Symmetric approximations of pseudo-Boolean functions with applications to influence indexes

We introduce an index for measuring the influence of the k-th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making.

💡 Research Summary

The paper introduces a novel framework for approximating pseudo‑Boolean functions by symmetric combinations of order‑statistic functions and uses this approximation to define an influence index for the k‑th smallest variable. A pseudo‑Boolean function f : {0,1}ⁿ → ℝ is first expressed in terms of order‑statistic functions OSFₖ(x) = 1_{