Transform Poisson-GP parameters into Point-Process
(PP) parameters. In the POT Poisson-GP framework the three
parameters are the rate lambda \(\lambda_u\)
of the Poisson process in time and the two GP parameters:
scale \(\sigma_u\) and shape
\(\xi\). The vector loc contains the fixed
threshold \(u\), and w the fixed block
duration. These parameters are converted into the vector of
three parameters of the GEV distribution for the maximum of
the marks \(Y_i\) on a time interval with duration
w, the number \(N\) of these marks being a r.v. with
Poisson distribution. More precisely, the GEV distribution
applies when \(N > 0\).
poisGP2PP(lambda, loc = 0.0, scale = 1.0, shape = 0.0, w =
1.0, deriv = FALSE)A numeric matrix with three columns representing the
Point-Process parameters loc
\(\mu^\star_w\), scale
\(\sigma^\star_w\) and shape
\(\xi^\star\).
A numeric vector containing the Poisson rate(s).
A numeric vector containing the Generalized Pareto location, i.e. the threshold in the POT framework.
Numeric vectors containing the Generalized Pareto scale and shape parameters.
The block duration. Its physical dimension is time and the product \(\lambda_u \times w\) is dimensionless.
Logical. If TRUE the derivative (Jacobian) of
the transformation is computed and returned as an attribute
named "gradient" of the attribute.
The three PP parameters \(\mu^\star_w\), \(\sigma^\star_w\) and \(\xi^\star\) relate to the Poisson-GP parameters according to $$\left\{ \begin{array}{c c l} \mu^\star_w &=& u + \frac{(\lambda_u w)^\xi - 1}{\xi} \, \sigma_u, \\ \sigma^\star_w &=& (\lambda_u w)^\xi \, \sigma_u,\\ \xi^\star &=& \xi, \end{array} \right.$$ the fraction \([(\lambda_u w)^\xi - 1] / \xi\) of the first equation being to be replaced for \(\xi = 0\) by its limit \(\log(\lambda_u w)\).
PP2poisGP for the reciprocal
transformation.