Steven N. Evans

Inverse problems as statistics

What mathematicians, scientists, engineers, and statisticians mean by “inverse problem” differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results.

Local properties of Lévy processes on a totally disconnected group

This paper is the first study of the sample path behavior of processes with stationary independent increments taking values in a nondiscrete, locally compact, metrizable, totally disconnected Abelian group. After some preparatory results of independent interest we give a general integral criterion for a deterministic function to be a local modulus of right-continuity for the paths of the process and then study the sets of “fast” and “slow” points where the local growth of the process is anomalously large or small. We establish the lim sup behavior for the sequence of first exit times from a collection of concentric balls for an arbitrary process and show that no deterministic function can act as an exact lower envelope. Under appropriate conditions similar results hold for the related sojourn time sequence. We consider various candidates for measuring the variation of the paths of the process, show that they exist and coincide in our situation, and then determine the common value for a general process. Using earlier results we calculate the Hausdorff and packing dimensions of the image of an interval, exhibit the correct Hausdorff measure for this set, and establish a dichotomy that classifies measure functions into those that lead to a zero packing measure for the image and those that lead to an infinite packing measure. Lastly, we prove some uniform dimension results, which bound the dimension of the image of a set in terms of the dimension of the set itself. These results hold almost surely for all sets simultaneously.

Probability and Real Trees

Around the Continuum Random Tree.- R-Trees and 0-Hyperbolic Spaces.- Hausdorff and Gromov-Hausdorff Distance.- Root Growth with Re-Grafting.- The Wild Chain and other Bipartite Chains.- Diffusions on a R-Tree without Leaves: Snakes and Spiders.- R-Trees from Coalescing Particle Systems.- Subtree Prune and Re-Graft.

Construction of markovian coalescents

Abstract Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m , and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x + y at rate к ( x,y ), for some non-negative, symmetric collision rate kernel к ( x,y ). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on к and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel к ( x,y ) = x + y . This process, which arises from the evolution of tree components in a random graph process, has asymptotic properties related to the stable subordinator of index 1/2.

Estimating Allele Age and Selection Coefficient from Time-Serial Data

Recent advances in sequencing technologies have made available an ever-increasing amount of ancient genomic data. In particular, it is now possible to target specific single nucleotide polymorphisms in several samples at different time points. Such time-series data are also available in the context of experimental or viral evolution. Time-series data should allow for a more precise inference of population genetic parameters and to test hypotheses about the recent action of natural selection. In this manuscript, we develop a likelihood method to jointly estimate the selection coefficient and the age of an allele from time-serial data. Our method can be used for allele frequencies sampled from a single diallelic locus. The transition probabilities are calculated by approximating the standard diffusion equation of the Wright–Fisher model with a one-step process. We show that our method produces unbiased estimates. The accuracy of the method is tested via simulations. Finally, the utility of the method is illustrated with an application to several loci encoding coat color in horses, a pattern that has previously been linked with domestication. Importantly, given our ability to estimate the age of the allele, it is possible to gain traction on the important problem of distinguishing selection on new mutations from selection on standing variation. In this coat color example for instance, we estimate the age of this allele, which is found to predate domestication.

SUBTREE PRUNE AND REGRAFT: A REVERSIBLE REAL TREE-VALUED MARKOV PROCESS

We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov–Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.

Collision local times and measure-valued processes

We consider two independent Dawson-Watanabe super-Brownian motions, Y and Y. These processes are diffusions taking values in the space of finite measures on R . We show that if d < 5 then with positive probability there exist times / such that the closed supports of Yj and Y intersect; whereas if d > 5 then no such intersections occur. For the case d < 5, we construct a continuous, non-decreasing measure-valued process L(Y, Y), the collision local time, such that the measure defined by [0, t] x B i—• U(Y, Y)(B), B £ #(R ), is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y, Y). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures Yt that are uniform in t. These theorems use a nonstandard description of Y and are of independent interest.

Edge Principal Components and Squash Clustering: Using the Special Structure of Phylogenetic Placement Data for Sample Comparison

Principal components analysis (PCA) and hierarchical clustering are two of the most heavily used techniques for analyzing the differences between nucleic acid sequence samples taken from a given environment. They have led to many insights regarding the structure of microbial communities. We have developed two new complementary methods that leverage how this microbial community data sits on a phylogenetic tree. Edge principal components analysis enables the detection of important differences between samples that contain closely related taxa. Each principal component axis is a collection of signed weights on the edges of the phylogenetic tree, and these weights are easily visualized by a suitable thickening and coloring of the edges. Squash clustering outputs a (rooted) clustering tree in which each internal node corresponds to an appropriate “average” of the original samples at the leaves below the node. Moreover, the length of an edge is a suitably defined distance between the averaged samples associated with the two incident nodes, rather than the less interpretable average of distances produced by UPGMA, the most widely used hierarchical clustering method in this context. We present these methods and illustrate their use with data from the human microbiome.

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 oo. 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.

Measure-valued Markov branching processes conditioned on non-extinction

We consider a particular class of measure-valued Markov branching processes that are constructed as “superprocesses” over some underlying Markov process. Such a processX dies out almost surely, so we introduce various conditioning schemes which keepX alive at large times. Under suitable hypotheses, which include the convergence of the semigroup for the underlying process to some limiting probability measureν, we show that the conditional distribution oft−1Xt converges to that ofZν ast → ∞, whereZ is some strictly positive, real random variable.