Rice distribution

In the 2D plane, pick a fixed point at distance ν from the origin. Generate a distribution of 2D points centered around that point, where the x and y coordinates are chosen independently from a gaussian distribution with standard deviation σ (blue region). If R is the distance from these points to the origin, then R has a 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	$\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)$
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	$2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)$
Skewness	(complicated)
Ex. kurtosis	(complicated)

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

Characterization

The probability density function is

f(x\mid\nu,\sigma) = \frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\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.

The characteristic function is:^[1]^[2]

\begin{align} &\chi_X(t\mid\nu,\sigma) \\ & \quad = \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. {} \qquad + 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+n}}{(\gamma)_m(\gamma')_n} \frac{x^m y^n}{m!n!},

where

(x)_n = x(x+1)\cdots(x+n-1) = \frac{\Gamma(x+n)}{\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+\nu^2\,

\mu_3^{'}= 3\sigma^3\sqrt{\pi/2}\,\,L_{3/2}(-\nu^2/2\sigma^2)

\mu_4^{'}= 8\sigma^4+8\sigma^2\nu^2+\nu^4\,

\mu_5^{'}=15\sigma^5\sqrt{\pi/2}\,\,L_{5/2}(-\nu^2/2\sigma^2)

\mu_6^{'}=48\sigma^6+72\sigma^4\nu^2+18\sigma^2\nu^4+\nu^6\,

and, in general, the raw moments are given by

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

Here L_q(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. When k is even, the raw moments become simple polynomials in σ and ν, as in the examples above.

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}

The second central moment, the variance, is

\mu_2= 2\sigma^2+\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 $L^{(2)}_{1/2}(\cdot).$

Differential equation

The pdf of the Rice distribution is a solution of the following differential equation:

\left\{\begin{array}{l} \sigma ^4 x^2 f''(x)+\left(2\sigma^2 x^3-\sigma^4 x\right) f'(x)+f(x) \left(\sigma ^4-v^2 x^2+x^4\right)=0 \\[10pt] f(1)=\frac{\exp\left(-\frac{v^2+1}{2\sigma^2}\right) I_0\left(\frac{v}{\sigma^2}\right)}{\sigma^2} \\[10pt] f'(1)=\frac{\exp\left(-\frac{v^2+1}{2 \sigma ^2}\right) \left(\left(\sigma^2-1\right) I_0\left(\frac{v}{\sigma ^2}\right)+v I_1\left(\frac{v}{\sigma^2}\right)\right)}{\sigma^4} \end{array}\right\}

Related distributions

$R \sim \mathrm{Rice}\left(\nu,\sigma\right)$ has a Rice distribution if $R = \sqrt{X^2 + Y^2}$ where $X \sim N\left(\nu\cos\theta,\sigma^2\right)$ and $Y \sim N\left(\nu \sin\theta,\sigma^2\right)$ are statistically independent normal random variables and $\theta$ is any real number.

Another case where $R \sim \mathrm{Rice}\left(\nu,\sigma\right)$ comes from the following steps:

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}.

If $R \sim \text{Rice}\left(\nu,1\right)$ then $R^2$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter $\nu^2$ .
If $R \sim \text{Rice}\left(\nu,1\right)$ then $R$ has a noncentral chi distribution with two degrees of freedom and noncentrality parameter $\nu$ .
If $R \sim \text{Rice}\left(0,\sigma\right)$ then $R \sim \text{Rayleigh}\left(\sigma\right)$ , i.e., for the special case of the Rice distribution given by ν = 0, the distribution becomes the Rayleigh distribution, for which the variance is $\mu_2= \frac{4-\pi}{2}\sigma^2$ .
If $R \sim \text{Rice}\left(0,\sigma\right)$ then $R^2$ has an exponential distribution.^[5]

Limiting cases

For large values of the argument, the Laguerre polynomial becomes^[6]

\lim_{x\rightarrow -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1+\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 parameters of the Rice distribution, (1) method of moments,^[7]^[8]^[9]^[10] (2) method of maximum likelihood,^[7]^[8]^[9] and (3) method of least squares. In the first two methods 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 μ₂^1/2.

