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

Nakagami

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

Characterization

Its probability density function (pdf) is[1]

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

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

Its cumulative distribution function is[1]

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

where P is the incomplete gamma function (regularized).

Parametrization

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

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

and

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

Parameter estimation

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

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

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

Its logarithm is

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

Therefore

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

These derivatives vanish only when

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

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, 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 ${\displaystyle k=m}$, ${\displaystyle \theta =\Omega /m}$, and taking the square root of ${\displaystyle Y}$:

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

Alternatively, 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.

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

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

Finally, there is also a more efficient generation method using efficient rejection-sampling.[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] and to study the impact of fading channels on wireless communications.[7]

Related distributions

• Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as Hoyt distribution.[8][9][10]

"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

1. 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. (2012). "Almost rejectionless sampling from Nakagami-m distributions (m≥1)". Electronics Letters (Submitted manuscript). 48 (24): 1559–1561. doi:10.1049/el.2012.3513. hdl:10016/16643.
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., doi:10.1016/B978-0-08-009306-2.50005-4
6. Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
7. 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.
8. Paris, J.F. (2009). "Nakagami-q (Hoyt) distribution function with applications". Electronics Letters. 45 (4): 210. doi:10.1049/el:20093427.
9. "HoytDistribution".
10. "NakagamiDistribution".