fbvevd {evd}

Maximum-likelihood Fitting of Bivariate Extreme Value Distributions

Description

Fit models for one of nine parametric bivariate extreme value distributions, including linear modelling of the marginal location parameters, and allowing any of the parameters to be held fixed if desired.

Usage

fbvevd(x, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog",
    "negbilog", "ct", "amix"), start, ..., sym = FALSE,
    nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc,
    cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS",
    warn.inf = TRUE)

Arguments

x

A matrix or data frame, ordinarily with two columns, which may contain missing values. A data frame may also contain a third column of mode logical, which itself may contain missing values (see More Details).

model

The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in rbvevd.

start

A named list giving the initial values for the parameters over which the likelihood is to be maximized. If start is omitted the routine attempts to find good starting values using marginal maximum likelihood estimators.

...

Additional parameters, either for the bivariate extreme value model or for the optimization function optim. If parameters of the model are included they will be held fixed at the values given (see Examples).

sym

Logical; if 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 warning is given).

nsloc1, nsloc2

A data frame with the same number of rows as x, 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 numeric vector can be given as an alternative to a single column data frame.

cshape

Logical; if TRUE, a common shape parameter is fitted to each margin.

cscale

Logical; if 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.

cloc

Logical; if 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 fitted.

std.err

Logical; if TRUE (the default), the standard errors are returned.

corr

Logical; if TRUE, the correlation matrix is returned.

method

The optimization method (see optim for details).

warn.inf

Logical; if TRUE (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.

Details

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 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 (symmetric) mixed model (e.g. Tawn, 1998) can be obtained as a special case of the asymmetric logistic or asymmetric mixed models (see Examples).

The value Dependence given in the printed output is 2(1-A(1/2)), where A is the estimated dependence function (see abvevd). It measures the strength of dependence, and lies in the interval [0,1]; at independence and complete dependence it is zero and one respectively (Coles, Heffernan and Tawn, 1999). See chiplot for further information.

Value

Returns an object of class 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

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

The estimate of 2(1-A(1/2)).

corr

The correlation matrix.

var.cov

The variance covariance matrix.

convergence, counts, message

Components taken from the list returned by optim.

data

The data passed to the argument x.

tdata

The data, transformed to stationarity (for non-stationary models).

nsloc1, nsloc2

The arguments nsloc1 and nsloc2.

n

The number of rows in x.

sym

The argument sym.

cmar

The vector c(cloc, cscale, cshape).

model

The argument model.

call

The call of the current function.

More Details

If 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: the density, dependence function and quantile curves plots contain fitted functions for observations where the logical case is unknown, and the conditional P-P plots condition on both the logical case and the given margin (which requires numerical integration at each data point).

Artificial Constraints

For numerical reasons parameters are subject to artificial constraints. Specifically, these constraints are: marginal scale parameters not less than 0.01; dep not less than [0.1] [0.2] [0.05] in [logistic] [Husler-Reiss] [negative logistic] models; dep not greater than [10] [5] in [Husler-Reiss] [negative logistic] models; asy1 and asy2 not less than 0.001; alpha and beta not less than [0.1] [0.1] [0.001] in [bilogistic] [negative bilogistic] [Coles-Tawn] models; alpha and beta not greater than [0.999] [20] [30] in [bilogistic] [negative bilogistic] [Coles-Tawn] models.

Warning

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).

References

Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339–365.

Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. Biometrika, 72, 67–90.

Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. Biometrika 92(1), 213–217.

Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397–415.

See Also

anova.evd, optim, plot.bvevd, profile.evd, profile2d.evd, rbvevd

Examples

bvdata <- 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)
## Not run: par(mfrow = c(1,1))
## Not run: M1P <- profile(M1, which = "dep")
## Not run: 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)
amixed <- fbvevd(bvdata, model = "amix")
mixed <- fbvevd(bvdata, model = "amix", beta = 0)
anova(amixed, mixed)