The following is an efficient method, known as the "Koay inversion technique".^[11] 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^[7]^[12] 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+r^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 + \theta^2 - \frac{\pi}{8} \exp{(-\theta^2/2)}\left[ (2+\theta^2) I_0 (\theta^2/4) + \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 $r = \sqrt{\pi/(4-\pi)}$ ^[11] (Notice that this is the $r=\mu^{'}_1/\mu^{1/2}_2$ of a Rayleigh distribution). 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, $g$ , $i$ 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 + \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.^[11] This particular approach is highly efficient.

Applications

The Euclidean norm of a bivariate normally distributed random vector.
Rician fading
Effect of sighting error on target shooting.^[13]

Notes

↑ Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).
↑ Annamalai 2000 (in a sum of infinite series).
↑ Erdelyi 1953.
↑ Srivastava 1985.
↑ Richards, M.A., Rice Distribution for RCS, Georgia Institute of Technology (Sep 2006)
↑ Abramowitz and Stegun (1968) §13.5.1
1 2 3 Talukdar et al. 1991
1 2 Bonny et al. 1996
1 2 Sijbers et al. 1998
↑ den Dekker and Sijbers 2014
1 2 3 Koay et al. 2006 (known as the SNR fixed point formula).
↑ Abdi 2001
↑ "Ballistipedia". Retrieved 4 May 2014.

References

Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. ISBN 0-486-61272-4
Rice, S. O., Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945) 46–156.
I. Soltani Bozchalooi and Ming Liang (20 November 2007). "A smoothness index-guided approach to wavelet parameter selection in signal de-noising and fault detection". Journal of Sound and Vibration 308 (1–2): 253–254. doi:10.1016/j.jsv.2007.07.038.
Liu, X. and Hanzo, L., A Unified Exact BER Performance Analysis of Asynchronous DS-CDMA Systems Using BPSK Modulation over Fading Channels, IEEE Transactions on Wireless Communications, Volume 6, Issue 10, October 2007, Pages 3504–3509.
Annamalai, A., Tellambura, C. and Bhargava, V. K., Equal-Gain Diversity Receiver Performance in Wireless Channels, IEEE Transactions on Communications,Volume 48, October 2000, Pages 1732–1745.
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, Volume 1. McGraw-Hill Book Company Inc., 1953.
Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series. Ellis Horwood Ltd., 1985.
Sijbers J., den Dekker A. J., Scheunders P. and Van Dyck D., "Maximum Likelihood estimation of Rician distribution parameters", IEEE Transactions on Medical Imaging, Vol. 17, Nr. 3, p. 357–361, (1998)
den Dekker, A.J., and Sijbers, J (December 2014). "Data distributions in magnetic resonance images: a review". Physica Medica 30 (7): 725–741. doi:10.1016/j.ejmp.2014.05.002.
Koay, C.G. and Basser, P. J., Analytically exact correction scheme for signal extraction from noisy magnitude MR signals, Journal of Magnetic Resonance, Volume 179, Issue = 2, p. 317–322, (2006)
Abdi, A., Tepedelenlioglu, C., Kaveh, M., and Giannakis, G. On the estimation of the K parameter for the Rice fading distribution, IEEE Communications Letters, Volume 5, Number 3, March 2001, Pages 92–94.
Talukdar, K.K., and Lawing, William D. (March 1991). "Estimation of the parameters of the Rice distribution". Journal of the Acoustical Society of America 89 (3): 1193–1197. doi:10.1121/1.400532.
Bonny,J.M., Renou, J.P., and Zanca, M. (November 1996). "Optimal Measurement of Magnitude and Phase from MR Data". Journal of Magnetic Resonance, Series B 113 (2): 136–144. doi:10.1006/jmrb.1996.0166.

External links

MATLAB code for Rice/Rician distribution (PDF, mean and variance, and generating random samples)

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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