638 research outputs found
The Characteristic Polynomial of a Random Permutation Matrix at Different Points
We consider the logarithm of the characteristic polynomial of random
permutation matrices, evaluated on a finite set of different points. The
permutations are chosen with respect to the Ewens distribution on the symmetric
group. We show that the behavior at different points is independent in the
limit and are asymptotically normal. Our methods enables us to study more
general matrices, closely related to permutation matrices, and multiplicative
class functions.Comment: 30 pages, 2 figures. Differences to Version 1: We have improved the
presentation and add some references Stochastic Processes and their
Applications, 201
Singular solutions of the diffusion equation of population genetics
The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0; 1) that excludes the frequencies 0 and 1. Previously, the probabilities if the absorbing frequencies 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0; 1) were found from the forward diffusion equation. Our uni fied approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability
Temporal and dimensional effects in evolutionary graph theory
The spread in time of a mutation through a population is studied analytically
and computationally in fully-connected networks and on spatial lattices. The
time, t_*, for a favourable mutation to dominate scales with population size N
as N^{(D+1)/D} in D-dimensional hypercubic lattices and as N ln N in
fully-connected graphs. It is shown that the surface of the interface between
mutants and non-mutants is crucial in predicting the dynamics of the system.
Network topology has a significant effect on the equilibrium fitness of a
simple population model incorporating multiple mutations and sexual
reproduction. Includes supplementary information.Comment: 6 pages, 4 figures Replaced after final round of peer revie
Central limit theorem for multiplicative class functions on the symmetric group
Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for
the characteristic polynomial of a permutation matrix with respect to the
uniform measure on the symmetric group. We generalize this result in several
ways. We prove here a central limit theorem for multiplicative class functions
on symmetric group with respect to the Ewens measure and compute the covariance
of the real and the imaginary part in the limit. We also estimate the rate of
convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but
there are several improvments in the presentation, including a more intuitve
name for the considered function
Shift in critical temperature for random spatial permutations with cycle weights
We examine a phase transition in a model of random spatial permutations which
originates in a study of the interacting Bose gas. Permutations are weighted
according to point positions; the low-temperature onset of the appearance of
arbitrarily long cycles is connected to the phase transition of Bose-Einstein
condensates. In our simplified model, point positions are held fixed on the
fully occupied cubic lattice and interactions are expressed as Ewens-type
weights on cycle lengths of permutations. The critical temperature of the
transition to long cycles depends on an interaction-strength parameter
. For weak interactions, the shift in critical temperature is expected
to be linear in with constant of linearity . Using Markov chain
Monte Carlo methods and finite-size scaling, we find .
This finding matches a similar analytical result of Ueltschi and Betz. We also
examine the mean longest cycle length as a fraction of the number of sites in
long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial
permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures
which appeared at the end of the PDF
Measuring the Sensitivity of Single-locus “Neutrality Tests” Using a Direct Perturbation Approach
A large number of statistical tests have been proposed to detect natural selection based on a sample of variation at a single genetic locus. These tests measure the deviation of the allelic frequency distribution observed within populations from the distribution expected under a set of assumptions that includes both neutral evolution and equilibrium population demography. The present study considers a new way to assess the statistical properties of these tests of selection, by their behavior in response to direct perturbations of the steady-state allelic frequency distribution, unconstrained by any particular nonequilibrium demographic scenario. Results from Monte Carlo computer simulations indicate that most tests of selection are more sensitive to perturbations of the allele frequency distribution that increase the variance in allele frequencies than to perturbations that decrease the variance. Simulations also demonstrate that it requires, on average, 4N generations (N is the diploid effective population size) for tests of selection to relax to their theoretical, steady-state distributions following different perturbations of the allele frequency distribution to its extremes. This relatively long relaxation time highlights the fact that these tests are not robust to violations of the other assumptions of the null model besides neutrality. Lastly, genetic variation arising under an example of a regularly cycling demographic scenario is simulated. Tests of selection performed on this last set of simulated data confirm the confounding nature of these tests for the inference of natural selection, under a demographic scenario that likely holds for many species. The utility of using empirical, genomic distributions of test statistics, instead of the theoretical steady-state distribution, is discussed as an alternative for improving the statistical inference of natural selection
Pupil mobility, attainment and progress in secondary school
This paper is the second of two articles arising from a study of the association between pupil mobility and attainment in national tests and examinations in an inner London borough. The first article (Strand & Demie, 2006) examined the association of pupil mobility with attainment and progress during primary school. It concluded that pupil mobility had little impact on performance in national tests at age 11, once pupils’ prior attainment at age 7 and other pupil background factors such as age, sex, special educational needs, stage of fluency in English and socio-economic disadvantage were taken into account. The present article reports the results for secondary schools (age 11-16). The results indicate that pupil mobility continues to have a significant negative association with performance in public examinations at age 16, even after including statistical controls for prior attainment at age 11 and other pupil background factors. Possible reasons for the contrasting results across school phases are explored. The implications for policy and further research are discussed
Structurama: Bayesian Inference of Population Structure
Structurama is a program for inferring population structure. Specifically, the program calculates the posterior probability of assigning individuals to different populations. The program takes as input a file containing the allelic information at some number of loci sampled from a collection of individuals. After reading a data file into computer memory, Structurama uses a Gibbs algorithm to sample assignments of individuals to populations. The program implements four different models: The number of populations can be considered fixed or a random variable with a Dirichlet process prior; moreover, the genotypes of the individuals in the analysis can be considered to come from a single population (no admixture) or as coming from several different populations (admixture). The output is a file of partitions of individuals to populations that were sampled by the Markov chain Monte Carlo algorithm. The partitions are sampled in proportion to their posterior probabilities. The program implements a number of ways to summarize the sampled partitions, including calculation of the ‘mean’ partition—a partition of the individuals to populations that minimizes the squared distance to the sampled partitions
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
The statistical mechanics of a polygenic characterunder stabilizing selection, mutation and drift
By exploiting an analogy between population genetics and statistical
mechanics, we study the evolution of a polygenic trait under stabilizing
selection, mutation, and genetic drift. This requires us to track only four
macroscopic variables, instead of the distribution of all the allele
frequencies that influence the trait. These macroscopic variables are the
expectations of: the trait mean and its square, the genetic variance, and of a
measure of heterozygosity, and are derived from a generating function that is
in turn derived by maximizing an entropy measure. These four macroscopics are
enough to accurately describe the dynamics of the trait mean and of its genetic
variance (and in principle of any other quantity). Unlike previous approaches
that were based on an infinite series of moments or cumulants, which had to be
truncated arbitrarily, our calculations provide a well-defined approximation
procedure. We apply the framework to abrupt and gradual changes in the optimum,
as well as to changes in the strength of stabilizing selection. Our
approximations are surprisingly accurate, even for systems with as few as 5
loci. We find that when the effects of drift are included, the expected genetic
variance is hardly altered by directional selection, even though it fluctuates
in any particular instance. We also find hysteresis, showing that even after
averaging over the microscopic variables, the macroscopic trajectories retain a
memory of the underlying genetic states.Comment: 35 pages, 8 figure
- …