Generalized Extreme Value Regression Family Function
Maximum likelihood estimation of the 3-parameter generalized extreme value (GEV) distribution.
gev(llocation = "identitylink", lscale = "loglink",
lshape = logofflink(offset = 0.5), percentiles = c(95, 99),
ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
gshape = (-5:5) / 11 + 0.01,
iprobs.y = NULL, tolshape0 = 0.001,
type.fitted = c("percentiles", "mean"),
zero = c("scale", "shape"))
gevff(llocation = "identitylink", lscale = "loglink",
lshape = logofflink(offset = 0.5), percentiles = c(95, 99),
ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
gshape = (-5:5) / 11 + 0.01,
iprobs.y = NULL, tolshape0 = 0.001,
type.fitted = c("percentiles", "mean"), zero = c("scale", "shape"))llocation, lscale, lshape |
Parameter link functions for mu, sigma and
xi respectively.
See For the shape parameter,
the default |
percentiles |
Numeric vector of percentiles used for the fitted values.
Values should be between 0 and 100.
This argument is ignored if |
type.fitted |
See |
ilocation, iscale, ishape |
Numeric. Initial value for the location parameter, sigma and
xi. A |
imethod |
Initialization method. Either the value 1 or 2.
If both methods fail then try using |
gshape |
Numeric vector.
The values are used for a grid search for an initial value
for xi.
See |
gprobs.y, gscale.mux, iprobs.y |
Numeric vectors, used for the initial values.
See |
tolshape0 |
Passed into |
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 |
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, reverse 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 = logofflink(offset = 0.5), say,
or lshape = extlogitlink(min = -0.5, max = 0.5), say.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Currently, if an estimate of xi is too close to 0 then
an error may occur for gev() with multivariate responses.
In general, gevff() 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.)
The VGAM family function gev can handle a multivariate
(matrix) response, cf. multiple responses.
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
gevff will give the same result but be faster and it handles
the xi = 0 case.
The function gev implements Tawn (1988) while
gevff implements Prescott and Walden (1980).
Function egev() has been replaced by the
new family function gevff(). It now
conforms to the usual VGAM philosophy of
having M1 linear predictors per (independent) response.
This is the usual way multiple responses are handled.
Hence vglm(cbind(y1, y2)..., gevff, ...) will have
6 linear predictors and it is possible to constrain the
linear predictors so that the answer is similar to gev().
Missing values in the response of gevff() will be deleted;
this behaviour is the same as with almost every other
VGAM family function.
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.
T. W. Yee
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.
## 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, gevff, 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)
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