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

Abstract : 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 f defined on the feature space (e.g. a normalized density of a data set). For a given point $$\varvec{w}$$ of concept space, we give explicitly a set of points in feature space that are mapped onto $$\varvec{w}$$ 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.
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Wladyslaw Homenda, Tomasz Penza. Mapping Points Back from the Concept Space with Minimum Mean Squared Error. 15th IFIP International Conference on Computer Information Systems and Industrial Management (CISIM), Sep 2016, Vilnius, Lithuania. pp.67-78, ⟨10.1007/978-3-319-45378-1_7⟩. ⟨hal-01637496⟩

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