pord_weakdom {agop} R Documentation

## 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≤ n_y and

2. for all i=1,…,n it holds x_{(n_x-i+1)}≤ 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

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`