Local Kendall's tau (local indicator of mobility association)

Decomposes Kendall's tau into a per-observation contribution. Each unit's local tau is the average sign of its pairwise rank concordance with all other units, \(\sum_{j \ne i} S_{ij} / (n - 1)\), where \(S_{ij} = \mathrm{sign}((x_i - x_j)(y_i - y_j))\). Units with local tau near 1 keep their relative position; near -1 they move against the grain. The overall (tau-a) statistic is the mean of the off-diagonal S.

Usage

tau_local(x, y)

Arguments

x, y

Numeric vectors of equal length.

Value

An object of class sddr_tau_local: a list with the per-observation tau_local vector, the global tau (tau-a), the sign matrix S, and n.

See also

tau()

Examples

set.seed(1)
x <- rnorm(20); y <- x + rnorm(20)
tau_local(x, y)$tau_local
#>  [1]  0.05263158  0.15789474  0.89473684 -0.36842105  0.15789474  0.89473684
#>  [7]  0.36842105 -0.15789474  0.05263158  0.26315789  0.89473684  0.26315789
#> [13]  0.36842105  1.00000000 -0.26315789  0.57894737  0.57894737  0.36842105
#> [19]  0.68421053  0.57894737