Simulates from the prior distribution for GEV parameters based on Crowder (1992), in which independent beta priors are specified for ratios of probabilities (which is equivalent to a Dirichlet prior on differences between these probabilities).
rprior_prob(n, quant, alpha, exc = FALSE, lb = NULL, lb_prob = 0.001)A numeric scalar. The size of sample required.
A numeric vector of length 3. Contains quantiles
\(q_1, q_2, q_3\). A prior distribution is placed on the
non-exceedance (exc = FALSE) or exceedance (exc = TRUE)
probabilities corresponding to these quantiles.
The values should increase with the index of the vector.
If not, the values in quant will be sorted into increasing order
without warning.
A numeric vector of length 4. Parameters of the Dirichlet distribution for the exceedance probabilities.
A logical scalar. Let \(M\) be the GEV variable,
\(r_q = P(M \leq q)\),
\(p_q = P(M > q) = 1 - r_q\) and
quant = (\(q_1, q_2, q_3\)).
If exc = FALSE then a Dirichlet(alpha) distribution is
placed on
\((r_{q_1}, r_{q_2} - r_{q_1}, r_{q_3} - r_{q_2}, 1 - r_{q_3})\), as in
Northrop et al. (2017).
If exc = TRUE then a Dirichlet(alpha) distribution
is placed on
\((1 - p_{q_1}, p_{q_1} - p_{q_2}, p_{q_2} - p_{q_3}, p_{q_3})\), where
\(p_q = P(M > q)\), as in Stephenson (2016).
A numeric scalar. If this is not NULL then the simulation
is constrained so that lb is an approximate lower bound on the
GEV variable. Specifically, only simulated GEV parameter values for
which the 100lb_prob% quantile is greater than lb are
retained.
A numeric scalar. The non-exceedance probability involved
in the specification of lb. Must be in (0,1). If lb=NULL
then lb_prob is not used.
An n by 3 numeric matrix.
The simulation is based on the way that the prior is constructed.
See
Stephenson (1996)
the evdbayes user guide or Northrop et al. (2017)
Northrop et al. (2017)
for details of the construction of the prior. First, differences between
probabilities are simulated from a Dirichlet distribution. Then the GEV
location, scale and shape parameters that correspond to these quantile
values are found, by solving numerically a set of three non-linear
equations in which the GEV quantile function evaluated at the simulated
probabilities is equated to the quantiles in quant.
This is reduced to a one-dimensional optimisation over the GEV shape
parameter.
Crowder, M. (1992) Bayesian priors based on parameter transformation using the distribution function. Ann. Inst. Statist. Math., 44(3), 405-416. https://link.springer.com/article/10.1007/BF00050695
Stephenson, A. 2016. Bayesian Inference for Extreme Value Modelling. In Extreme Value Modeling and Risk Analysis: Methods and Applications, edited by D. K. Dey and J. Yan, 257-80. London: Chapman and Hall. https://doi.org/10.1201/b19721
Northrop, P. J., Attalides, N. and Jonathan, P. (2017) Cross-validatory extreme value threshold selection and uncertainty with application to ocean storm severity. Journal of the Royal Statistical Society Series C: Applied Statistics, 66(1), 93-120. https://doi.org/10.1111/rssc.12159
prior.prob to set this prior using
the evdbayes package.
posterior: evdbayes function that can
sample from this prior distribution (using MCMC) if the argument
lh = "none" is given.
rpost and rpost_rcpp for sampling
from an extreme value posterior distribution.
# NOT RUN {
quant <- c(85, 88, 95)
alpha <- c(4, 2.5, 2.25, 0.25)
x <- rprior_prob(n = 1000, quant = quant, alpha = alpha, exc = TRUE)
x <- rprior_prob(n = 1000, quant = quant, alpha = alpha, exc = TRUE, lb = 0)
# }
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