# Browsing by Author "Mattingly, Jonathan"

###### Results Per Page

###### Sort Options

Item Open Access A dimensionless number for understanding the evolutionary dynamics of antigenically variable RNA viruses.(Proc Biol Sci, 2011-12-22) Koelle, Katia; Ratmann, Oliver; Rasmussen, David A; Pasour, Virginia; Mattingly, JonathanAntigenically variable RNA viruses are significant contributors to the burden of infectious disease worldwide. One reason for their ubiquity is their ability to escape herd immunity through rapid antigenic evolution and thereby to reinfect previously infected hosts. However, the ways in which these viruses evolve antigenically are highly diverse. Some have only limited diversity in the long-run, with every emergence of a new antigenic variant coupled with a replacement of the older variant. Other viruses rapidly accumulate antigenic diversity over time. Others still exhibit dynamics that can be considered evolutionary intermediates between these two extremes. Here, we present a theoretical framework that aims to understand these differences in evolutionary patterns by considering a virus's epidemiological dynamics in a given host population. Our framework, based on a dimensionless number, probabilistically anticipates patterns of viral antigenic diversification and thereby quantifies a virus's evolutionary potential. It is therefore similar in spirit to the basic reproduction number, the well-known dimensionless number which quantifies a pathogen's reproductive potential. We further outline how our theoretical framework can be applied to empirical viral systems, using influenza A/H3N2 as a case study. We end with predictions of our framework and work that remains to be done to further integrate viral evolutionary dynamics with disease ecology.Item Open Access A Merge-Split Proposal for Reversible Monte Carlo Markov Chain Sampling of Redistricting PlansCarter, Daniel; Hunter, Zach; Herschlag, Gregory; Mattingly, JonathanWe describe a Markov chain on redistricting plans that makes relatively global moves. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number elements which each correspond to a different district. The partitions satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. When these constraints and criteria are chosen to align well with classical legal redistricting criteria, the algorithm can be used to generate a collection of non-partisan, neutral plans. This collection of plans can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection plans, the given plan may have been gerrymandered and is labeled as an outlier.Item Open Access A stochastic-Lagrangian particle system for the Navier-Stokes equations(Nonlinearity, 2008-11-01) Iyer, Gautam; Mattingly, JonathanThis paper is based on a formulation of the Navier-Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not just the motion of N particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly. © 2008 IOP Publishing Ltd and London Mathematical Society.Item Open Access Absolute Continuity of Singular SPDEs and Bayesian Inference on Dynamical Systems(2023) Su, LangxuanWe explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

Item Open Access Analyzing the Effects of Partisan Correlation on Election Outcomes Using Order Statistics(2019-04-24) Wiebe, ClaireThe legislative representation of political parties in the United States is dependent not only on way that legislative district boundaries are drawn, but also on the way in which people are distributed across a state. That is, there exists a level of partisan correlation within the spacial distribution of an electorate that affects legislative outcomes. This work aims to study the effect of this partisan clustering on election outcomes and related metrics using analytic models and order statistics. Two models of North Carolina, one with a uniformly distributed electorate and one with a symmetrically clustered electorate, are considered both independently and in comparison. These models are used to study expected election outcomes, the proportionality of legislative representation for given state-wide vote fraction, and the sensitivity of vote share to seat share across different correlation length scales. The findings provide interesting insight into the relationship between district proportionality and legislative proportionality, the extent to which the minority party is expected to be underrepresented in seat share for given state-wide vote share and correlation length, and the extent to which the responsiveness of seat share is dependent on state wide vote share and correlation length.Item Open Access Expert report in Common Cause v Rucho(2017-03-06) Mattingly, JonathanItem Open Access Expert Report on the North Carolina State Legislature(2019-04) Mattingly, JonathanItem Open Access Expert Report on the North Carolina State Legislature and Congressional Redistricting (Corrected Version)(2021-12-23) Mattingly, JonathanItem Open Access Mathematically Quantifying Gerrymandering and the Non-responsiveness of the 2021 Georgia Congressional Districting Plan(2022-03-12) Zhao, Zhanzhan; Hettle, Cyrus; Gupta, Swati; Mattingly, Jonathan; Randall, Dana; Herschlag, GregoryItem Open Access MCMC Sampling Geospatial Partitions for Linear Models(2021) Wyse, Evan TGeospatial statistical approaches must frequently confront the problem of correctlypartitioning a group of geographical sub-units, such as counties, states, or precincts,into larger blocks which share information. Since the space of potential partitions isquite large, sophisticated approaches are required, particularly when this partitioninginteracts with other parts of a larger model, as is frequent with Bayesian inference.Authors such as Balocchi et al. (2021) provide stochastic search algorithms whichprovide certain theoretical guarantees about this partition in the context of Bayesianmodel averaging. We borrow tools from Herschlag et al. (2020) to examine a potentialapproach to sampling these clusters efficiently using a Markov Chain Monte Carlo(MCMC) approach.

