GevModelling {fExtremes} | R Documentation |
A collection and description functions to compute
the generalized extreme value distribution and to
estimate it parameters. The functions compute
density, distribution function, quantile function
and generate random deviates for the GEV, for the
Frechet, Gumbel, and Weibull distributions. To model
the GEV three types of approaches for parameter
estimation are provided: Maximum likelihood
estimation, probability weighted moment method,
and estimation by the MDA approach. MDA includes
functions for the Pickands, Einmal-Decker-deHaan,
and Hill estimators together with several plot
variants.
The GEV distribution functions are:
dgev | density of the GEV Distribution, |
pgev | probability function of the GEV Distribution, |
qgev | quantile function of the GEV Distribution, |
rgev | random variates from the GEV Distribution. |
[dpqr]frechet | Frechet Distribution, |
[dpqr]gumbel | Gumbel Distribution, |
[dpqr]weibull | Weibull Distribution, |
[dpqr]evd | an alternative call for the GEV Distribution. |
The GEV modelling functions are:
gevSim | generates data from the GEV, |
gevFit | fits empirical or simulated data to the distribution, |
print | print method for a fitted GEV object, |
plot | plot method for a fitted GEV object, |
summary | summary method for a fitted GEV object, |
gevrlevelPlot | k-block return level with confidence intervals. |
Maximum Domain of Attraction Estimators:
hillPlot | shape parameter and Hill estimate of the tail index, |
shaparmPlot | variation of shape parameter with tail depth. |
dgev(x, xi = 1, mu = 0, sigma = 1, log = FALSE) pgev(q, xi = 1, mu = 0, sigma = 1, lower.tail = TRUE) qgev(p, xi = 1, mu = 0, sigma = 1, lower.tail = TRUE) rgev(n, xi = 1, mu = 0, sigma = 1) devd(x, loc = 0, scale = 1, shape = 0, log = FALSE) pevd(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) qevd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE) revd(n, loc = 0, scale = 1, shape = 0) gevSim(model = list(shape = 0.25, location = 0, scale = 1), n = 1000) gevFit(x, block = NA, type = c("mle", "pwm"), gumbel = FALSE, ...) ## S3 method for class 'gevFit': print(x, ...) ## S3 method for class 'gevFit': plot(x, which = "all", ...) ## S3 method for class 'gevFit': summary(object, doplot = TRUE, which = "all", ...) gevrlevelPlot(object, k.blocks = 20, add = FALSE, ...) hillPlot(x, option = c("alpha", "xi", "quantile"), start = 15, end = NA, reverse = FALSE, p = NA, ci = 0.95, autoscale = TRUE, labels = TRUE, ...) shaparmPlot(x, revert = FALSE, standardize = FALSE, tails = 0.01*(1:10), doplot = c(FALSE, FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE), which = c(TRUE, TRUE, TRUE), doprint = TRUE, both.tails = TRUE, xi.range = c(-0.5, 1.5), alpha.range = c(0, 10))
add |
[gevrlevelPlot] - whether the return level should be added graphically to a time series plot; if FALSE a graph of the profile
likelihood curve showing the return level and its confidence
interval is produced.
|
alpha.range, xi.range |
[saparmPlot] - plotting ranges. |
autoscale |
[hillPlot] - whether or not plot should be automatically scaled; if not, xlim and ylim graphical
parameters may be entered.
|
block |
[gevFit] - the block size. Only used if method="mle" is selected.
A numeric value is interpreted as the
number of data values in each successive block. All the data is
used, so the last block may not contain block observations.
If the data has a times attribute containing (in
an object of class "POSIXct" , or an object that can be
converted to that class; see as.POSIXct ) the
times/dates of each observation, then block may instead
take the character values "month" , "quarter" ,
"semester" or "year" .
|
both.tails |
[shaparmPlot] - a logical, decides whether or not both tails should be investigated. By default TRUE. If FALSE only the lower tail will be investigated. |
ci |
[hillPlot] - probability for asymptotic confidence band; for no confidence band set ci to zero.
|
doplot |
a logical. Should the results be plotted?
[shaparmPlot] - a vector of logicals of the same lengths as tails defining for wich tail depths plots should be created, by default plots will be generated for a tail depth of 5 percent. By default c(FALSE, FALSE, FALSE, FALSE,
TRUE, FALSE, FALSE, FALSE, FALSE, FALSE) .
|
doprint |
[shaparmPlot] - a logical, decides whether or not for all tail depths the result for the shape parameter 1/alpha should be
printed.
|
gumbel |
[gevFit] - a logical, by default FALSE. To fit a Gumbel model with fixed shape=0 set gumbel=TRUE .
|
k.blocks |
[gevrlevelPlot] - specifies the particular return level to be estimated; default set arbitrarily to 20. |
labels |
[hillPlot] - whether or not axes should be labelled. |
loc, scale, shape |
loc is the location parameter,
scale the scale parameter,
and shape is the shape parameter.
