Browsing by Subject "Stochastic processes"

Evolution by natural selection can occur at multiple biological scales. This is particularly the case for host-pathogen systems, where selection occurs both within each infected host as well as through transmission between hosts. Despite there being established mathematical models for understanding evolution at a single biological scale, fewer tractable models exist for multiscale evolutionary dynamics. Here I present mathematical approaches using tools from probability and stochastic processes as well as dynamical systems to handle multiscale evolutionary systems. The first problem I address concerns the antigenic evolution of influenza. Using a combination of ordinary differential equations and inhomogeneous Poisson processes, I study how immune selection pressures at the within-host level impact population-level evolutionary dynamics. The second problem involves the more general question of evolutionary dynamics when selection occurs antagonistically at two biological scales. In addition to host-pathogen systems, such situations arise naturally in the evolution of traits such as the production of a public good and the use of a common resource. I introduce a model for this general phenomenon that is intuitively visualized as a a stochastic ball-and-urn system and can be used to systematically obtain general properties of antagonistic multiscale evolution. Lastly, this ball-and-urn framework is in itself an interesting mathematical object which can studied as either a measure-valued process or an interacting particle system. In this mathematical context, I show that under different scalings, the measure-valued process can have either a propagation of chaos or Fleming-Viot limit.

We are interested in using probability theory to characterize the evolution of different phenomena and the distribution of rare events within these processes with particular emphasis on high dimensional data that appear in biomedical applications.Inferring and simulating stochastic processes are important for many aspects of health care, such as describing the trajectory of a patient's prognosis. However, a lack of mature computational frameworks that are both flexible and mathematically interpretable limits the broad application of such techniques within a health care setting. In this thesis, we make strides towards developing computational methods for achieving this goal. In Chapters 2 and 3, we begin by describing interpretable machine learning frameworks for inferring differential equations from data. In Chapters 4 and 5, we then discuss how to extend this to the case of nonlinear stochastic differential equations and their related partial differential equations. In Chapter 6, we shift our focus to developing methods that infer high dimensional tail events. We develop a particular neural network architecture that preserves the properties of extreme value distributions. In Chapter 7, we connect the work in Chapters 3 and 6 by describing arrivals of tail events as the excursions of a latent stochastic differential equation. In Chapter 8, we then describe how we can infer representations of high dimensional tail data. Throughout the chapters, we apply the proposed methods to several biomedical applications such as disease forecasting, neural spike inference, and modeling electroencephalography data. The methods in this thesis provide a direction for describing stochastic processes and tail distributions through flexible frameworks that scale to high dimensional covariates.

Cloud computing appears to be a paradigm shift in service oriented computing. Massively scalable Cloud architectures are spawned by new business and social applications as well as Internet driven economics. Besides being inherently large scale and highly distributed, Cloud systems are almost always virtualized and operate in automated shared environments. The deployed Cloud services are still in their infancy and a variety of research challenges need to be addressed to predict their long-term behavior. Performance and dependability of Cloud services are in general stochastic in nature and they are affected by a large number of factors, e.g., nature of workload and faultload, infrastructure characteristics and management policies. As a result, developing scalable and predictive analytics for Cloud becomes difficult and non-trivial. This dissertation presents the research framework needed to develop high fidelity stochastic models for large scale enterprise systems using Cloud computing as an example. Throughout the dissertation, we show how the developed models are used for: (i) performance and availability analysis, (ii) understanding of power-performance trade-offs, (ii) resiliency quantification, (iv) cost analysis and capacity planning, and (v) risk analysis of Cloud services. In general, the models and approaches presented in this thesis can be useful to a Cloud service provider for planning, forecasting, bottleneck detection, what-if analysis or overall optimization during design, development, testing and operational phases of a Cloud.

We present novel model frameworks and inference procedures for stochastic point processes on dynamic networks. The point process can be defined for a random phenomenon that spreads among the network nodes, and for the temporally evolving network itself. Methods development is motivated by the needs of health and social science data, where partial observations or latent structures are common and create challenges to likelihood-based inference. In this dissertation, we will discuss parameter estimation techniques that can handle these latent variables and make effective use of the complete data likelihood for efficient inference. We start with developing individualized continuous time Markov chain models for stochastic epidemics on a dynamic contact network. Data-augmentation algorithms are designed to address partial observations (such as missing infection and recovery times) in epidemic data while accommodating the network dynamics. We apply the frameworks to the study of non-pharmaceutical interventions in a college population. Next, we move on to study the higher-order latent structures of dynamic inter-personal interactions by combining a multi-resolution spatio-temporal stochastic process with a latent factor model for a dynamic social network. We apply it to analyzing basketball data where the discovered latent structure defines a metric that helps evaluate the quality of game play. Finally, we discuss extensions to a non-Markovian setting of self and mutually exciting point processes (Hawkes processes). We utilize the branching structure of the Hawkes processes to uncover the latent replying structure of a group conversation, which can be further employed to quantitatively measure individual social impact.

We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.

Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.

Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.

Tree-like structures are fundamental in nature. A wide variety of two-dimensional imaging techniques allow us to image trees. However, an image of a tree typically includes spurious branch crossings and the original relationships of ancestry among edges may be lost. We present a methodology for estimating the most likely topology of a rooted, directed, three-dimensional tree given a single two-dimensional image of it. We regularize this inverse problem via a prior parametric tree-growth model that realistically captures the morphology of a wide variety of trees. We show that the problem of estimating the optimal tree has linear complexity if ancestry is known, but is NP-hard if it is lost. For the latter case, we present both a greedy approximation algorithm and a heuristic search algorithm that effectively explore the space of possible trees. Experimental results on retinal vessel, plant root, and synthetic tree datasets show that our methodology is both accurate and efficient.