qad (version 1.0.5)

D1: Calculate the D1 distance between two dependence structures

Description

Computation of the D1 distance between two checkerboard copulas A and B, corresponding to the random vectors (X1,Y1) and (X2,Y2), respectively. The function D1() computes the difference between the dependence structures of two random vectors. The function D1.ECBC() computes the D1-distance between two checkerboard copulas with the same resolution. The function zeta1() is defined as 3D1(A,Pi), where Pi denotes the independence copula and returns the dependence measure computed in qad.

Usage

D1(x1, y1, x2, y2, resolution = NULL)
D1.ECBC(A, B)
zeta1(X, Y, resolution = NULL)

Value

D1() returns the D1 distance, introduced in (Trutschnig, 2011).

D1.ECBC() returns the D1-distance between to checkerboard copulas A and B with same resolution

zeta1() returns the directed dependence from x to y.

Arguments

x1

a (non-empty) numeric vector of data values for the first random vector (first coordinate)

y1

a (non-empty) numeric vector of data values for the first random vector (second coordinate)

x2

a (non-empty) numeric vector of data values for the second random vector (first coordinate)

y2

a (non-empty) numeric vector of data values for the second random vector (second coordinate)

resolution

integer indicating the resolution of the checkerboard copula. (default = NULL)

A

Numeric matrix of dimension NxN indicating the mass of the first N-checkerboad copula

B

Numeric matrix of dimension NxN indicating the mass of the second N-checkerboad copula

X

Numeric vector of values in the first coordinate

Y

Numeric vector of values in the second coordinate

References

Trutschnig, W. (2011). On a strong metric on the space of copulas and its induced dependence measure. Journal of Mathematical Analysis and Applications. 384 (2), 690-705.

Junker, R.R., Griessenberger, F. and Trutschnig, W. (2021). Estimating scale-invariant directed dependence of bivariate distributions. Computational Statistics and Data Analysis, 153, 107058.

Examples

n <- 100
x1 <- runif(n)
y1 <- x1
x2 <- runif(n)
y2 <- 1-x2
D1(x1,y1,x2,y2)

n <- 1000
x <- runif(n, 0, 1)
y1 <- ifelse(x < 0.5, runif(length(x < 0.5), 0,0.5), runif(length(x >= 0.5), 0.5, 1))
y2 <- ifelse(x > 0.5, runif(length(x < 0.5), 0,0.5), runif(length(x >= 0.5), 0.5, 1))
A <- ECBC(x,y1, resolution = 50)
B <- ECBC(x,y2, resolution = 50)
#plot_density(A)
#plot_density(B)
D1.ECBC(A,B)