Rice {shotGroups}

The Rice Distribution

Description

Density, distribution function, quantile function, and random deviate generation for the Rice distribution. The radius around the origin in a bivariate uncorrelated normal random variable with equal variances and an offset mean, re-written in polar coordinates (radius and angle), follows a Rice distribution.

Usage

dRice(x, nu, sigma)
pRice(q, nu, sigma, lower.tail = TRUE)
qRice(p, nu, sigma, lower.tail = TRUE)
rRice(n, nu, sigma, method = c('eigen', 'chol', 'cdf'))

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

nu

vector of Rice shape parameters nu. See details.

sigma

vector of Rice scale parameter sigma. See details.

method

string indicating which method to use for generating random deviates. See details.

lower.tail

logical. If TRUE (default), probabilities are P[X \le x] otherwise, P[X > x].

Details

dRice involves the modified Bessel function of the first kind and order 0. pRice and qRice are implemented using the Marcum Q-function. The Marcum Q-function is the cdf of a non-central chi^2 variable (Nuttall, 1975).

rRice with method='eigen' or with method='chol' simulates 2D normal deviates based on the diagonal covariance matrix with entries sigma^2, and then determines the radius. rRice with method='cdf' uses qRice with simulated quantiles from a uniform random variable in (0,1).

See Hoyt for the distribution of radial error around the true center of correlated bivariate normal variables with unequal variances. See Rayleigh for the distribution of radial error around the true center of uncorrelated bivariate normal variables with equal variances. See mvnEll for the distribution of radial error around an offset center for correlated normal variables with unequal variances.

For very large signal-to-noise ratios (nu/sigma), a normal approximation is used.

Value

dRice gives the density, pRice gives the cumulative distribution function, qRice gives the quantile function, rRice generates random deviates.

The length of the result is determined by n for rRice, and is the maximum of the lengths of the numerical parameters for the other functions.

In dRice, pRice and qRice, the numerical parameters are recycled to the length of the result. Only the first element of the logical parameters is used. In rRice, only the first element of nu and sigma is used.

References

Nuttall, AH. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96

https://reference.wolfram.com/language/ref/RiceDistribution.html

See Also

Rayleigh, Hoyt, mvnEll, Bessel, Chisquare, uniroot

Examples

dRice(1, nu=c(0.1, 0.5, 0.9), sigma=10)
pRice(c(0.1, 0.5, 0.9), nu=0.5, sigma=10)
qRice(0.5, nu=0.5, sigma=c(5, 10, 15))
rRice(5, nu=0.5, sigma=10)