MLE of mixtures of Mallows models with Spearman distance via EM algorithms
Perform the MLE of mixtures of Mallows model with Spearman distance on full and partial rankings via EM algorithms. Partial rankings with missing data in arbitrary positions are supported.
print method for class "emMSmix".
fitMSmix( rankings, n_clust = 1, n_start = 1, n_iter = 200, mc_em = FALSE, eps = 10^(-6), init = list(list(rho = NULL, theta = NULL, weights = NULL))[rep(1, n_start)], plot_log_lik = FALSE, comp_log_lik_part = FALSE, plot_log_lik_part = FALSE, parallel = FALSE, theta_max = 3, theta_tol = 1e-05, theta_tune = 1, subset = NULL, item_names = NULL ) ## S3 method for class 'emMSmix' print(x, ...)
rankings: Integer N$$x$$n matrix or data frame with partial rankings in each row. Missing positions must be coded as NA.n_clust: Number of mixture components. Defaults to 1.n_start: Number of starting points. Defaults to 1.n_iter: Maximum number of EM iterations. Defaults to 200.mc_em: Logical: whether the Monte Carlo EM algorithm must be used for MLE on partial rankings completion, see Details. Ignored when rankings does not contain any partial sequence. Defaults to FALSE.eps: Positive tolerance value for the convergence of the EM algorithm. Defaults to .init: List of n_start lists with the starting values of the parameters to initialize the EM algorithm. Each list must contain three named objects, namely: 1) rho: integer G$$x$$n matrix with the component-specific consensus rankings in each row; 2) theta: numeric vector of non-negative component-specific precision parameters; 3) weights: numeric vector of positive mixture weights. Defaults to NULL, meaning that the starting points are automatically generated from the uniform distribution.plot_log_lik: Logical: whether the iterative log-likelihood values (based on full or augmented rankings) must be plotted. Defaults to FALSE.comp_log_lik_part: Logical: whether the maximized observed-data log-likelihood value (based on partial rankings) must be returned. Ignored when rankings does not contain any partial sequence or data_augmentation cannot be applied. See Details. Defaults to FALSE.plot_log_lik_part: Logical: whether the iterative observed-data log-likelihood values (based on partial rankings) must be plotted. Ignored when rankings does not contain any partial sequence. In the presence of partial rankings, this argument is ignored when comp_log_lik_part = FALSE or data_augmentation cannot be applied. Defaults to FALSE.parallel: Logical: whether parallelization over multiple initializations must be used. Defaults to FALSE.theta_max: Positive upper bound for the precision parameters. Defaults to 3.theta_tol: Positive convergence tolerance for the M-step of the precision parameters. Defaults to .theta_tune: Positive tuning constant affecting the precision parameters in the Monte Carlo step. Ignored when rankings does not contain any partial sequence or mc_em = FALSE. Defaults to 1.subset: Optional logical or integer vector specifying the subset of observations, i.e. rows of the rankings, to be kept. Missing values are taken as FALSE. Defaults to NULL meaning that all the rows are considered.item_names: Character vector with the names to be used for the items. Defaults to NULL, meaning that colnames(rankings) is used and, if not available, item_names is set equal to "Item1","Item2",....x: An object of class "emMSmix" returned by fitMSmix....: Further arguments passed to or from other methods (not used).An object of class "emMSmix", namely a list with the following named components:
mod: List of named objects describing the best fitted model in terms of maximized log-likelihood over the n_start initializations. See Details.max_log_lik: Maximized log-likelihood values for each initialization.partial_data: Logical: whether the dataset includes some partially-ranked sequences.convergence: Binary convergence indicators of the EM algorithm for each initialization: 1 = convergence has been achieved, 0 = otherwise.record: Best log-likelihood values sequentially achieved over the n_start initializations.em_settings: List of settings used to fit the model.call: The matched call.The mod sublist contains the following named objects:
