The Generalized Moment-Based Filter

Can we solve the filtering problem from the only knowledge of few moments of the noise terms? In this technical note, by exploiting set of distributions based filtering, we solve this problem without introducing additional assumptions on the distributions of the noises (e.g., Gaussianity) or on the final form of the estimator (e.g., linear estimator). Given the moments (e.g., mean and variance) of random variable X, it is possible to define the set of all distributions that are compatible with the moments information. This set can be equivalently characterized by its extreme distributions: a family of mixtures of Dirac´s deltas. The lower and upper expectation of any function g of X are obtained in correspondence of these extremes and can be computed by solving a linear programming problem. The filtering problem can then be solved by running iteratively this linear programming problem. In this technical note, we discuss theoretical properties of this filter, we show the connection with set-membership estimation and its practical applications.