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Open AccessArticle

Representational Rényi Heterogeneity

1
Department of Psychiatry, Dalhousie University, Halifax, NS B3H 2E2, Canada
2
Faculty of Computer Science, Dalhousie University, Halifax, NS B3H 4R2, Canada
3
Department of Physics and Atmospheric Sciences, Dalhousie University, Halifax, NS B3H 4R2, Canada
*
Authors to whom correspondence should be addressed.
Current address: 5909 Veterans Memorial Lane (8th Floor), Abbie J. Lane Memorial Building, QE II. Health Sciences Centre, Halifax, NS B3H 2E2, Canada.
Entropy 2020, 22(4), 417; https://doi.org/10.3390/e22040417
Received: 26 March 2020 / Revised: 3 April 2020 / Accepted: 4 April 2020 / Published: 7 April 2020
(This article belongs to the Special Issue Entropy in Data Analysis)
A discrete system’s heterogeneity is measured by the Rényi heterogeneity family of indices (also known as Hill numbers or Hannah–Kay indices), whose units are the numbers equivalent. Unfortunately, numbers equivalent heterogeneity measures for non-categorical data require a priori (A) categorical partitioning and (B) pairwise distance measurement on the observable data space, thereby precluding application to problems with ill-defined categories or where semantically relevant features must be learned as abstractions from some data. We thus introduce representational Rényi heterogeneity (RRH), which transforms an observable domain onto a latent space upon which the Rényi heterogeneity is both tractable and semantically relevant. This method requires neither a priori binning nor definition of a distance function on the observable space. We show that RRH can generalize existing biodiversity and economic equality indices. Compared with existing indices on a beta-mixture distribution, we show that RRH responds more appropriately to changes in mixture component separation and weighting. Finally, we demonstrate the measurement of RRH in a set of natural images, with respect to abstract representations learned by a deep neural network. The RRH approach will further enable heterogeneity measurement in disciplines whose data do not easily conform to the assumptions of existing indices. View Full-Text
Keywords: heterogeneity; diversity; Rényi heterogeneity; representation learning; variational autoencoder; functional diversity indices; Hill numbers; Leinster–Cobbold Index; Rao’s quadratic entropy heterogeneity; diversity; Rényi heterogeneity; representation learning; variational autoencoder; functional diversity indices; Hill numbers; Leinster–Cobbold Index; Rao’s quadratic entropy
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MDPI and ACS Style

Nunes, A.; Alda, M.; Bardouille, T.; Trappenberg, T. Representational Rényi Heterogeneity. Entropy 2020, 22, 417. https://doi.org/10.3390/e22040417

AMA Style

Nunes A, Alda M, Bardouille T, Trappenberg T. Representational Rényi Heterogeneity. Entropy. 2020; 22(4):417. https://doi.org/10.3390/e22040417

Chicago/Turabian Style

Nunes, Abraham; Alda, Martin; Bardouille, Timothy; Trappenberg, Thomas. 2020. "Representational Rényi Heterogeneity" Entropy 22, no. 4: 417. https://doi.org/10.3390/e22040417

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