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)Returns a single logical value
indicating whether x is weakly
dominated by y.
numeric vector with nonnegative elements
numeric vector with nonnegative elements
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()