posPoiMlink | R Documentation |

Computes the `posPoiMlink`

transformation,
its inverse and the first two derivatives.

```
posPoiMlink(theta, bvalue = NULL,
alg.roots = c("Newton-Raphson", "bisection")[1],
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)
```

`theta` |
Numeric or character. This is |

`bvalue` |
Details at |

`alg.roots` |
Character. The iterative method to find the inverse of this link
function. Same as |

`inverse, deriv, short, tag` |
Details at |

This is a link function for the mean of the positive Poisson distribution. It is defined as

```
\eta = \tt{posPoiMlink}(\lambda) =
-\log (\lambda^{-1} - \lambda^{-1} e^{-\lambda}),
```

where `\lambda > 0`

stands for the single parameter of
`pospoisson`

, i.e.
`theta`

in the VGLM/VGAM context.

Notice, the mean of the positive Poisson is given by

` \frac{\lambda}{1 - e^{-\lambda}}. `

This link function comes up by taking the logarithm on both sides of this equation.

The domain set for `\lambda`

is `(0, \infty)`

.
Hence, non–positive values of `\lambda`

will result in `NaN`

or `NA`

. Use argument `bvalue`

to properly replace
them before computing the link function.

`posPoiMlink`

tends to infinity as `\lambda`

increases. Specially, its inverse grows at a higher rate. Therefore,
large values of `\lambda`

will result in `Inf`

accordingly.
See example 2 below.

If `theta`

is a character, arguments `inverse`

and
`deriv`

are disregarded.

For `deriv = 0`

, the `posPoiMlink`

transformation of
`theta`

, if `inverse = FALSE`

. When `inverse = TRUE`

,
`theta`

becomes `\eta`

and the inverse of
`posPoiMlink`

is needed but cannot be written in
closed–form. Instead this link function returns the approximated
inverse image of `\eta`

, say `\theta_\eta`

,
such that

` \tt{posPoiMlink}(\theta_\eta) = \eta.`

Here, `\theta_\eta`

is iteratively computed
as the unique root of the auxiliary function

` f(\theta; \eta) = \tt{posPoiMlink}(\theta) - \eta,`

as a function of `\theta`

.
This work is performed via Newton–Raphson or bisection,
as per argument `alg.roots`

.

For `deriv = 1`

, *d* `eta`

/ *d* `theta`

as a function of `theta`

if `inverse = FALSE`

, else
the reciprocal *d* `theta`

/ *d* `eta`

.

Similarly, when `deriv = 2`

the second order derivatives
are returned in terms of `theta`

.

This link function is monotonic increasing in `(0, \infty)`

so that
the horizontal axis is an asymptote. Then, in order to assure the root
of the auxiliary

` f(\theta; \eta) = \tt{posPoiMlink}(\theta) - \eta`

to be real, `\eta`

must be positive. As a result,
`posPoiMlink`

is *shited–down* and hence intersecting
the horizontal axis uniquely.

This link function is useful to model any parameter
in `(0, \infty)`

. Some numerical issues may arise if there are
covariates causing negative values the parameter.
Try `identitylink`

alternatively.

V. Miranda and T. W. Yee.

`pospoisson`

,
`newtonRaphson.basic`

,
`bisection.basic`

,
`Links`

,
`identitylink`

.

```
## Example 1. Special values for theta (or eta, accordingly) ##
m.lambda <- c(0, 0.5, 1, 10, 20, 25, 1e2, 1e3, Inf, -Inf, NaN, NA)
# The 'posPoiMlink' transformation and the first two derivatives.
print(rbind(m.lambda,
deriv1 = posPoiMlink(theta = m.lambda, inverse = FALSE, deriv = 1),
deriv2 = posPoiMlink(theta = m.lambda, inverse = FALSE, deriv = 2)),
digits = 2)
# The inverse of 'posPoiMlink' and the first two derivatives.
print(rbind(m.lambda,
Invderiv1 = posPoiMlink(theta = m.lambda, inverse = TRUE, deriv = 1),
Invderiv2 = posPoiMlink(theta = m.lambda, inverse = TRUE, deriv = 2)),
digits = 2)
## Example 2. The inverse of 'posPoiMlink' ##
m.lambda <- c(0, 1, 5, 10, 1e2, 1e3)
posPoiMlink(theta = posPoiMlink(m.lambda, inverse = TRUE))
pospoi.inv <- posPoiMlink(posPoiMlink(m.lambda, inverse = TRUE)) - m.lambda
summary(pospoi.inv) ## Should be zero.
## Example 3. Plot of 'posPoiMlink' and its first two derivatives ##
## inverse = FALSE, deriv = 0, 1, 2. ##
m.lambda <- seq(0, 35, by = 0.01)[-1]
y.lambda <- posPoiMlink(theta = m.lambda, deriv = 0)
der.1 <- posPoiMlink(theta = m.lambda, deriv = 1)
der.2 <- posPoiMlink(theta = m.lambda, deriv = 2)
plot(y.lambda ~ m.lambda, col = "black",
main = "log(mu), mu = E[Y], Y ~ pospoisson(lambda).",
ylim = c(-1, 10), xlim = c(-1, 26),
lty = 1, type = "l", lwd = 3)
abline(v = 0, h = 0, col = "gray50", lty = "dashed")
lines(m.lambda, der.1, col = "blue", lty = 5, lwd = 3)
lines(m.lambda, der.2, col = "chocolate", lty = 4, lwd = 3)
legend(5, 9, legend = c("posPoiMlink", "deriv = 1", "deriv = 2"),
col = c("black", "blue", "chocolate"), lty = c(1, 5, 4), lwd = c(3, 3, 3))
```

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