A collection and description of functions to investigate
bivariate elliptical copulae.
Elliptical Copulae Functions:
ellipticalTau | Computes Kendall's tau for elliptical copulae, |
ellipticalRho | computes Spearman's rho for elliptical copulae, |
ellipticalTailCoeff | computes tail dependence for elliptical copulae, |
ellipticalTailPlot | plots tail dependence for elliptical copulae. |
ellipticalTau(rho)
ellipticalRho(rho, param = NULL, type = ellipticalList(), subdivisions = 500)
ellipticalTailCoeff(rho, param = NULL, type = c("norm", "cauchy", "t"))
ellipticalTailPlot(param = NULL, type = c("norm", "cauchy", "t"),
tail = c("Lower", "Upper"))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.
[*ellipticalCopula] -
is the numeric value setting the correlation strength, ranging
between minus one and one.
[*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.
[ellipticalRho] -
an integer value with the number of subdivisons in each direction
on the two dimensional unit square to compute the mean value of
Spearman's Rho. By default 500 subdivisions are used.
[ellipticalTailPlot] -
a character string, either "Upper" or "Lower" denoting
which of the two tails should be displayed. By default the upper
tail dependence will be considered.
[*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".
Diethelm Wuertz for the Rmetrics R-port.
## [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)
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