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

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• 作者：Maciej Pietrzak ; Grzegorz A. Rempała ; Michał Seweryn ; Jacek Wesołowski
• 关键词：Hill number ; Central limit theorem ; Next ; generation sequencing ; Triangular arrays ; T ; cell receptors
• 刊名：TEST
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：25
• 期：4
• 页码：654-673
• 全文大小：797 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Statistical Theory and Methods
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1863-8260
• 卷排序：25

文摘

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.