Probability density function Cumulative distribution function ${\displaystyle m\ or\ \mu >=0.5}$ shape (real) ${\displaystyle \Omega \ or\ \omega >0}$ spread (real) ${\displaystyle x>0\!}$ ${\displaystyle {\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right)}$ ${\displaystyle {\frac {\gamma \left(m,{\frac {m}{\Omega }}x^{2}\right)}{\Gamma (m)}}}$ ${\displaystyle {\frac {\Gamma (m+{\frac {1}{2}})}{\Gamma (m)}}\left({\frac {\Omega }{m}}\right)^{1/2}}$ No simple closed form ${\displaystyle {\frac {\sqrt {2}}{2}}\left({\frac {(2m-1)\Omega }{m}}\right)^{1/2}}$ ${\displaystyle \Omega \left(1-{\frac {1}{m}}\left({\frac {\Gamma (m+{\frac {1}{2}})}{\Gamma (m)}}\right)^{2}\right)}$

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

## Characterization

Its probability density function (pdf) is[1]

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

where P is the incomplete gamma function (regularized).

Differential equation

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

## Parameter estimation

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

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

and

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

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

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

and

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

It is reported by authors[who?] 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 ${\displaystyle Y\,\sim {\textrm {Gamma}}(k,\theta )}$, 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 ${\displaystyle Y}$:

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

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:

${\displaystyle X={\sqrt {(\Omega /2m)}}Y.}$
Currently, the more efficient generation method is provided in.[4]

## History and applications

The Nakagami distribution is relatively new, being first proposed in 1960.[5] It has been used to model attenuation of wireless signals traversing multiple paths.[6]

## Related distributions

• Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as Hoyt distribution.[7][8][9]

"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."[10]

## References

1. ^ a b Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
2. ^ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
3. ^ Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9–12.
4. ^ Luengo, D.; Martino, L. "Almost rejectionless sampling from Nakagami-m distributions (m≥1)". Electronics Letters 48 (24): 1559–1561. doi:10.1049/el.2012.3513.
5. ^ Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18–20, 1958, pp 3-36. Pergamon Press.
6. ^ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
7. ^ "Nakagami-q (Hoyt) distribution function with applications". doi:10.1049/el:20093427.
8. ^
9. ^
10. ^ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package ‘shotGroups’ version 0.6.2)".

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Nakagami_distribution — Please support Wikipedia.

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