This function estimates the parameters of extremal dependence models.
fExtDep(method="PPP", data, model, par.start = NULL,
c = 0, optim.method = "BFGS", trace = 0, sig = 3,
Nsim, Nbin = 0, Hpar, MCpar, seed = NULL)
When method == "PPP"
or "Composite"
, a list is returned including
par
: The estimated parameters.
LL
: The maximised log-likelihood.
SE
: The standard errors.
TIC
: The Takeuchi Information Criterion.
When method == "BayesianPPP"
, a list is returned including
stored.vales
: A \((Nsim-Nbin)*d\) matrix, where \(d\) is the dimension of the parameter space
llh
: A vector of size \((Nsim-Nbin)\) containing the log-likelihoods evaluadted at each parameter
of the posterior sample.
lprior
: A vector of size \((Nsim-Nbin)\) containing the logarithm of the prior densities evaluated
at each parameter of the posterior sample.
arguments
: The specifics of the algorithm.
elapsed
: The time elapsed, as given by proc.time
between the start and end of the run.
Nsim
: The same as the passed argument.
Nbin
: Idem.
n.accept
: The total number of accepted proposals.
n.accept.kept
: The number of accepted proposals after the burn-in period.
emp.mean
: The estimated posterior parameters mean.
emp.sd
: The empirical posterior sample standard deviation.
BIC
: The Bayesian Information Criteria.
A character string indicating the estimation method inlcuding "PPP"
, "BayesianPPP"
and "Composite"
.
A matrix containing the data.
A character string with the name of the model. When method="PPP"
or "BayesianPPP"
, this includes "PB"
, "HR"
, "ET"
, "EST"
, TD
and AL
whereas when method="composite"
it is restricted to "HR"
, "ET"
and "EST"
.
A vector representing the initial parameters values for the optimization algorithm.
A real value in \([0,1]\) required when method="PPP"
or "BayesianPPP"
and model="ET"
, "EST"
and "AL"
. See dExtDep
for more details.
A character string indicating the optimization algorithm. Required when method="PPP"
or "Composite"
. See optim
for more details.
A non-negative integer, tracing the progress of the optimization. Required when method="PPP"
or "Composite"
. See optim
for more details.
An integer indicating the number of significant digits when reporting outputs.
An integer indicating the number of MCMC simulations. Required when method="BayesianPPP"
.
An integer indicating the length of the burn-in period. Required when method="BayesianPPP"
.
A list of hyper-parameters. See 'details'. Required when method="BayesianPPP"
.
A positive real representing the variance of the proposal distirbution. See 'details'. Required when method="BayesianPPP"
.
An integer indicating the seed to be set for reproducibility, via the routine set.seed
.
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com;
When method="PPP"
the approximate likelihood is used to estimate the model parameters. It relies on the dExtDep
function with argument method="Parametric"
and angular=TRUE
.
When method="BayesianPPP"
a Bayesian estimation procedure of the spatral measure is considered, following Sabourin et al. (2013) and Sabourin & Naveau (2014). The argument Hpar
is required to specify the hyper-parameters of the prior distributions, taking the following into consideration:
For the Pairwise Beta model, the parameters components are independent, log-normal.
The vector of parameters is of size choose(dim,2)+1
with positive components. The first elements are the
pairiwse dependence parameters b
and the last one is the global dependence parameter alpha
.
The list of hyper-parameters should be of the form
mean.alpha=, mean.beta=, sd.alpha=, sd.beta=
;
For the Husler-Reiss model, the parameters are independent, log-normally distributed.
The elements correspond to the lambda
parameter. The list of hyper-parameters should be of the form mean.lambda=, sd.lambda=
;
For the Dirichlet model, the parameters are independent, log-normally distributed.
The elements correspond to the alpha
parameter. The list of hyper-parameters should be of the form mean.alpha=, sd.alpha=
;
For the Extremal-t model, the parameters are independent, logit-squared for rho
and log-normal for mu
. The first elements correspond to the correlation parameters rho
and the last parameter is the global dependence parameter mu
. The list of hyper-parameters should be of the form mean.rho=, mean.mu=, sd.rho=, sd.mu=
;
For the Extremal skewt-t model, the parameters are independent, logit-squared for rho
, normal for alpha
and log-normal for mu
. The first elements correspond to the correlation parameters rho
, then the skewness parameters alpha
and the last parameter is the global dependence parameter mu
. The list of hyper-parameters should be of the form mean.rho=, mean.alpha=, mean.mu=, sd.rho=, sd.alpha=, sd.mu=
;
For the Asymmetric Logistic model, the parameters' components are independent, log-normal for alpha
and logit for beta
. The list of hyper-parameters should be of the form mean.alpha=, mean.beta=, sd.alpha=, sd.beta=
.
The proposal distribution for each (transformed) parameter is a normal distribution centred on the (transformed) current parameter value, with variance MCpar
.
When method="Composite"
, the pairwise composite likelihood is applied, based on the dExtDep
function with argument method="Parametric"
and angular=FALSE
.
Beranger, B. and Padoan, S. A. (2015). Extreme dependence models, chapater of the book Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman Hall/CRC.
Sabourin, A., Naveau, P. and Fougeres, A-L (2013) Bayesian model averaging for multivariate extremes Extremes, 16, 325-350.
Sabourin, A. and Naveau, P. (2014) Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization Computational Statistics & Data Analysis, 71, 542-567.
dExtDep
, pExtDep
, rExtDep
, fExtDep.np
# Example using the Poisson Point Proce Process appraoch
data(pollution)
if (interactive()){
f.hr <- fExtDep(method="PPP", data=PNS, model="HR",
par.start = rep(0.5, 3), trace=2)
}
# Example using the pairwise composite (full) likelihood
if (interactive()){
set.seed(1)
data <- rExtDep(n=300, model="ET", par=c(0.6,3))
f.et <- fExtDep(method="Composite", data=data, model="ET",
par.start = c(0.5, 1), trace=2)
}
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