Rice distribution

Probability density function
Cumulative distribution function
Parameters ν ≥ 0 — distance between the reference point and the center of the bivariate distribution,
σ ≥ 0 — scale
Support x ∈ [0, +∞)
PDF \frac{x}{\sigma^2}\exp\left(\frac{-(x^2%2B\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)
CDF 1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)

where Q1 is the Marcum Q-function
Mean \sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)
Variance 2\sigma^2%2B\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)
Skewness (complicated)
Ex. kurtosis (complicated)

In probability theory, the Rice distribution or Rician distribution is the probability distribution of the absolute value of a circular bivariate normal random variable with potentially non-zero mean. It was named after Stephen O. Rice.

Contents

Characterization

The probability density function is

f(x|\nu,\sigma) = \frac{x}{\sigma^2}\exp\left(\frac{-(x^2%2B\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right),

where I0(z) is the modified Bessel function of the first kind with order zero. When v = 0, the distribution reduces to a Rayleigh distribution.

The characteristic function is:[1][2]

\begin{align} \chi_X(t|\nu,\sigma) & = \exp \left( -\frac{\nu^2}{2\sigma^2} \right) \left[ \Psi_2 \left( 1; 1, \frac{1}{2}; \frac{\nu^2}{2\sigma^2}, -\frac{1}{2} \sigma^2 t^2 \right) \right. \\[8pt] & \left. {} \quad %2B i \sqrt{2} \sigma t \Psi_2 \left( \frac{3}{2}; 1, \frac{3}{2}; \frac{\nu^2}{2\sigma^2}, -\frac{1}{2} \sigma^2 t^2 \right) \right], \end{align}

where \Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right) is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of x and y. It is given by:[3][4]

\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right) = \sum_{n=0}^{\infty}\sum_{m=0}^\infty \frac{(\alpha)_{m%2Bn}}{(\gamma)_m(\gamma')_n} \frac{x^m y^n}{m!n!},

where

(x)_n = x(x%2B1)\cdots(x%2Bn-1) = \frac{\Gamma(x%2Bn)}{\Gamma(x)}

is the rising factorial.

Properties

Moments

The first few raw moments are:

\mu_1^'= \sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)
\mu_2^'= 2\sigma^2%2B\nu^2\,
\mu_3^'= 3\sigma^3\sqrt{\pi/2}\,\,L_{3/2}(-\nu^2/2\sigma^2)
\mu_4^'= 8\sigma^4%2B8\sigma^2\nu^2%2B\nu^4\,
\mu_5^'=15\sigma^5\sqrt{\pi/2}\,\,L_{5/2}(-\nu^2/2\sigma^2)
\mu_6^'=48\sigma^6%2B72\sigma^4\nu^2%2B18\sigma^2\nu^4%2B\nu^6\,

where Lq(x) denotes a Laguerre polynomial:

L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\,_1F_1(-q;1;x)

where M(a,b,z) = _1F_1(a;b;z) is the confluent hypergeometric function of the first kind.

For the case q = 1/2:

\begin{align} L_{1/2}(x) &=\,_1F_1\left( -\frac{1}{2};1;x\right) \\ &= e^{x/2} \left[\left(1-x\right)I_0\left(\frac{-x}{2}\right) -xI_1\left(\frac{-x}{2}\right) \right]. \end{align}

Generally the moments are given by

\mu_k^'=s^k2^{k/2}\,\Gamma(1\!%2B\!k/2)\,L_{k/2}(-\nu^2/2\sigma^2), \,

where s = σ1/2.

When k is even, the moments become actual polynomials in σ and ν.

The second central moment, equals the variance equation below (which is listed to the right):

\mu_2= 2\sigma^2%2B\nu^2-(\pi\sigma^2/2)\,L^2_{1/2}(-\nu^2/2\sigma^2)

Note that L^2_{1/2}(\cdot) indicates the square of the Laguerre polynomial L_{1/2}(\cdot), not the generalized Laguerre polynomial L^{(2)}_{1/2}(\cdot).

When the Rice distribution parameter ν = 0, the distribution becomes the Rayleigh distribution.

\mu_2= 2\sigma^2%2B0^2-(\pi\sigma^2/2)\,L^2_{1/2}(0)
L^2_{1/2}(0) = (e^0[(1-0)I_0(0)-0I_1(0)])^2
L^2_{1/2}(0) = (1 \cdot [I_0(0)])^2
L^2_{1/2}(0) = (1 \cdot 1)^2=1
\mu_2= 2\sigma^2-(\pi\sigma^2/2)\,
\mu_2= (4\sigma^2-\pi\sigma^2)/2\,
\mu_2= \frac{4-\pi}{2}\sigma^2\,

which is the variance of the Rayleigh distribution.

