Rice distribution

{{short description|Probability distribution}} [[File:Rice distribution motivation.svg|thumb|300px|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 distribution | type = continuous | pdf_image = 325px|Rice probability density functions ''σ'' = 1.0 | cdf_image = 325px|Rice cumulative distribution functions ''σ'' = 1.0 | notation = <math>\mathrm{Rice}(\nu,\sigma)</math> | parameters = <math>\nu\ge 0</math>, distance between the reference point and the center of the bivariate distribution, <math>\sigma\ge 0</math>, scale | support = <math>x \in [0,\infty)</math> | pdf = <math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)</math> | cdf = <math>1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)</math> where ''Q''1 is the Marcum Q-function | mean = <math>\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math> | median = | mode = | variance = <math>2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)</math> | skewness = (complicated) | kurtosis = (complicated) | entropy = | mgf = | cf = }}

In probability theory, the '''Rice distribution''' or '''Rician distribution''' (or, less commonly, '''Ricean distribution''') is the probability distribution of the magnitude of a circularly symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice (1907–1986).

== Characterization == The probability density function is : <math> 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)H(x),</math> where ''I''0(''z'') is the modified Bessel function of the first kind with order zero, and H(x) is the Heaviside unit step.<ref>Johnson, Norman L., Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous univariate distributions, volume 1. Vol. 1. John wiley & sons, 1994.</ref>

In the context of Rician fading, the distribution is often also rewritten using the ''shape parameter'' <math>K = \frac{\nu^2}{2\sigma^2}</math>, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the ''scale parameter'' <math> \Omega = \nu^2+2\sigma^2 </math>, defined as the total power received in all paths.<ref>Abdi, A. and Tepedelenlioglu, C. and Kaveh, M. and Giannakis, G., [https://dx.doi.org/10.1109/4234.913150 "On the estimation of the K parameter for the Rice fading distribution]", ''IEEE Communications Letters'', March 2001, p. 92–94</ref>

The characteristic function of the Rice distribution is given as:<ref>Liu 2007 (in one of Horn's confluent hypergeometric functions with two variables).</ref><ref>Annamalai 2000 (in a sum of infinite series).</ref> : <math> \begin{align} \chi_X(t\mid\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. {} + 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} </math> where <math>\Psi_2 \left( \alpha; \gamma, \gamma'; x, y \right)</math> is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of <math>x</math> and {{tmath| y }}. It is given by:<ref>Erdelyi 1953.</ref><ref>Srivastava 1985.</ref> : <math>\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!},</math> where : <math>(x)_n = x(x+1)\cdots(x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)}</math> is the rising factorial.

== Properties ==

=== Moments === The first few raw moments are: : <math>\begin{align} \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 \end{align}</math> and, in general, the raw moments are given by : <math>\mu_k^{'}=\sigma^k2^{k/2}\,\Gamma(1\!+\!k/2)\,L_{k/2}(-\nu^2/2\sigma^2).</math>

Here {{tmath| L_q(x) }} denotes a Laguerre polynomial: : <math>L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\,_1F_1(-q;1;x)</math> where <math>M(a,b,z) = _1F_1(a;b;z)</math> is the confluent hypergeometric function of the first kind. When {{tmath| k }} is even, the raw moments become simple polynomials in {{tmath| \sigma }} and {{tmath| \nu }}, as in the examples above.

For the case {{tmath|1= q = 1/2}}: : <math> \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} </math>

The second central moment, the variance, is : <math>\mu_2= 2\sigma^2+\nu^2-(\pi\sigma^2/2)\,L^2_{1/2}(-\nu^2/2\sigma^2) .</math>

