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A robust unscented transformation for uncertain moments. (English) Zbl 1456.65180
In numerous problems of statistics and stochastic filtering, one is often interested in calculating the posterior expectation of a continuous random variable that undergoes a nonlinear transform.
The authors propose a robust version of the unscented transform (UT) for one-dimensional random variables. The principle behind UT is to approximate the continuous distribution by the discrete distribution by equating the first \(m\) moments of these distributions. UT is a deterministic sampling technique and has less computational burden than the Monte Carlo integration method.
In the paper, it is proposed to use the Chebyshev center of the semialgebraic set defined by the possible choices of sigma points and their weights as a robust UT. As the moments are not usually exactly known in practical situations, it is assumed that they lie in some intervals. Two approaches for generating robust sigma points are proposed. The moment matching equations of UT are reformulated as a system of polynomial equations with polynomial inequalities. As this system can have more than one solution, it is possible to choose a set of sigma points which minimizes a given cost function by formulating the problem as a polynomial optimization problem solved by using Lasserre’s hierarchy of semidefinite programming relaxations.
The main contribution of the paper is the introduction of the concept of UT robustness in the sense of exploiting the upper and lower bounds for moments. Robustness is achieved by matching precisely known high order moments. Lasserre’s hierarchy of relaxations is applied to polynomial equations with polynomial inequalities.
MSC:
60E10 Characteristic functions; other transforms
90C23 Polynomial optimization
90C17 Robustness in mathematical programming
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