pord_weakdom {agop}

Weak Dominance Relation (Preorder) in the Producer Assessment Problem

Description

Checks whether a given numeric vector of arbitrary length is (weakly) dominated by another vector, possibly of different length, in terms of (sorted) elements' values and their number.

Usage

pord_weakdom(x, y)

Arguments

x

numeric vector with nonnegative elements

y

numeric vector with nonnegative elements

Details

We say that a numeric vector x of length n_x is weakly dominated by y of length n_y iff

1. n_x\le n_y and

2. for all i=1,\dots,n it holds x_{(n_x-i+1)}\le y_{(n_y-i+1)}.

This relation is a preorder: it is reflexive (see rel_is_reflexive) and transitive (see rel_is_transitive), but not necessarily total (see rel_is_total). See rel_graph for a convenient function to calculate the relationship between all pairs of elements of a given set.

Note that this dominance relation gives the same value for all permutations of input vectors' element. Such a preorder is tightly related to symmetric impact functions: each impact function is a morphism between weak-dominance-preordered set of vectors and the set of reals equipped with standard linear ordering (see Gagolewski, Grzegorzewski, 2011 and Gagolewski, 2013).

Value

Returns a single logical value indicating whether x is weakly dominated by y.

References

Gagolewski M., Grzegorzewski P., Possibilistic Analysis of Arity-Monotonic Aggregation Operators and Its Relation to Bibliometric Impact Assessment of Individuals, International Journal of Approximate Reasoning 52(9), 2011, pp. 1312-1324.

Gagolewski M., Scientific Impact Assessment Cannot be Fair, Journal of Informetrics 7(4), 2013, pp. 792-802.

Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7

See Also

Other binary_relations: check_comonotonicity(), pord_nd(), pord_spread(), rel_graph(), rel_is_antisymmetric(), rel_is_asymmetric(), rel_is_cyclic(), rel_is_irreflexive(), rel_is_reflexive(), rel_is_symmetric(), rel_is_total(), rel_is_transitive(), rel_reduction_hasse()

Other impact_functions: index_g(), index_h(), index_lp(), index_maxprod(), index_rp(), index_w()