A Fuzzy Set Estimation Using Interval Contractors: Application to Localization

This paper proposes a new approach to characterize fuzzy sets defined by membership functions using interval analysis. The goal is to combine pieces of information (the granules) to build a fuzzy set providing a representation of the knowledge we have on the parameter vector we want to estimate. Then the different α-cuts of this fuzzy set can be approximated by an interval procedure. The proposed formalism can handle efficiently various situations since there is a lot of freedom to define the wanted combination. The information contained in a random vector is represented by a membership function, which is issued from the composition of a score function and characteristic functions associated with some elementary epistemic sets, considered as the elementary granules of knowledge. Each granule is attached to a given measurement or any other elementary information we have on the vector to be estimated. The score function is an aggregation operator that defines the weighting of these granules, simplifying their combination. Thus, it is possible to deal with complex outliers-related situations in a context where uncertainties are only partially known. The proposed approach makes it possible to obtain an efficient interval-based algorithm able to find an inner and an outer approximation of the α-cut to be characterized. An application related to the localization of an underwater robot is presented to illustrate the efficiency of the approach.