expected_spear_dist() R function from [MSmix]

Expectation of the Spearman distance

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


expected_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.

Returns

Either the exact or the approximate (log-)expected value of the Spearman distance under the Mallows model with Spearman distance.

Details

When $n\leq 20$ , the expectation 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.

The expected Spearman distance is independent of the consensus ranking of the Mallows model with Spearman distance due to the right-invariance of the metric. When $\theta=0$ , this is equal to $\frac{n^3-n}{6}$ , which is the expectation of the Spearman distance under the uniform (null) model.

Examples


## Example 1. Expected Spearman distance under the uniform (null) model,
## coinciding with (n^3-n)/6.
n_items <- 10
expected_spear_dist(theta = 0, n_items = n_items, log = FALSE)
(n_items^3-n_items)/6

## Example 2. Expected Spearman distance.
expected_spear_dist(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-expected Spearman distance as a function of theta.
expected_spear_dist_vec <- Vectorize(expected_spear_dist, vectorize.args = "theta")
curve(expected_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(3, 5.5),
  xlab = expression(theta), ylab = expression(log(E[theta](D))),
  main = "Log-expected Spearman distance")

## Example 4. Log-expected Spearman distance for varying number of items
# and values of the concentration parameter.
expected_spear_dist_vec <- Vectorize(expected_spear_dist, vectorize.args = "theta")
curve(expected_spear_dist_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2, ylim = c(3, 9),
  xlab = expression(theta), ylab = expression(log(E[theta](D))),
  main = "Log-expected Spearman distance")
curve(expected_spear_dist_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(expected_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")

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 MG (1970). Rank correlation methods. 4th ed. Griffin London.

expected_spear_dist function