GevModelling            package:fExtremes            R Documentation

_G_e_n_e_r_a_l_i_z_e_d _E_x_t_r_e_m_e _V_a_l_u_e _M_o_d_e_l_l_i_n_g

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description functions to estimate  the parameters
     of the GEV distribution. 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 modelling functions are:

       'gevSim'         generates data from the GEV distribution,
       'gumbelSim'      generates data from the Gumbel distribution,
       'gevFit'         fits data to the GEV distribution,
       'gumbelFit'      fits data to the Gumbel 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.

_U_s_a_g_e:

     gevSim(model = list(xi = -0.25, mu = 0, beta = 1), n = 1000, seed = NULL)
     gumbelSim(model = list(mu = 0, beta = 1), n = 1000, seed = NULL)

     gevFit(x, block = 1, type = c("mle", "pwm"), title = NULL, description = NULL, ...)
     gumbelFit(x, block = 1, type = c("mle", "pwm"), title = NULL, description = NULL, ...)

     ## S4 method for signature 'fGEVFIT':
     show(object)
     ## S3 method for class 'fGEVFIT':
     plot(x, which = "ask", ...)
     ## S3 method for class 'fGEVFIT':
     summary(object, doplot = TRUE, which = "all", ...)

_A_r_g_u_m_e_n_t_s:

   block: block size. 

description: a character string which allows for a brief description. 

  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)'. 

   model: [gevSim][gumbelSim] - 
           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][gumbelSim] - 
           number of generated data points, an integer value. 
           [rgev] - 
                   the number of observations. 

  object: [summary][grlevelPlot] - 
           a fitted object of class '"gevFit"'. 

    seed: [gevSim] - 
           an integer value to set the seed for the random number
          generator. 

   title: [gevFit] - 
           a character string which allows for a project title. 

    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: [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, beta: [*gev] - 
           'xi' is the shape parameter, 'mu' the location parameter,
          and 'sigma' is the scale parameter. The default values are 
          'xi=1', 'mu=0', and 'beta=1'. Note, if 'xi=0'  the
          distribution is of type Gumbel. 

     ...: [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. 

_D_e_t_a_i_l_s:

     *Parameter Estimation:* 

      'gevFit' and 'gumbelFit' estimate the parameters either  by the
     probability weighted moment method, 'method="pwm"' or  by maximum
     log likelihood estimation 'method="mle"'. 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.

_V_a_l_u_e:

     '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.

_N_o_t_e:

     *GEV Parameter Estimation:* 

      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'.

_A_u_t_h_o_r(_s):

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

_R_e_f_e_r_e_n_c_e_s:

     Coles S. (2001); _Introduction to Statistical Modelling of Extreme
     Values_, Springer.

     Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); _Modelling
     Extremal Events_,  Springer.

_E_x_a_m_p_l_e_s:

     ## gevSim -
        # Simulate GEV Data, use default length n=1000
        x = gevSim(model = list(xi = 0.25, mu = 0 , beta = 1), n = 1000)
        head(x)

     ## gumbelSim -
        # Simulate GEV Data, use default length n=1000
        x = gumbelSim(model = list(xi = 0.25, mu = 0 , beta = 1))
          
     ## gevFit -
        # Fit GEV Data by Probability Weighted Moments:
        fit = gevFit(x, type = "pwm") 
        print(fit)
        
     ## summary -
        # Summarize Results:
        par(mfcol = c(2, 2))
        summary(fit)

