EllipticalCopulae          package:fCopulae          R Documentation

_B_i_v_a_r_i_a_t_e _E_l_l_i_p_t_i_c_a_l _C_o_p_u_l_a_e

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

     A collection and description of functions to investigate 
     bivariate elliptical copulae.  

     Elliptical Copulae Functions:

       'rellipticalCopula'  Generates elliptical copula variates,
       'pellipticalCopula'  computes elliptical copula probability,
       'dellipticalCopula'  computes elliptical copula density,
       'rellipticalSlider'  displays interactive plots of variates,
       'pellipticalSlider'  displays interactive plots of probability,
       'dellipticalSlider'  displays interactive plots of density.

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

         
     rellipticalCopula(n, rho = 0.75, param = NULL, type = c("norm", "cauchy", 
         "t"))
     pellipticalCopula(u = 0.5, v = u, rho = 0.75, param = NULL, 
         type = ellipticalList(), output = c("vector", "list"), border = TRUE)
     dellipticalCopula(u = 0.5, v = u, rho = 0.75, param = NULL, 
         type = ellipticalList(), output = c("vector", "list"), border = TRUE)

     rellipticalSlider(B = 100)
     pellipticalSlider(type = c("persp", "contour"), B = 20)
     dellipticalSlider(type = c("persp", "contour"), B = 20)

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

       B: [*Slider] - 
           the maximum slider menu value when the boundary value is
          infinite.  By default this is set to 10. 

  border: [pellipticalCopula][dellipticalCopula] - 
           a logical flag. If the argument 'u' is an integer, say 'N', 
          greater than one than all points on a square grid
          '[(0:N)/N]^2' are computed. If border is FALSE than the
          border points are removed  from the returned value, by
          default this is not the case. 

       n: [rellipticalCopula][ellipticalCopulaSim] - 
                    the number of random deviates to be generated, an
          integer value. 

  output: [pellipticalCopula][dellipticalCopula] - 
           a character string specifying how the output should be
          formatted.  By default a vector of the same length as 'u' and
          'v'  is returned. If specified as '"list"' then 'u' and 'v' 
          are expected to span a two-dimensional grid as outputted by
          the  function 'grid2d' and the function returns a list with
          elements '$x', 'y', and 'z' which can be directly  used for
          example by 2D plotting functions. For the grid version, when
          'u' is specified as an integer greater than one, always  the
          output in form of a list will be returned. 

     rho: [*ellipticalCopula] - 
           is the numeric value setting the correlation strength,
          ranging between minus one and one. 

   param: [*ellipticalCopula][gfunc] - 
           additional distributional parameters: for the Sudent-t
          distribution this is "nu", for the Kotz distribution this is
          "r", and for the Exponential Power distribution these are "r"
          and "s". If the argument 'param=NULL' then default values are
          taken. These are for the Student-t 'param=c(nu=4))', for the
          Kotz distribution 'param=c(r=1))', and for the exponential
          power distribution 'param=c(r=1,s=1)'. Note, that the Kotz
          and exponential power copulae are independent of 'r', and
          that 'r' only enters the generator, the density, the
          probability and the quantile  functions. 

    type: [*ellipticalCopula][gfunc] - 
           the type of the elliptical copula. A character string
          selected  from: "norm", "cauchy", "t", "logistic", "laplace",
          "kotz",  or "epower". [*ellipticalSlider] - 
           a character string which indicates what kind of plot should
          be  displayed, either a perspective plot if 'type="persp"',
          the default value, or a contour plot if 'type="contour"'. 

    u, v: [*ellipticalCopula] - 
           two numeric values or vectors of the same length at which
          the copula will be computed. If 'u' is a list then the the
          '$x' and '$y' elements will be used as 'u'  and 'v'. If 'u'
          is a two column matrix then the  first column will be used as
          'u' and the the second  as 'v'. If 'u' is an integer value
          greater than one, say 'N', than the values for all points on
          the  '[(0:N)/N]^2' grid spanning the unit square will be
          returned. 

     ...: [ellipticalCopulaFit] - 
           arguments passed to the optimization function 'nlminb'. 

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

     *Copula Functions:* 

     The functions '[rpd]ellipticalCopula' return a numeric vector of
     random variates, probabilities, or densities for the specified 
     copula computed at grid coordinates 'u'|'v'. 

     The functions '[rpd]ellipticalSlider' display an interactive graph
     of an perspective copula plot either for random variates,
     probabilities or densities. Alternatively, an image underlayed
     contour plot can be shown. 

     *Copula Dependence Measures:* 

     The functions 'ellipticalTau' and 'ellipticalRho' return a
     numericc value for Kendall's Tau and Spearman's Rho. 

     *Copula Tail Coefficient:* 

     The function 'ellipticalTailCoeff' returns the coefficient of tail
     dependence for a specified copula. The function 
     'ellipticalTailPlot' displays a whole plot for the upper or
     alternatively for the lower tail dependence as a function of  'u'
     for a set of nine 'rho' values. 

     *Copula Generator Function:* 

     The function 'gfunc' computes the generator function for the
     specified copula, by default the normal copula. If the argument
     'x' is missing, then the normalization constand lambda will be
     returned, otherwise if 'x' is specified the values for the
     function _g(x)_ will be freturned. The selected type of copula is
     added to the output as an attribute named '"control"'.  The
     function 'gfuncSlider' allows to display interactively  the
     generator function, the marginal density, the marginal 
     probability, and the contours of the the bivariate density. 

     *Copula Simulation and Parameter Fitting:* 

     The function 'ellipticalCopulaSim' returns a numeric two-column
     matrix with randomly generated variates for the specified copula. 

     The function 'ellipticalCopulaFit' returns a fit to empirical data
     for the specified copula. The returned object is a list with
     elements from the function 'nlminb'.

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

     Diethelm Wuertz for the Rmetrics R-port.

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

     ## Not run: 
     ## [rp]ellipticalCopula -
        # Default Normal Copula:
        rellipticalCopula(10)
        pellipticalCopula(10)

     ## [rp]ellipticalCopula -   
        # Student-t Copula Probability and Density:
        u = grid2d(x = (0:25)/25)
        pellipticalCopula(u, rho = 0.75, param = 4, 
          type = "t", output = "list")
        d = dellipticalCopula(u, rho = 0.75, param = 4, 
          type = "t", output = "list")   
        persp(d, theta = -40, phi = 30, col = "steelblue")
        
     ## ellipticalTau -
     ## ellipticalRho -
        # Dependence Measures:
        ellipticalTau(rho = -0.5)
        ellipticalRho(rho = 0.75, type = "logistic", subdivisions = 100)
        
     ## ellipticalTailCoeff -
        # Student-t Tail Coefficient:
        ellipticalTailCoeff(rho = 0.25, param = 3, type = "t")

     ## gfunc -
        # Generator Function:
        plot(gfunc(x = 0:10), main = "Generator Function")
        
     ## ellipticalCopulaSim -
     ## ellipticalCopulaSim -
        # Simualtion and Parameter Fitting:
        rv = ellipticalCopulaSim(n = 100, rho = 0.75)
        ellipticalCopulaFit(rv)
     ## End(Not run)  

