Computes the (Ordered) Weighted Maximum/Minimum.
owmax(x, w = rep(Inf, length(x)))owmin(x, w = rep(-Inf, length(x)))
wmax(x, w = rep(Inf, length(x)))
wmin(x, w = rep(-Inf, length(x)))
These functions return a single numeric value.
numeric vector to be aggregated
numeric vector of the same length as x; weights
The OWMax operator is given by
$$
\mathsf{OWMax}_\mathtt{w}(\mathtt{x})=\bigvee_{i=1}^{n} w_{i}\wedge x_{(i)}
$$
where \(x_{(i)}\) denotes the \(i\)-th smallest
value in x.
The OWMin operator is given by $$ \mathsf{OWMin}_\mathtt{w}(\mathtt{x})=\bigwedge_{i=1}^{n} w_{i}\vee x_{(i)} $$
The WMax operator is given by $$ \mathsf{WMax}_\mathtt{w}(\mathtt{x})=\bigvee_{i=1}^{n} w_{i}\wedge x_{i} $$
The WMin operator is given by $$ \mathsf{WMin}_\mathtt{w}(\mathtt{x})=\bigwedge_{i=1}^{n} w_{i}\vee x_{i} $$
OWMax and WMax by default return the greatest value in x
and OWMin and WMin - the smallest value in x.
Classically, it is assumed that if we aggregate vectors with elements in \([a,b]\), then the largest weight for OWMax should be equal to \(b\) and the smallest for OWMin should be equal to \(a\).
There is a strong connection between the OWMax/OWMin operators and the Sugeno integrals w.r.t. some monotone measures. Additionally, it may be shown that the OWMax and OWMin classes are equivalent.
Moreover, index_h for integer data
is a particular OWMax operator.
Dubois D., Prade H., Testemale C., Weighted fuzzy pattern matching, Fuzzy Sets and Systems 28, 1988, pp. 313-331.
Dubois D., Prade H., Semantics of quotient operators in fuzzy relational databases, Fuzzy Sets and Systems 78(1), 1996, pp. 89-93.
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
Sugeno M., Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Technology, 1974.
Other aggregation_operators:
owa()