R: Generalized Extreme Value Distribution Family Function

gev {VGAM}

R Documentation

Generalized Extreme Value Distribution Family Function

Description

Maximum likelihood estimation of the 3-parameter generalized extreme value (GEV) distribution.

Usage

gev(llocation = "identitylink", lscale = "loge", lshape = logoff(offset = 0.5),
    percentiles = c(95, 99), iscale=NULL, ishape = NULL,
    imethod = 1, gshape = c(-0.45, 0.45), tolshape0 = 0.001,
    type.fitted = c("percentiles", "mean"), giveWarning = TRUE,
    zero = c("scale", "shape"))
egev(llocation = "identitylink", lscale = "loge", lshape = logoff(offset = 0.5),
     percentiles = c(95, 99), iscale=NULL,  ishape = NULL,
     imethod = 1, gshape = c(-0.45, 0.45), tolshape0 = 0.001,
     type.fitted = c("percentiles", "mean"), giveWarning = TRUE,
     zero = c("scale", "shape"))

Arguments

llocation, lscale, lshape

Parameter link functions for mu, sigma and xi respectively. See Links for more choices.

For the shape parameter, the default logoff link has an offset called A below; and then the linear/additive predictor is log(xi+A) which means that xi > -A. For technical reasons (see Details) it is a good idea for A = 0.5.

`percentiles`	Numeric vector of percentiles used for the fitted values. Values should be between 0 and 100. This argument is ignored if `type.fitted = "mean"`.
`type.fitted`	See `CommonVGAMffArguments` for information. The default is to use the `percentiles` argument. If `"mean"` is chosen, then the mean mu + sigma (gamma(1-xi)-1)/xi* is returned as the fitted values, and these are only defined for xi<1.
`iscale, ishape`	Numeric. Initial value for sigma and xi. A `NULL` means a value is computed internally. The argument `ishape` is more important than the other two because they are initialized from the initial xi. If a failure to converge occurs, or even to obtain initial values occurs, try assigning `ishape` some value (positive or negative; the sign can be very important). Also, in general, a larger value of `iscale` is better than a smaller value.

`imethod`	Initialization method. Either the value 1 or 2. Method 1 involves choosing the best xi on a course grid with endpoints `gshape`. Method 2 is similar to the method of moments. If both methods fail try using `ishape`.
`gshape`	Numeric, of length 2. Range of xi used for a grid search for a good initial value for xi. Used only if `imethod` equals 1.
`tolshape0, giveWarning`	Passed into `dgev` when computing the log-likelihood.
`zero`	A specifying which linear/additive predictors are modelled as intercepts only. The values can be from the set {1,2,3} corresponding respectively to mu, sigma, xi. If `zero = NULL` then all linear/additive predictors are modelled as a linear combination of the explanatory variables. For many data sets having `zero = 3` is a good idea. See `CommonVGAMffArguments` for information.

Details

The GEV distribution function can be written

G(y) = exp( -[ (y- mu)/ sigma ]_{+}^{- 1/ xi})

where sigma > 0, -Inf < mu < Inf, and 1 + xi*(y-mu)/sigma > 0. Here, x_+ = max(x,0). The mu, sigma, xi are known as the location, scale and shape parameters respectively. The cases xi>0, xi<0, xi = 0 correspond to the Frechet, Weibull, and Gumbel types respectively. It can be noted that the Gumbel (or Type I) distribution accommodates many commonly-used distributions such as the normal, lognormal, logistic, gamma, exponential and Weibull.

For the GEV distribution, the kth moment about the mean exists if xi < 1/k. Provided they exist, the mean and variance are given by mu + sigma { Gamma(1-xi)-1} / xi and sigma^2 { Gamma(1-2 xi) - Gamma^2 (1- xi) } / xi^2 respectively, where Gamma is the gamma function.

Smith (1985) established that when xi > -0.5, the maximum likelihood estimators are completely regular. To have some control over the estimated xi try using lshape = logoff(offset = 0.5), say, or lshape = extlogit(min = -0.5, max = 0.5), say.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Currently, if an estimate of xi is too close to 0 then an error will occur for gev() with multivariate responses. In general, egev() is more reliable than gev().

Fitting the GEV by maximum likelihood estimation can be numerically fraught. If 1 + xi*(y-mu)/sigma <= 0 then some crude evasive action is taken but the estimation process can still fail. This is particularly the case if vgam with s is used; then smoothing is best done with vglm with regression splines (bs or ns) because vglm implements half-stepsizing whereas vgam doesn't (half-stepsizing helps handle the problem of straying outside the parameter space.)

Note

The VGAM family function gev can handle a multivariate (matrix) response. If so, each row of the matrix is sorted into descending order and NAs are put last. With a vector or one-column matrix response using egev will give the same result but be faster and it handles the xi = 0 case. The function gev implements Tawn (1988) while egev implements Prescott and Walden (1980).

The shape parameter xi is difficult to estimate accurately unless there is a lot of data. Convergence is slow when xi is near -0.5. Given many explanatory variables, it is often a good idea to make sure zero = 3. The range restrictions of the parameter xi are not enforced; thus it is possible for a violation to occur.

Successful convergence often depends on having a reasonably good initial value for xi. If failure occurs try various values for the argument ishape, and if there are covariates, having zero = 3 is advised.

Author(s)

T. W. Yee

References

Yee, T. W. and Stephenson, A. G. (2007) Vector generalized linear and additive extreme value models. Extremes, 10, 1–19.

Tawn, J. A. (1988) An extreme-value theory model for dependent observations. Journal of Hydrology, 101, 227–250.

Prescott, P. and Walden, A. T. (1980) Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika, 67, 723–724.

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

Examples

## Not run: 
# Multivariate example
fit1 <- vgam(cbind(r1, r2) ~ s(year, df = 3), gev(zero = 2:3),
             data = venice, trace = TRUE)
coef(fit1, matrix = TRUE)
head(fitted(fit1))
par(mfrow = c(1, 2), las = 1)
plot(fit1, se = TRUE, lcol = "blue", scol = "forestgreen",
     main = "Fitted mu(year) function (centered)", cex.main = 0.8)
with(venice, matplot(year, depvar(fit1)[, 1:2], ylab = "Sea level (cm)",
     col = 1:2, main = "Highest 2 annual sea levels", cex.main = 0.8))
with(venice, lines(year, fitted(fit1)[,1], lty = "dashed", col = "blue"))
legend("topleft", lty = "dashed", col = "blue", "Fitted 95 percentile")

# Univariate example
(fit <- vglm(maxtemp ~ 1, egev, data = oxtemp, trace = TRUE))
head(fitted(fit))
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
vcov(fit, untransform = TRUE)
sqrt(diag(vcov(fit)))  # Approximate standard errors
rlplot(fit)

## End(Not run)

[Package VGAM version 1.0-1 Index]