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

== Related distributions == * <math>R \sim \mathrm{Rice}\left(|\nu|,\sigma\right)</math> if <math>R = \sqrt{X^2 + Y^2}</math> where <math>X \sim N\left(\nu\cos\theta,\sigma^2\right)</math> and <math>Y \sim N\left(\nu \sin\theta,\sigma^2\right)</math> are statistically independent normal random variables and <math>\theta</math> is any real number. * Another case where <math>R \sim \mathrm{Rice}\left(\nu,\sigma\right)</math> comes from the following steps: *# Generate <math>P</math> having a Poisson distribution with parameter (also mean, for a Poisson) {{tmath|1= \textstyle \lambda = \frac{\nu^2}{2\sigma^2} }}. *# Generate <math>X</math> having a chi-squared distribution with {{tmath| 2P + 2 }} degrees of freedom. *# Set <math>R = \sigma\sqrt{X}.</math> * If <math>R \sim \operatorname{Rice}(\nu,1)</math> then <math>R^2</math> has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter {{tmath| \nu^2 }}. * If <math>R \sim \operatorname{Rice}(\nu,1)</math> then <math>R</math> has a noncentral chi distribution with two degrees of freedom and noncentrality parameter {{tmath| \nu }}. * If <math>R \sim \operatorname{Rice}(0,\sigma)</math> then {{tmath| R \sim \operatorname{Rayleigh}(\sigma) }}, i.e., for the special case of the Rice distribution given by <math>\nu = 0</math>, the distribution becomes the Rayleigh distribution, for which the variance is {{tmath|1= \textstyle \mu_2= \frac{4-\pi}{2}\sigma^2 }}. * If <math>R \sim \operatorname{Rice}(0,\sigma)</math> then <math>R^2</math> has an exponential distribution.<ref>Richards, M.A., [http://users.ece.gatech.edu/mrichard/Rice%20power%20pdf.pdf Rice Distribution for RCS], Georgia Institute of Technology (Sep 2006)</ref> * If <math>R \sim \operatorname{Rice}\left(\nu,\sigma\right)</math> then <math>1/R</math> has an Inverse Rician distribution.<ref>Jones, Jessica L., Joyce McLaughlin, and Daniel Renzi. [https://iopscience.iop.org/article/10.1088/1361-6420/aa6163/ampdf "The noise distribution in a shear wave speed image computed using arrival times at fixed spatial positions."], Inverse Problems 33.5 (2017): 055012.</ref> * The folded normal distribution is the univariate restriction of the Rice distribution.

== Limiting cases == For large values of the argument, the Laguerre polynomial becomes<ref>Abramowitz and Stegun (1968) [http://www.math.sfu.ca/~cbm/aands/page_508.htm §13.5.1]</ref> : <math>\lim_{x \to -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1+\nu)}.</math>

It is seen that as {{tmath| \nu }} becomes large or {{tmath| \sigma }} becomes small, the mean becomes {{tmath| \nu }} and the variance becomes {{tmath| \sigma^2 }}.

The transition to a Gaussian approximation proceeds as follows. From Bessel function theory we have : <math> I_\alpha(z) \to \frac{e^z}{\sqrt{2\pi z}} \left(1 - \frac{4 \alpha^2 - 1}{8z} + \cdots \right) \text { as } z \rightarrow \infty </math> so, in the large <math> x\nu/\sigma^2 </math> region, an asymptotic expansion of the Rician distribution: : <math> \begin{align} f(x,\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) \\ \text{ is } \\ & \frac{x}{\sigma^2}\exp\left(\frac{-(x^2 + \nu^2)} {2\sigma^2}\right) \sqrt{\frac{\sigma^2}{2\pi x \nu}} \exp \left( \frac {2x \nu}{2\sigma^2} \right) \left(1 + \frac{\sigma^2}{8x\nu} + \cdots \right)\\ \rightarrow {} & \frac{1}{\sigma \sqrt{2 \pi}}\exp\left(-\frac{(x - \nu)^2} {2\sigma^2}\right) \sqrt{ \frac{x}{\nu} } , \;\;\; \text{ as } \frac{x\nu}{\sigma^2} \rightarrow \infty \end{align} </math>

Moreover, when the density is concentrated around <math display="inline"> \nu </math> and <math display="inline">|x - \nu| \ll \sigma </math> because of the Gaussian exponent, we can also write <math display="inline"> \sqrt{ {x}/{\nu} } \approx 1 </math> and finally get the Normal approximation : <math> f(x,\nu,\sigma) \approx \frac{1}{\sigma \sqrt{2\pi}} \exp\left(- \frac{(x - \nu)^2}{2\sigma^2}\right) , \;\;\; \frac{\nu}{\sigma} \gg 1</math> The approximation becomes usable for {{tmath| {\nu} / {\sigma} > 3 }}.

== Parameter estimation (Koay inversion technique) == There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,<ref name=T>Talukdar et al. 1991</ref><ref name=B>Bonny et al. 1996</ref><ref name=S>Sijbers et al. 1998</ref><ref name=S2>den Dekker and Sijbers 2014</ref> (2) method of maximum likelihood,<ref name=T/><ref name=B/><ref name=S/><ref name=V>Varadarajan and Haldar 2015</ref> and (3) method of least squares.{{citation needed|date=June 2012}} In the first two methods the interest is in estimating the parameters of the distribution, {{tmath| \nu }} and {{tmath| \sigma }}, 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 {{tmath| \mu_1^{'} }} and the sample standard deviation is an estimate of {{tmath| \mu_2^{1/2} }}.

