| get_cardinalities {BayesMallows} | R Documentation |
Get cardinalities for each distance
Description
The partition function for the Mallows model can be defined in a computationally efficient manner as
Z_{n}(\alpha) = \sum_{d_{n} \in
\mathcal{D}_{n}} N_{m,n} e^{-(\alpha/n) d_{m}}.
In this equation, \mathcal{D}_{n} a set containing all possible
distances at the given number of items, and d_{m} is on element of
this set. Finally, N_{m,n} is the number of possible configurations
of the items that give the particular distance. See
Irurozki et al. (2016),
Vitelli et al. (2018), and
Crispino et al. (2023) for details.
For footrule distance, the cardinalities come from entry A062869 in the On-Line Encyclopedia of Integer Sequences (OEIS) (Sloane and Inc. 2020). For Spearman distance, they come from entry A175929, and for Ulam distance from entry A126065.
Usage
get_cardinalities(n_items, metric = c("footrule", "spearman", "ulam"))
Arguments
n_items |
Number of items. |
metric |
Distance function, one of "footrule", "spearman", or "ulam". |
Value
A dataframe with two columns, distance which contains each distance
in the support set at the current number of items, i.e., d_{m}, and
value which contains the number of values at this particular distances,
i.e., N_{m,n}.
References
Crispino M, Mollica C, Astuti V, Tardella L (2023).
“Efficient and accurate inference for mixtures of Mallows models with Spearman distance.”
Statistics and Computing, 33(5).
ISSN 1573-1375, doi:10.1007/s11222-023-10266-8, http://dx.doi.org/10.1007/s11222-023-10266-8.
Irurozki E, Calvo B, Lozano JA (2016).
“PerMallows: An R Package for Mallows and Generalized Mallows Models.”
Journal of Statistical Software, 71(12), 1–30.
doi:10.18637/jss.v071.i12.
Sloane NJA, Inc. TOF (2020).
“The on-line encyclopedia of integer sequences.”
https://oeis.org/.
Vitelli V, Sørensen, Crispino M, Arjas E, Frigessi A (2018).
“Probabilistic Preference Learning with the Mallows Rank Model.”
Journal of Machine Learning Research, 18(1), 1–49.
https://jmlr.org/papers/v18/15-481.html.
See Also
Other partition function:
compute_exact_partition_function(),
estimate_partition_function()
Examples
# Extract the cardinalities for four items with footrule distance
n_items <- 4
dat <- get_cardinalities(n_items)
# Compute the partition function at alpha = 2
alpha <- 2
sum(dat$value * exp(-alpha / n_items * dat$distance))
#'
# We can confirm that it is correct by enumerating all possible combinations
all <- expand.grid(1:4, 1:4, 1:4, 1:4)
perms <- all[apply(all, 1, function(x) length(unique(x)) == 4), ]
sum(apply(perms, 1, function(x) exp(-alpha / n_items * sum(abs(x - 1:4)))))
# We do the same for the Spearman distance
dat <- get_cardinalities(n_items, metric = "spearman")
sum(dat$value * exp(-alpha / n_items * dat$distance))
#'
# We can confirm that it is correct by enumerating all possible combinations
sum(apply(perms, 1, function(x) exp(-alpha / n_items * sum((x - 1:4)^2))))