# EllipticalModelling: Bivariate Elliptical Copulae In fCopulae: Rmetrics - Bivariate Dependence Structures with Copulae

## Description

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

Elliptical Copulae Functions:

 `ellipticalCopulaSim` simulates an elliptical copula, `ellipticalCopulaFit` fits the parameters of an elliptical copula.

## Usage

 ```1 2 3``` ``` ellipticalCopulaSim(n, rho = 0.75, param = NULL, type = c("norm", "cauchy", "t")) ellipticalCopulaFit(u, v, type = c("norm", "cauchy", "t"), ...) ```

## Arguments

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

## Value

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`.

## Author(s)

Diethelm Wuertz for the Rmetrics R-port.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```## [rp]ellipticalCopula - # Default Normal Copula: rellipticalCopula(10) pellipticalCopula(10) ## [rp]ellipticalCopula - # Student-t Copula Probability and Density: u = grid2d(x = (0:25)/25) # CHECK ERROR # pellipticalCopula(u, rho = 0.75, param = 4, # type = "t", output = "list") # CHECK ERROR DONE 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) ```

fCopulae documentation built on Nov. 17, 2017, 2:31 p.m.