scores {relations}R Documentation

Relation Scores

Description

Compute scores for the tuples of an endorelation.

Usage

relation_scores(x,
                method = c("Barthelemy/Monjardet", "Wei", "Borda",
                           "Kendall", "differential", "rankB"),
                normalize = FALSE)

Arguments

x an object inheriting from class relation, representing an endorelation.
method character string indicating the method (see details).
normalize logical indicating whether the score vector should be normalized to sum up to 1.

Details

In the following, consider the pairs of relation R, represented by x. Available built-in methods are as follows:

"Barthelemy/Monjardet"
for each a in the domain of R, (M(a) + N(a) - 1) / 2, where M(a) and N(a) are the numbers of objects b such that b R a, and b R a and not a R b, respectively. If R is a <= preference relation, we get the number of dominated objects plus half the number of the equivalent objects minus 1, i.e., the usual average ranks minus 1. See Barthélemy and Monjardet (1981).
"Wei"
the eigenvector corresponding to the greatest eigenvalue of the incidence matrix of the complement of R. See Wei (1952).
"Kendall", "Borda"
the column sums of the incidence matrix, i.e., for each object a, the number of objects b such that b R a. See Borda (1770) and Kendall (1955).
"differential"
the column sums of the incidence matrix minus the row sums. For each object a, this is the difference of the numbers of objects b such that b R a and a R b, respectively. In the case of a <= preference relation, this is the difference between the indegree and the outdegree of each object/node in the digraph corresponding to the relation incidence. See Regenwetter and Rykhlevskaia (2004).
"rankB"
a linear transformation of the differential D(R), so that the scores of the n objects start with 1: (n + 1 + D(R)) / 2.

Value

A vector of scores, with names taken from the relation domain labels.

References

J.-P. Barthélemy and B. Monjardet (1981), The median procedure in cluster analysis and social choice theory. Mathematical Social Sciences, 1:235–267.

J. C. Borda (1781), Mémoire sur les élections au scrutin. Histoire de l'Académie Royale des Sciences.

M. Kendall (1955), Further contributions to the theory of paired comparisons. Biometrics, 11:43–62.

M. Regenwetter and E. Rykhlevskaia (2004), On the (numerical) ranking associated with any finite binary relation. Journal of Mathematical Psychology, 48:239–246.

T. H. Wei (1952). The algebraic foundation of ranking theory. Unpublished thesis, Cambridge University.

Examples

## Example taken from Cook and Cress (1992, p.74)
I <- matrix(c(0, 0, 1, 1, 1,
              1, 0, 0, 0, 1,
              0, 1, 0, 0, 1,
              0, 1, 1, 0, 0,
              0, 0, 0, 1, 0),
            ncol = 5,
            byrow = TRUE)
R <- relation(domain = letters[1:5], incidence = I)

## Note that this is a "preference matrix", so take complement:
R <- !R

## Compare Kendall and Wei scores
cbind(
      Kendall = relation_scores(R, method = "Kendall", normalize = TRUE),
      Wei = relation_scores(R, method = "Wei", normalize = TRUE)
     )


[Package relations version 0.2-0 Index]