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
and a second parameter controlling spread
.
Characterization
Its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
(pdf) is
:
where
Its
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
is
:
where ''P'' is the regularized (lower)
incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which ...
.
Parametrization
The parameters
and
are
:
and
:
Parameter estimation
An alternative way of fitting the distribution is to re-parametrize
and ''m'' as ''σ'' = Ω/''m'' and ''m''.
Given
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
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
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
.
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
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
of maximum-likelihood estimation, one then obtains the MLE for Ω as well.
Generation
The Nakagami distribution is related to the
gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
.
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
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
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.
History and applications
The Nakagami distribution is relatively new, being first proposed in 1960. It has been used to model attenuation of
wireless
Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...
signals
traversing multiple paths and to study the impact of
fading
In wireless communications, fading is variation of the attenuation of a signal with various variables. These variables include time, geographical position, and radio frequency. Fading is often modeled as a random process. A fading channel is a ...
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
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
around the true mean in a bivariate normal
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One ...
random variable, re-written in polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
(radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable does."
* With ''2m = k'', the Nakagami distribution gives a scaled
chi distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
.
* With
, the Nakagami distribution gives a scaled
half-normal distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the hal ...
.
* A Nakagami distribution is a particular form of
generalized gamma distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many dis ...
, with ''p = 2'' and ''d = 2m''
See also
*
Normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
*
Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
*
Modified half-normal distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the hal ...
*
*
Reciprocal normal distribution
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for sc ...
*
Ratio normal distribution
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions.
Given two (usually independent) random variables ''X'' ...
*
Standard normal table A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is used to find the probability that ...
*
Sub-Gaussian distribution In probability theory, a sub-Gaussian distribution is a probability distribution with strong tail decay. Informally, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gi ...
References
{{ProbDistributions, continuous-semi-infinite
Continuous distributions