fbvpot | R Documentation |
Fit models for one of nine parametric bivariate extreme-value distributions using threshold exceedances, allowing any of the parameters to be held fixed if desired.
fbvpot(x, threshold, model = c("log", "bilog", "alog", "neglog",
"negbilog", "aneglog", "ct", "hr", "amix"), likelihood =
c("censored", "poisson"), start, ..., sym = FALSE, cshape =
cscale, cscale = FALSE, std.err = TRUE, corr = FALSE, method =
"BFGS", warn.inf = TRUE)
x |
A matrix or data frame with two columns. If this contains missing values, those values are treated as if they fell below the corresponding marginal threshold. |
threshold |
A vector of two thresholds. |
model |
The specified model; a character string. Must be
either |
likelihood |
The likelihood model; either |
start |
A named list giving the initial values for all of the
parameters in the model. If |
... |
Additional parameters, either for the bivariate extreme
value model or for the optimization function |
sym |
Logical; if |
cshape |
Logical; if |
cscale |
Logical; if |
std.err |
Logical; if |
corr |
Logical; if |
method |
The optimization method (see |
warn.inf |
Logical; if |
For the "censored"
method bivariate peaks over threshold models
are fitted by maximizing the censored likelihood as given in e.g. Section
8.3.1 of Coles(2001). For the "poisson"
method models are fitted
using Equation 5.4 of Coles and Tawn (1991), see also Joe, Smith and
Weissman (1992). This method is only available for models whose spectral
measure does not contain point masses (see hbvevd). It is not
recommended as in practice it can produce poor estimates.
For either likelihood the margins are modelled using a generalized Pareto
distribution for points above the threshold and an empirical model for
those below. For the "poisson"
method data lying below both thresholds
is not used. For the "censored"
method the number of points lying
below both thresholds is used, but the locations of the those points are
not.
The dependence parameter names are one or more of dep
,
asy1
, asy2
, alpha
and beta
, depending on
the model selected (see rbvevd
).
The marginal parameter names are scale1
and shape1
for the first margin, and scale2
and shape2
for the
second margin.
If cshape
is true, the models are constrained so that
shape2 = shape1
. The parameter shape2
is then
taken to be specified, so that e.g. the common shape
parameter can only be fixed at zero using shape1 = 0
,
since using shape2 = 0
gives an error. Similar
comments apply for cscale
.
If sym
is TRUE
, the asymmetric logistic and
asymmetric negative logistic models are constrained so that
asy2 = asy1
, and the Coles-Tawn model is constrained
so that beta = alpha
. The parameter asy2
or
beta
is then taken to be specified, so that e.g.
the parameters asy1
and asy2
can only
be fixed at 0.8
using asy1 = 0.8
, since
using asy2 = 0.8
gives an error.
Bilogistic and negative bilogistic models constrained to symmetry are logistic and negative logistic models respectively. The (symmetric) mixed model (e.g. Tawn, 1998) can be obtained as a special case of the asymmetric logistic or asymmetric mixed models (see fbvevd).
For numerical reasons the parameters of each model are subject the
artificial constraints given in fbvevd
.
Returns an object of class c("bvpot","evd")
.
The generic accessor functions fitted
(or
fitted.values
), std.errors
,
deviance
, logLik
and
AIC
extract various features of the
returned object.
The functions profile
and profile2d
can be
used to obtain deviance profiles.
The function anova
compares nested models, and the
function AIC
compares non-nested models.
There is currently no plot method available.
An object of class c("bvpot","evd")
is a list containing
the following components
estimate |
A vector containing the maximum likelihood estimates. |
std.err |
A vector containing the standard errors. |
fixed |
A vector containing the parameters that have been fixed at specific values within the optimization. |
fixed2 |
A vector containing the parameters that have been set to be equal to other model parameters. |
param |
A vector containing all parameters (those optimized, those fixed to specific values, and those set to be equal to other model parameters). |
deviance |
The deviance at the maximum likelihood estimates. |
dep.summary |
A value summarizing the strength of dependence in the fitted model (see fbvevd). |
corr |
The correlation matrix. |
var.cov |
The variance covariance matrix. |
convergence , counts , message |
Components taken from the
list returned by |
data |
The data passed to the argument |
threshold |
The argument |
n |
The number of rows in |
nat |
The vector of length three containing the number of exceedances on the first, second and both margins respectively. |
likelihood |
The argument |
sym |
The argument |
cmar |
The vector |
model |
The argument |
call |
The call of the current function. |
The standard errors and the correlation matrix in the returned
object are taken from the observed information, calculated by a
numerical approximation.
They must be interpreted with caution when either of the
marginal shape parameters are less than -0.5
, because
the usual asymptotic properties of maximum likelihood estimators
do not then hold (Smith, 1985).
Chris Ferro and Alec Stephenson
Coles, S. G. (2001) An Introduction to Statistical Modelling of Extreme Values, London: Springer–Verlag.
Coles, S. G. and Tawn, J. A. (1991) Modelling multivariate extreme events. J. R. Statist. Soc. B, 53, 377–392.
Joe, H., Smith, R. L. and Weissman, I. (1992) Bivariate threshold methods for extremes. J. R. Statist. Soc. B, 54, 171–183.
Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. Biometrika, 72, 67–90.
abvevd
, anova.evd
,
fbvevd
, optim
, rbvevd
bvdata <- rbvevd(1000, dep = 0.5, model = "log")
u <- apply(bvdata, 2, quantile, probs = 0.9)
M1 <- fbvpot(bvdata, u, model = "log")
M2 <- fbvpot(bvdata, u, "log", dep = 0.5)
anova(M1, M2)
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