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Entropy 2018, 20(7), 498; https://doi.org/10.3390/e20070498

Assessing Information Transmission in Data Transformations with the Channel Multivariate Entropy Triangle

Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Leganés 28911, Spain
These authors contributed equally to this work.
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Received: 3 May 2018 / Revised: 11 June 2018 / Accepted: 20 June 2018 / Published: 27 June 2018
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Abstract

Data transformation, e.g., feature transformation and selection, is an integral part of any machine learning procedure. In this paper, we introduce an information-theoretic model and tools to assess the quality of data transformations in machine learning tasks. In an unsupervised fashion, we analyze the transformation of a discrete, multivariate source of information X¯ into a discrete, multivariate sink of information Y¯ related by a distribution PX¯Y¯. The first contribution is a decomposition of the maximal potential entropy of (X¯,Y¯), which we call a balance equation, into its (a) non-transferable, (b) transferable, but not transferred, and (c) transferred parts. Such balance equations can be represented in (de Finetti) entropy diagrams, our second set of contributions. The most important of these, the aggregate channel multivariate entropy triangle, is a visual exploratory tool to assess the effectiveness of multivariate data transformations in transferring information from input to output variables. We also show how these decomposition and balance equations also apply to the entropies of X¯ and Y¯, respectively, and generate entropy triangles for them. As an example, we present the application of these tools to the assessment of information transfer efficiency for Principal Component Analysis and Independent Component Analysis as unsupervised feature transformation and selection procedures in supervised classification tasks. View Full-Text
Keywords: entropy, entropy visualization; entropy balance equation; Shannon-type relations; multivariate analysis; machine learning evaluation; data transformation entropy, entropy visualization; entropy balance equation; Shannon-type relations; multivariate analysis; machine learning evaluation; data transformation
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Valverde-Albacete, F.J.; Peláez-Moreno, C. Assessing Information Transmission in Data Transformations with the Channel Multivariate Entropy Triangle. Entropy 2018, 20, 498.

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