yulesimonMlink transformation, its inverse
and the first two derivatives.
Numeric or character. This is theta by default,
or eta depending upon other arguments.
Assume Y ~ Yule-Simon(rho),
where rho is a shape parameter as in
Then, the mean of Y
is given by
μ = rho / (rho - 1) = (1 - rho^(-1))^(-1),
provided rho > 1.
This link function may be conceived as a natural link function
for the mean of the Yule–Simon distribution which comes up by
taking the logarithm on both sides of this equation. More precisely,
yulesimonMlink tranformation for rho > 1
is given by
yulesimonMlink(rho) = - log (1 - rho^(-1)).
While this link function can be used to model any parameter lying in (1, ∞), it is particularly useful for event-rate data where the mean, μ, can be written in terms of some rate of events, say λ, and the timeframe observed t. Specifically,
μ = λ t.
Assuming that additional covariates might be available to linearly model λ (or log(λ)), this model can be treated as a VGLM with one parameter where the time t (as log(t)) can be easily incorporated in the analysis as an offset.
Under this link function the domain set for rho
is (1, ∞). Hence, values of rho too
close to 1 from the right, or out of range will result
bvalue to adequately replace them before
computing the link function.
zetaffMlink, the inverse of
this link function can be written in close form.
theta is a character, arguments
deriv are disregarded.
deriv = 0, the
yulesimonMlink transformation of
inverse = FALSE, and if
inverse = TRUE then
exp(theta) / (exp(theta) - 1).
deriv = 1, d
eta / d
as a function of
inverse = FALSE, else
the reciprocal d
theta / d
deriv = 2 the second order derivatives
are correspondingly returned.
yulesimon, the domain
set for rho is (0, ∞). However, in order for
yulesimonMlink to be a real number, rho must be greater
then 1.0. Then, when a VGLM is fitted via
yulesimon using this link function,
numerical instability will occur if the estimated or the true value of
rho lies between 0 and 1, or if the initial values for rho
yulesimon fail to meet
rho > 1. Alternatively, try
loglink if this happens.
If the underlying assumption rho > 1 is not met,
then this function returns
This is equivalent to claim that the mean is infinite or negative
and, consequently, its logarithm will not be real.
The vertical line
theta = 1 is an asymptote for this link
function. As a result, it may return
Inf for values of
rho too close to 1 from the right.
V. Miranda and T. W. Yee
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
## Example 1 ## Shapes <- 1:10 yulesimonMlink(theta = Shapes, deriv = 1) ## d eta/d theta, as function of theta yulesl.inv <- # The inverse minus actual values yulesimonMlink(theta = yulesimonMlink(theta = Shapes), inverse = TRUE) - Shapes summary(yulesl.inv) ## zero ## Example 2. Special values of theta (rho) ## rhos <- c(-Inf, -2, -1, 0.0, 0.5, 1, 5, 10, 100, Inf, NaN, NA) rbind(rho = rhos, yuleslink = yulesimonMlink(theta = rhos), inv.yulesl =yulesimonMlink(theta = rhos, inverse = TRUE)) ## Example 3 The yulesimonMlink transformation and the first two derivatives ## rhos <- seq(1, 20, by = 0.01)[-1] y.rhos <- yulesimonMlink(theta = rhos, deriv = 0) der.1 <- yulesimonMlink(theta = rhos, deriv = 1) der.2 <- yulesimonMlink(theta = rhos, deriv = 2) plot(y.rhos ~ rhos, col = "black", main = "log(mu), mu = E[Y], Y ~ Yule-Simon(rho).", ylim = c(-5, 10), xlim = c(-1, 5), lty = 1, type = "l", lwd = 3) abline(v = 1.0, col = "orange", lty = 2, lwd = 3) abline(v = 0, h = 0, col = "gray50", lty = "dashed") lines(rhos, der.1, col = "blue", lty = 5) lines(rhos, der.2, col = "chocolate", lty = 4) legend(2, 7, legend = c("yulesimonMlink", "deriv = 1", "deriv = 2"), col = c("black", "blue", "chocolate"), lty = c(1, 5, 4))
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