sqrtlink {VGAM}

Square Root and Folded Square Root Link Functions

Description

Computes the square root and folded square root transformations, including their inverse and their first two derivatives.

Usage

foldsqrtlink(theta, min = 0, max = 1, mux = sqrt(2),
     inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)
sqrtlink(theta, inverse = FALSE, deriv = 0, short = TRUE,
         tag = FALSE, c10 = c(2, -2))

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.

c10

Numeric, 2-vector c(c1, c0) for a linear transformation. The plain link is multiplied by c1 and then c0 is added so that c1 = 1:0 is simply sqrt. The default is intended to match lcsloglink for poissonff at lambda (theta) equal to 1.

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.

More general information can be found at alogitlink.

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 sqrtlink with deriv = 0 and c10 = 1:0: \sqrt{\theta} when inverse = FALSE, and if inverse = TRUE then the square is returned.

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

For foldsqrtlink, 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, poissonff, sloglink, hdeff.

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)),
          col = "blue", ylab = "transformation",
          main = ifelse(d == 0, "Some probability links",
          "First derivative"), type = "n", las = 1)
  lines(p,    logitlink(p, deriv = d), col = "green")
  lines(p,   probitlink(p, deriv = d), col = "blue")
  lines(p,  clogloglink(p, deriv = d), col = "red")
  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("green", "blue",
                            "red", "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 = "blue", 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="green")
  lines(y,   probitlink(y, deriv=d, inverse=TRUE), col="blue")
  lines(y,  clogloglink(y, deriv=d, inverse=TRUE), col="red")
  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("green", "blue", "red", "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)