GevGlmFit {fExtremes} | R Documentation

## Modelling the GEV Distribution including GLM

### Description

A collection and description of functions to model the Generalized Extreme Value, GEV, distribution by maximum likelihood approximation based on R's 'ismev' package. In addition to the function gevFit the parameter estimation allows to include generalized linear modelling, GLM, of each parameter.

The functions are:

`gevglmFit` | fits empirical or simulated data to the distribution,
`print` | print method for a fitted GEV object of class ...,
`plot` | plot method for a fitted GEV object,
`summary` | summary method for a fitted GEV object,
`gevglmprofPlot` | profile log-likelihoods for return levels,
`gevglmprofxiPlot` | profile log-likelihoods for shape parameters.

### Usage

    gevglmFit(x, y = NULL, gumbel = FALSE, mul = NULL, sigl = NULL, shl = NULL,
        mulink = identity, siglink = identity, shlink = identity, show = FALSE,
        method = "Nelder-Mead", maxit = 10000, ...)

    ## S3 method for class 'gevglmFit':
    print(x, ...)
    ## S3 method for class 'gevglmFit':
    plot(x, which = "ask", ...)
    ## S3 method for class 'gevglmFit':
    summary(object, doplot = TRUE, which = "all", ...)

    gevglmprofPlot(object, m, xlow, xup, conf = 0.95, nint = 100)
    gevglmprofxiPlot(object, xlow, xup, conf = 0.95, nint = 100)

### Arguments

`conf` | [gevglmprof*Plot] - the confidence coefficient of the plotted profile confidence interval.
`doplot` | a logical. Should the results be plotted?
`gumbel` | [gevglmFit] - a logical, should a Gumbel fit be created? In this case the shape parameter equals zero, and the arguments `shl` and and `shlink` are omitted.
`m` | [gevglmprofPlot] - the return level; i.e. the profile likelihood is for the value that is exceeded with probability 1/`m`.
`maxit` | [gevglmFit] - the maximum number of iterations.
`method` | [gevglmFit] - the optimization method (see `optim` for details).
`mul, sigl, shl` | [gevglmFit] - numeric vectors of integers, giving the columns of `ydat` that contain covariates for generalized linear modelling of the location, scale and shape parameters repectively (or `NULL` (the default) if the corresponding parameter is stationary).
`mulink, siglink, shlink` | [gevglmFit] - inverse link functions for generalized linear modelling of the location, scale and shape parameters repectively.
`nint` | [gevglmprof*Plot] - the number of points at which the profile likelihood is evaluated.
`object` | [summary][grlevelPlot] - a fitted object of class `"gevglmFit"`.
`show` | [gevglmFit] - a logical; if `TRUE` (the default), print details of the fit.
`x` | [gevglmFit] - a numeric vector of data to be fitted. [print][plot] - a fitted object of class `"gevglmFit"`.
`xlow, xup` | [gevglmprof*Plot] - the least and greatest value at which to evaluate the profile likelihood.
`y` | [gevglmFit] - a matrix of covariates for generalized linear modelling of the parameters (or `NULL` (the default) for stationary fitting). The number of rows should be the same as the length of `xdat`.
`which` | [plot][summary] - a vector of logicals, one for each plot, denoting which plot should be displayed.
`...` | [gevglmFit] - control parameters optionally passed to the optimization function. Parameters for the optimization function are passed to components of the `control` argument of `optim`. [plot][summary] - arguments passed to the plot function.

### Details

Simulation:
To simulate a GEV series use the function `gevSim`.

Parameter Estimation:
`gevglmFit` fits by the Maximum-likelihood approach the generalized extreme value distribution, including generalized linear modelling of each parameter.

Methods:
`print.gevglm`, `plot.gevglm` and `summary.gevglm` are print, plot, and summary methods for a fitted object of class `gevglmFit`.

Profile Likelihood:
`gevglmprofPlot` and `gevglmprofxiPlot` produce profile log-likelihoods for shape parameters and `m` year/block return levels for stationary GEV models using the output of the function `gevFit`.

Nonstationary Models:
For non-stationary fitting it is recommended that the covariates within the generalized linear models are (at least approximately) centered and scaled (i.e. the columns of `ydat` should be approximately centered and scaled).

### Value

`gevglmFit`
returns a list containing the following components. A subset of these components are printed after the fit. If `show` is `TRUE`, then assuming that successful convergence is indicated, the components `nllh`, `mle` and `se` are always printed.

`trans` | an logical indicator for a non-stationary fit.
`model` | a list with components `mul`, `sigl` and `shl`.
`link` | a character vector giving inverse link functions.
`conv` | the convergence code, taken from the list returned by `optim`. A zero indicates successful convergence.
`nllh` | the negative logarithm of the likelihood evaluated at the maximum likelihood estimates.
`data` | the data that has been fitted. For non-stationary models, the data is standardized.
`mle` | a vector containing the maximum likelihood estimates.
`cov` | the covariance matrix.
`se` | a vector containing the standard errors.
`vals` | a matrix with three columns containing the maximum likelihood estimates of the location, scale and shape parameters at each data point.

For stationary models four plots are produced; a probability plot, a quantile plot, a return level plot and a histogram of data with fitted density. For non-stationary models two plots are produced; a residual probability plot and a residual quantile plot.
`gevglmprofPlot`
`gevglmprofxiPlot`
a plot of the profile likelihood is produced, with a horizontal line representing a profile confidence interval with confidence coefficient `conf`.

### Author(s)

Alec Stephenson for the code implemented from R's ismev package,
Stuart Scott for the original code, and Diethelm Wuertz for this R-port.

### References

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

### Examples

    ## SOURCE("fExtremes.54A-ExtremesGlm")

    ## Simulated GEV Data:
       xmpExtremes("\nStart: Simulte GEV Sample >")
       x = gevSim(model = list(shape = 0.25, location = 0, scale = 1), n = 1000)
       par(mfrow = c(2, 2))
       plot(x, main = "Simulated GEV Data")
       # Fit GEV Data:
       fit = gevglmFit(x)
       print(fit)
       # Summarize Results:
       summary(fit, which = c(TRUE, TRUE, TRUE, FALSE))

    ## Simulated GEV Data:
       xmpExtremes("\nNext: Estimate Parameters >")
       ##> x = gevSim(model = list(shape = 0, location = 0, scale = 1), n = 1000)
       ##> par(mfrow = c(2, 2))
       ##> plot(x, main = "Simulated Gumbel Data")
       # Fit GEV Data:
       ##> fit = gevglmFit(x, gumbel = TRUE)
       ##> print(fit)
       # Summarize Results:
       ##> summary(fit, which = c(TRUE, TRUE, TRUE, FALSE))

    ## Portpirie Data:
       xmpExtremes("\nNext: Fit Portpirie Data >")
       par(mfrow = c(2, 1))
       data(portpirie)
       fit = gevglmFit(portpirie[, 2])
       gevglmprofPlot(fit, m = 10, 4.1, 5)
       title(main = "Portpirie")
       gevglmprofxiPlot(fit, -0.3, 0.3)
       title(main = "Portpirie")

    ## Interactive Plot:
       ##> par(mfrow = c(2, 2))
       ##> plot(fit)

[Package fExtremes version 221.10065 Index]