Given a threshold which defines excesses above that threshold, estimate the extremal index of a dependent sequence by using the method of Ferro and Segers, 2003. The extremal index estimate can then be used to carry out automatic declustering of the sequence to identify independent clusters and estimate the GPD for cluster maxima. Graphical diagnostics of model fit are available.
extremalIndex(y, data = NULL, threshold)extremalIndexRangeFit(y, data = NULL, umin = quantile(y,.5), umax =
quantile(y, 0.95), nint = 10, nboot = 100, alpha = .05, estGPD=TRUE,
verbose = TRUE, trace = 10, ...)
bootExtremalIndex(x)
declust(y, r=NULL, data = NULL, ...)
# S3 method for extremalIndex
declust(y, r=NULL,...)
# S3 method for declustered
plot(x, ylab = "Data",...)
# S3 method for declustered
evm(y, data=NULL, family=gpd, ...)
# S3 method for extremalIndexRangeFit
plot(x,addNexcesses=TRUE,estGPD=TRUE,...)
# S3 method for extremalIndex
print(x,...)
# S3 method for declustered
print(x,...)
# S3 method for extremalIndexRangeFit
ggplot(data=NULL, mapping, xlab, ylab, main,
ylim = "auto",ptcol="dark blue",col="dark blue",fill="orange",
textsize=4,addNexcesses=TRUE,estGPD=TRUE,..., environment)
The function extremalIndex returns a list of class
"extremalIndex":
Estimate of the extremal index by using the intervals estimator of Ferro and Segers.
threshold for declustering and estimation
length of original data series
number of exceedances of threshold in original
series.
probablity of threshold exceedance in original series.
the original function call
times between threshold exceedances
observation from the original series which are above
threshold
times of occurrance of threshold exceedances
original dependent series
data frame or NULL
The function declust returns a list of type "declustered":
integer labels assigning threshold exceedances to clusters
number of exceedances in each cluster
vector made up of the largest observation from each distinct cluster. In the case of ties, the first value is taken.
logical; length equal to number of threshold
exceedances, value is TRUE for threshold exceedances which correspond
to cluster maxima
see entry for object of class "extremalIndex" above
see entry for object of class "extremalIndex" above
see entry for object of class "extremalIndex" above
see entry for object of class "extremalIndex" above
see entry for object of class "extremalIndex" above
times between threshold exceedances, length is one less than the number of threshold exceedances
logical:
indicates inter exceedance times larger than r the run length used
for declustering
see entry for object of class "extremalIndex" above
see entry for object of class "extremalIndex" above
run length used for declustering
Number of indenendent clusters identified
Method used for declustering (either "intervals" or "runs")
The function bootExtremalIndex return a single vector corersponding
to a bootstrap sample from the original series: observations are censored at
threshold so that values below this threshold are indicated by the
value -1.
The method evm for class "declustered" returns an object of type
"evmOpt" or "evmSim" depending on the precise function call - see
documentation for evm.
Argument to function extremalIndex: either a numeric vector
or the name of a variable in data.
A data frame containing y and any covariates. In
evm.declustered, it should be NULL and is included to match the
arguments of generic evm.
The threshold for y, exceedances above which will be
used to estimate the extremal index and carry out automatic declustering.
The type of extreme value model. The user should not change
this from its default in evm.declustered.
Objects passed to methods.
Positivie integer: run length to be used under "runs" declustering. If specified then so-called "runs" declustering will be carried out, otherwise defaults to NULL in which case the automatic "intervals" declustering method of Ferro and Segers is used.
The minimum threshold above which to esimate the parameters.
The maximum threshold above which to esimate the parameters.
The number of thresholds at which to perform the estimation.
Number of bootstrap samples to simulate at each threshold for estimation.
100(1 - alpha)% confidence intervals will be plotted with the
point estimates. Defaults to alpha = 0.05.
Label for the x-axis (ggplot).
Label for the y-axis (ggplot).
Whether to annotate the top axis of plots with the
number of excesses above the corresponding threhsold. Defaults to
TRUE.
Whether to estimate GPD parameters at each choice of
thereshold -- defaults to TRUE in which case the GPD parameters are
estimated.
Whether to report on progress in RangeFit calculations.
Defaults to TRUE.
How frequently to report bootstrap progress in RangeFit calculations. Defaults to 10.
Further arguments to ggplot method.
Further arguments to methods.
Janet E. Heffernan
The function extremalIndex estimates the extremal index of a
dependent series of observations above a given threshold threshold,
returning an object of class "extremalIndex". Plot and print methods are
available for this class. A graphical diagnostic akin to Figure 1 in Ferro
and Segers (2003) is produced by the plot method for this class.
This plot is used to test the model assumption underpinning the estimation,
with good fit being indicated by interexceedance times which correspond to
inter-cluster times lying close to the diagonal line indicated.
In addition to good model fit, an appropriate choice of threshold is one
above which the estimated extremal index is stable over further, higher
thresholds (up to estimation uncertainty). This can be assessed by using
the function extremalIndexRangeFit, which examines a range of
threshold values. At each threshold, the extremal index is estimated; that
estimate is used to decluster the series and the parameters of the GPD are
optionally estimated for the resulting declustered series. Uncertainty in
the estimation of the extremal index and GPD parameters is assessed by using
a bootstrap scheme which accounts for uncertainty in the extremal index
estimation, and the corresponding uncertainty in the declustering of the
series. There are plot and ggplot methods for output of this function, which is of class extremalIndexRangeFit.
The function declust returns an object of class "declustered",
identifying independent clusters in the original series. Print, plot and
show methods are available for this class. The GPD model can be fitted to
objects of this class, including the use of covariates in the linear
predictors for the parameters of the GPD. See examples below.
Ferro, C.A.T. and Segers, J., (2003) "Inference for clusters of Extreme Values", JRSS B 65, Part 2, pp 545--556.
evm
par(mfrow=c(2,2));
extremalIndexRangeFit(summer$O3,nboot=10)
ei <- extremalIndex(summer$O3,threshold=45)
plot(ei)
d <- declust(ei)
plot(d)
evm(d)
## fitting with covariates:
so2 <- extremalIndex(SO2,data=winter,threshold=15)
plot(so2)
so2 <- extremalIndex(SO2,data=winter,threshold=20)
plot(so2) ## fits better
so2.d <- declust(so2)
par(mfrow=c(1,1)); plot(so2.d)
so2.d.gpd <- evm(so2.d) # AIC 661.1
evm(so2.d,phi=~NO)
evm(so2.d,phi=~NO2)
evm(so2.d,phi=~O3) # better AIC 651.9
evm(so2.d,phi=~PM10)
so2.d.gpd.o3 <- evm(so2.d,phi=~O3)
par(mfrow=c(2,2)); plot(so2.d.gpd.o3)
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