经验Rényi熵和散度的极限定理及其在分子多样性分析中的应用
Limit Theorems for Empirical Rényi Entropy and Divergence with Applications to Molecular Diversity Analysis
摘要：
生物多样性研究的定量方法传统上植根于有限频率表分析的经典理论。然而，在现代实验工具的帮助下，如高通量测序，我们现在开始解锁植物和动物基因组数据的杰出多样性，反映了我们星球的漫长进化历史。这种分子数据通常违背经典的频率/列联表假设，并且似乎需要具有非常大数量的类别和高度不平衡的细胞计数的稀疏表，例如，遵循重尾分布（例如幂律）。受分子多样性研究的启发，我们在这里提出了一个基于频率的框架，生物多样性分析的渐近制度，类别的数量增长与样本量（无限列联表）。我们的方法是植根于信息论和基于高斯极限结果的有效物种数（希尔数）和经验Renyi熵和分歧。我们认为，当应用到分子生物多样性分析，我们的方法可以适当地占复杂的数据频率模式，一方面和实际的样本量的限制。我们用两个具体的RNA测序例子来说明这一原理：T细胞受体群体的比较研究和一些预选的分子肝细胞癌（HCC）标志物的验证。
英文摘要：
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.
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