PP2poisGP: Transform Point-Process Parameters into Poisson-GP Parameters

In nieve: Miscellaneous Utilities for Extreme Value Analysis

Transform Point-Process Parameters into Poisson-GP Parameters

Description

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.

Usage

PP2poisGP(locStar = 0.0, scaleStar = 1.0, shapeStar = 0.0,
          threshold,
          w = 1.0, deriv = FALSE)

Arguments

locStar, scaleStar, shapeStar	Numeric vectors containing the GEV location, scale and shape parameters.
threshold	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.
w	The block duration. Its physical dimension is time and the product \lambda \times w is dimensionless.
deriv	Logical. If TRUE the derivative (Jacobian) of the transformation is computed and returned as an attribute named "gradient" of the attribute.

Details

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.

Value

A matrix with three columns representing the Poisson-GP parameters lambda, scale and shape.

Note

This function is essentially a re-implementation in C of the function gev2Ren of Renext. As a major improvement, this function is "vectorized" w.r.t. the parameters so it can transform efficiently a large number of PP parameter vectors as it can be required e.g. in a MCMC Bayesian inference. Note also that this function copes with values near zero for the shape parameter: it suitably computes then both the function value and its derivatives.

See Also

poisGP2PP for the reciprocal transformation.

nieve documentation built on Oct. 6, 2023, 1:07 a.m.