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Limit theorems for empirical Rényi entropy and divergence with applications to molecular diversity analysis

Author

Listed:
  • Maciej Pietrzak

    (The Ohio State University)

  • Grzegorz A. Rempała

    (The Ohio State University)

  • Michał Seweryn

    (University of Łódz)

  • Jacek Wesołowski

    (Politechnika Warszawska)

Abstract

Quantitative methods for studying biodiversity have been traditionally rooted in the classical theory of finite frequency tables analysis. However, with the help of modern experimental tools, like high-throughput sequencing, we now begin to unlock the outstanding diversity of genomic data in plants and animals reflective of the long evolutionary history of our planet. This molecular data often defies the classical frequency/contingency tables assumptions and seems to require sparse tables with very large number of categories and highly unbalanced cell counts, e.g., following heavy-tailed distributions (for instance, power laws). Motivated by the molecular diversity studies, we propose here a frequency-based framework for biodiversity analysis in the asymptotic regime where the number of categories grows with sample size (an infinite contingency table). Our approach is rooted in information theory and based on the Gaussian limit results for the effective number of species (the Hill numbers) and the empirical Renyi entropy and divergence. We argue that when applied to molecular biodiversity analysis, our methods can properly account for the complicated data frequency patterns on one hand and the practical sample size limitations on the other. We illustrate this principle with two specific RNA sequencing examples: a comparative study of T-cell receptor populations and a validation of some preselected molecular hepatocellular carcinoma (HCC) markers.

Suggested Citation

  • Maciej Pietrzak & Grzegorz A. Rempała & Michał Seweryn & Jacek Wesołowski, 2016. "Limit theorems for empirical Rényi entropy and divergence with applications to molecular diversity analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 654-673, December.
  • Handle: RePEc:spr:testjl:v:25:y:2016:i:4:d:10.1007_s11749-016-0489-z
    DOI: 10.1007/s11749-016-0489-z
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    References listed on IDEAS

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    1. de Andrade Mariza & Wang Xin, 2011. "Entropy Based Genetic Association Tests and Gene-Gene Interaction Tests," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-21, August.
    2. Anna Cebula & Michal Seweryn & Grzegorz A. Rempala & Simarjot Singh Pabla & Richard A. McIndoe & Timothy L. Denning & Lynn Bry & Piotr Kraj & Pawel Kisielow & Leszek Ignatowicz, 2013. "Thymus-derived regulatory T cells contribute to tolerance to commensal microbiota," Nature, Nature, vol. 497(7448), pages 258-262, May.
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    Cited by:

    1. W. V. Félix de Lima & A. D. C. Nascimento & G. J. A. Amaral, 2021. "Entropy-based pivotal statistics for multi-sample problems in planar shape," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 153-178, March.
    2. Agnieszka M Borys & Michał Seweryn & Tomasz Gołąbek & Łukasz Bełch & Agnieszka Klimkowska & Justyna Totoń-Żurańska & Julita Machlowska & Piotr Chłosta & Krzysztof Okoń & Paweł P Wołkow, 2019. "Patterns of gene expression characterize T1 and T3 clear cell renal cell carcinoma subtypes," PLOS ONE, Public Library of Science, vol. 14(5), pages 1-19, May.
    3. Rempała, Grzegorz A. & Wesołowski, Jacek, 2016. "Double asymptotics for the chi-square statistic," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 317-325.

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