The default values are loc=0 , scale=1 , and
shape=0 .
|
log |
a logical, if TRUE , the log density is returned.
|
lower.tail |
a logical, if TRUE , the default, then
probabilities are P[X <= x] , otherwise, P[X > x] .
|
model |
[gevSim] - a list with components shape , location and
scale giving the parameters of the GEV distribution.
By default the shape parameter has the value 0.25, the
location is zero and the scale is one.
To fit random deviates from a Gumbel distribution set
shape=0 .
|
n |
[gevSim] - number of generated data points, an integer value. [rgev][revd] - the number of observations. |
object |
[summary][grlevelPlot] - a fitted object of class "gevFit" .
|
option |
[hillPlot] - whether alpha , xi (1/alpha) or
quantile (a quantile estimate) should be plotted.
|
p |
[qgev][qevs] - a numeric vector of probabilities. [hillPlot] - probability required when option quantile is
chosen.
|
q |
[pgev][pevs] - a numeric vector of quantiles. |
reverse |
[hillPlot] - whether plot is to be by increasing threshold, TRUE , or
increasing number of order statistics FALSE .
|
revert |
[shaparmPlot] - a logical value, by default FALSE, if set to TRUE the sign of the vector will be reverted: x = -x .
|
start, end |
[hillPlot] - lowest and highest number of order statistics at which to plot a point. |
standardize |
[shaparmPlot] - a logical value, by default FALSE, if set to TRUE the vector x will be standardized:
x = (x-mean(x))/sqrt(var(x)) .
|
tails |
[shaparmPlot] - a numeric vector of tail depths to be considered; by default ten values ranging from 0.1 to 1.0 in steps of 0.1 corresponding to values ranging from 1 to 10 percent. |
type |
a character string denoting the type of parameter estimation,
either by maximum likelihood estimation "mle" , the
default value, or by the probability weighted moment menthod
"pwm" .
|
which |
[shaparmPlot] - a vector of 3 logicals indicating which plots from the three methods will be created. The first entry decides for the Pickands estimator, the second for the Hill estimator, and the last for the Deckers-Einmahl-deHaan estimator. By default all three will be created. By default c(TRUE, TRUE, TRUE) .
[plot][summary] - a vector of logicals, one for each plot, denoting which plot should be displayed. Alkternatively if which="ask" the
user will be interactively asked which of the plots should be
desplayed. By default which="all" .
|
x |
[dgev][devd] - a numeric vector of quantiles. [gevFit] - data vector. In the case of method="mle" the interpretation
depends on the value of block: if no block size is specified then
data are interpreted as block maxima; if block size is set, then data
are interpreted as raw data and block maxima are calculated.
[hillPlot][shaparmPlot] - the data from which to calculate the shape parameter, a numeric vector. [print][plot] - a fitted object of class "gevFit" .
|
xi, mu, sigma |
xi is the shape parameter,
mu the location parameter,
and sigma is the scale parameter.
The default values are xi=1 , mu=0 , and
sigma=1 .
|
... |
[gevFit] - control parameters optionally passed to the optimization function. Parameters for the optimization function are passed to components of the control argument of
optim .
[hillPlot] - other graphics parameters. [plot][summary] - arguments passed to the plot function. |
Generalized Extreme Value Distribution:
Computes density, distribution function, quantile function and
generates random variates for the Generalized Extreme Value
Distribution, GEV, for the Frechet, Gumbel, and Weibull
distributions.
Parameter Estimation:
gevFit
estimates the parameters either by the probability
weighted moment method, method="pwm"
or by maximum log
likelihood estimation method="mle"
.
As a limiting case the Gumbel distribution can be selected. The
summary method produces diagnostic plots for fitted GEV or Gumbel
models.
Methods:
print.gev
, plot.gev
and summary.gev
are
print, plot, and summary methods for a fitted object of class
gev
. Concerning the summary method, the data are
converted to unit exponentially distributed residuals under null
hypothesis that GEV fits. Two diagnostics for iid exponential data
are offered. The plot method provides two different residual plots
for assessing the fitted GEV model. Two diagnostics for
iid exponential data are offered.
Return Level Plot:
gevrlevelPlot
calculates and plots the k-block return level
and 95% confidence interval based on a GEV model for block maxima,
where k
is specified by the user. The k-block return level
is that level exceeded once every k
blocks, on average. The
GEV likelihood is reparameterized in terms of the unknown return
level and profile likelihood arguments are used to construct a
confidence interval.
