Efficient Computation of the Joint Sample Frequency Spectra for Multiple Populations

A wide range of studies in population genetics have employed the sample frequency spectrum (SFS), a summary statistic which describes the distribution of mutant alleles at a polymorphic site in a sample of DNA sequences and provides a highly efficient dimensional reduction of large-scale population genomic variation data. Recently, there has been much interest in analyzing the joint SFS data from multiple populations to infer parameters of complex demographic histories, including variable population sizes, population split times, migration rates, admixture proportions, and so on. SFS-based inference methods require accurate computation of the expected SFS under a given demographic model. Although much methodological progress has been made, existing methods suffer from numerical instability and high computational complexity when multiple populations are involved and the sample size is large. In this article, we present new analytic formulas and algorithms that enable accurate, efficient computation of the expected joint SFS for thousands of individuals sampled from hundreds of populations related by a complex demographic model with arbitrary population size histories (including piecewise-exponential growth). Our results are implemented in a new software package called momi (MOran Models for Inference). Through an empirical study, we demonstrate our improvements to numerical stability and computational complexity.     

Ancestral population genomics using coalescence hidden Markov models and heuristic optimisation algorithms

With full genome data from several closely related species now readily available, we have the ultimate data for demographic inference. Exploiting these full genomes, however, requires models that can explicitly model recombination along alignments of full chromosomal length. Over the last decade a class of models, based on the sequential Markov coalescence model combined with hidden Markov models, has been developed and used to make inference in simple demographic scenarios. To move forward to more complex demographic modelling we need better and more automated ways of specifying these models and efficient optimisation algorithms for inferring the parameters in complex and often high-dimensional models.In this paper we present a framework for building such coalescence hidden Markov models for pairwise alignments and present results for using heuristic optimisation algorithms for parameter estimation. We show that we can build more complex demographic models than our previous frameworks and that we obtain more accurate parameter estimates using heuristic optimisation algorithms than when using our previous gradient based approaches.Our new framework provides a flexible way of constructing coalescence hidden Markov models almost automatically. While estimating parameters in more complex models is still challenging we show that using heuristic optimisation algorithms we still get a fairly good accuracy.     

Random covering of an interval and a variation of Kingman's coalescent

We consider the covering of [0, 1] by a large number of small random intervals. We show that a simple variation of Kingman's coalescent describes the emergence of macroscopic connected components. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

The Convergence to Equilibrium of Neutral Genetic Models

This article is concerned with the long-time behavior of neutral genetic population models with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.     

The Distribution of F st for the Island Model in the Large Population,Weak Mutation Limit

Populations are often divided into subpopulations. Biologists use the statistic F _st to perform hypothesis tests for the existence of population subdivision and to estimate migration rates between different subpopulations. The distribution of F _st is not known. In this article, we use coalescent theory methods to find the limiting distribution of F _st in the large population, weak mutation limit under the island model of migration. Our analysis uses the scattering-collection decomposition of the island model coalescent introduced by Wakeley.     

Gamma-Dirichlet algebra and applications

The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief survey of several existing results concerning this structure. New results are then obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. We finish the paper with the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration by exploring the Gamma-Dirichlet algebra embedded in these processes. This last result is motivated by an open R. C. Gritfiths. problem proposed by S. N. Ethier and     

The coalescent point process of multi-type branching trees

We define a multi-type coalescent point process of a general branching process with countably many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population providing types along ancestral lineages of all individuals in the standing population. We show that the coalescent process is a functional of a certain Markov chain defined by the planar embedding of the multi-type branching process.