Related distributions

1. Generate P having a Poisson distribution with parameter (also mean, for a Poisson) \lambda = \frac{\nu^2}{2\sigma^2}.
2. Generate X having a chi-squared distribution with 2P + 2 degrees of freedom.
3. Set R = \sigma\sqrt{X}.

Limiting cases

For large values of the argument, the Laguerre polynomial becomes (see Abramowitz and Stegun §13.5.1)

\lim_{x\rightarrow -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1%2B\nu)}.

It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ2.

Parameter estimation (the Koay inversion technique)

There are three different methods for estimating the Rice parameters, (1) method of moments, (2) method of maximum likelihood, and (3) method of least squares. The first two methods have been investigated by Talukdar et al.[6] and Bonny et al.[7] and Sijbers et al.[8]

Here the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ1' and the sample standard deviation is an estimate of μ21/2.

The following is an efficient method, known as the "Koay inversion technique", published by Koay et al.[9] for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier works [10][11] on the method of moments usually use a root-finding method to solve the problem, which is not efficient.

First, the ratio of the sample mean to the sample standard deviation is defined as r, i.e., r=\mu^{'}_1/\mu^{1/2}_2. The fixed point formula of SNR is expressed as

g(\theta) = \sqrt{ \xi{(\theta)} \left[ 1%2Br^2\right] - 2},

where \theta is the ratio of the parameters, i.e., \theta = \frac{\nu}{\sigma}, and \xi{\left(\theta\right)} is given by:

\xi{\left(\theta\right)} = 2 %2B \theta^2 - \frac{\pi}{8} \exp{(-\theta^2/2)}\left[ (2%2B\theta^2) I_0 (\theta^2/4) %2B \theta^2 I_1(\theta^{2}/4)\right]^2,

where I_0 and I_1 are modified Bessel functions of the first kind.

Note that \xi{\left(\theta\right)} is a scaling factor of \sigma and is related to \mu_{2} by:

\mu_2 = \xi{\left(\theta\right)} \sigma^2.\,

To find the fixed point, \theta^{*} , of g , an initial solution is selected, {\theta}_{0} , that is greater than the lower bound, which is {\theta}_{\mathrm{lower bound}} = 0 and occurs when r = \sqrt{\pi/(4-\pi)} (Notice that this is the r=\mu^{'}_1/\mu^{1/2}_2 of a Rayleigh). This provides a starting point for the iteration, which uses functional composition, and this continues until \left|g^{i}\left(\theta_{0}\right)-\theta_{i-1}\right| is less than some small positive value. Here, g^{i} denotes the composition of the same function, g, i-th times. In practice, we associate the final \theta_{n} for some integer n as the fixed point, \theta^{*}, i.e., \theta^{*} = g\left(\theta^{*}\right).

Once the fixed point is found, the estimates \nu and \sigma are found through the scaling function, \xi{\left(\theta\right)} , as follows:

\sigma = \frac{\mu^{1/2}_{2}}{\sqrt{\xi\left(\theta^{*}\right)}} ,

and

\nu = \sqrt{\left( \mu^{'~2}_{1} %2B \left(\xi\left(\theta^{*}\right) - 2\right)\sigma^{2} \right)} .

To speed up the iteration even more, one can use the Newton's method of root-finding as presented by Koay et al.[12] This particular approach is highly efficient.

The author has also provided an online calculator for computing the fixed point, which is also known as the underlying SNR from r=\mu^{'}_{1}/\mu^{1/2}_{2}, the magnitude SNR. See the link here under the subtitle called HI-SPEED SNR Analysis I. Note that the number of combined channel is 1 for the Rician distribution.

See also

Notes

  1. ^ Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).
  2. ^ Annamalai 2000 (in a sum of infinite series).
  3. ^ Erdelyi 1953.
  4. ^ Srivastava 1985.
  5. ^ Richards, M.A., Rice Distribution for RCS, Georgia Institute of Technology (Sep 2006)
  6. ^ Talukdar 1991
  7. ^ Bonny 1996
  8. ^ Sijbers 1998
  9. ^ Koay 2006 (known as the SNR fixed point formula).
  10. ^ Talukdar 1991
  11. ^ Abdi 2001
  12. ^ Koay 2006 (in Appendix A).

References

External links
 
Benini · Benktander 1st kind · Benktander 2nd kind · Beta prime · Bose–Einstein · Burr · chi-squared · chi · Coxian · Dagum · Davis · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-squared · hyper-exponential · hypoexponential · inverse chi-squared (scaled-inverse-chi-squared) · inverse Gaussian · inverse gamma · Kolmogorov · Lévy · log-Cauchy · log-Laplace · log-logistic · log-normal · Maxwell–Boltzmann · Maxwell speed · Mittag–Leffler · Nakagami · noncentral chi-squared · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda