lmomco (version 1.4.3)

pdfrice: Probability Density Function of the Rice Distribution

Description

This function computes the probability density of the Rice distribution given parameters ($\nu$ and $\mathrm{SNR}$) of the distribution computed by parrice. The probability density function of the distribution is $$f(x) = \frac{x}{\alpha^2}\mathrm{exp}(-(x^2+\nu^2)/(2\alpha^2))I_o(x\nu/\alpha^2)$$ 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 $\mathrm{SNR}$. 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 $\mu=\alpha*\mathrm{SNR}$ and $\sigma = A$.

Usage

pdfrice(x, para)

Arguments

x

A real value.

para

The parameters from parrice or similar.

Value

• Probability density ($f$) for $x$.

See Also

cdfrice, quarice, 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")

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)
}

# 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 code triggered


# For SNR=v/a > 52 or 521/10 > 52, the Normal distribution is
# 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)-.01):10^(log10(520)+.01)
plot( x, pdfrice(x, rice1), type="l", lwd=5, lty=3)
lines(x, pdfrice(x, rice2), lwd=1, col=3) # NORMAl code triggered