Statistical inference for space-time models involving stochastic differential equations

许多领域都经历了最近的增长，在使用随机微分方程（SDES）模型的科学现象随着时间的推移。例子包括海洋学、生态学和公共卫生方面的应用。PDE可以同时捕获感兴趣的变量的已知确定性动态（例如，海洋流动、水体的化学和物理特性、疾病的存在、不存在和传播），同时使建模者能够在随机环境中捕捉未知的动态和测量过程。该建议开发了用于构建、拟合和诊断多元和空间变化的SDEs拟合的统计方法。这些模型通常受到机械建模的启发，可以包含感兴趣的变量的复杂动态。虽然使用单一地点的数据拟合和分析空间微分方程模型的统计方法正得到越来越广泛的使用，但用于多变量空间微分方程和空间索引的空间微分方程的精确统计方法还远远没有发展起来。这个项目将推导出改进的一维和多维SDES的近似推理方法，这些方法比通常使用的但幼稚的欧拉近似更准确。将建立空间变化的空间数据模型，用于模拟在空间和时间上可能不规则观察到的时空数据。这项研究将应用于解决应用学科的问题。对学生进行统计方法方面的教育将是本项目的一个重要目标。此外，还将利用一些外联方案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.