RVinePIT | R Documentation |
This function applies the probability integral transformation (PIT) for R-vine copula models to given copula data.
RVinePIT(data, RVM)
data |
An N x d data matrix (with uniform margins). |
RVM |
|
The multivariate probability integral transformation (PIT) of Rosenblatt
(1952) transforms the copula data u = (u_1,\ldots,u_d)
with a given
multivariate copula C into independent data in [0,1]^d
, where d is the
dimension of the data set.
Let u = (u_1,\ldots,u_d)
denote copula data of dimension d. Further
let C be the joint cdf of u = (u_1,\ldots,u_d)
. Then Rosenblatt's
transformation of u, denoted as y = (y_1,\ldots,y_d)
, is defined as
y_1 := u_1,\ \ y_2 := C(u_2|u_1), \ldots\ y_d :=
C(u_d|u_1,\ldots,u_{d-1}),
where C(u_k|u_1,\ldots,u_{k-1})
is the
conditional copula of U_k
given U_1 = u_1,\ldots, U_{k-1} =
u_{k-1}, k = 2,\ldots,d
. The data vector y = (y_1,\ldots,y_d)
is now
i.i.d. with y_i \sim U[0, 1]
. The algorithm for the R-vine PIT is
given in the appendix of Schepsmeier (2015).
An N
x d matrix of PIT data from the given R-vine copula
model.
Ulf Schepsmeier
Rosenblatt, M. (1952). Remarks on a Multivariate Transformation. The Annals of Mathematical Statistics 23 (3), 470-472.
Schepsmeier, U. (2015) Efficient information based goodness-of-fit tests for vine copula models with fixed margins. Journal of Multivariate Analysis 138, 34-52.
RVineGofTest()
# load data set
data(daxreturns)
# select the R-vine structure, families and parameters
RVM <- RVineStructureSelect(daxreturns[,1:3], c(1:6))
# PIT data
pit <- RVinePIT(daxreturns[,1:3], RVM)
par(mfrow = c(1,2))
plot(daxreturns[,1], daxreturns[,2]) # correlated data
plot(pit[,1], pit[,2]) # i.i.d. data
cor(pit, method = "kendall")
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