fExtDep {ExtremalDep}

Extremal dependence estimation

Description

This function estimates the parameters of extremal dependence models.

Usage

fExtDep(method="PPP", data, model, par.start = NULL, 
        c = 0, optim.method = "BFGS", trace = 0, sig = 3,
        Nsim, Nbin = 0, Hpar, MCpar, seed = NULL)

Arguments

method

A character string indicating the estimation method inlcuding "PPP", "BayesianPPP" and "Composite".

data

A matrix containing the data.

model

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".

par.start

A vector representing the initial parameters values for the optimization algorithm.

c

A real value in [0,1] required when method="PPP" or "BayesianPPP" and model="ET", "EST" and "AL". See dExtDep for more details.

optim.method

A character string indicating the optimization algorithm. Required when method="PPP" or "Composite". See optim for more details.

trace

A non-negative integer, tracing the progress of the optimization. Required when method="PPP" or "Composite". See optim for more details.

sig

An integer indicating the number of significant digits when reporting outputs.

Nsim

An integer indicating the number of MCMC simulations. Required when method="BayesianPPP".

Nbin

An integer indicating the length of the burn-in period. Required when method="BayesianPPP".

Hpar

A list of hyper-parameters. See 'details'. Required when method="BayesianPPP".

MCpar

A positive real representing the variance of the proposal distirbution. See 'details'. Required when method="BayesianPPP".

seed

An integer indicating the seed to be set for reproducibility, via the routine set.seed.

Details

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.

Value

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.

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail] https://www.borisberanger.com;

References

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.

See Also

dExtDep, pExtDep, rExtDep, fExtDep.np

Examples

# Example using the Poisson Point Proce Process appraoch
data(pollution)

  f.hr <- fExtDep(method="PPP", data=PNS, model="HR", 
                  par.start = rep(0.5, 3), trace=2)


# Example using the pairwise composite (full) likelihood

  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)