rho: Integer G$$x$$n matrix with the MLEs of the component-specific consensus rankings in each row.theta: Numeric vector with the MLEs of the component-specific precision parameters.weights: Numeric vector with the MLEs of the mixture weights.z_hat: Numeric N$$x$$G matrix of the estimated posterior component membership probabilities. Returned when n_clust > 1, otherwise NULL.map_classification: Integer vector of mixture component memberships based on the MAP allocation from the z_hat matrix. Returned when n_clust > 1, otherwise NULL.log_lik: Numeric vector of the log-likelihood values (based on full or augmented rankings) at each iteration.best_log_lik: Maximized log-likelihood value (based on full or augmented rankings) of the fitted model.bic: BIC value of the fitted model based on best_log_lik.log_lik_part: Numeric vector of the observed-data log-likelihood values (based on partial rankings) at each iteration. Returned when rankings contains some partial sequences that can be completed with data_augmentation and plot_log_lik_part = TRUE, otherwise NULL. See Details.best_log_lik_part: Maximized observed-data log-likelihood value (based on partial rankings) of the fitted model. Returned when rankings contains some partial sequences that can be completed with data_augmentation, otherwise NULL. See Details.bic_part: BIC value of the fitted model based on best_log_lik_part. Returned when rankings contains some partial sequences that can be completed with data_augmentation, otherwise NULL. See Details.conv: Binary convergence indicator of the best fitted model: 1 = convergence has been achieved, 0 = otherwise.augmented_rankings: Integer N$$x$$n matrix with rankings completed through the Monte Carlo step in each row. Returned when rankings contains some partial sequences and mc_em = TRUE, otherwise NULL.The EM algorithms are launched from n_start initializations and the best solution in terms of maximized log-likelihood value (based on full or augmented rankings) is returned.
When mc_em = FALSE, the scheme introduced by Crispino et al. (2023) is performed, where partial rankings are augmented with all compatible full rankings. This type of data augmentation is supported up to 10 missing positions in the partial rankings.
When mc_em = TRUE, the - computationally more efficient - Monte Carlo EM algorithm introduced by Crispino et al. (2025+) is implemented. In the case of a large number of censored positions and sample sizes, the mc_em = TRUE must be preferred.
Regardless of the fitting method adopted for inference on partial rankings, note that setting the argument comp_log_lik_part = TRUE for the computation of the observed-data log-likelihood values (based on partial rankings) can slow down the procedure in the case of a large number of censored positions and sample sizes.
## Example 1. Fit the 3-component mixture of Mallows models with Spearman distance ## to the Antifragility dataset. r_antifrag <- ranks_antifragility[, 1:7] set.seed(123) mms_fit <- fitMSmix(rankings = r_antifrag, n_clust = 3, n_start = 10) mms_fit$mod$rho; mms_fit$mod$theta; mms_fit$mod$weights ## Example 2. Fit the Mallows model with Spearman distance ## to simulated partial rankings through data augmentation. rank_data <- rbind(c(NA, 4, NA, 1, NA), c(NA, NA, NA, NA, 1), c(2, NA, 1, NA, 3), c(4, 2, 3, 5, 1), c(NA, 4, 1, 3, 2)) mms_fit <- fitMSmix(rankings = rank_data, n_start = 10) mms_fit$mod$rho; mms_fit$mod$theta ## Example 3. Fit the Mallows model with Spearman distance ## to the Reading genres dataset through Monte Carlo EM. top5_read <- ranks_read_genres[, 1:11] mms_fit <- fitMSmix(rankings = top5_read, n_start = 10, mc_em = TRUE) mms_fit$mod$rho; mms_fit$mod$theta
Crispino M, Mollica C and Modugno L (2025+). MSmix: An R Package for clustering partial rankings via mixtures of Mallows Models with Spearman distance. (submitted)
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.
Beckett LA (1993). Maximum likelihood estimation in Mallows’s model using partially ranked data. In Probability models and statistical analyses for ranking data, pages 92--107. Springer New York.
summary.emMSmix, plot.emMSmix
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