fbvevd(x, model = "log", start, ..., sym = FALSE, nsloc1 = NULL,
nsloc2 = NULL, cshape = cscale, cscale = cloc, cloc = FALSE,
std.err = TRUE, dsm = TRUE, corr = FALSE, method = "BFGS",
warn.inf = TRUE)logical, which
itself may contain missing values (see More Details)."log" (the default), "alog", "hr",
"neglog", "aneglog", "bilog",
"negbilog" or "ct"start is omitted the routine attempts to find good
starting values using marginal maximum likelihood estimators.optim. If
parameters of the model are included they will be held fixed at
the values given (see Examples).TRUE, the dependence structure
of the models "alog", "aneglog" or "ct" are
constrained to be symmetric (see Details). For all other
models, the argument is ignored (and a wx, for linear modelling of the location parameter on the
first/second margin (see Details).
The data frames are treated as covariate matrices, excluding the
intercept. A numeTRUE, a common shape parameter is
fitted to each margin.TRUE, a common scale parameter is
fitted to each margin, and the default value of cshape
is then TRUE, so that under this default common scale
and shape parameters are fitted.TRUE, a common location parameter is
fitted to each margin, and the default values of cshape
and cscale are then TRUE, so that under these
defaults common marginal parameters are TRUE (the default), the standard
errors are returned.TRUE (the default), summaries of the
dependence structure are returned.TRUE, the correlation matrix is
returned.optim for
details).TRUE (the default), a warning is
given if the negative log-likelihood is infinite when evaluated at
the starting values.c("bvevd","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.
The function plot produces diagnostic plots.
An object of class c("bvevd","evd") is a list containing
the following components
optim.x.nsloc1 and nsloc2.x.sym.c(cloc, cscale, cshape).model.x is a data frame with a third column of mode
logical, then the model is fitted using the likelihood
derived by Stephenson and Tawn (2004). This is appropriate
when each bivariate data point comprises componentwise maxima
from some underlying bivariate process, and where the
corresponding logical value denotes whether or not the maxima
were caused by the same event within that process.
Under this scheme the diagnostic plots that are produced
using plot are somewhat different to those described
in plot.bvevd. In particular, there is no
comparative non-parametric dependence function estimate,
and the conditional P-P plots condition on both the logical
case and the given margin (which requires numerical integration
at each data point).dep,
asy1, asy2, alpha and beta, depending on
the model selected (see rbvevd). The marginal parameter
names are loc1, scale1 and shape1 for the first
margin, and loc2, scale2 and shape2 for the
second margin.
If nsloc1 is not NULL, so that a linear model is
implemented for the first marginal location parameter, the parameter
names for the first margin are loc1, loc1x1,
..., loc1xn, scale and shape, where
x1, ..., xn are the column names of nsloc1,
so that loc1 is the intercept of the linear model, and
loc1x1, ..., loc1xn are the
ncol(nsloc1) coefficients.
When nsloc2 is not NULL, the parameter names for the
second margin are constructed similarly.
It is recommended that the covariates within the linear models for
the location parameters are (at least approximately) centered and
scaled (i.e. that the columns of nsloc1 and nsloc2
are centered and scaled), particularly if automatic starting values
are used, since the starting values for the associated parameters are
then zero. If cloc is TRUE, both nsloc1 and
nsloc2 must be identical, since a common linear model is
then implemented on both margins. 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 and cloc.
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 mixed model (e.g. Tawn, 1998)
is obtained by the asymmetric negative logistic model upon
setting the dependence parameter to be one, and constraining
the asymmetry parameters to be equal to each other. It can
therefore be fitted using model = "anegl" with
dep = 1 and sym = TRUE (see Examples).
If dsm is TRUE, three values are returned which
summarize the dependence structure, based on the fitted
dependence function $A$ (see abvpar).
Two are measures of the strength of dependence.
The first (Dependence One) is given by $2(1-A(1/2))$.
The second (Dependence Two) is the integral of $4(1 - A(x))$,
taken over $0\leq x\leq1$.
Both measures are zero at independence and one at complete dependence.
The third value (Asymmetry) is a measure of asymmetry, given by
the integral of
$4(A(x) - A(1-x))/(3 - 2\sqrt2)$,
taken over $0 \leq x \leq 0.5$.
This lies in the closed interval [-1,1], with
larger absolute values representing stronger asymmetry.
For the logistic, Husler-Reiss and negative logistic models
$A(x) = A(1-x)$ for all $0 \leq x \leq 0.5$,
so the value will be zero.
For numerical reasons the parameters of each model are subject the artificial constraints given in Table 1 of the User's Guide.
Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. Biometrika (To Appear).
Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397--415.
anova.evd, optim,
plot.bvevd, profile.evd,
profile2d.evd, rbvevdbvdata <- rbvevd(100, dep = 0.6, model = "log", mar1 = c(1.2,1.4,0.4))
M1 <- fbvevd(bvdata, model = "log")
M2 <- fbvevd(bvdata, model = "log", dep = 0.75)
anova(M1, M2)
par(mfrow = c(2,2))
plot(M1)
plot(M1, mar = 1)
plot(M1, mar = 2)
par(mfrow = c(1,1))
M1P <- profile(M1, which = "dep")
plot(M1P)
trend <- (-49:50)/100
rnd <- runif(100, min = -.5, max = .5)
fbvevd(bvdata, model = "log", nsloc1 = trend)
fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend
= trend, random = rnd))
fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend
= trend, random = rnd), loc2random = 0)
bvdata <- rbvevd(100, dep = 1, asy = c(0.5,0.5), model = "anegl")
anlog <- fbvevd(bvdata, model = "anegl")
mixed <- fbvevd(bvdata, model = "anegl", dep = 1, sym = TRUE)
anova(anlog, mixed)Run the code above in your browser using DataLab