GevModelling {fExtremes} | R Documentation

## Generalized Extreme Value Modelling

### Description

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.

### Usage

    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))

### Arguments

`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.

### Details

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.

### Value

`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.

### Note

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`.

### Author(s)

Alec Stephenson for R's `evd` and `evir` package, and
Diethelm Wuertz for this R-port.

### References

Coles S. (2001); Introduction to Statistical Modelling of Extreme Values, Springer.

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.

### Examples

    ## 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

[Package fExtremes version 221.10065 Index]