gev {VGAM} | R Documentation |
Maximum likelihood estimation of the 3-parameter generalized extreme value (GEV) distribution.
gev(llocation = "identity", lscale = "loge", lshape = "logoff", elocation = list(), escale = list(), eshape = if(lshape=="logoff") list(offset=0.5) else if(lshape=="elogit") list(min=-0.5, max=0.5) else list(), percentiles = c(95, 99), iscale=NULL, ishape = NULL, method.init = 1, gshape=c(-0.45, 0.45), tshape0=0.001, zero = 3) egev(llocation = "identity", lscale = "loge", lshape = "logoff", elocation = list(), escale = list(), eshape = if(lshape=="logoff") list(offset=0.5) else if(lshape=="elogit") list(min=-0.5, max=0.5) else list(), percentiles = c(95, 99), iscale=NULL, ishape = NULL, method.init=1, gshape=c(-0.45, 0.45), tshape0=0.001, zero = 3)
llocation, lscale, lshape |
Parameter link functions for mu, sigma and
xi respectively.
See Links for more choices.
|
elocation, escale, eshape |
List. Extra argument for the respective links.
See earg in Links for general information.
For the shape parameter,
if the logoff link is chosen then the offset is
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.
However, if percentiles=NULL , then the mean
mu + sigma * (gamma(1-xi)-1)/xi
is returned, and this is only defined if 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.
|
method.init |
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 method.init equals 1.
|
tshape0 |
Positive numeric.
Threshold/tolerance value for resting whether xi is zero.
If the absolute value of the estimate of xi is less than
this value then it will be assumed zero and Gumbel derivatives etc. will
be used.
|
zero |
An integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
The values must 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.
|
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"
and the eshape=list(offset=0.5)
, say,
or lshape="elogit"
and eshape=list(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 zero 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.)
The VGAM family function gev
can handle a multivariate
(matrix) response. If so, each row of the matrix is sorted into
descending order and NA
s 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.
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.
rgev
,
gumbel
,
egumbel
,
guplot
,
rlplot.egev
,
gpd
,
elogit
,
oxtemp
,
venice
.
# Multivariate example data(venice) y = as.matrix(venice[,paste("r", 1:10, sep="")]) fit1 = vgam(y[,1:2] ~ s(year, df=3), gev(zero=2:3), venice, trace=TRUE) coef(fit1, matrix=TRUE) fitted(fit1)[1:4,] ## Not run: par(mfrow=c(1,2), las=1) plot(fit1, se=TRUE, lcol="blue", scol="forestgreen", main="Fitted mu(year) function (centered)") attach(venice) matplot(year, y[,1:2], ylab="Sea level (cm)", col=1:2, main="Highest 2 annual sealevels and fitted 95 percentile") lines(year, fitted(fit1)[,1], lty="dashed", col="blue") detach(venice) ## End(Not run) # Univariate example data(oxtemp) (fit = vglm(maxtemp ~ 1, egev, data=oxtemp, trace=TRUE)) fitted(fit)[1:3,] coef(fit, mat=TRUE) Coef(fit) vcov(fit) vcov(fit, untransform=TRUE) sqrt(diag(vcov(fit))) # Approximate standard errors ## Not run: rlplot(fit)