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

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)
# ## End(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") ## End(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) ## End(Not run)