Statistical Approaches for Spatio-Temporal Stochastic Population Models

数学模型被用于描述和探索许多领域的复杂过程，包括种群疾病动力学的研究和电厂污染物风传传播的研究。遥感和数据收集方面的不断进步，使人们能够以一代人以前不可能的分辨率收集有关类似过程的数据。统计学和数据科学是专注于数据分析的领域，为决策和科学探究提供信息，但最常用的方法，如线性回归或机器学习方法，不能轻易地在其分析方法中使用数学模型。在这项工作中，PI将开发方法，使其更容易分析来自系统的数据，其中数学模型有助于描述所讨论的过程。所开发的方法包括对常见形式的数据进行建模的统计方法，例如空间上的年平均污染浓度，或目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。可能的结果包括更好地估计流行病学参数，例如个人感染某种疾病但没有表现出症状的比率，这对于预测流行病的未来至关重要。该项目将为研究生提供研究培训机会。机械过程模型的使用，如ode、SDEs和pde，是生态和流行病学过程数学分析的核心。然而，它们在统计推断中的应用相对有限。在这项工作中，PI将开发用于分析数据的统计方法，当科学地知道控制过程遵循机械过程模型时。作为这项工作的一部分，PI将开发方法来联合推断个人层面的数据（如个体动物运动数据）和群体层面的数据（如群体层面的动物丰度计数），并在这两个数据流和单一过程模型之间建立正式的联系。此外，PI将开发建模数据的方法，这些数据来自一个假设的随机过程，如扩散模型或空间疾病传播模型，但收集的数据要么是及时的快照，要么是一段时间的平均值（即年平均污染物浓度）。总之，这些项目将提供更大的能力，以具体说明和拟合机械统计模型的数据，共同在各种科学学科。这项工作将提高科学家利用具有可解释参数的机制模型对各种常见格式获得的数据进行建模的能力。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.