fExtDep.np | R Documentation |
This function estimates the bivariate extremal dependence structure using a non-parametric approach based on Bernstein Polynomials.
fExtDep.np(method, data, cov1=NULL, cov2=NULL, u, mar.fit=TRUE,
mar.prelim=TRUE, par10, par20, sig10, sig20, param0=NULL,
k0=NULL, pm0=NULL, prior.k="nbinom", prior.pm="unif",
nk=70, lik=TRUE,
hyperparam = list(mu.nbinom=3.2, var.nbinom=4.48),
nsim, warn=FALSE, type="rawdata")
method |
A character string indicating the estimation method inlcuding |
data |
A matrix containing the data. |
cov1, cov2 |
A covariate vector/matrix for linear model on the location parameter of the marginal distributions. |
u |
When |
mar.fit |
A logical value indicated whether the marginal distributions should be fitted. When |
rawdata |
A character string specifying if the data is |
mar.prelim |
A logical value indicated whether a preliminary fit of marginal distributions should be done prior to estimating the margins and dependence. Required when |
par10, par20 |
Vectors of starting values for the marginal parameter estimation. Required when |
sig10, sig20 |
Positive reals representing the initial value for the scaling parameter of the multivariate normal proposal distribution for both margins. Required when |
param0 |
A vector of initial value for the Bernstein polynomial coefficients. It should be a list with elements |
k0 |
An integer indicating the initial value of the polynomial order. Required when |
pm0 |
A list of initial values for the probability masses at the boundaries of the simplex. It should be a list with two elements |
prior.k |
A character string indicating the prior distribution on the polynomial order. By default |
prior.pm |
A character string indicating the prior on the probability masses at the endpoints of the simplex. By default |
nk |
An integer indicating the maximum polynomial order. Required when |
lik |
A logical value; if |
hyperparam |
A list of the hyper-parameters depending on the choice of |
nsim |
An integer indicating the number of iterations in the Metropolis-Hastings algorithm. Required when |
warn |
A logical value. If |
type |
A character string indicating whther the data are |
When method="Bayesian"
, the vector of hyper-parameters is provided in the argument hyperparam
. It should include:
If prior.pm="unif"
requires hyperparam$a.unif
and hyperparam$b.unif
.
If prior.pm="beta"
requires hyperparam$a.beta
and hyperparam$b.beta
.
If prior.k="pois"
requires hyperparam$mu.pois
.
If prior.k="nbinom"
requires hyperparam$mu.nbinom
and hyperparam$var.nbinom
or hyperparam$pnb
and hyperparam$rnb
. The relationship is pnb = mu.nbinom/var.nbinom
and rnb = mu.nbinom^2 / (var.nbinom-mu.nbinom)
.
When u
is specified Algorithm 1 of Beranger et al. (2021) is applied whereas when it is not specified Algorithm 3.5 of Marcon et al. (2016) is considered.
When method="Frequentist"
, if type="rawdata"
then pseudo-polar coordinates are extracted and only observations with a radial component above some high threshold (the quantile equivalent of u
for the raw data) are retained. The inferential approach proposed in Marcon et al. (2017) based on the approximate likelihood is applied.
When method="Empirical"
, the empirical estimation procedure presented in Einmahl et al. (2013) is applied.
Outputs take the form of list including:
method
: The argument.
type
: whether it is "maxima"
or "rawdata"
(in the broader sense that a threshold exceedance model was taken).
If method="Bayesian"
the list also includes:
mar.fit
: The argument.
pm
: The posterior sample of probability masses.
eta
: The posterior sample for the coeficients of the Bernstein polynomial.
k
: The posterior sample for the Bernstein polynoial order.
accepted
: A binary vector indicating if the proposal was accepted.
acc.vec
: A vector containing the acceptance probabilities for the dependence parameters at each iteration.
prior
: A list containing hyperparam
, prior.pm
and prior.k
.
nsim
: The argument.
data
: The argument.
