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.

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 {
head(predict(fit.h, hunua, type = "response"))


# }
# NOT RUN {
# The following fails.
pneumo <- transform(pneumo, let = log(exposure.time))
fit <- vglm(cbind(normal, mild, severe) ~ let,
            cumulative(link = foldsqrt(mux = 10), par = TRUE, rev = TRUE),
            data = pneumo, trace = TRUE, maxit = 200) 
# }