fsqrt: Folded Square Root Link Function

Description

Computes the folded square root transformation, including its inverse and the first two derivatives.

Usage

fsqrt(theta, earg = list(min=0, max=1, mux=sqrt(2)),
      inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

earg

List with components min, max and mux. These are called $L$, $U$ and $K$ below.

inverse

Logical. If TRUE the inverse function is computed.

deriv

Order of the derivative. Integer with value 0, 1 or 2.

short

Used for labelling the blurb slot of a vglmff-class object.

tag

Used for labelling the linear/additive predictor in the initialize slot of a vglmff-class object. Contains a little more information if TRUE.

Value

For fsqrt with deriv = 0: $K (\sqrt{\theta-L} - \sqrt{U-\theta})$ or mux * (sqrt(theta-min) - sqrt(max-theta)) when inverse = FALSE, and if inverse = TRUE then some more complicated function that returns a NA unless theta is between -mux*sqrt(max-min) and mux*sqrt(max-min).
For deriv = 1, then the function returns d theta / d eta as a function of theta if inverse = FALSE, else if inverse = TRUE then it returns the reciprocal.

Details

The folded square root link function can be applied to parameters that lie between $L$ and $U$ inclusive. Numerical values of theta out of range result in NA or NaN.

The arguments short and tag are used only if theta is character.

Examples

Run this code

p = seq(0.01, 0.99, by=0.01)
fsqrt(p)
max(abs(fsqrt(fsqrt(p), inverse=TRUE) - p)) # Should be 0

p = c(seq(-0.02, 0.02, by=0.01), seq(0.97, 1.02, by=0.01))
fsqrt(p)  # Has NAs

p = seq(0.01, 0.99, by=0.01)
par(mfrow=c(2,2))
y = seq(-4, 4, length=100)
for(d in 0:1) {
    matplot(p, cbind(logit(p, deriv=d), fsqrt(p, deriv=d)),
            type="n", col="purple", ylab="transformation",
            lwd=2, las=1,
            main = if (d == 0) "Some probability link functions"
            else "First derivative")
    lines(p, logit(p, deriv=d), col="limegreen", lwd=2)
    lines(p, probit(p, deriv=d), col="purple", lwd=2)
    lines(p, cloglog(p, deriv=d), col="chocolate", lwd=2)
    lines(p, fsqrt(p, deriv=d), col="tan", lwd=2)
    if (d == 0) {
        abline(v=0.5, h=0, lty="dashed")
        legend(0, 4.5, c("logit", "probit", "cloglog", "fsqrt"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    } else
        abline(v=0.5, lty="dashed")
}

for(d in 0) {
    matplot(y, cbind(logit(y, deriv=d, inverse=TRUE),
                     fsqrt(y, deriv=d, inverse=TRUE)),
            type="n", col="purple", xlab="transformation", ylab="p",
            lwd=2, las=1,
            main = if (d == 0) "Some inverse probability link functions"
            else "First derivative")
    lines(y, logit(y, deriv=d, inverse=TRUE), col="limegreen", lwd=2)
    lines(y, probit(y, deriv=d, inverse=TRUE), col="purple", lwd=2)
    lines(y, cloglog(y, deriv=d, inverse=TRUE), col="chocolate", lwd=2)
    lines(y, fsqrt(y, deriv=d, inverse=TRUE), col="tan", lwd=2)
    if (d == 0) {
        abline(h=0.5, v=0, lty="dashed")
        legend(-4, 1, c("logit", "probit", "cloglog", "fsqrt"),
               col=c("limegreen","purple","chocolate", "tan"), lwd=2)
    }
}

# This is lucky to converge
earg = list(min=0, max=1, mux=5)
fit.h = vglm(agaaus ~ bs(altitude),
             fam= binomialff(link="fsqrt", earg=earg),
             data=hunua, trace=TRUE, crit="d")
plotvgam(fit.h, se=TRUE, lcol="red", scol="red",
         main="Red is Hunua, Blue is Waitakere")
head(predict(fit.h, hunua, type="response"))


# The following fails.
pneumo = transform(pneumo, let=log(exposure.time))
earg = list(min=0, max=1, mux=10)
fit = vglm(cbind(normal, mild, severe) ~ let,
           cumulative(link="fsqrt", earg=earg, par=TRUE, rev=TRUE),
           data = pneumo, trace=TRUE, maxit=200)

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