# A Robust Unscented Transformation for Uncertain Moments

## TL;DR

This paper introduces a robust version of the unscented transform that accounts for uncertain moments within intervals, using polynomial optimization and relaxation techniques to improve robustness in one-dimensional stochastic analysis.

## Contribution

It develops a robust UT framework based on Chebychev centers for uncertain moments, reformulating the problem as a polynomial optimization with proposed relaxation algorithms.

## Key findings

- Provides a robust UT method for uncertain moments.
- Formulates the problem as a polynomial optimization.
- Proposes algorithms to relax NP-hard problems.

## Abstract

This paper proposes a robust version of the unscented transform (UT) for one-dimensional random variables. It is assumed that the moments are not exactly known, but are known to lie in intervals. In this scenario, the moment matching equations are reformulated as a system of polynomial equations and inequalities, and it is proposed to use the Chebychev center of the solution set as a robust UT. This method yields a parametrized polynomial optimization problem, which in spite of being NP-Hard, can be relaxed by some algorithms that are proposed in this paper.