Lin Wan1, Xin Kang2, Jie Ren3, Fengzhu Sun3. 1. NCMIS, LSC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. 2. School of Mathematical Sciences, Fudan University, Shanghai 200433, China. 3. Quantitative and Computational Biology Program, University of Southern California, Los Angeles, CA 90089, USA.
Abstract
BACKGROUND: Markov chains (MC) have been widely used to model molecular sequences. The estimations of MC transition matrix and confidence intervals of the transition probabilities from long sequence data have been intensively studied in the past decades. In next generation sequencing (NGS), a large amount of short reads are generated. These short reads can overlap and some regions of the genome may not be sequenced resulting in a new type of data. Based on NGS data, the transition probabilities of MC can be estimated by moment estimators. However, the classical asymptotic distribution theory for MC transition probability estimators based on long sequences is no longer valid. METHODS: In this study, we present the asymptotic distributions of several statistics related to MC based on NGS data. We show that, after scaling by the effective coverage d defined in a previous study by the authors, these statistics based on NGS data approximate to the same distributions as the corresponding statistics for long sequences. RESULTS: We apply the asymptotic properties of these statistics for finding the theoretical confidence regions for MC transition probabilities based on NGS short reads data. We validate our theoretical confidence intervals using both simulated data and real data sets, and compare the results with those by the parametric bootstrap method. CONCLUSIONS: We find that the asymptotic distributions of these statistics and the theoretical confidence intervals of transition probabilities based on NGS data given in this study are highly accurate, providing a powerful tool for NGS data analysis.
BACKGROUND: Markov chains (MC) have been widely used to model molecular sequences. The estimations of MC transition matrix and confidence intervals of the transition probabilities from long sequence data have been intensively studied in the past decades. In next generation sequencing (NGS), a large amount of short reads are generated. These short reads can overlap and some regions of the genome may not be sequenced resulting in a new type of data. Based on NGS data, the transition probabilities of MC can be estimated by moment estimators. However, the classical asymptotic distribution theory for MC transition probability estimators based on long sequences is no longer valid. METHODS: In this study, we present the asymptotic distributions of several statistics related to MC based on NGS data. We show that, after scaling by the effective coverage d defined in a previous study by the authors, these statistics based on NGS data approximate to the same distributions as the corresponding statistics for long sequences. RESULTS: We apply the asymptotic properties of these statistics for finding the theoretical confidence regions for MC transition probabilities based on NGS short reads data. We validate our theoretical confidence intervals using both simulated data and real data sets, and compare the results with those by the parametric bootstrap method. CONCLUSIONS: We find that the asymptotic distributions of these statistics and the theoretical confidence intervals of transition probabilities based on NGS data given in this study are highly accurate, providing a powerful tool for NGS data analysis.
Entities:
Keywords:
Markov chains; confidence intervals; next generation sequencing; transition probabilities