Mapping Points Back from the Concept Space with Minimum Mean Squared Error

摘要

In this article we present a method to map points from the concept space, associated with the fuzzy c-means algorithm, back to the feature space. We assume that we have a probability density function / defined on the feature space (e.g. a normalized density of a data set). For a given point ω of concept space, we give explicitly a set of points in feature space that are mapped onto ω and we give a formula for a reverse mapping to the feature space which results in minimum mean squared error, with respect to density f, of the operation of mapping a point of feature space into the concept space and back. We characterize the circumstances under which points can be mapped back into the feature space unambiguously and provide a formula for the inverse mapping.