rMSmix function

Random samples from a mixture of Mallows models with Spearman distance

Draw random samples of full rankings from a mixture of Mallows models with Spearman distance.

rMSmix(
  sample_size = 1,
  n_items,
  n_clust = 1,
  rho = NULL,
  theta = NULL,
  weights = NULL,
  uniform = FALSE,
  mh = TRUE
)

Arguments

• sample_size: Number of full rankings to be sampled. Defaults to 1.
• n_items: Number of items.
• n_clust: Number of mixture components. Defaults to 1.
• rho: Integer G$$x$$n matrix with the component-specific consensus rankings in each row. Defaults to NULL, meaning that the consensus rankings are randomly generated according to the sampling scheme indicated by the uniform argument. See Details.
• theta: Numeric vector of $G$ non-negative component-specific precision parameters. Defaults to NULL, meaning that the concentrations are uniformly generated from an interval containing typical values for the precisions. See Details.
• weights: Numeric vector of $G$ positive mixture weights (normalization is not necessary). Defaults to NULL, meaning that the mixture weights are randomly generated according to the sampling scheme indicated by the uniform argument. See Details.
• uniform: Logical: whether rho or weights have to be sampled uniformly on their support. When uniform = FALSE they are sampled, respectively, to ensure separation among mixture components and populated weights. Used when $G>1$ and either rho or weights are NULL (see Details). Defaults to FALSE.
• mh: Logical: whether the samples must be drawn with the Metropolis-Hastings (MH) scheme implemented in the BayesMallows package, rather by direct sampling from the Mallows probability distribution. For n_items > 10, the MH is always applied to speed up the sampling procedure. Defaults to TRUE.

Returns

A list of the following named components:

• samples: Integer N$$x$$n matrix with the sample_size simulated full rankings in each row.

• rho: Integer G$$x$$n matrix with the component-specific consensus rankings used for the simulation in each row.

• theta: Numeric vector of the $G$ component-specific precision parameters used for the simulation.

• weights: Numeric vector of the $G$ mixture weights used for the simulation.

• classification: Integer vector of the sample_size component membership labels.

Details

When n_items > 10 or mh = TRUE, the random samples are obtained by using the Metropolis-Hastings algorithm, described in Vitelli et al. (2018) and implemented in the sample_mallows function of the package BayesMallows package.

When theta = NULL, the concentration parameters are randomly generated from a uniform distribution on the interval $(1/n^{2},3/n^{1.5})$ containing typical values for the precisions.

When uniform = FALSE, the mixing weights are sampled from a symmetric Dirichlet distribution with shape parameters all equal to $2G$, to favor populated and balanced clusters, and the consensus parameters are sampled to favor well-separated clusters, i. e., at least at Spearman distance equal to $\frac{2}{G}\binom{n+1}{3}$ from each other.

Examples

## Example 1. Drawing from a mixture with randomly generated parameters of separated clusters.
set.seed(12345)
rMSmix(sample_size = 50, n_items = 25, n_clust = 5)

## Example 2. Drawing from a mixture with uniformly generated parameters.
set.seed(12345)
rMSmix(sample_size = 100, n_items = 9, n_clust = 3, uniform = TRUE)

## Example 3.  Drawing from a mixture with customized parameters.
r_par <- rbind(1:5, c(4, 5, 2, 1, 3))
t_par <- c(0.01, 0.02)
w_par <- c(0.4, 0.6)
set.seed(12345)
rMSmix(sample_size = 50, n_items = 5, n_clust = 2, theta = t_par, rho = r_par, weights = w_par)

References

Vitelli V, Sørensen Ø, Crispino M, Frigessi A and Arjas E (2018). Probabilistic Preference Learning with the Mallows Rank Model. Journal of Machine Learning Research, 18 (158), pages 1--49, ISSN: 1532-4435, https://jmlr.org/papers/v18/15-481.html.

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.

Chenyang Zhong (2021). Mallows permutation model with L1 and L2 distances I: hit and run algorithms and mixing times. arXiv: 2112.13456.