relations (version 0.3-3)

scores: Relation Scores

Description

Compute scores for the tuples of an endorelation.

Usage

relation_scores(x,
                method = c("ranks", "Barthelemy/Monjardet", "Borda",
                           "Kendall", "Wei", "differential"),
                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.

...

further arguments to be passed to methods.

Value

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

encoding

UTF-8

Details

In the following, consider an endorelation $R$ on $n$ objects. Let the in-degree $I(a)$ and out-degree $O(a)$ of an object $a$ be defined as the numbers of objects $b$ such that $b R a$ and, respectively, $a R b$, and let $D(a) = I(a) - O(a)$ be the differential of $a$ (see Regenwetter and Rykhlevskaia (2004)). Note that $I$ and $O$ are given by the column sums and row sums of the incidence matrix of $R$. If $R$ is a preference relation with a $\le$ intepretation, $D(a)$ is the difference between the numbers of objects dominated by $a$ (i.e., $< a$) and dominating $a$ (i.e., $> a$), as ``ties'' cancel out.

Available built-in score methods are as follows: [object Object],[object Object],[object Object],[object Object],[object Object]

Definitions of scores for preference relations $R$ are somewhat ambiguous because $R$ can encode $\le$ or $\ge$ (or strict variants thereof) relationships (and all such variants are used in the literature). Package relations generally assumes a $\le$ encoding, and that scores in the strict sense should increase with preference (the most preferred get the highest scores) whereas ranks decrease with preference (the most preferred get the lowest ranks).

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.

D. Bouyssou (1992), Ranking methods based on valued preference relations: A characterization of the net flow network. European Journal of Operational Research, 60:61--67. 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)
     )