spear_dist_distr {MSmix}

Spearman distance distribution under the uniform ranking model

Description

Provide (either the exact or the approximate) frequency distribution of the Spearman distance under the uniform (null) ranking model.

Usage

spear_dist_distr(n_items, log = TRUE)

Arguments

n_items

Number of items.

log

Logical: whether the frequencies must be reported on the log scale. Defaults to TRUE.

Details

When n\leq 20, the exact distribution provided by OEIS Foundation Inc. (2023) is returned by relying on a call to the get_cardinalities routine of the BayesMallows package. When n>20, the approximate distribution introduced by Crispino et al. (2023) is returned. If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

A list of two named objects:

distances

All the possible Spearman distance values (i.e., the support of the distribution).

logcard

(Log)-frequencies corresponding to each value in distances.

References

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

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.

Sørensen Ø, Crispino M, Liu Q and Vitelli V (2020). BayesMallows: An R Package for the Bayesian Mallows Model. The R Journal, 12(1), pages 324–342, DOI: 10.32614/RJ-2020-026.

See Also

spear_dist, expected_spear_dist, partition_fun_spear

Examples

## Example 1. Exact Spearman distance distribution for n=20 items.
distr <- spear_dist_distr(n_items = 20, log = FALSE)
plot(distr$distances,distr$logcard,type='l',ylab = 'Frequency',xlab='d',
main='Distribution of the Spearman distance\nunder the null model')


## Example 2. Approximate Spearman distance distribution for n=50 items with log-frequencies.
distr <- spear_dist_distr(n_items = 50)
plot(distr$distances,distr$logcard,type='l',ylab = 'Log-frequency',xlab='d',
    main='Log-distribution of the Spearman distance\nunder the null model')