Copula {copula}

Density, Evaluation, and Random Number Generation for Copula Functions

Description

Density (dCopula), distribution function (pCopula), and random generation (rCopula) for a copula object.

Usage

dCopula(u, copula, log=FALSE, ...)
pCopula(u, copula, ...)
rCopula(n, copula, ...)

Arguments

copula

an R object of class "Copula", (i.e., "copula" or "nacopula").

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(\cdot)) should be returned instead of f(\cdot).

n

(for rCopula():) number of observations to be generated.

...

further optional arguments for some methods, e.g., method.

Details

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.

If an u_j is outside (0,1) we declare the density to be zero, and this is true even when another u_k, k \ne j is NA or NaN; see also the “outside” example.

The distribution function of a t copula uses pmvt from package mvtnorm; similarly, the density (dCopula) calls dmvt from CRAN package 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.

Value

dCopula() gives the density, pCopula() gives the distribution function, and rCopula() generates random variates.

References

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.

See Also

the copula and acopula classes, the acopula families, acopula-families. Constructor functions such as ellipCopula, archmCopula, fgmCopula.

Examples

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])