This function estimates the parameters of extremal dependence models.
fExtDep(x, method="PPP", model, par.start = NULL,
c = 0, optim.method = "BFGS", trace = 0,
Nsim, Nbin = 0, Hpar, MCpar, seed = NULL)
# S3 method for ExtDep_Freq
plot (x, type, log=TRUE, contour=TRUE, style, labels,
cex.dat=1, cex.lab=1, cex.cont=1, Q.fix, Q.range,
Q.range0, cond=FALSE,...)
# S3 method for ExtDep_Freq
logLik (object, ...)# S3 method for ExtDep_Bayes
plot (x, type, log=TRUE, contour=TRUE, style, labels,
cex.dat=1, cex.lab=1, cex.cont=1, Q.fix, Q.range,
Q.range0, cond=FALSE, cred.ci=TRUE, subsamp, ...)
# S3 method for ExtDep_Bayes
summary (object, cred=0.95, plot=FALSE, ...)
fExtDep:
When method == "PPP" or "Composite", a list of class ExtDep_Freq is returned including
The argument model.
The estimated parameters.
The maximised log-likelihood.
The standard errors.
The Takeuchi Information Criterion.
The argument x.
When method == "BayesianPPP", a list of class ExtDep_Bayes is returned including
A \((Nsim-Nbin)*d\) matrix, where \(d\) is the dimension of the parameter space
A vector of size \((Nsim-Nbin)\) containing the log-likelihoods evaluated at each parameter of the posterior sample.
A vector of size \((Nsim-Nbin)\) containing the logarithm of the prior densities evaluated at each parameter of the posterior sample.
The specifics of the algorithm.
The time elapsed, as given by proc.time between the start and end of the run.
The same as the passed argument.
Idem.
The total number of accepted proposals.
The number of accepted proposals after the burn-in period.
The estimated posterior parameters mean.
The empirical posterior sample standard deviation.
The Bayesian Information Criteria.
logLik: method function: A numerical value indicating the value of the maximized log-likelihood.
fExtDep.np: A matrix containing the data.
plot method functions: any object returned by fExtDep.
summary.ExtDep_Bayes method function: A list object of class ExtDep_Bayes.
logLik method function: any object returned by fExtDep.
A character string indicating the estimation method inlcuding "PPP", "BayesianPPP" and "Composite".
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 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.
For plot method functions: a character string indicating the type of plot to be displayed. Can take value angular, pickands or returns.
Required for plot method functions with type angular or pickands. See angular.plot and pickands.plot.
Required for plot method functions with type angular or pickands. See angular.plot and pickands.plot.
Required for plot method functions with type angular. A character string indicating the plotting style of the 2-dimensional data. Takes value "hist" or "ticks" (default). See details.
Required for plot method functions. See angular.plot, pickands.plot or returns.plot.
Required for plot method functions with type angular. A positive real indicating the size of the 3-dimensional data points.
Required for plot method functions. See angular.plot, pickands.plot or returns.plot.
Required for plot method functions with type angular or pickands. See angular.plot and pickands.plot.
Required for plot method functions with type returns. See returns.plot.
Required for plot method functions with type returns. See returns.plot.
Required for plot method functions with type returns. See returns.plot.
Required for plot method functions with type returns. See returns.plot.
Required for plot method functions with type returns. If TRUE, selects a subsample from the posterior to compute \(95\%\)credible bands.
Required for plot method functions with type returns and pred.ci=TRUE. A percentage indicating the size of the posterior subsample to be considered.
A probability indicating the coverage of the credible interval.
A logical value. If TRUE kernel density plots of the posterior distribution of each parameter is displayed.
Additional graphical arguments for the plot method functions: hist() function when type="angular" with style="hist", and plot() and contour() functions when type="returns". For the summary method function: additional parameters to the density() function. For the logLik() function: can provide a digits argument, an integer indicating the number of decimal places to be reported.
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com;
Regarding the fExtDep.np function:
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.
Regarding the code plot method function:
Refer to the angular.plot, pickands.plot or returns.plot functions.
When displaying the bivariate angular density, there is the choice to summarise the data using a histogram (style="hist") or to display the observations using tick marks (style="ticks").
When displaying the trivariate angular density, the size of the data points can be controlled using cex.dat.
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)
# \donttest{
f.hr <- fExtDep(x=PNS, method="PPP", model="HR",
par.start = rep(0.5, 3), trace=2)
plot(x=f.hr, type="angular",
labels=c(expression(PM[10]), expression(NO), expression(SO[2])),
cex.lab=2)
plot(x=f.hr, type="pickands",
labels=c(expression(PM[10]), expression(NO), expression(SO[2])),
cex.lab=2) # Takes time!
# }
# Example using the pairwise composite (full) likelihood
# \donttest{
set.seed(1)
data <- rExtDep(n=300, model="ET", par=c(0.6,3))
f.et <- fExtDep(x=data, method="Composite", model="ET",
par.start = c(0.5, 1), trace=2)
plot(x=f.et, type="angular", cex.lab=2)
# }
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