VGAM (version 1.0-4)

# foldsqrt: Folded Square Root Link Function

## Description

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

## Usage

foldsqrt(theta, 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.

min, max, mux

These are called $$L$$, $$U$$ and $$K$$ below.

inverse, deriv, short, tag

Details at Links.

## Value

For foldsqrt 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 eta / d theta 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.

Links.

## Examples

Run this code
# NOT RUN {
p <- seq(0.01, 0.99, by = 0.01)
foldsqrt(p)
max(abs(foldsqrt(foldsqrt(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))
foldsqrt(p)  # Has NAs

# }
# NOT RUN {
p <- seq(0.01, 0.99, by = 0.01)
par(mfrow = c(2, 2), lwd = (mylwd <- 2))
y <- seq(-4, 4, length = 100)
for (d in 0:1) {
matplot(p, cbind(logit(p, deriv = d), foldsqrt(p, deriv = d)),
type = "n", col = "purple", ylab = "transformation", las = 1,
main = if (d == 0) "Some probability link functions"
else "First derivative")
lines(p, logit(p, deriv = d), col = "limegreen")
lines(p, probit(p, deriv = d), col = "purple")
lines(p, cloglog(p, deriv = d), col = "chocolate")
lines(p, foldsqrt(p, deriv = d), col = "tan")
if (d == 0) {
abline(v = 0.5, h = 0, lty = "dashed")
legend(0, 4.5, c("logit", "probit", "cloglog", "foldsqrt"), lwd = 2,
col = c("limegreen","purple","chocolate", "tan"))
} else
abline(v = 0.5, lty = "dashed")
}

for (d in 0) {
matplot(y, cbind(logit(y, deriv = d, inverse = TRUE),
foldsqrt(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")
lines(y, probit(y, deriv = d, inverse = TRUE), col = "purple")
lines(y, cloglog(y, deriv = d, inverse = TRUE), col = "chocolate")
lines(y, foldsqrt(y, deriv = d, inverse = TRUE), col = "tan")
if (d == 0) {
abline(h = 0.5, v = 0, lty = "dashed")
legend(-4, 1, c("logit", "probit", "cloglog", "foldsqrt"), lwd = 2,
col = c("limegreen","purple","chocolate", "tan"))
}
}
par(lwd = 1)
# }
# NOT RUN {
# This is lucky to converge
fit.h <- vglm(agaaus ~ sm.bs(altitude), binomialff(link = foldsqrt(mux = 5)),
data = hunua, trace = TRUE)
# }
# NOT RUN {
plotvgam(fit.h, se = TRUE, lcol = "orange", scol = "orange",
main = "Orange is Hunua, Blue is Waitakere")
# }
# NOT RUN {