Density, Evaluation, and Random Number Generation for Copula Functions
Density (dCopula), distribution function (pCopula), and
random generation (rCopula) for a copula object.
dCopula(u, copula, log=FALSE, ...) pCopula(u, copula, ...) rCopula(n, copula, ...)
copula |
an R object of class |
u |
a vector of the copula dimension d or a matrix with d columns, giving the points where the density or distribution function needs to be evaluated. Note that in all cases, values outside of the cube [0,1]^d are treated equivalently to those on the cube boundary. So, e.g., the density is zero. |
log |
logical indicating if the log(f(.)) should be returned instead of f(.). |
n |
(for |
... |
further optional arguments for some methods, e.g.,
|
The density (dCopula) and distribution function
(pCopula) methods for Archimedean copulas now use the
corresponding function slots of the Archimedean copula objects, such
as copClayton, copGumbel, etc.
The distribution function of a t copula uses pmvt from
package mvtnorm; similarly, the density (dCopula) calls
dmvt from mvtnorm.
The normalCopula methods use dmvnorm and pmvnorm
from the same package.
The random number generator for an Archimedean copula uses the conditional approach for the bivariate case and the Marshall-Olkin (1988) approach for dimension greater than 2.
dCopula() gives the density, pCopula() gives the
distribution function, and rCopula() generates random variates.
Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. North American Actuarial Journal 2, 1–25.
Genest, C. and Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering 12, 347–368.
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
Marshall, A. W. and Olkin, I. (1988) Families of multivariate distributions. Journal of the American Statistical Association 83, 834–841.
Nelsen, R. B. (2006) An introduction to Copulas. Springer, New York.
the copula and acopula
classes, the acopula families, acopula-families.
Constructor functions such as
ellipCopula, archmCopula,
fgmCopula.
norm.cop <- normalCopula(0.5)
norm.cop
## one d-vector =^= 1-row matrix, works too :
dCopula(c(0.5, 0.5), norm.cop)
pCopula(c(0.5, 0.5), norm.cop)
u <- rCopula(100, norm.cop)
plot(u)
dCopula(u, norm.cop)
pCopula(u, norm.cop)
persp (norm.cop, dCopula)
contour(norm.cop, pCopula)
## a 3-dimensional normal copula
u <- rCopula(1000, normalCopula(0.5, dim = 3))
if(require(scatterplot3d))
scatterplot3d(u)
## a 3-dimensional clayton copula
cl3 <- claytonCopula(2, dim = 3)
v <- rCopula(1000, cl3)
pairs(v)
if(require(scatterplot3d))
scatterplot3d(v)
## Compare with the "nacopula" version :
fu1 <- dCopula(v, cl3)
fu2 <- copClayton@dacopula(v, theta = 2)
Fu1 <- pCopula(v, cl3)
Fu2 <- pCopula(v, onacopula("Clayton", C(2.0, 1:3)))
## The density and cumulative values are the same:
stopifnot(all.equal(fu1, fu2, tolerance= 1e-14),
all.equal(Fu1, Fu2, tolerance= 1e-15))
## NA and "outside" u[]
u <- v[1:12,]
## replace some by values outside (0,1) and some by NA/NaN
u[1, 2:3] <- c(1.5, NaN); u[2, 1] <- 2; u[3, 1:2] <- c(NA, -1)
u[cbind(4:9, 1:3)] <- c(NA, NaN)
f <- dCopula(u, cl3)
cbind(u, f) # note: f(.) == 0 at [1] and [3] inspite of NaN/NA
stopifnot(f[1:3] == 0, is.na(f[4:9]), 0 < f[10:12])Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.