var_spear_dist {MSmix}

Variance of the Spearman distance

Description

Compute (either the exact or the approximate) (log-)variance of the Spearman distance under the Mallow model with Spearman distance.

Usage

var_spear_dist(theta, n_items, log = TRUE)

Arguments

theta

Non-negative precision parameter.

n_items

Number of items.

log

Logical: whether the expected Spearman distance on the log scale must be returned. Defaults to TRUE.

Details

When n\leq 20, the variance is exactly computed by relying on the Spearman distance distribution provided by OEIS Foundation Inc. (2023). When n>20, it is approximated with the method introduced by Crispino et al. (2023) and, if n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

When \theta=0, this is equal to \frac{n^2(n-1)(n+1)^2}{36}, which is the variance of the Spearman distance under the uniform (null) model.

The variance of the Spearman distance is independent of the consensus ranking of the Mallow model with Spearman distance due to the right-invariance of the metric.

Value

Either the exact or the approximate (log-)variance of the Spearman distance under the Mallow model with Spearman distance.

References

Crispino M., Mollica C., Astuti V. and Tardella L. (2023). Efficient and accurate inference for mixtures of Mallows models with Spearman distance. Statistics and Computing, 33(98), DOI: 10.1007/s11222-023-10266-8.

OEIS Foundation Inc. (2023). The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org

Kendall, M. G. (1970). Rank correlation methods. 4th ed. Griffin London.

Examples

## Example 1. Variance of the Spearman distance under the uniform (null) model,
## coinciding with n^2(n-1)(n+1)^2/36.
n_items <- 10
var_spear_dist(theta = 0, n_items= n_items, log = FALSE)
n_items^2*(n_items-1)*(n_items+1)^2/36

## Example 2. Variance of the Spearman distance.
var_spear_dist(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-variance of the Spearman distance as a function of theta.
var_spear_dist_vec <- Vectorize(var_spear_dist, vectorize.args = "theta")
curve(var_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2,
  xlab = expression(theta), ylab = expression(log(V[theta](D))),
  main = "Log-variance of the Spearman distance")

## Example 4. Log--variance of the Spearman distance for varying number of items
# and values of the concentration parameter.
var_spear_dist_vec <- Vectorize(var_spear_dist, vectorize.args = "theta")
curve(var_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(5, 14),
  xlab = expression(theta), ylab = expression(log(V[theta](D))),
  main = "Log-variance of the Spearman distance")
curve(var_spear_dist_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(var_spear_dist_vec(x, n_items = 30), add = TRUE, col = 4, lwd = 2)
legend("topright", legend = c(expression(n == 10), expression(n == 20), expression(n == 30)),
  col = 2:4, lwd = 2, bty = "n")