The following is an efficient method, known as the "Koay inversion technique".<ref name=K>Koay et al. 2006 (known as the SNR fixed point formula).</ref> 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<ref name=T/><ref>Abdi 2001</ref> 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 {{tmath| r }}, i.e., {{tmath|1= r=\mu^{'}_1/\mu^{1/2}_2 }}. The fixed point formula of SNR is expressed as : <math> g(\theta) = \sqrt{ \xi{(\theta)} \left[ 1+r^2\right] - 2},</math> where <math> \theta</math> is the ratio of the parameters, i.e., {{tmath|1= \theta = {\nu}/{\sigma} }}, and <math>\xi{\left(\theta\right)}</math> is given by: : <math> \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,</math> where <math>I_0</math> and <math>I_1</math> are modified Bessel functions of the first kind.

Note that <math> \xi{\left(\theta\right)} </math> is a scaling factor of <math>\sigma</math> and is related to <math>\mu_{2}</math> by: : <math> \mu_2 = \xi{\left(\theta\right)} \sigma^2. </math>

To find the fixed point, {{tmath| \theta^{*} }}, of {{tmath| g }}, an initial solution is selected, {{tmath| {\theta}_{0} }}, that is greater than the lower bound, which is <math> {\theta}_{\text{lower bound}} = 0 </math> and occurs when <math display="inline">r = \sqrt{\pi/(4-\pi)}</math><ref name="K"/> (Notice that this is the <math>r=\mu^{'}_1/\mu^{1/2}_2</math> of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition,{{clarify|reason=is this worth saying if meaning is not defined|date=June 2012}} and this continues until <math>\left|g^{i}\left(\theta_{0}\right)-\theta_{i-1}\right|</math> is less than some small positive value. Here, <math>g^{i}</math> denotes the composition of the same function, {{tmath| g }}, <math>i</math> times. In practice, we associate the final <math>\theta_{n}</math> for some integer <math>n</math> as the fixed point, {{tmath| \theta^{*} }}, i.e., {{tmath|1= \theta^{*} = g\left(\theta^{*}\right) }}.

Once the fixed point is found, the estimates <math>\nu</math> and <math>\sigma</math> are found through the scaling function, {{tmath| \xi{\left(\theta\right)} }}, as follows: : <math> \sigma = \frac{\mu^{1/2}_2}{\sqrt{\xi\left(\theta^{*}\right)}}, </math> and : <math> \nu = \sqrt{\left( \mu^{'~2}_1 + \left(\xi\left(\theta^{*}\right) - 2\right)\sigma^2 \right)}. </math>

To speed up the iteration even more, one can use the Newton's method of root-finding.<ref name=K/> This particular approach is highly efficient.

== Applications == * The Euclidean norm of a bivariate circularly symmetric normally distributed random vector. * Rician fading (for multipath interference)) * Effect of sighting error on target shooting.<ref>{{cite web|title=Ballistipedia|url=http://ballistipedia.com/index.php?title=Closed_Form_Precision#How_many_sighter_shots_do_you_need.3F |access-date=4 May 2014}}</ref> * Analysis of diversity receivers in radio communications.<ref>{{Cite journal|last1=Beaulieu|first1=Norman C|last2=Hemachandra|first2=Kasun|date=September 2011 |title=Novel Representations for the Bivariate Rician Distribution|journal=IEEE Transactions on Communications|volume=59|issue=11|pages=2951–2954 |doi=10.1109/TCOMM.2011.092011.090171|s2cid=1221747 }}</ref><ref>{{Cite journal|last1=Dharmawansa|first1=Prathapasinghe| last2=Rajatheva|first2=Nandana| last3=Tellambura|first3=Chinthananda| date=March 2009|title=New Series Representation for the Trivariate Non-Central Chi-Squared Distribution| url=http://www.ece.ualberta.ca/~chintha/resources/papers/2009/4799042.pdf|journal=IEEE Transactions on Communications|volume=57 |issue=3|pages=665–675| doi=10.1109/TCOMM.2009.03.070083|citeseerx=10.1.1.582.533|s2cid=15706035 }}</ref> * Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.<ref>{{Cite journal |last=Laskar |first=J. |date=2008-07-01 |title=Chaotic diffusion in the Solar System |url=https://www.sciencedirect.com/science/article/pii/S0019103508001097 |journal=Icarus |language=en |volume=196 |issue=1 |pages=1–15 |doi=10.1016/j.icarus.2008.02.017 |arxiv=0802.3371 |bibcode=2008Icar..196....1L |s2cid=11586168 |issn=0019-1035}}</ref> * Distribution of noise in magnetic resonance imaging images is rician<ref>{{cite journal |last1=Gudbjartsson |first1=HáKon |last2=Patz |first2=Samuel |title=The rician distribution of noisy mri data |journal=Magnetic Resonance in Medicine |date=December 1995 |volume=34 |issue=6 |pages=910–914 |doi=10.1002/mrm.1910340618}}</ref>

