poisGP2PP: Transform Poisson-GP Parameters into Point-Process Parameters

In nieve: Miscellaneous Utilities for Extreme Value Analysis

Transform Poisson-GP Parameters into Point-Process Parameters

Description

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.

Usage

poisGP2PP(lambda, loc = 0.0, scale = 1.0, shape = 0.0, w =
    1.0, deriv = FALSE)

Arguments

lambda	A numeric vector containing the Poisson rate(s).
loc	A numeric vector containing the Generalized Pareto location, i.e. the threshold in the POT framework.
scale, shape	Numeric vectors containing the Generalized Pareto scale and shape parameters.
w	The block duration. Its physical dimension is time and the product \lambda_u \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 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).

Value

A numeric matrix with three columns representing the Point-Process parameters loc \mu^\star_w, scale \sigma^\star_w and shape \xi^\star.

Note

This function is essentially a re-implementation in C of the function Ren2gev of Renext. As a major improvement, this function is "vectorized" w.r.t. the parameters so it can transform efficiently a large number of Poisson-GP parameter vectors as 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

PP2poisGP for the reciprocal transformation.

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