Transform Point Process (PP) parameters into
Poisson-GP parameters. The provided parameters are GEV
parameters: location \(\mu^\star\), scale
\(\sigma^\star_w\) and shape
\(\xi^\star\). They are assumed to describe (the
tail of) the distribution for a maximum on a time-interval
with given duration \(w\). For a given threshold \(u\)
chosen to be in the interior of the support of the GEV
distribution, there exists a unique vector of three Poisson-GP
parameters such that the maximum \(M\) of the marks on an
interval with duration w has the prescribed GEV
tail. Remind that the three Poisson-GP parameters are the rate
of the Poisson process in time: \(\lambda_u\), and the two
GP parameters: scale \(\sigma_u\) and shape
\(\xi\). The shape parameters \(\xi^\star\) and
\(\xi\) are identical.
PP2poisGP(locStar = 0.0, scaleStar = 1.0, shapeStar = 0.0,
threshold,
w = 1.0, deriv = FALSE)A matrix with three columns representing the Poisson-GP
parameters lambda, scale and shape.
Numeric vectors containing the GEV location, scale and shape parameters.
Numeric vector containing the thresholds of the Poisson-GP model, i.e. the location of the Generalised Pareto Distribution. The threshold must be an interior point of the support of the corresponding GEV distribution.
The block duration. Its physical dimension is time and the product \(\lambda \times w\) is dimensionless.
Logical. If TRUE the derivative (Jacobian) of
the transformation is computed and returned as an attribute named
"gradient" of the attribute.
The Poisson-GP parameters are obtained by $$\left\{ \begin{array}{c c l} \sigma_u &=& \sigma_w^\star + \xi^\star \left[ u - \mu_w^\star \right],\\ \lambda_u &=& w^{-1} \, \left[\sigma_u / \sigma_w^\star \right]^{-1/ \xi^\star},\\ \xi &=& \xi^\star, \end{array}\right.$$ the second equation becomes \(\lambda_u = w^{-1}\) for \(\xi^\star = 0\).
poisGP2PP for the reciprocal
transformation.