RVinePIT {VineCopula}

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.

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 (2015).

Value

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

Author(s)

Ulf Schepsmeier

References

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.

See Also

RVineGofTest()

Examples

# 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")