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Steven N. Evans

Publications and source records attributed to Steven N. Evans.

64 records · Page 4Linked to original sources
arXiv2006

Non-equilibrium theory of the allele frequency spectrum

A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection and varying population size, the boundary condition is $\lim_{x \downarrow 0} x f(x,t)=θρ(t)$, where $f(\cdot,t)$ is the frequency spectrum of derived alleles at independent loci at time $t$ and $ρ(t)$ is the relative population size at time $t$. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but the forward equation allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, we derive a set of ordinary differential equations for the moments of $f(\cdot,t)$ and express the expected spectrum of a finite sample in terms of those moments. We illustrate the use of the forward equation by considering neutral and selected alleles in a highly simplified model of human history. For example, we show that approximately 30% of the expected heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last $150,000$ years.

q-bio.PE↗
arXiv2006

Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials

There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such representations involve a specific labeling of the vertices or at least the leaves, and so it is natural to attempt to identify trees by some feature of the associated matrices that is invariant under relabeling. An obvious candidate is the spectrum of eigenvalues (or, equivalently, the characteristic polynomial). We show for any of these choices of matrix that the fraction of binary trees with a unique spectrum goes to zero as the number of leaves goes to infinity. We investigate the rate of convergence of the above fraction to zero using numerical methods. For the adjacency and Laplacian matrices, we show that that the {\em a priori} more informative immanantal polynomials have no greater power to distinguish between trees.

q-bio.PE↗
arXiv2005

Quasistationary distributions for one-dimensional diffusions with killing

We extend some results on the convergence of one-dimensional diffusions killed at the boundary, conditioned on extended survival, to the case of general killing on the interior. We show, under fairly general conditions, that a diffusion conditioned on long survival either runs off to infinity almost surely, or almost surely converges to a quasistationary distribution given by the lowest eigenfunction of the generator. In the absence of internal killing, only a sufficiently strong inward drift can keep the process close to the origin, to allow convergence in distribution. An alternative, that arises when general killing is allowed, is that the conditioned process is held near the origin by a high rate of killing near infinity. We also extend, to the case of general killing, the standard result on convergence to a quasistationary distribution of a diffusion on a compact interval.

math.PR↗
arXiv2004

Unidentifiable divergence times in rates-across-sites models

The rates-across-sites assumption in phylogenetic inference posits that the rate matrix governing the Markovian evolution of a character on an edge of the putative phylogenetic tree is the product of a character-specific scale factor and a rate matrix that is particular to that edge. Thus, evolution follows basically the same process for all characters, except that it occurs faster for some characters than others. To allow estimation of tree topologies and edge lengths for such models, it is commonly assumed that the scale factors are not arbitrary unknown constants, but rather unobserved, independent, identically distributed draws from a member of some parametric family of distributions. A popular choice is the gamma family. We consider an example of a clock-like tree with three taxa, one unknown edge length, and a parametric family of scale factor distributions that contain the gamma family. This model has the property that, for a generic choice of unknown edge length and scale factor distribution, there is another edge length and scale factor distribution which generates data with exactly the same distribution, so that even with infinitely many data it will be typically impossible to make correct inferences about the unknown edge length.

q-bio.PE↗
arXiv2004

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.

math.PR↗
arXiv2004

A generalized model of mutation-selection balance with applications to aging

A probability model is presented for the dynamics of mutation-selection balance in a haploid infinite-population infinite-sites setting sufficiently general to cover mutation-driven changes in full age-specific demographic schedules. The model accommodates epistatic as well as additive selective costs. Closed form characterizations are obtained for solutions in finite time, along with proofs of convergence to stationary distributions and a proof of the uniqueness of solutions in a restricted case. Examples are given of applications to the biodemography of aging, including instabilities in current formulations of mutation accumulation.

q-bio.PE↗
arXiv1998

Continuum-sites stepping-stone models, coalescing exhcangeable partitions, and random trees

Analogues of stepping--stone models are considered where the site--space is continuous, the migration process is a general Markov process, and the type--space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum--sites stepping--stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle--1/2 Cantor set (and hence also to the Brownian zero set).

math.PR↗
arXiv1998

Local fields, Gaussian measures, and Brownian motions

These are lecture notes from a course given at the CRM in Montreal in 1992. They survey the author's attempts to find and understand canonical probabilistic entities in a local field (e.g. p-adic) setting. We propose answers to the related questions: ``What are the analogues for Gaussian measures?'' and ``What are the analogues for Brownian motion and its multiparameter relatives?'' There is a suitable concept of orthogonality in the local field setting, and one can mimic a classical abstract definition of Euclidean Gaussian measures. The resulting theory is, in some sense, an L^\infty rather than an L^2 one; and so although there are many similarities to the Euclidean case there are also some striking differences. Local field Brownian motion is the most natural local field Gaussian process that takes values in a local field vector space and is indexed by another local field vector space. Characterising the polar sets for such processes lead us to study the notion of additive functionals or homogeneous random measures and a related Riesz-like potential theory. We study local times, show they can be recovered by counterparts of classical intrinsic contstructions using dilation and Hausdorff measures, and establish an analogue of Trotter's theorem. We finish with a discussion of random series with local field Gaussian coefficients. Although we show that there are broad classes of random series which are stationary, we also find that there is no obvious counterpart of the representation of a general stationary Gaussian process on the circle as a random Fourier series.

math.PR↗
arXiv1998

Transition operators of diffusions reduce zero-crossing

If $u(t,x)$ is a solution of a one--dimensional, parabolic, second--order, linear partial differential equation (PDE), then it is known that, under suitable conditions, the number of zero--crossings of the function $u(t,\cdot)$ decreases (that is, does not increase) as time $t$ increases. Such theorems have applications to the study of blow--up of solutions of semilinear PDE, time dependent Sturm Liouville theory, curve shrinking problems and control theory. We generalise the PDE results by showing that the transition operator of a (possibly time--inhomogenous) one--dimensional diffusion reduces the number of zero--crossings of a function or even, suitably interpreted, a signed measure. Our proof is completely probabilistic and depends in a transparent manner on little more than the sample--path continuity of diffusion processes.

math.AP↗