foldsqrtlink: Folded Square Root Link Function

In VGAM: Vector Generalized Linear and Additive Models

Folded Square Root Link Function

Description

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

Usage

foldsqrtlink(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.

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.

Value

For foldsqrtlink 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.

Note

The default has, if theta is 0 or 1, the link function value is -sqrt(2) and +sqrt(2) respectively. These are finite values, therefore one cannot use this link function for general modelling of probabilities because of numerical problem, e.g., with binomialff, cumulative. See the example below.

Author(s)

Thomas W. Yee

See Also

Links.

Examples

p <- seq(0.01, 0.99, by = 0.01)
foldsqrtlink(p)
max(abs(foldsqrtlink(foldsqrtlink(p), inverse = TRUE) - p))  # 0

p <- c(seq(-0.02, 0.02, by = 0.01), seq(0.97, 1.02, by = 0.01))
foldsqrtlink(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(logitlink(p, deriv = d),
                foldsqrtlink(p, deriv = d)), las = 1,
          type = "n", col = "purple", ylab = "transformation",
          main = if (d == 0) "Some probability link functions"
          else "First derivative")
  lines(p, logitlink(p, deriv = d), col = "limegreen")
  lines(p, probitlink(p, deriv = d), col = "purple")
  lines(p, clogloglink(p, deriv = d), col = "chocolate")
  lines(p, foldsqrtlink(p, deriv = d), col = "tan")
  if (d == 0) {
    abline(v = 0.5, h = 0, lty = "dashed")
    legend(0, 4.5, c("logitlink", "probitlink",
                     "clogloglink", "foldsqrtlink"),
           lwd = 2, col = c("limegreen", "purple",
                            "chocolate", "tan"))
  } else
    abline(v = 0.5, lty = "dashed")
}

for (d in 0) {
  matplot(y, cbind(logitlink(y, deriv = d, inverse = TRUE),
                   foldsqrtlink(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, logitlink(y, deriv=d, inverse=TRUE), col = "limegreen")
  lines(y, probitlink(y, deriv=d, inverse=TRUE), col = "purple")
  lines(y, clogloglink(y, deriv=d, inverse=TRUE), col = "chocolate")
  lines(y, foldsqrtlink(y, deriv=d, inverse = TRUE), col = "tan")
  if (d == 0) {
    abline(h = 0.5, v = 0, lty = "dashed")
    legend(-4, 1, c("logitlink", "probitlink",
                    "clogloglink", "foldsqrtlink"), 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(foldsqrtlink(mux = 5)),
              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(foldsqrtlink(mux = 10), par = TRUE, rev = TRUE),
       data = pneumo, trace = TRUE, maxit = 200) 
## End(Not run)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.