Statistical Approaches for Spatio-Temporal Stochastic Population Models

数学模型被用来描述和探索许多领域的复杂过程，包括研究人口中的疾病动态和研究发电厂污染物的风生传播。遥感和数据收集方面的不断进步使以上一代人以前不可能实现的分辨率收集类似过程的数据成为可能。统计和数据科学是专注于为决策和科学查询提供信息的数据分析领域，但最常用的方法，如线性回归或机器学习方法，在其分析方法中不容易使用数学模型。在这项工作中，PI将开发一些方法，使其更容易分析来自系统的数据，在这些系统中，数学模型有助于描述所讨论的过程。开发的方法包括以常见形式对数据进行建模的统计方法，例如空间上的年平均污染浓度，或当前因某种疾病住院的人数。这些方法将提高我们在空间流行病学、疾病建模和生态学中理解和预测复杂行为的能力。潜在的结果包括更好地估计流行病学参数，如个人感染疾病但没有表现出症状的速度，这对预测疫情的未来至关重要。该项目将为研究生提供研究培训机会。机制过程模型的使用，如常微分方程组、随机微分方程和偏微分方程，是生态学和流行病学过程的数学分析的核心。然而，它们在统计推断中的使用相对有限。在这项工作中，当科学地知道管理过程遵循机械过程模型时，PI将开发有助于分析数据的统计方法。作为这项工作的一部分，PI将开发方法，将个体水平的数据(如个体动物运动数据)与种群水平的数据(如动物丰度的种群水平计数)进行联合推断，并在这两个数据流和单一过程模型之间建立正式联系。此外，PI将开发建模方法，用于模拟来自假设的随机过程的数据，如扩散模型或空间疾病传播模型，但收集的数据要么是时间快照，要么是时间的平均值(即年度平均污染物浓度)。这些项目将共同提高指定和适应各种科学学科中常见数据的机械性统计模型的能力。这项工作将提高科学家使用带有可解释参数的机械模型来模拟以各种常见格式获得的数据的能力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical models are used to describe and explore complex processes in many fields, including the study of disease dynamics in a population and the study of the wind-born spread of pollutants from power plants. Continuing advances in remote sensing and data collection have made it possible to collect data on similar processes at resolutions that were impossible a generation ago. Statistics and Data Science are fields focused on the analysis of data to inform decision making and scientific inquiry, but the most common methods used, such as linear regression or machine learning methods, cannot easily use mathematical models in their analysis approach. In this work, the PI will develop methods that make it easier to analyze data from systems where mathematical models are useful to describe the process in question. The methods developed include statistical approaches for modeling data in common forms, such as yearly averaged pollution concentrations over space, or the current number of individuals hospitalized with a disease. These methods will improve our ability to understand and predict complex behavior in spatial epidemiology, disease modeling, and ecology. Potential results include better estimates of epidemiological parameters such as the rate at which individuals contract a disease but do not show symptoms, which is critical for predicting the future of an epidemic. The project will provide research training opportunities for graduate students. The use of mechanistic process models, like ODEs, SDEs, and PDEs, is central to the mathematical analysis of ecological and epidemiological processes. However, their use in statistical inference is relatively limited. In this work, the PI will develop statistical methods useful for analyzing data when the governing process is scientifically known to follow a mechanistic process model. As part of this work, the PI will develop methods for joint inference of individual-level data (like individual animal movement data) with population-level data (like population-level counts of animal abundance) with formal links between these two data streams and a single process model. In addition, the PI will develop methods for modeling data that come from an assumed stochastic process, like a diffusion model or a spatial disease spread model, but are collected as either a snapshot in time or an average over time (i.e., yearly average pollutant concentration). Together these projects will provide increased ability to specify and fit mechanistic statistical models to data common in a wide variety of scientific disciplines. This work will advance the ability of scientists to model data obtained in a variety of common formats using mechanistic models with interpretable parameters.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.