MSmix (version 2.0.0)

partition_fun_spear: Partition function of the Mallows model with Spearman distance

Description

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

Usage

partition_fun_spear(theta, n_items, log = TRUE)

Value

Either the exact or the approximate (log-)partition function of the Mallows model with Spearman distance.

Arguments

theta

Non-negative precision parameter.

n_items

Number of items.

log

Logical: whether the partition function must be returned on the log scale. Defaults to TRUE.

Details

When \(n\leq 20\), the partition function 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 partition function is independent of the consensus ranking of the Mallows model with Spearman distance due to the right-invariance of the metric. When \(\theta=0\), the partition function is equivalent to \(n!\), which is the normalizing constant of 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.

See Also

spear_dist_distr, expected_spear_dist

Examples

## Example 1. Partition function of the uniform (null) model, coinciding with n!.
partition_fun_spear(theta = 0, n_items = 10, log = FALSE)
factorial(10)

## Example 2. Partition function of the Mallows model with Spearman distance.
partition_fun_spear(theta = 0.5, n_items = 10, log = FALSE)

## Example 3. Log-partition function of the Mallows model with Spearman distance
## as a function of theta.
partition_fun_spear_vec <- Vectorize(partition_fun_spear, vectorize.args = "theta")
curve(partition_fun_spear_vec(x, n_items = 10), from = 0, to = 0.1, lwd = 2,
  xlab = expression(theta), ylab = expression(log(Z(theta))),
  main = "Log-partition function of the Mallows model with Spearman distance",
  ylim = c(7, log(factorial(10))))

## Example 4. Log-partition function of the Mallows model with Spearman distance
## for varying number of items and values of the concentration parameter.
partition_fun_spear_vec <- Vectorize(partition_fun_spear, vectorize.args = "theta")
curve(partition_fun_spear_vec(x, n_items = 10),
  from = 0, to = 0.1, lwd = 2, col = 2,
  xlab = expression(theta), ylab = expression(log(Z(theta))),
  main = "Log-partition function of the Mallows model with Spearman distance",
  ylim = c(0, log(factorial(30))))
curve(partition_fun_spear_vec(x, n_items = 20), add = TRUE, col = 3, lwd = 2)
curve(partition_fun_spear_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")