This simple form of a hyperbola is not well known. I find it useful for change point
models: since the derivative is continuous it often behaves better in a maximizer.
h1 <- function(x, b, k=3) .5 * b * (x + sqrt(x^2 + k^2))
Function h1() has asymptotes of y=0 to the left of 0 and y=x to the right of 0. The
parameter k controls how tightly the curve bends at zero.
A generalization is
h2 <- function (x, b, k=3) {
z <- x - b[4]
b[1] + b[2]*z + .5*b[3]* (z + sqrt(z^2 + k^2))
}
b[1] and b[2] are the intercept and slope of the left portion of the curve, b[4] is the
inflection point, and b[3] is the change in slope at the inflection point.
The main downside is that k is arbitrary. I simply choose it to "look right", though it
too could be part of an optimization.
Terry Therneau
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