empCopula  R Documentation 
Computes the empirical copula (according to a provided method) and auxiliary tools.
empCopula(X, smoothing = c("none", "beta", "checkerboard", "schaake.shuffle"), 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 

b 

offset 
used in scaling the result which is of the form

... 
additional arguments passed to 
Given pseudoobservations from a distribution with continuous margins
and copula C, the empirical copula is the (default) empirical
distribution function of these pseudoobservations. 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 firstorder 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
pseudoobservations, that is,
(1/n) sum(I_{U_i<=u}, i=1, .., n),
where U_i,
i=1,..,n, denote the pseudoobservations 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
pseudoobservations.
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[j1], min(u[j]+b, 1), u[j+1], .., u[d])  C[n](u[1], .., u[j1], 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 pseudoobservations
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 firstorder 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.
Clark, M., Gangopadhyay, S., Hay, L., Rajagopalan, B. and Wilby, R. (2004). The Schaake Shuffle: A Method for Reconstructing SpaceTime Variability in Forecasted Precipitation and Temperature Fields. Journal of Hydrometeorology, pages 243262.
Rémillard, B. and Scaillet, O. (2009). Testing for equality between two copulas. Journal of Multivariate Analysis, 100(3), pages 377386.
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.
Kiriliouk, A., Segers, J. and Tsukahara, H. (2020). Resampling Procedures with Empirical Beta Copulas. https://arxiv.org/abs/1905.12466.
pobs()
for computing pseudoobservations.
## Generate data X (from a metaGumbel 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)) # metaGumbel 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) # pseudoobservations 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(der23der23.))) ## 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) ## 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 X
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