== See also == * Hoyt distribution * Rayleigh distribution

== References == {{reflist}}

== Further reading == * 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. * {{cite journal |author1=I. Soltani Bozchalooi |author2=Ming Liang | doi = 10.1016/j.jsv.2007.07.038 | title = A smoothness index-guided approach to wavelet parameter selection in signal de-noising and fault detection | journal = Journal of Sound and Vibration | volume = 308 | issue = 1–2 | date = 20 November 2007 | pages = 253–254 |ref=refBozchalooi2007| bibcode = 2007JSV...308..246B }} * {{cite journal|doi=10.1016/j.jsv.2017.02.013 |title=On the distribution of the modulus of Gabor wavelet coefficients and the upper bound of the dimensionless smoothness index in the case of additive Gaussian noises: Revisited |journal=Journal of Sound and Vibration |volume=395 |pages=393–400 |year=2017 |last1=Wang |first1=Dong |last2=Zhou |first2=Qiang |last3=Tsui |first3=Kwok-Leung }} * <cite id=refLiu2007>Liu, X. and Hanzo, L., [https://ieeexplore.ieee.org/document/4350297/ 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, pp. 3504–3509.</cite> * <cite id=refAnnamalai2000>Annamalai, A., Tellambura, C. and Bhargava, V. K., [https://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/26/18877/00871398.pdf?temp=x Equal-Gain Diversity Receiver Performance in Wireless Channels], IEEE Transactions on Communications, Volume 48, October 2000, pp. 1732–1745.</cite> * <cite id=refErdelyi1953>Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., [http://apps.nrbook.com/bateman/Vol1.pdf Higher Transcendental Functions, Volume 1.] {{Webarchive|url=https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf |date=11 August 2011 }} McGraw-Hill Book Company Inc., 1953.</cite> * <cite id=refSrivastava1985>Srivastava, H. M. and Karlsson, P. W., Multiple Gaussian Hypergeometric Series. Ellis Horwood Ltd., 1985.</cite> * <cite id=refSijbers1998>Sijbers J., den Dekker A. J., Scheunders P. and Van Dyck D., [http://webh01.ua.ac.be/visielab/papers/sijbers/ieee98.pdf "Maximum Likelihood estimation of Rician distribution parameters"] {{Webarchive|url=https://web.archive.org/web/20111019045550/http://webh01.ua.ac.be/visielab/papers/sijbers/ieee98.pdf |date=19 October 2011 }}, IEEE Transactions on Medical Imaging, Vol. 17, Nr. 3, pp. 357–361, (1998)</cite> * <cite id=refVaradarajan2015>Varadarajan D. and Haldar J. P., [https://ieeexplore.ieee.org/abstract/document/7097060 "A Majorize-Minimize Framework for Rician and Non-Central Chi MR Images"], IEEE Transactions on Medical Imaging, Vol. 34, no. 10, pp. 2191–2202, (2015)</cite> * {{cite journal |author1=den Dekker, A.J. |author2=Sijbers, J | doi =10.1016/j.ejmp.2014.05.002 | pmid =25059432 | journal= Physica Medica | title = Data distributions in magnetic resonance images: a review | volume = 30 | issue = 7 | date = December 2014 | pages = 725–741|ref=RefDekker2014}} * <cite id=refKoay2006> Koay, C.G. and Basser, P. J., [https://doi.org/10.1016/j.jmr.2006.01.016 Analytically exact correction scheme for signal extraction from noisy magnitude MR signals], Journal of Magnetic Resonance, Volume 179, Issue = 2, p. 317–322, (2006)</cite> * <cite id=RefAbdi>Abdi, A., Tepedelenlioglu, C., Kaveh, M., and Giannakis, G. [https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=913150 On the estimation of the K parameter for the Rice fading distribution], IEEE Communications Letters, Volume 5, Number 3, March 2001, pp. 92–94.</cite> * {{cite journal |author1=Talukdar, K.K. |author2=Lawing, William D. | doi = 10.1121/1.400532 | title = Estimation of the parameters of the Rice distribution | journal = Journal of the Acoustical Society of America | volume = 89 | issue = 3 | date = March 1991 | pages = 1193–1197 |ref=RefTalukdar| bibcode = 1991ASAJ...89.1193T }} * {{cite journal |author=Bonny, J.M. |author2=Renou, J.P. |author3=Zanca, M. | doi = 10.1006/jmrb.1996.0166 | pmid = 8954899 | title = Optimal Measurement of Magnitude and Phase from MR Data | journal = Journal of Magnetic Resonance, Series B | volume = 113 | issue = 2 | date = November 1996 | pages = 136–144 |ref=RefBonny| bibcode = 1996JMRB..113..136B }}

== External links == * [https://au.mathworks.com/matlabcentral/fileexchange/14237-rice-rician-distribution MATLAB code for Rice/Rician distribution] (PDF, mean and variance, and generating random samples)

{{DEFAULTSORT:Rice Distribution}} Category:Continuous distributions

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