Hill Plot:
The function hillPlot
investigates the shape parameter and
plots the Hill estimate of the tail index of heavy-tailed data, or
of an associated quantile estimate. This plot is usually calculated
from the alpha perspective. For a generalized Pareto analysis of
heavy-tailed data using the gpdFit
function, it helps to
plot the Hill estimates for xi
.
Shape Parameter Plot:
The function shaparmPlot
investigates the shape parameter and
plots for the upper and lower tails the shape parameter as a function
of the taildepth. Three approaches are considered, the Pickands
estimator, the Hill estimator, and the
Decker-Einmal-deHaan estimator.
d*
returns the density,
p*
returns the probability,
q*
returns the quantiles, and
r*
generates random variates.
All values are numeric vectors.
gevSim
returns a vector of data points from the simulated series.
gevFit
returns an object of class gev
describing the fit.
print.summary
prints a report of the parameter fit.
summary
performs diagnostic analysis. The method provides two different
residual plots for assessing the fitted GEV model.
gevrlevelPlot
returns a vector containing the lower 95% bound of the confidence
interval, the estimated return level and the upper 95% bound.
hillPlot
displays a plot.
shaparmPlot
returns a list with one or two entries, depending on the
selection of the input variable both.tails
. The two
entries upper
and lower
determine the position of
the tail. Each of the two variables is again a list with entries
pickands
, hill
, and dehaan
. If one of the
three methods will be discarded the printout will display zeroes.
Generalized Extreme Value Distribution:
Here we implemented the notation for the arguments as used
by the GEV functions in the EVIS package or SPlus/FinMetrics
module. Additional arguments to these packages are the log
and the lower.tail
arguments, underlying the code
from R's evd
package.
An alternative usage is proposed by the evd
package.
There the following arguments are used:
*gev(x, loc = 0, scale = 1, shape = 0, ...)
What you prefer is a matter of taste. The GEV functions from
the evd
package are renamed from *gev
to *evd
so that both versions are available.
In addition functions for the density, probability, quantiles,
and the generation of random variates for the
Frechet [dpqr]frechet
,
Gumbel [dpqr]gumbel
, and
Weibull [dpqr]weibull
are also available.
If you stay with both versions you can access the help page for
evds
's function dgev
by help(dgev, package="evd")
.
Generalized Extreme Value Distribution:
If method "mle"
is selected the parameter fitting in gevFit
is passed to the internal function gev.mle
or gumbel.mle
depending on the value of gumbel
, FALSE
or TRUE
.
On the other hand, if method "pwm"
is selected the parameter
fitting in gevFit
is passed to the internal function
gev.pwm
or gumbel.pwm
again depending on the value of
gumbel
, FALSE
or TRUE
.
Alec Stephenson for R's evd
and evir
package, and
Diethelm Wuertz for this R-port.
Coles S. (2001); Introduction to Statistical Modelling of Extreme Values, Springer.
Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.
## SOURCE("fExtremes.52A-GevModelling") ## *gev - xmpExtremes("\nStart: GEV Frechet >") # Create and plot 1000 GEV/Frechet distributed rdv: par(mfrow = c(3, 3)) r = rgev(n = 1000, xi = 1) plot(r, type = "l", main = "GEV/Frechet Series") ## Plot empirical density and compare with true density: ## Omit values greater than 500 from plot hist(r[r<10], n = 25, probability = TRUE, xlab = "r", xlim = c(-5, 5), ylim = c(0, 1.1), main = "Density") x = seq(-5, 5, by=0.01) lines(x, dgev(x, xi = 1), col = 2) ## Plot df and compare with true df: plot(sort(r), (1:length(r)/length(r)), xlim = c(-3, 6), ylim = c(0, 1.1), cex = 0.5, ylab = "p", xlab = "q", main = "Probability") q = seq(-5,5, by=0.1) lines(q, pgev(q, xi=1), col=2) ## Compute quantiles, a test: qgev(pgev(seq(-5, 5, 0.25), xi = 1), xi = 1) ## *gev - xmpExtremes("\nNext: GEV Gumbel >") # Create and plot 1000 Gumbel distributed rdv: ##> r = rgev(n = 1000, xi = 0) ##> plot(r, type = "l", main = "Gumbel Series") ## Plot empirical density and compare with true density: ##>hist(r[abs(r)<10], nclass = 25, freq = FALSE, xlab = "r", ##> xlim = c(-5,5), ylim = c(0,1.1), main = "Density") ##>x = seq(-5, 5, by = 0.01) ##>lines(x, dgev(x, xi = 0), col=2) ## Plot df and compare with true df: ##>plot(sort(r), (1:length(r)/length(r)), ##> xlim = c(-3, 6), ylim = c(0, 1.1), ##> cex=0.5, ylab = "p", xlab="q", main="Probability") ##>q = seq(-5, 5, by = 0.1) ##>lines(q, pgev(q, xi = 0), col = 2) ## Compute quantiles, a test: ##>qgev(pgev(seq(-5, 5, 0.25), xi = 0), xi = 0) ## *gev - xmpExtremes("\nNext: GEV Weibull >") # Create and plot 1000 Weibull distributed rdv: r = rgev(n = 1000, xi = -1) plot(r, type = "l", main = "Weibull Series") ## Plot empirical density and compare with true density: hist(r[abs(r)<10], nclass = 25, freq = FALSE, xlab = "r", xlim=c(-5,5), ylim=c(0,1.1), main="Density") x = seq(-5, 5, by=0.01) lines(x, dgev(x, xi = -1), col = 2) ## Plot df and compare with true df: plot(sort(r), (1:length(r)/length(r)), xlim = c(-3, 6), ylim = c(0, 1.1), cex = 0.5, ylab = "p", xlab = "q", main = "Probability") q=seq(-5, 5, by = 0.1) lines(q, pgev(q, xi = -1), col = 2) ## Compute quantiles, a test: qgev(pgev(seq(-5, 5, 0.25), xi = -1), xi = -1) ## gevSim - ## gevFit - # Simulate GEV Data: xmpExtremes("\nStart: Simulte GEV Sample >") # Use default length n=1000 x = gevSim(model = list(shape = 0.25, location =0 , scale = 1)) # Fit GEV Data by Probability Weighted Moments: fit = gevFit(x, type = "pwm") print(fit) # Summarize Results: par(mfcol = c(3, 2)) summary(fit) ## gevFit - # Fit GEV Data by Max Log Likelihood Method: xmpExtremes("\nNext: Estimate Parameters >") fit = gevFit(x, type = "mle") print(fit) # Summarize Results: summary(fit) ## gevSim - ## gevFit - # Simulate Gumbel Data: xmpExtremes("\nNext: Simulte Gumbel Sample >") # Use default length n=1000 ##> x = gevSim(model = list(shape = 0, location = 0, scale = 1)) # Fit Gumbel Data by Probability Weighted Moments: ##> fit = gevFit(x, type = "pwm", gumbel = TRUE) ##> print(fit) # Summarize Results: ##> par(mfcol = c(3, 2)) ##> summary(fit) ## Fit Gumbel Data by Max Log Likelihood Method: xmpExtremes("\nNext: Estimate Parameters >") ##> fit = gevFit(x, type = "mle", gumbel = TRUE) ##> print(fit) # Summarize Results: ##> summary(fit) ##> xmpExtremes("Press any key to continue >") ## Return levels based on GEV Fit: # BMW Stock Data: xmpExtremes("\nNext: Compute BMW Return Levels >") par(mfrow = c(2, 1)) data(bmw) # Fit GEV to monthly Block Maxima: fit = gevFit(-bmw, block = "month") # Calculate the 40 month return level gevrlevelPlot(fit, k.block = 40, main = "BMW: Return Levels") ## Return levels based on GEV Fit: xmpExtremes("\nNext: Compute SIEMENS Return Levels >") # Siemens Stock Data: data(siemens) # Fit GEV to monthly Block Maxima: fit = gevFit(-siemens, block = "month") # Calculate the 40 month return level gevrlevelPlot(fit, k.block = 40, main = "SIEMENS: Return Levels") ## Interactive Plot: ##> par(mfrow = c(1, 1), ask = TRUE) ##> plot(fit) ## hillPlot - xmpExtremes("\nStart: Hill Estimator >") # Hill plot of heavy-tailed Danish fire insurance data # and BMW stock data for estimated 0.999 quantile par(mfrow = c(2, 2)) data(bmw) hillPlot(bmw) hillPlot(bmw, option = "quantile", end = 500, p = 0.999) data(danish) hillPlot(danish) hillPlot(danish, option = "quantile", end = 500, p = 0.999) ## shaparmPlot - xmpExtremes("\nNext: Shape Parameter Plots >") par(mfcol = c(3, 2), cex = 0.6) data(bmw) shaparmPlot(bmw) ## shaparmPlot - xmpExtremes("\nNext: Simulated Frechet Data >") par(mfcol = c(3, 2), cex = 0.6) set.seed(4711) x = rgev(10000, xi = 1/4) shaparmPlot(x, revert = TRUE, both.tails = FALSE) lines(c(0.01, 0.1), c(4, 4), col = "steelblue3") # True Value