Statistical inference for space-time models involving stochastic differential equations

近年来，许多领域越来越多地使用随机微分方程 (SDE) 来模拟随时间变化的科学现象。例子包括海洋学、生态学和公共卫生领域的应用。 SDE 可以同时捕获感兴趣变量的已知确定性动态（例如，洋流、水体的化学和物理特征、疾病的存在、不存在和传播），同时使建模者能够捕获随机环境中的未知动态和测量过程。该提案开发了用于构建、拟合和诊断多元和空间变化 SDE 拟合的统计方法。此类模型通常受到机械建模的启发，可以结合感兴趣变量的复杂动态。尽管使用单一位置数据拟合和分析 SDE 模型的统计方法越来越广泛使用，但用于多元 SDE 和空间索引 SDE 的精确统计方法还远未开发出来。该项目将为一维和多维 SDE 推导改进的近似方法，这些方法比常用但朴素的欧拉近似更准确。将构建空间变化的 SDE 模型，用于对在空间和时间上可能不规则观察到的时空数据进行建模。这项研究将用于解决应用学科的问题。对学生进行 SDE 统计方法教育将是该项目的一个重要目标。此外，还将利用一些外展计划来教育更广泛的受众。

Many fields have experienced a recent growth in the use of stochastic differential equations (SDEs) to model scientific phenomena over time. Examples include applications in oceanography, ecology, and public health. SDEs can simultaneously capture the known deterministic dynamics of the variables of interest (e.g., ocean flow, the chemical and physical characteristics of a body of water, the presence, absence and spread of a disease), while enabling a modeler to capture the unknown dynamics and measurement processes in a stochastic setting. This proposal develops statistical methodology for building, fitting, and diagnosing the fit of multivariate and spatially-varying SDEs. Such models, which are often inspired by mechanistic modeling, can incorporate the complex dynamics of the variables of interest. Although statistical methods for the fitting and analysis of SDEs models using data at a single location are becoming more widely used, accurate statistical methods for multivariate SDEs and SDEs indexed in space are far less developed. This project will derive improved approximate methods of inference for one- and multi-dimensional SDEs that are more accurate than the commonly used, but naive, Euler approximations. Spatially-varying SDE models will be built for modeling spatio-temporal data observed potentially irregularly in space and time. This research will be applied to address problems in applied disciplines. Education of students in statistical methods for SDEs will be an important goal of this project. In addition, a number of outreach programs will be used to educate a broader audience.