The Empirical Copula
Computes the empirical copula (according to a provided method) and auxiliary tools.
empCopula(X, smoothing = c("none", "beta", "checkerboard"), offset = 0,
ties.method = c("max", "average", "first", "last", "random", "min"))
C.n(u, X, smoothing = c("none", "beta", "checkerboard"), offset = 0,
ties.method = c("max", "average", "first", "last", "random", "min"))
dCn(u, U, j.ind = 1:d, b = 1/sqrt(nrow(U)), ...)
F.n(x, X, offset = 0, smoothing = c("none", "beta", "checkerboard"))
Cn(x, w) ## <-- deprecated! use C.n(w, x) instead!
toEmpMargins(U, x, ...)X |
an (n, d)- |
u,w |
an (m, d)- |
U |
an (n,d)- |
x |
an (m, d)- |
smoothing |
|
ties.method |
|
j.ind |
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b |
|
offset |
used in scaling the result which is of the form
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... |
additional arguments passed to |
Given pseudo-observations from a distribution with continuous margins
and copula C, the empirical copula is the (default) empirical
distribution function of these pseudo-observations. It is thus a
natural nonparametric estimator of C. The function C.n()
computes the empirical copula or two alternative smoothed versions of
it: the empirical beta copula or the empirical
checkerboard copula; see Eqs. (2.1) and (4.1) in Segers, Sibuya and
Tsukahara (2017), and the references therein. empCopula()
is the constructor of an object of class empCopula.
The function dCn() approximates first-order partial derivatives
of the unknown copula using the empirical copula.
The function F.n() computes the empirical distribution function
of a multivariate sample. Note that C.n(u, X, smoothing="none",
*) simply calls F.n(u, pobs(X), *) after checking u.
There are several asymptotically equivalent definitions of the
empirical copula. C.n(, smoothing = "none") is simply defined
as the empirical distribution function computed from the
pseudo-observations, that is,
(1/n) sum(I_{U_i<=u}, i=1, .., n),
where U_i,
i=1,..,n, denote the pseudo-observations and
n the sample size. Internally, C.n(,smoothing = "none")
is just a wrapper for F.n() and is expected to be fed with the
pseudo-observations.
The approximation for the jth partial derivative of the unknown copula C is implemented as, for example, in Rémillard and Scaillet (2009), and given by
hat(C.)[jn](u) = (C[n](u[1], .., u[j-1], min(u[j]+b, 1), u[j+1], .., u[d]) - C[n](u[1], .., u[j-1], max(u[j]-b, 0), u[j+1], .., u[d])) / (2b),
where b denotes the bandwidth and C[n] the empirical copula.
empCopula() is the constructor for objects of class
empCopula.
C.n() returns the empirical copula of the pseudo-observations
X evaluated at u (or a smoothed version of it).
dCn() returns a vector (length(j.ind) is 1) or a matrix
(with number of columns equal to length(j.ind)), containing
the approximated first-order partial derivatives of the unknown
copula at u with respect to the arguments in j.ind.
F.n() returns the empirical distribution function of X
evaluated at x if smoothing = "none", the empirical beta
copula evaluated at x if smoothing = "beta" and the empirical
checkerboard copula evaluated at x if smoothing =
"checkerboard".
toEmpMargins() transforms the copula sample U to the
empirical margins based on the sample x.
The first version of our empirical copula implementation, Cn(),
had its two arguments reversed compared to C.n(), and is
deprecated now. You must swap its arguments to transform to
new code.
The use of the two smoothed versions assumes implicitly no ties in the component samples of the data.
Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics, Annals of Statistics 4, 912–923.
Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: un test non paramétrique d'indépendance, Acad. Roy. Belg. Bull. Cl. Sci., 5th Ser. 65, 274–292.
Deheuvels, P. (1981). A non parametric test for independence, Publ. Inst. Statist. Univ. Paris 26, 29–50.
Rémillard, B. and Scaillet, O. (2009). Testing for equality between two copulas. Journal of Multivariate Analysis, 100(3), pages 377-386.
Segers, J., Sibuya, M. and Tsukahara, H. (2017). The Empirical Beta Copula. Journal of Multivariate Analysis, 155, pages 35–51, https://arxiv.org/abs/1607.04430.
pobs() for computing pseudo-observations.
