Maximum information preservable by nominal lumping

Calculates the maximum amount of mutual information that can be preserved by lumping a nominal variable.

Usage

maximum_mutual_information_nominal(
  counts,
  threshold,
  adj_matrix = NULL,
  verbose = FALSE,
  alternative_metric = c("mutual information", "bin count", "surplus")
)

Arguments

counts

Named numeric vector containing the number of times each level is observed.

threshold

Minimum number of samples each level must contain.

adj_matrix

Adjancency matrix of the preference graph. Default: a complete graph, allowing all lumpings.

verbose

Whether to print diagnostic messages or not. Default: FALSE.

alternative_metric

The metric that should be optimised for, if it is different from the default, the mutual information. For an explanation of the metrics see vignette("metrics").

Value

A list containing information about the optimal lumping:

mutual_information

Double representing the mutual information between the lumped and unlumped variable.

loss

Double representing the amount of entropy lost in the lumping process.

lumping

A list of character vectors, where each vector contains the names of the original levels that have been lumped together.

Details

Since these two pursuits are equivalent, the actual quantity optimized for is not the mutual information, but the empirical entropy of the lumped levels.

Be advised that, since the problem is NP-hard, the implementation here has time complexity \(O\left(2^{2^m}\right)\), where \(m\) is the number of levels in the nominal variable.

See also

lump_nominal() for a more user-friendly wrapper around this function that actually carries out the lumping.

maximum_mutual_information_nominal_heuristic() to approximate this function when the number of levels is too large.

maximum_mutual_information_hierarchical() for a version of this function that takes advantage of hierarchical structure to speed up the execution time.

Author

Daan Koning

Examples

counts = c(A = 3, B = 1, C = 3, D = 2, E = 2)
threshold <- 3
maximum_mutual_information_nominal(counts, threshold)
#> $mutual_information
#> [1] 1.09006
#> 
#> $loss
#> [1] 0.456539
#> 
#> $lumping
#> $lumping[[1]]
#> [1] "D" "E"
#> 
#> $lumping[[2]]
#> [1] "B" "C"
#> 
#> $lumping[[3]]
#> [1] "A"
#> 
#> 

# Or ban certain pairings:
preference_graph <- adjacency_from_edge_list(
 names(counts),
 disallow = list(c("B", "E"), c("D", "E"))
)
maximum_mutual_information_nominal(counts, threshold, preference_graph)
#> $mutual_information
#> [1] 1.06709
#> 
#> $loss
#> [1] 0.4795092
#> 
#> $lumping
#> $lumping[[1]]
#> [1] "A"
#> 
#> $lumping[[2]]
#> [1] "C" "E"
#> 
#> $lumping[[3]]
#> [1] "B" "D"
#> 
#> 

# Or only allow certain pairings
preference_graph <- adjacency_from_edge_list(
 names(counts),
 allow = list(c("B", "E"), c("C", "D"), c("D", "E"))
)
maximum_mutual_information_nominal(counts, threshold, preference_graph)
#> $mutual_information
#> [1] 1.06709
#> 
#> $loss
#> [1] 0.4795092
#> 
#> $lumping
#> $lumping[[1]]
#> [1] "A"
#> 
#> $lumping[[2]]
#> [1] "C" "D"
#> 
#> $lumping[[3]]
#> [1] "B" "E"
#> 
#>