The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It is used to model physical phenomena, such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.
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The family of Nakagami distributions has two parameters: a shape parameter and a second parameter controlling spread .
Its probability density function (pdf) is[1]
where and .
Its cumulative distribution function (CDF) is[1]
where P is the regularized (lower) incomplete gamma function.
An alternative way of fitting the distribution is to re-parametrize as σ = Ω/m.[3]
Given independent observations from the Nakagami distribution, the likelihood function is
Its logarithm is
Therefore
These derivatives vanish only when
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 , it is possible to obtain a random variable , by setting , , and taking the square root of :
Alternatively, the Nakagami distribution can be generated from the chi distribution with parameter set to and then following it by a scaling transformation of random variables. That is, a Nakagami random variable is generated by a simple scaling transformation on a Chi-distributed random variable as below.
For a Chi-distribution, the degrees of freedom must be an integer, but for Nakagami the 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]
Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04. 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.