Probability density function
Cumulative distribution function
|Parameters|| shape (real)
The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter and a second parameter controlling spread, .
where P is the incomplete gamma function (regularized).
The parameters and are
An alternative way of fitting the distribution is to re-parametrize and m as σ = Ω/m and m. 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:
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.
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 :
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:
History and applications
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- 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.
- Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.