Nakagami distribution#Hoyt

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{{Probability distribution |

name =Nakagami|

type =density|

pdf_image =325px|

cdf_image =325px|

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 = $\Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)$ |

skewness =|

kurtosis =|

entropy =|

mgf =|

char =|

}}

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.

Characterization

Its probability density function (pdf) is{{cite web

| last =Laurenson

| first =Dave

| title =Nakagami Distribution

| work =Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques

| date =1994

| url =https://www.era.lib.ed.ac.uk/bitstream/handle/1842/12397/Laurensen1994.Pdf?sequence=1&isAllowed=y

| access-date = 2007-08-04 }}

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

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 $\Omega$ areR. Kolar, R. Jirik, J. Jan (2004) [http://www.radioeng.cz/fulltexts/2004/04_01_08_12.pdf "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography"], Radioengineering, 13 (1), 8–12

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

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.{{cite journal|last=Mitra|first=Rangeet|author2=Mishra, Amit Kumar |author3=Choubisa, Tarun |title=Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution|journal=International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012 |date=2012|pages=9–12}}

Given independent observations $X_1=x_1,\ldots,X_n=x_n$ 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

: $\begin{align}
\frac{\partial\ell}{\partial\sigma} = \frac{-nm\sigma+\sum_{i=1}^n x_i^2}{\sigma^2} \quad \text{and} \quad \frac{\partial\ell}{\partial m} = -n\frac{\Gamma'(m)}{\Gamma(m)} -n \log\sigma + 2\sum_{i=1}^n \log x_i.
\end{align}$

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.

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

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.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., {{doi|10.1016/B978-0-08-009306-2.50005-4}} It has been used to model attenuation of wireless signals traversing multiple pathsParsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley. and to study the impact of fading channels on wireless communications.{{Cite book|author1=Ramon Sanchez-Iborra |author2=Maria-Dolores Cano |author3=Joan Garcia-Haro |title=2013 World Congress on Computer and Information Technology (WCCIT) |chapter=Performance evaluation of QoE in VoIP traffic under fading channels |pages=1–6 |year=2013|doi=10.1109/WCCIT.2013.6618721 |isbn=978-1-4799-0462-4 |s2cid=16810288 }}

Related distributions

Restricting m to the unit interval (q = m; 0 < q < 1){{dubious|reason=above it is said that Nakagami first parameter should be larger than 1/2, while here it is allowed to run below 1/2, being it set equal to q and q let to lie in the unit interval|date=June 2022}} defines the Nakagami-q distribution, also known as {{vanchor|Hoyt}} distribution, first studied by R.S. Hoyt in the 1940s.{{cite journal |title=Nakagami-q (Hoyt) distribution function with applications |journal=Electronics Letters |volume=45 |issue=4 |pages=210–211 |doi=10.1049/el:20093427 |year=2009 |last1=Paris|first1=J.F. |bibcode=2009ElL....45..210P }}{{cite web |title=HoytDistribution |url=https://reference.wolfram.com/language/ref/HoytDistribution.html}}{{cite web |title=NakagamiDistribution |url=https://reference.wolfram.com/language/ref/NakagamiDistribution.html}} 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.

References

Category:Continuous distributions