# 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.
Type de document :
Communication dans un congrès
Khalid Saeed; Władysław Homenda. 15th IFIP International Conference on Computer Information Systems and Industrial Management (CISIM), Sep 2016, Vilnius, Lithuania. Springer International Publishing, Lecture Notes in Computer Science, LNCS-9842, pp.67-78, 2016, Computer Information Systems and Industrial Management. 〈10.1007/978-3-319-45378-1_7〉
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https://hal.inria.fr/hal-01637496
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### Citation

Wladyslaw Homenda, Tomasz Penza. Mapping Points Back from the Concept Space with Minimum Mean Squared Error. Khalid Saeed; Władysław Homenda. 15th IFIP International Conference on Computer Information Systems and Industrial Management (CISIM), Sep 2016, Vilnius, Lithuania. Springer International Publishing, Lecture Notes in Computer Science, LNCS-9842, pp.67-78, 2016, Computer Information Systems and Industrial Management. 〈10.1007/978-3-319-45378-1_7〉. 〈hal-01637496〉

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