The pairwise coefficient
Example 1
In this example, we demonstrate how to use
coverage_correlation() with a simple simulation. We compute
the coverage correlation coefficient between two one-dimensional Normal
random variables, X and Y, and then vary the
strength of their relationship to observe how the statistic changes.
set.seed(1)
n <- 100
X <- rnorm(n)
Y <- rnorm(n)
result <- coverage_correlation(Y, X, visualise = TRUE)
result
#>
#> Coverage correlation
#>
#> statistic = 0.03622657, p-value = 0.1151168
#> n = 100, dimension = 2, method = exactIn the example above, X and Y are
independent.
The parameter visualise defaults to FALSE,
but setting it to TRUE produces a plot that illustrates the
intuition behind the coverage correlation coefficient. The coverage
correlation coefficient first transforms X and
Y into their Monge-Kantorovich ranks, denoted by
X_rank and Y_rank, which are uniformly
distributed on \([0, 1]\). The plot
displays the pairs \((X_{\text{rank}_i},
Y_{\text{rank}_i})\) along with cubes of volume \(n^{-1}\).
Inside the function coverage_correlation(), we compute
\(V_n\), the total uncovered area after
taking the union of these cubes. The coverage correlation coefficient is
then defined as \[
\kappa_n^{X, Y} := \frac{V_n - e^{-1}}{1 - e^{-1}}.
\]
The function returns an object of class "covercorr" (a
list) with the following elements:
$stat: the value of the coverage correlation coefficient.$pval: the p-value of this statistic under the null hypothesis of independence.$method: the method used to compute the statistic (e.g.,"exact"or"approx").$mc_se: if method"approx"was used,mc_seis the standard error of the Monte Carlo approximation, otherwise it is 0.
Printing the result (as above) uses the print method for
the "covercorr" class; use str(result) to see
the full list structure.
By default, method = "auto". In this mode, if the
total dimension of X and Y
(i.e., ncol(X) + ncol(Y), treating vectors as
one-dimensional) is at most 6, the method is set to
"exact"; otherwise, it uses "approx".
Next we can see how the result changes as we introduce dependence
between X and Y.
set.seed(2)
n <- 100
X <- rnorm(n)
Z <- rnorm(n)
rho <- 0.9
Y <- rho * X + sqrt(1 - rho^2) * Z
result <- coverage_correlation(Y, X, visualise = TRUE)
result
#>
#> Coverage correlation
#>
#> statistic = 0.3591986, p-value = < 2.2204e-16
#> n = 100, dimension = 2, method = exactYou may notice parts of some cubes appearing at the corners of the plot. This happens because we treat \([0, 1]^2\) as a torus. If a cube centered at one of the rank points lies partially outside \([0, 1]^2\), we wrap it around so that the plot reflects this topology.
Example 2
The coverage correlation coefficient can handle multidimensional random vectors as well.
set.seed(3)
n <- 100
p <- 2
X <- matrix(rnorm(p * n), ncol = p)
Y <- matrix(0, nrow = n, ncol = p)
Y[, 1] <- X[, 1]^2
Y[, 2] <- X[, 1] * X[, 2]
result <- coverage_correlation(Y, X)
result
#>
#> Coverage correlation
#>
#> statistic = 0.273216, p-value = < 2.2204e-16
#> n = 100, dimension = 4, method = exactIn this case we cannot visualise the whole plot as X and
Y are not one-dimensional.
Example 3
In the example below, X and Y are
independent and 2-dimensional. We set the method parameter
equal to "approx".
set.seed(4)
n <- 50
p <- 2
X <- matrix(rnorm(p * n), ncol = p)
Y <- matrix(rnorm(p * n), ncol = p)
result <- coverage_correlation(Y, X, method = "approx")
result
#>
#> Coverage correlation
#>
#> statistic = 0.03919494, p-value = 0.0682784
#> n = 50, dimension = 4, method = approx
#> Monte Carlo standard error = 0.02595504