qad {qad}

Measure of (asymmetric and directed) dependence

Description

Quantification of (asymmetric and directed) dependence structures between two random variables X and Y.

Usage

qad(x, ...)

## S3 method for class 'data.frame'
qad(
  x,
  resolution = NULL,
  p.value = TRUE,
  nperm = 1000,
  p.value_asymmetry = FALSE,
  nboot = 1000,
  print = TRUE,
  remove.00 = FALSE,
  ...
)

## S3 method for class 'numeric'
qad(
  x,
  y,
  resolution = NULL,
  p.value = TRUE,
  nperm = 1000,
  p.value_asymmetry = FALSE,
  nboot = 1000,
  print = TRUE,
  remove.00 = FALSE,
  ...
)

Arguments

x

a data.frame containing two columns with the observations of the bi-variate sample or a (non-empty) numeric vector of data values

...

Further arguments passed to 'qad' will be ignored

resolution

an integer indicating the number of strips for the checkerboard aggregation (see ECBC). We recommend to use the default value (resolution = NULL)

p.value

a logical indicating whether to return a p-value of rejecting independence (based on permutation).

nperm

an integer indicating the number of permutation runs (if p.value = TRUE)

p.value_asymmetry

a logical indicating whether to return a (heuristic) p-value for the measure of asymmetry (based on bootstrap).

nboot

an integer indicating the number of runs for the bootstrap.

print

a logical indicating whether the result of qad is printed.

remove.00

a logical indicating whether double 0 entries should be excluded (default = FALSE)

y

a (non-empty) numeric vector of data values.

Details

qad is the implementation of a strongly consistent estimator of the copula based dependence measure zeta_1 introduced in Trutschnig 2011. We first compute the empirical copula of a two-dimensional sample, aggregate it to the so called empirical checkerboard copula (ECBC), and calculate zeta_1 of the ECBC and its transpose. In order to test for independence (in both directions), a built-in p-value is implemented (a permutation test with nperm permutation runs to estimate the p-value). Furthermore, a (heuristic) bootstrap test with nboot runs can be applied to estimate a p-value for the measure of asymmetry a.

Value

qad returns an object of class qad containing the following components:

data

a data.frame containing the input data.

q(X, Y)

influence of X on Y

q(Y, X)

influence of Y on X

max.dependence

maximal dependence

results

a data.frame containing the results of the dependence measures.

mass_matrix

a matrix containing the mass distribution of the empirical checkerboard copula.

resolution

an integer containing the used resolution of the checkerboard aggregation.

n

Sample size.

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, 690-705.

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

See Also

A tutorial can be found at http://www.trutschnig.net/software.html.

Examples

#Example 1 (independence)

n <- 100
x <- runif(n,0,1)
y <- runif(n,0,1)
sample <- data.frame(x,y)
qad(sample)

###

#Example 2 (mutual complete dependence)

n <- 500
x <- runif(n,0,1)
y <- x^2
sample <- data.frame(x,y)
qad(sample)

#Example 3 (complete dependence)

n <- 1000
x <- runif(n,-10,10)
y <- sin(x)
sample <- data.frame(x,y)
qad(sample)

#Example 4 (Asymmetry)

n <- 100
x <- runif(n,0,1)
y <- (2*x) %% 1
qad(x, y)