## Generate data X (from a meta-Gumbel model with N(0,1) margins)
n <- 100
d <- 3
family <- "Gumbel"
theta <- 2
cop <- onacopulaL(family, list(theta=theta, 1:d))
set.seed(1)
X <- qnorm(rCopula(n, cop)) # meta-Gumbel data with N(0,1) margins
## Evaluate empirical copula
u <- matrix(runif(n*d), n, d) # random points were to evaluate the empirical copula
ec <- C.n(u, X = X)
## Compare the empirical copula with the true copula
pc <- pCopula(u, copula = cop)
mean(abs(pc - ec)) # ~= 0.012 -- increase n to decrease this error
## The same for the two smoothed versions
beta <- C.n(u, X, smoothing = "beta")
mean(abs(pc - beta))
check <- C.n(u, X, smoothing = "checkerboard")
mean(abs(pc - check))
## Compare the empirical copula with F.n(pobs())
U <- pobs(X) # pseudo-observations
stopifnot(identical(ec, F.n(u, X = pobs(U)))) # even identical
## Compare the empirical copula based on U at U with the Kendall distribution
## Note: Theoretically, C(U) ~ K, so K(C_n(U, U = U)) should approximately be U(0,1)
plot(ecdf(pK(C.n(U, X), cop = cop@copula, d = d)), asp = 1, xaxs="i", yaxs="i")
segments(0,0, 1,1, col=adjustcolor("blue",1/3), lwd=5, lty = 2)
abline(v=0:1, col="gray70", lty = 2)
## Compare the empirical copula and the true copula on the diagonal
C.n.diag <- function(u) C.n(do.call(cbind, rep(list(u), d)), X = X) # diagonal of C_n
C.diag <- function(u) pCopula(do.call(cbind, rep(list(u), d)), cop) # diagonal of C
curve(C.n.diag, from = 0, to = 1, # empirical copula diagonal
main = paste("True vs empirical diagonal of a", family, "copula"),
xlab = "u", ylab = quote("True C(u,..,u) and empirical"~C[n](u,..,u)))
curve(C.diag, lty = 2, add = TRUE) # add true copula diagonal
legend("bottomright", lty = 2:1, bty = "n", inset = 0.02,
legend = expression(C, C[n]))
## Approximate partial derivatives w.r.t. the 2nd and 3rd component
j.ind <- 2:3 # indices w.r.t. which the partial derivatives are computed
## Partial derivatives based on the empirical copula and the true copula
der23 <- dCn(u, U = pobs(U), j.ind = j.ind)
der23. <- copula:::dCdu(archmCopula(family, param=theta, dim=d), u=u)[,j.ind]
## Approximation error
summary(as.vector(abs(der23-der23.)))
## For an example of using F.n(), see help(mvdc)% ./Mvdc.Rd
## Generate a bivariate empirical copula object (various smoothing methods)
n <- 10 # sample size
d <- 2 # dimension
set.seed(271)
X <- rCopula(n, copula = claytonCopula(3, dim = d))
ecop.orig <- empCopula(X) # smoothing = "none"
ecop.beta <- empCopula(X, smoothing = "beta")
ecop.check <- empCopula(X, smoothing = "checkerboard")
## Sample from these (smoothed) empirical copulas
m <- 50
U.orig <- rCopula(m, copula = ecop.orig)
U.beta <- rCopula(m, copula = ecop.beta)
U.check <- rCopula(m, copula = ecop.check)
## Note: They all obviously just resample from the original n = 10 observations
## Plot
wireframe2(ecop.orig, FUN = pCopula, draw.4.pCoplines = FALSE)
wireframe2(ecop.beta, FUN = pCopula)
wireframe2(ecop.check, FUN = pCopula)
## Density (only exists when smoothing = "beta")
wireframe2(ecop.beta, FUN = dCopula)
## Transform a copula sample to empirical margins
set.seed(271)
X <- qexp(rCopula(1000, copula = claytonCopula(2))) # multivariate distribution
U <- rCopula(917, copula = gumbelCopula(2)) # new copula sample
X. <- toEmpMargins(U, x = X) # tranform U to the empirical margins of X
plot(X.) # Gumbel sample with empirical margins of XPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.