Limit theorems for empirical Rényi entropy and divergence with applications to molecular diversity analysis
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Maciej Pietrzak [1] ; Grzegorz A. Rempała [1] ; Michał Seweryn [2] ; Jacek Wesołowski [3]
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[1] Ohio State University
Ohio State University
City of Columbus, Estados Unidos
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[2] University of Łódź
University of Łódź
Łódź, Polonia
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[3] Warsaw University of Technology
Warsaw University of Technology
Warszawa, Polonia
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- Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 25, Nº. 4, 2016, págs. 654-673
- Idioma: inglés
- DOI: 10.1007/s11749-016-0489-z
- Texto completo no disponible (Saber más ...)
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Resumen
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.
