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.
pord_weakdom(x, y)numeric vector with nonnegative elements
numeric vector with nonnegative elements
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
\(n_x\le n_y\) and
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).
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
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