arXiv subjects

Steven N. Evans

Publications and source records attributed to Steven N. Evans.

At least 55 records · Page 3Linked to original sources

arXiv2010

Non-existence of Markovian time dynamics for graphical models of correlated default

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be re-rewritten as a standard Ising model on the graph by identifying the set of defaulting firms in the default model with the set of sites in the Ising model for which the spin is +1. We ask whether there is a suitable continuous time Markov chain taking values in the subsets of the vertex set such that the initial state of the chain is the empty set, each jump of the chain involves the inclusion of a single extra vertex, the distribution of the chain at some fixed time horizon time is the one given by the default model, and the distribution of the chain for other times is described by a probability distribution in the same family as the default model. We show for three simple but financially natural special cases that this is not possible outside of the trivial case where there is complete independence between the firms.

q-fin.CP↗

arXiv2010

Shape-based peak identification for ChIP-Seq

We present a new algorithm for the identification of bound regions from ChIP-seq experiments. Our method for identifying statistically significant peaks from read coverage is inspired by the notion of persistence in topological data analysis and provides a non-parametric approach that is robust to noise in experiments. Specifically, our method reduces the peak calling problem to the study of tree-based statistics derived from the data. We demonstrate the accuracy of our method on existing datasets, and we show that it can discover previously missed regions and can more clearly discriminate between multiple binding events. The software T-PIC (Tree shape Peak Identification for ChIP-Seq) is available at http://math.berkeley.edu/~vhower/tpic.html

q-bio.GN↗

arXiv2010

Coverage statistics for sequence census methods

Background: We study the statistical properties of fragment coverage in genome sequencing experiments. In an extension of the classic Lander-Waterman model, we consider the effect of the length distribution of fragments. We also introduce the notion of the shape of a coverage function, which can be used to detect abberations in coverage. The probability theory underlying these problems is essential for constructing models of current high-throughput sequencing experiments, where both sample preparation protocols and sequencing technology particulars can affect fragment length distributions. Results: We show that regardless of fragment length distribution and under the mild assumption that fragment start sites are Poisson distributed, the fragments produced in a sequencing experiment can be viewed as resulting from a two-dimensional spatial Poisson process. We then study the jump skeleton of the the coverage function, and show that the induced trees are Galton-Watson trees whose parameters can be computed. Conclusions: Our results extend standard analyses of shotgun sequencing that focus on coverage statistics at individual sites, and provide a null model for detecting deviations from random coverage in high-throughput sequence census based experiments. By focusing on fragments, we are also led to a new approach for visualizing sequencing data that should be of independent interest.

q-bio.GN↗

arXiv2010

Commuting birth-and-death processes

We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition.

math.PR↗

arXiv2010

Dynamics of the time to the most recent common ancestor in a large branching population

If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting "MRCA age" process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright--Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on $\mathbb{R}_{++}\times\mathbb{R}_{++}$ with intensity measures of the form $λ\otimesμ$ where $λ$ is Lebesgue measure, and $μ$ (the "family lifetime measure") is an arbitrary, absolutely continuous measure satisfying $μ((0,\infty))=\infty$ and $μ((x,\infty))<\infty$ for all $x>0$. Special cases of this construction describe the time evolution of the MRCA age in $(1+β)$-stable continuous state branching processes conditioned on nonextinction--a particular case of which, $β=1$, is Feller's continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist.

math.PR↗

arXiv2009

Vital rates from the action of mutation accumulation

New models for evolutionary processes of mutation accumulation allow hypotheses about the age-specificity of mutational effects to be translated into predictions of heterogeneous population hazard functions. We apply these models to questions in the biodemography of longevity, including proposed explanations of Gompertz hazards and mortality plateaus, and use them to explore the possibility of melding evolutionary and functional models of aging.

q-bio.PE↗

arXiv2009

A zero-one law for linear transformations of Levy noise

A Lévy noise on $\mathbb{R}^d$ assigns a random real "mass" $Π(B)$ to each Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure. The distribution of $Π(B)$ only depends on the Lebesgue measure of $B$, and if $B_1, ..., B_n$ is a finite collection of pairwise disjoint sets, then the random variables $Π(B_1), ..., Π(B_n)$ are independent with $Π(B_1 \cup >... \cup B_n) = Π(B_1) + ... + Π(B_n)$ almost surely. In particular, the distribution of $Π\circ g$ is the same as that of $Π$ when $g$ is a bijective transformation of $\mathbb{R}^d$ that preserves Lebesgue measure. It follows from the Hewitt--Savage zero--one law that any event which is almost surely invariant under the mappings $Π\mapsto Π\circ g$ for every Lebesgue measure preserving bijection $g$ of $\mathbb{R}^d$ must have probability 0 or 1. We investigate whether certain smaller groups of Lebesgue measure preserving bijections also possess this property. We show that if $d \ge 2$, the Lévy noise is not purely deterministic, and the group consists of linear transformations and is closed, then the invariant events all have probability 0 or 1 if and only if the group is not compact.

