pdfrice {lmomco}

Probability Density Function of the Rice Distribution

Description

This function computes the probability density of the Rice distribution given parameters (\nu and \mathrm{SNR}) computed by parrice. The probability density function is

f(x) = \frac{x}{\alpha^2}\,\exp\!\left(\frac{-(x^2+\nu^2)}{2\alpha^2}\right)\,I_0(x\nu/\alpha^2)\mbox{,}

where f(x) is the nonexceedance probability for quantile x, \nu is a parameter, and \nu/\alpha is a form of signal-to-noise ratio \mathrm{SNR}, and I_k(x) is the modified Bessel function of the first kind, which for integer k=0 is seen under LaguerreHalf. If \nu=0, then the Rayleigh distribution results and pdfray is used. If 24 < \mathrm{SNR} < 52 is used, then the Normal distribution functions are used with appropriate parameter estimation for \mu and \sigma that include the Laguerre polynomial LaguerreHalf. If \mathrm{SNR} > 52, then the Normal distribution functions continue to be used with \mu=\alpha\times\mathrm{SNR} and \sigma = A.

Usage

pdfrice(x, para)

Arguments

x

A real value vector.

para

The parameters from parrice or vec2par.

Value

Probability density (f) for x.

Note

The VGAM package provides a pdf of the Rice for reference:

"drice" <- function(x, vee, sigma, log = FALSE) { # From the VGAM package
    if(!is.logical(log.arg <- log)) stop("bad input for argument 'log'")
    rm(log)
    N = max(length(x), length(vee), length(sigma))
    x = rep(x, len=N); vee = rep(vee, len=N); sigma = rep(sigma, len=N)
    logdensity = rep(log(0), len=N)
    xok = (x > 0)
    x.abs = abs(x[xok]*vee[xok]/sigma[xok]^2)
    logdensity[xok] = log(x[xok]) - 2 * log(sigma[xok]) +
                      (-(x[xok]^2+vee[xok]^2)/(2*sigma[xok]^2)) +
                      log(besselI(x.abs, nu=0, expon.scaled = TRUE)) + x.abs
    logdensity[sigma <= 0] <- NaN; logdensity[vee < 0] <- NaN
    if(log.arg) logdensity else exp(logdensity)
}

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

cdfrice, quarice, lmomrice, parrice

Examples

lmr <- lmoms(c(10, 43, 27, 26, 49, 26, 62, 39, 51, 14))
rice <- parrice(lmr)
x <- quarice(nonexceeds(),rice)
plot(x,pdfrice(x,rice), type="b")


# For SNR=v/a > 24 or 240.001/10 > 24, the Normal distribution is
# used by the Rice as implemented here.
rice1 <- vec2par(c(239.9999,10), type="rice")
rice2 <- vec2par(c(240.0001,10), type="rice")
x <- 200:280
plot( x, pdfrice(x, rice1), type="l", lwd=5, lty=3) # still RICIAN code
lines(x, dnorm(  x, mean=240.0001, sd=10), lwd=3, col=2) # NORMAL obviously
lines(x, pdfrice(x, rice2), lwd=1, col=3) # NORMAL distribution code is triggered

# For SNR=v/a > 52 or 521/10 > 52, the Normal distribution
# used by the Rice as implemented here with simple parameter estimation
# because this high of SNR is beyond limits of Bessel function in Laguerre
# polynomial
rice1 <- vec2par(c(520,10), type="rice")
rice2 <- vec2par(c(521,10), type="rice")
x <- 10^(log10(520) - 0.05):10^(log10(520) + 0.05)
plot( x, pdfrice(x, rice1), type="l", lwd=5, lty=3)
lines(x, pdfrice(x, rice2), lwd=1, col=3) # NORMAL code triggered