Mathematics Colloquia and Seminars
The marginal and joint distributions of the total tree lengths across loci in populations with variable sizePDE and Applied Math Seminar
|Speaker:||Alexey Miroshnikov, UCLA|
|Start time:||Fri, Nov 18 2016, 4:10PM|
In recent years, a number of methods have been developed to infer demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. A particular focus has been on recent exponential growth in humans. This recent growth has had a strong impact on the distribution of rare genetic variants, which are of particular importance when studying disease related genetic variation.
Due to the Markovian structure of Coalescent Hidden Markov Models, an essential building block is the joint distribution of local genealogies, or statistics of these genealogies, at two linked loci in populations of variable size. This joint distribution of local genealogies, has received little attention in the literature, especially under variable population size. In this talk, we present a novel method to compute the joint distribution of the total length of the genealogical trees at two linked loci for samples of arbitrary size. We show that the joint distribution is can be obtained by solving a certain system of hyperbolic PDEs (in several space dimensions) and present a numerical algorithm that can be used to efficiently and accurately solve the system and compute this distribution. We demonstrate its utility to study properties of the joint distribution. Our flexible method can be straightforwardly extended to other statistics and structured populations. This is a joint work with Matthias Steinrücken (UMass).
Please join us at Bistro 33 for free hors d’oeuvres. All are welcome.