bicMSmix {MSmix}

BIC and AIC for mixtures of Mallows models with Spearman distance

Description

bicMSmix and aicMSmix compute, respectively, the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) for a mixture of Mallow models with Spearman distance fitted on partial rankings.

Usage

bicMSmix(rho, theta, weights, rankings)

aicMSmix(rho, theta, weights, rankings)

Arguments

rho

Integer G\timesn matrix with the component-specific consensus rankings in each row.

theta

Numeric vector of G non-negative component-specific precision parameters.

weights

Numeric vector of G positive mixture weights (normalization is not necessary).

rankings

Integer N\timesn matrix or data frame with partial rankings in each row. Missing positions must be coded as NA.

Details

The (log-)likelihood evaluation is performed by augmenting the partial rankings with the set of all compatible full rankings (see data_augmentation), and then the marginal likelihood is computed.

When n\leq 20, the (log-)likelihood is exactly computed, otherwise it is approximated with the method introduced by Crispino et al. (2023). If n>170, the approximation is also restricted over a fixed grid of values for the Spearman distance to limit computational burden.

Value

The BIC or AIC value.

References

Crispino M, Mollica C and Modugno L (2024+). 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.

Schwarz G (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), pages 461–464, DOI: 10.1002/sim.6224.

Sakamoto Y, Ishiguro M, and Kitagawa G (1986). Akaike Information Criterion Statistics. Dordrecht, The Netherlands: D. Reidel Publishing Company.

See Also

likMSmix, data_augmentation

Examples

## Example 1. Simulate rankings from a 2-component mixture of Mallow models
## with Spearman distance.
set.seed(12345)
rank_sim <- rMSmix(sample_size = 50, n_items = 12, n_clust = 2)
str(rank_sim)
rankings <- rank_sim$samples
# Fit the true model.
set.seed(12345)
fit <- fitMSmix(rankings = rankings, n_clust = 2, n_start = 10)
# Comparing the BIC at the true parameter values and at the MLE.
bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit$mod$rho, theta = fit$mod$theta, weights = fit$mod$weights,
       rankings = rank_sim$samples)


## Example 2. Simulate rankings from a basic Mallow model with Spearman distance.
set.seed(54321)
rank_sim <- rMSmix(sample_size = 50, n_items = 8, n_clust = 1)
str(rank_sim)
# Let us censor the observations to be top-5 rankings.
rank_sim$samples[rank_sim$samples > 5] <- NA
rankings <- rank_sim$samples
# Fit the true model with the two EM algorithms.
set.seed(54321)
fit_em <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10)
set.seed(54321)
fit_mcem <- fitMSmix(rankings = rankings, n_clust = 1, n_start = 10, mc_em = TRUE)
# Compare the BIC at the true parameter values and at the MLEs.
bicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights,
       rankings = rank_sim$samples)
bicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = rank_sim$rho, theta = rank_sim$theta, weights = rank_sim$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit_em$mod$rho, theta = fit_em$mod$theta, weights = fit_em$mod$weights,
       rankings = rank_sim$samples)
aicMSmix(rho = fit_mcem$mod$rho, theta = fit_mcem$mod$theta, weights = fit_mcem$mod$weights,
       rankings = rank_sim$samples)