VineCopula (version 1.3)

RVinePIT: Probability integral transformation for R-vine copula models

Description

This function applies the probability integral transformation (PIT) for R-vine copula models to given copula data.

Usage

RVinePIT(data,RVM)

Arguments

data

An N x d data matrix (with uniform margins).

RVM

RVineMatrix objects of the R-vine model.

Value

• An N x d matrix of PIT data from the given R-vine copula model.

Details

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 (2013).

References

Rosenblatt, M. (1952). Remarks on a Multivariate Transformation. The Annals of Mathematical Statistics 23 (3), 470-472. Schepsmeier, U. (2013) Efficient goodness-of-fit tests in multi-dimensional vine copula models. http://arxiv.org/abs/1309.5808

See Also

RVineGofTest

Examples

# load data set
data(daxreturns)

# select the R-vine structure, families and parameters
RVM = RVineStructureSelect(daxreturns[,1:5],c(1:6))

# PIT data
pit=RVinePIT(daxreturns[,1:5],RVM)

par(mfrow=c(1,2))
plot(daxreturns[,1],daxreturns[,2])	# correlated data
plot(pit[,1],pit[,2])	# i.i.d. data