OWA weighting strategies

An ordered weighted averaging (OWA) operator first sorts its input values and then computes a weighted sum. For values sorted in descending order, \(z_1\ge z_2\ge\cdots\ge z_n\), and normalized weights \(w_i\ge0\) with \(\sum_i w_i=1\),

\[ \operatorname{OWA}(z)=\sum_{i=1}^{n} w_i z_i. \]

In frsutils, OWAFRS removes the self-comparison, sorts the remaining evidence in descending order, and applies:

• ascending weights for the lower approximation, emphasizing smaller evidence values and approximating an infimum;
• descending weights for the upper approximation, emphasizing larger evidence values and approximating a supremum.

Implemented weight families

Name	Raw weight family before normalization	Parameters	Registered aliases
Linear	\(r_i=i\)	None	linear, additive
Exponential	\(r_i\propto b^i\)	\(b>1\)	exponential, exp, gp
Harmonic	\(r_i=1/i\)	None	harmonic, harm, inv_add

For linear weights in ascending order, normalization gives

\[ w_i=\frac{2i}{n(n+1)}. \]

The weights(n, order=...) method normalizes the raw values and explicitly sorts them into ascending or descending order. The exponential implementation shifts its exponents before normalization to avoid overflow while preserving the same normalized mathematical weights.

Configuration example

from frsutils import compute_approximations

result = compute_approximations(
    X,
    y,
    model="owafrs",
    ub_owa_method_name="linear",
    lb_owa_method_name="linear",
)

References

1. Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Transactions on Systems, Man, and Cybernetics, 18(1), 183–190. https://doi.org/10.1109/21.87068
2. Cornelis, C., Verbiest, N., & Jensen, R. (2010). Ordered weighted average based fuzzy rough sets. In Rough Set and Knowledge Technology, 78–85. https://doi.org/10.1007/978-3-642-16248-0_16
3. Vluymans, S., Mac Parthaláin, N., Cornelis, C., & Saeys, Y. (2019). Weight selection strategies for ordered weighted average based fuzzy rough sets. Information Sciences, 501, 155–171. https://doi.org/10.1016/j.ins.2019.05.085