In addition if the marginal parameters are estimated (mar.fit=TRUE
):
cov1
, cov2
: The arguments.
accepted.mar
, accepted.mar2
: Binary vectors indicating if the marginal proposals were accepted.
straight.reject1
, straight.reject2
: Binary vectors indicating if the marginal proposals were rejected straight away as not respecting existence conditions (proposal is multivariate normal).
acc.vec1
, acc.vec2
: Vectors containing the acceptance probabilities for the marginal parameters at each iteration.
sig1.vec
, sig2.vec
: Vectors containing the values of the scaling parameter in the marginal proposal distributions.
Finally, if the argument u
is provided, the list also contains:
threshold
: A bivariate vector indicating the threshold level for both margins.
kn
: The empirical estimate of the probability of being greater than the threshold.
When method="Frequentist"
, the list includes:
When method="Empirical"
, the list includes:
hhat
: An estimate of the angular density.
Hhat
: An estimate of the angular measure.
p0
, p1
: The estimates of the probability mass at 0 and 1.
Ahat
: An estimate of the PIckands dependence function.
w
: A sequence of value on the bivariate unit simplex.
q
: A real in [0,1]
indicating the quantile associated with the threshold u
. Takes value NULL
if type="maxima"
.
data
: The data on the unit Frechet scale (empirical transformation) if type="rawdata"
and mar.fit=TRUE
. Data on the original scale otherwise.
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com;
Beranger, B., Padoan, S. A. and Sisson, S. A. (2021). Estimation and uncertainty quantification for extreme quantile regions. Extremes, 24, 349-375.
Einmahl, J. H. J., de Haan, L. and Krajina, A. (2013). Estimating extreme bivariate quantile regions. Extremes, 16, 121-145.
Marcon, G., Padoan, S. A. and Antoniano-Villalobos, I. (2016). Bayesian inference for the extremal dependence. Electronic Journal of Statistics, 10, 3310-3337.
Marcon, G., Padoan, S.A., Naveau, P., Muliere, P. and Segers, J. (2017) Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials. Journal of Statistical Planning and Inference, 183, 1-17.
dExtDep
, pExtDep
, rExtDep
, fExtDep
# Example Bayesian estimation,
# Threshold exceedances approach, threshold set by default
# Joint estimation margins + dependence
# Default uniform prior on pm
# Default negative binomial prior on polynomial order
# Quadratic relationship between location and max temperature
## Not run:
data(MilanPollution)
data <- Milan.winter[,c("NO2", "SO2", "MaxTemp")]
data <- data[complete.cases(data),]
covar <- cbind(rep(1,nrow(data)), data[,3], data[,3]^2)
hyperparam <- list(mu.binom=6, var.binom=8, a.unif=0, b.unif=0.2)
pollut <- fExtDep.np(method="Bayesian", data = data[,-3], u=TRUE,
cov1 = covar, cov2 = covar, mar.prelim=FALSE,
par10 = c(100,0,0,35,1), par20 = c(20,0,0,20,1),
sig10 = 0.1, sig20 = 0.1, k0 = 5,
hyperparam = hyperparam, nsim = 5e+4)
# Warning: This is slow!
## End(Not run)
# Example Frequentist estimation
# Data are maxima
data(WindSpeedGust)
years <- format(ParcayMeslay$time, format="%Y")
attach(ParcayMeslay[which(years %in% c(2004:2013)),])
WS_th <- quantile(WS,.9)
DP_th <- quantile(DP,.9)
pars.WS <- evd::fpot(WS, WS_th, model="pp")$estimate
pars.DP <- evd::fpot(DP, DP_th, model="pp")$estimate
data_uf <- trans2UFrechet(cbind(WS,DP), type="Empirical")
rdata <- rowSums(data_uf)
r0 <- quantile(rdata, probs=.90)
extdata <- data_uf[rdata>=r0,]
SP_mle <- fExtDep.np(method="Frequentist", data=extdata, k0=10,
type="maxima")
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