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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 second parameter controlling spread $$\Omega >0}.$$

Characterization

Its probability density function (pdf) is

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

where $$(m\geq 1/2,{\text{ and }}\Omega >0)}$$

Its cumulative distribution function is

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

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

The parameters m {\displaystyle m} m and Ω {\displaystyle \Omega } \Omega are

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

and

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

Parameter estimation

An alternative way of fitting the distribution is to re-parametrize $$\Omega$$ and m as σ = Ω/m and m.

Given independent observations $${\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}}$$ 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{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

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

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, one then obtains the MLE for Ω as well.
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 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.

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

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. 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.

"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."

References

Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
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.
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
Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
Ramon Sanchez-Iborra; Maria-Dolores Cano; Joan Garcia-Haro (2013). Performance evaluation of QoE in VoIP traffic under fading channels. World Congress on Computer and Information Technology (WCCIT). pp. 1–6. doi:10.1109/WCCIT.2013.6618721. ISBN 978-1-4799-0462-4.
Paris, J.F. (2009). "Nakagami-q (Hoyt) distribution function with applications". Electronics Letters. 45 (4): 210. doi:10.1049/el:20093427.
"HoytDistribution".
"NakagamiDistribution".

Mathematics Encyclopedia