MSmix (version 2.0.0)

expected_spear_dist: Expectation of the Spearman distance

Description

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

Usage

expected_spear_dist(theta, n_items, log = TRUE)

Value

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

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 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.

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.

See Also

spear_dist_distr, partition_fun_spear

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")