lmomco (version 2.3.1)

pdfemu: Probability Density Function of the Eta-Mu Distribution

Description

This function computes the probability density of the Eta-Mu (\(\eta:\mu\)) distribution given parameters (\(\eta\) and \(\mu\)) computed by paremu. The probability density function is $$ f(x) = \frac{4\sqrt{\pi}\mu^{\mu - 1/2}h^\mu}{\gamma(\mu)H^{\mu - 1/2}}\,x^{2\mu}\,\exp(-2\mu h x^2)\,I_{\mu-1/2}(2\mu H x^2)\mbox{,} $$ where \(f(x)\) is the nonexceedance probability for quantile \(x\), and the modified Bessel function of the first kind is \(I_k(x)\), and the \(h\) and \(H\) are $$ h = \frac{1}{1-\eta^2}\mbox{,} $$ and $$ H = \frac{\eta}{1-\eta^2}\mbox{,} $$ for “Format 2” as described by Yacoub (2007). This format is exclusively used in the algorithms of the lmomco package.

If \(\mu=1\), then the Rice distribution results, although pdfrice is not used. If \(\kappa \rightarrow 0\), then the exact Nakagami-m density function results with a close relation to the Rayleigh distribution.

Define \(m\) as $$m = 2\mu\biggl[1 + {\biggr(\frac{H}{h}\biggl)}^2 \biggr]\mbox{,}$$ where for a given \(m\), the parameter \(\mu\) must lie in the range $$m/2 \le \mu \le m\mbox{.}$$

The \(I_k(x)\) for real \(x > 0\) and noninteger \(k\) is $$I_k(x) = \frac{1}{\pi} \int_0^\pi \exp(x\cos(\theta)) \cos(k \theta)\; \mathrm{d}\theta - \frac{\sin(k\pi)}{\pi}\int_0^\infty \exp(-x \mathrm{cosh}(t) - kt)\; \mathrm{d}t\mbox{.}$$

Usage

pdfemu(x, para, paracheck=TRUE)

Arguments

x

A real value vector.

para

The parameters from paremu or vec2par.

paracheck

A logical controlling whether the parameters and checked for validity.

Value

Probability density (\(f\)) for \(x\).

References

Yacoub, M.D., 2007, The kappa-mu distribution and the eta-mu distribution: IEEE Antennas and Propagation Magazine, v. 49, no. 1, pp. 68--81

See Also

cdfemu, quaemu, lmomemu, paremu

Examples

# NOT RUN {
x <- seq(0,4, by=.1)
para <- vec2par(c(.5, 1.4), type="emu")
F <- cdfemu(x, para);         X <- quaemu(F, para)
plot(F, X, type="l", lwd=8);  lines(F, x, col=2)

delx <- 0.005
x <- seq(0,3, by=delx)
plot(c(0,3), c(0,1), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
mu <- 0.6
# Note that in order to produce the figure correctly using the etas
# shown in the figure that it must be recognized that these are the etas
# for format1, but all of the algorithms in lmomco are built around
# format2
etas.format1 <- c(0, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5, 1)
etas.format2 <- (1 - etas.format1)/(1+etas.format1)
H <- etas.format2 / (1 - etas.format2^2)
h <-            1 / (1 - etas.format2^2)
for(eta in etas.format2) {
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")),
         col=rgb(eta^2,0,0))
}
mtext("Yacoub (2007, figure 5)")

plot(c(0,3), c(0,2), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
eta.format1 <- 0.5
eta.format2 <- (1 - eta.format1)/(1 + eta.format1)
mus <- c(0.25, 0.3, 0.5, 0.75, 1, 1.5, 2, 3)
for(mu in mus) {
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")))
}
mtext("Yacoub (2007, figure 6)")

plot(c(0,3), c(0,1), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
m <- 0.75
mus <- c(0.7425, 0.75, 0.7125, 0.675, 0.45, 0.5, 0.6)
for(mu in mus) {
   eta <- sqrt((m / (2*mu))^-1 - 1)
   print(eta)
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")))
}
mtext("Yacoub (2007, figure 7)") #
# }