# zeta1: Multivariate dependence measure In qmd: Quantification of Multivariate Dependence

## Description

Given a d-dimensional random vector X and a uni-variate random variable Y, the multivariate copula-based dependence measure ζ1 (zeta 1) quantifies the influence of the random vector X on the random variable Y. More precisely, ζ1 fulfills the following properties:

• [N] ζ1(X,Y) attains values in [0,1] (normalization).

• [I] ζ1(X,Y) = 0 if and only if X and Y are independent (independence).

• [C] ζ1(X,Y) = 1 if and only if Y is a function of X (complete dependence).

• [S] ζ1(X,Y) is scale-invariant.

• [IG] ζ1(X,Y) fulfills the information gain inequality, i.e., adding variables to X can not decrease the dependence measure.

For more information, see Griessenberger, Junker, Trutschnig (2021), On a multivariate copula-based dependence measure and its estimation, <arXiv:2109.12883> .

The derived checkerboard estimator is proved to be strongly consistent and implemented in the function zeta1. Note, that the estimator only attains positive values (within [0,1]). Therefore, interpretation of low values have to be done with care and always under consideration of the sample size.

## Usage

 `1` ```zeta1(X, Y, ties.correction = FALSE, approx = FALSE, resolution = NULL) ```

## Arguments

 `X` a numeric matrix of dimension d indicating the conditioning (predictor) variables `Y` a numeric vector indicating the response (uni-variate) `ties.correction` logical indicating if the dependence measure uses a correction term w.r.t. ties mentioned in the paper. Default = FALSE. `approx` logical indicating if an approximated version of the dependence measure is computed (fast version). It should be only applied on data with no (or only a few) ties. Default = FALSE. `resolution` an integer indicating the resolution N of the checkerboard aggregation. Default = NULL, which is recommended (N(n) = floor(n^(1/(d+1)))).

## Value

A numeric value indicating the dependence between X and Y (or, equivalently, the influence of X on Y).

## References

Griessenberger, Junker, Trutschnig (2021), On a multivariate copula-based dependence measure and its estimation, <arXiv:2109.12883>.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```#(complete dependence for dimension 4) n <- 300 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- x1 + x2 + x3 zeta1(X = cbind(x1,x2,x3), Y = y) zeta1(X = cbind(x1,x2,x3), Y = y, approx = TRUE) #(independence for dimension 4) n <- 500 x1 <- runif(n) x2 <- runif(n) x3 <- x1 + x2 + rnorm(n) y <- runif(n) zeta1(X = cbind(x1,x2,x3), Y = y) zeta1(X = cbind(x1,x2,x3), Y = y, approx = TRUE) #(binary output for dimension 3) n <- 500 x1 <- runif(n) x2 <- runif(n) y <- ifelse(x1 + x2 < 1, 0, 1) zeta1(X = cbind(x1,x2), Y = y, ties.correction = FALSE) zeta1(X = cbind(x1,x2), Y = y, ties.correction = TRUE) ```

qmd documentation built on Sept. 28, 2021, 5:08 p.m.