Item Open Access On SDEs with Partial Damping Inspired by the Navier-Stokes Equations(2019) Williamson, BrendanThe solution to the Navier Stokes equations on the 2D torus with stochastic forcing that is white noise in time, coloured in space has a Fourier series representation whose coordinates satisfy a countable system of Stochastic Differential Equations. Inspired by the structure of these equations, we construct a finite system of stochastic differential equations with a similar structure and explore the conditions under which the system has an invariant distribution. \\

Our main tool to prove existence of invariant distributions are Lyapunov functions, or more generally Lyapunov pairs. In particular, we construct the Lyapunov pairs piecewise over different regions and then use mollifiers to unify these disparate characterisations. We also apply some results from Algebraic Geometry and Matrix Perturbation Theory to study and exploit the geometry of the problem in high dimensions. \\

The combination of these methods allowed us to prove that a large class of the equations we constructed have an invariant distributions. Furthermore we have explicit tail estimates for these invariant distributions.

Item Open Access Optical Precursor Behavior(2007-05-07T19:07:14Z) LeFew, William R.Controlling and understanding the propagation of optical pulses through dispersive media forms the basis for optical communication, medical imaging, and other modern technological advances. Integral to this control and understanding is the ability to describe the transients which occur immediately after the onset of a signal. This thesis examines the transients of such a system when a unit step function is applied. The electromagnetic field is described by an integral resulting from Maxwell’s Equations. It was previously believed that optical precursors, a specific transient effect, existed only for only a few optical cycles and contributed only small magnitudes to the field. The main results of this thesis show that the transients arising from this integral are entirely precursors and that they may exist on longer time scales and contribute larger magnitudes to the field. The experimental detection of precursors has previously been recognized only through success comparison to the transient field resulting from an application of the method of steepest descent to that field integral. For any parameter regime where steepest descents may be applied, this work gives iterative methods to determine saddle points which are both more accurate than the accepted results and to extend into regimes where the current theory has failed. Furthermore, asymptotic formulae have been derived for regions where previous attempts at steepest descent have failed. Theory is also presented which evaluates the applicability of steepest descents in the represention of precursor behavior for any set of parameters. Lastly, the existence of other theoretical models for precursor behavior who may operate beyond the reach of steepest descent is validated through successful comparisons of the transient prediction of those methods to the steepest descent based results of this work.Item Open Access Optimal Legislative County Clustering in North Carolina(2019-11-22) Carter, Daniel; Zach, Hunter; Herschlag, Gregory; Mattingly, JonathanNorth Carolina's constitution requires that state legislative districts should not split counties. However, counties must be split to comply with the "one person, one vote" mandate of the U.S. Supreme Court. Given that counties must be split, the North Carolina legislature and courts have provided guidelines that seek to reduce counties split across districts while also complying with the "one person, one vote" criteria. Under these guidelines, the counties are separated into clusters. The primary goal of this work is to develop, present, and publicly release an algorithm to optimally cluster counties according to the guidelines set by the court in 2015. We use this tool to investigate the optimality and uniqueness of the enacted clusters under the 2017 redistricting process. We verify that the enacted clusters are optimal, but find other optimal choices. We emphasize that the tool we provide lists all possible optimal county clusterings. We also explore the stability of clustering under changing statewide populations and project what the county clusters may look like in the next redistricting cycle beginning in 2020/2021.Item Open Access Probabilistic Fréchet means for time varying persistence diagrams(Electronic Journal of Statistics, 2015-01-01) Munch, Elizabeth; Bendich, Paul; Turner, Katharine; Mukherjee, Sayan; Mattingly, Jonathan; Harer, John© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (Dp)^{N}→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.Item Open Access Rebuttal Of Defendant’s Expert Reports For Common Cause V. Lewis(2019-06) Mattingly, JonathanItem Open Access Scaling Limit: Exact and Tractable Analysis of Online Learning Algorithms with Applications to Regularized Regression and PCAWang, Chuang; Mattingly, Jonathan; Lu, Yue MWe present a framework for analyzing the exact dynamics of a class of online learning algorithms in the high-dimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression and principal component analysis. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measures of the target feature vector and its estimates provided by the algorithms will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE can be efficiently obtained. These solutions lead to precise predictions of the performance of the algorithms, as many practical performance metrics are linear functionals of the joint empirical measures. In addition to characterizing the dynamic performance of online learning algorithms, our asymptotic analysis also provides useful insights. In particular, in the high-dimensional limit, and due to exchangeability, the original coupled dynamics associated with the algorithms will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight for nonconvex optimization problems may prove an interesting line of future research.Item Open Access Stratified MCMC Sampling of non-Reversible Dynamics(2020) Earle, Gabriel JosephThe study of stratified sampling is of interest in systems which canbe solved accurately on small scales, or which depend heavily on rare transitions of particles from one subspace to another. We present a new form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. The method has potential usefulness in that many systems of interest are non-reversible, and can also benefit from stratification at the same time. It may also be useful for sampling on complex manifolds, and hence manifold learning. Our method is a generalization of previous stratified or nested sampling schemes which extend QSD sampling schemes. It can also be viewed as a generalization of the exact milestoning method previously studied by D. Aristoff. The primary advantages of our new results over such previous studies are generalization to non-reversible processes, expressions for the convergence rate in terms of the process's behavior within each stratum and large scale behavior between strata, and less restrictive assumptions for convergence. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the speeds of two versions of the new algorithm, one with an extra eigenvalue problem step and one without, relate to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by J. Weare. Finally, we will propose a way to relate the accuracy of finite approximations of a process using our stratified scheme to its expected exit times from each stratum and its approximation of the true process's generator, by means of a Poisson equation argument.

Item Open Access The Signature of Gerrymandering in Rucho v. Common Cause(South Carolina Law Review, 2019) Chin, Andrew; Herschlag, Gregory; Mattingly, JonathanItem Open Access