math.PR↗

arXiv2009

Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}ρ_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $ρ_n((k,k+1))$ are obtained by applying an irreducible unitary representation $ρ_n$ of the symmetric group on $\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $ρ_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on $\{1,2,...,n\}$ are indexed by partitions $λ_n$ of $n$. A consequence of the results we establish is that if $λ_{n,1}\geλ_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $ρ_n$, $μ_{n,1}\geμ_{n,2}\ge >...\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $μ_n$ is the transpose of the Young diagram of $λ_n$), $\lim_{n\to\infty}\frac{λ_{n,i}}{n}=p_i$ for each $i\ge1$, and $\lim_{n\to\infty}\frac{μ_{n,j}}{n}=q_j$ for each $j\ge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $θZ$ and variance $1-θ^2$, where $θ$ is the constant $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.

math.PR↗

arXiv2009

Mutation-selection balance with recombination: convergence to equilibrium for polynomial selection costs

We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, added mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite ``degree''. Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.

q-bio.PE↗

arXiv2008

The Age-Specific Force of Natural Selection and Walls of Death

W. D. Hamilton's celebrated formula for the age-specific force of natural selection furnishes predictions for senescent mortality due to mutation accumulation, at the price of reliance on a linear approximation. Applying to Hamilton's setting the full non-linear demographic model for mutation accumulation of Evans et al. (2007), we find surprising differences. Non-linear interactions cause the collapse of Hamilton-style predictions in the most commonly studied case, refine predictions in other cases, and allow Walls of Death at ages before the end of reproduction. Haldane's Principle for genetic load has an exact but unfamiliar generalization.

q-bio.PE↗

arXiv2008

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator $f\mapsto{1/2}f''$. We then consider the special case where the state space is the self-similar set $\{\pm q^k:k\in \mathbb{Z}\}\cup\{0\}$ for some $q>1$. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's ``lost'' notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, $q$-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local time -- both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve $q$-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

math.PR↗

arXiv2008

To what extent does genealogical ancestry imply genetic ancestry?

Recent statistical and computational analyses have shown that a genealogical most recent common ancestor (MRCA) may have lived in the recent past. However, coalescent-based approaches show that genetic most recent common ancestors for a given non-recombining locus are typically much more ancient. It is not immediately clear how these two perspectives interact. This paper investigates relationships between the number of descendant alleles of an ancestor allele and the number of genealogical descendants of the individual who possessed that allele for a simple diploid genetic model extending the genealogical model of Joseph Chang.

q-bio.PE↗

arXiv2008

Hyperdeterminantal point processes

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional "hypercubic" arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayley's first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.

math.PR↗

arXiv2007

Damage segregation at fissioning may increase growth rates: A superprocess model

A fissioning organism may purge unrepairable damage by bequeathing it preferentially to one of its daughters. Using the mathematical formalism of superprocesses, we propose a flexible class of analytically tractable models that allow quite general effects of damage on death rates and splitting rates and similarly general damage segregation mechanisms. We show that, in a suitable regime, the effects of randomness in damage segregation at fissioning are indistinguishable from those of randomness in the mechanism of damage accumulation during the organism's lifetime. Moreover, the optimal population growth is achieved for a particular finite, non-zero level of combined randomness from these two sources. In particular, when damage accumulates deterministically, optimal population growth is achieved by a moderately unequal division of damage between the daughters. Too little or too much division is sub-optimal. Connections are drawn both to recent experimental results on inheritance of damage in protozoans, to theories of the evolution of aging, and to models of resource division between siblings.

q-bio.PE↗

arXiv2007

Asymptotic evolution of acyclic random mappings

An acyclic mapping from an $n$ element set into itself is a mapping $ϕ$ such that if $ϕ^k(x) = x$ for some $k$ and $x$, then $ϕ(x) = x$. Equivalently, $ϕ^\ell = ϕ^{\ell+1} = ...$ for $\ell$ sufficiently large. We investigate the behavior as $n \to \infty$ of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Heuristic calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $n \to \infty$ to a real tree ($\R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. The limit process, which we construct using Dirichlet form methods, is a Hunt process with respect to a suitable Gromov-Hausdorff-like metric. This process is similar to one that appears in earlier work by Evans and Winter as the limit of chains involving the subtree prune and regraft tree (SPR) rearrangements from phylogenetics.

math.PR↗

arXiv2006

The expected number of zeros of a random system of $p$-adic polynomials

We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold Cartesian product of the $p$-adic integers. Considering models in which the maximum degree that each variable appears is $N$, this expected value is \[ p^{d \lfloor \log_p N \rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for the simplest such model.

math.PR↗

arXiv2006

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.

math.PR↗

arXiv2006

Expectation, Conditional Expectation and Martingales in Local Fields

We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.

math.PR↗