owmax {agop} R Documentation

## WMax, WMin, OWMax, and OWMin Operators

### Description

Computes the (Ordered) Weighted Maximum/Minimum.

### Usage

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)))


### Arguments

 x numeric vector to be aggregated w numeric vector of the same length as x; weights

### Details

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.

### Value

These functions return a single numeric value.

### References

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()