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

OWMax_w(x) = max_i{ min{w_i, x_(i)} }

where x_(i) denotes the i-th smallest value in `x`.

The OWMin operator is given by

OWMin_w(x) = min_i{ max{w_i, x_(i)} }

The WMax operator is given by

WMax_w(x) = max_i{ min{w_i, x_i} }

The WMin operator is given by

WMin_w(x) = min_i{ max{w_i, 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`