Checks whether a given numeric vector of arbitrary length is (weakly) dominated (elementwise) by another vector of the same length.
pord_nd(x, y, incompatible_lengths = NA)numeric vector with nonnegative elements
numeric vector with nonnegative elements
single logical value,
value to return iff lengths of x and y differ
Returns a single logical value
indicating whether x is weakly
dominated by y.
We say that a numeric vector x of length \(n_x\) is weakly dominated by y of length \(n_y\) iff
\(n_x=n_y\) and
for all \(i=1,\dots,n_x\) it holds \(x_i\le y_i\).
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.
Such a preorder is tightly related to classical aggregation functions: each aggregation function is a morphism between weak-dominance-preordered set of vectors and the set of reals equipped with standard linear ordering.
Grabisch M., Marichal J.-L., Mesiar R., Pap E., Aggregation functions, Cambridge University Press, 2009.
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
Other binary_relations: check_comonotonicity,
pord_spread, pord_weakdom,
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