Bayesian Nonparametric Point Processes: New Methods and Applications to Extreme Value Analysis

本研究项目的重点是开发灵活的贝叶斯统计方法来对点过程进行建模和推理。该研究将在广泛扩展的贝叶斯非参数领域中发展方法，为涉及点模式数据的大类问题提供一个通用的基于模型的推理框架。该项目将研究时间、空间和时空中的非齐次泊松过程的推理方法，包括扩展以合并时间和空间变化的协变量以及标记点过程。它还将考虑模型检查技术以及扩展到非泊松点过程的建模。这种通用方法将围绕极值分析领域发展，极值分析包括对概率分布尾部发生的事件的探索。关键应用出现在金融、精算科学和气候学等不同领域。点过程建模提供了一种解决极端研究中重要科学问题的通用方法。这种方法的理论已经得到了广泛的发展，但是现有的有限的统计方法的工作依赖于限制性的参数假设。贝叶斯非参数方法将为更灵活的推理和预测提供一个自然框架，在提高我们量化与相对不太可能发生的事件相关的风险的能力方面具有重要的实际意义。特别令人感兴趣的将是评估可能受气候变化影响的环境变量的极端行为。在时间、空间或时空中观察到的物理过程的极值（非常大或非常小的值）的研究在几个领域至关重要，包括计量经济学、地球科学和环境政策制定。极值分析的一个强大的统计建模方法来自于点过程理论，点过程是随时间和/或空间随机事件的随机模型。本研究将通过点过程建模和贝叶斯非参数（一个快速发展的贝叶斯统计领域）的新综合方法，制定一个分析极端的一般统计框架。由于其通用性，本研究项目将开发的统计方法有可能影响许多应用点过程的科学领域。在极值分析的背景下，该方法将侧重于对罕见但灾难性事件（如暴雨、严重干旱或股指崩盘）的不确定性进行适当量化。对于这些和相关的应用，改进对极端事件发生概率的预测和对相关因素的理解可以对有效的决策产生重要影响。

The focus of this research project is on development of flexible Bayesian statistical approaches to modeling and inference for point processes. The research will develop methods within the widely expanding field of Bayesian nonparametrics to provide a general model-based inference framework for a large class of problems involving point pattern data. The project will study inferential methods for non-homogeneous Poisson processes in time, space, and space-time, including extensions to incorporate time- and space-varying covariates as well as marked point processes. It also will consider techniques for model checking as well as extensions to modeling for non-Poisson point processes. This general methodology will be developed around the area of extreme value analysis, which consists of the exploration of events that occur in the tails of probability distributions. Key applications arise in fields as diverse as finance, actuarial sciences and climatology. Point process modeling provides a general approach to addressing scientifically important questions in the study of extremes. The theory for this approach has been extensively developed, but the limited existing work on statistical methods relies on restrictive parametric assumptions. The Bayesian nonparametric methodology will provide a natural framework for more flexible inference and prediction with important practical implications in enhancing our ability to quantify the risks associated with the occurrence of relatively unlikely events. Of particular interest will be assessment of the extreme behavior of environmental variables that are likely to be affected by climate change.The study of extremes (very large or very small values) of a physical process observed in time, space, or space-time is of critical importance in several fields, including econometrics, geosciences, and environmental policy making. A powerful approach to statistical modeling for extreme value analysis draws from the theory of point processes, which are stochastic models for random events over time and/or space. This research will formulate a general statistical framework for analysis of extremes through a novel synthesis of methods from point process modeling and Bayesian nonparametrics, a rapidly growing area of Bayesian statistics. Due to their generality, the statistical methods that will be developed under this research project have the potential of impacting many scientific fields where point processes are applied. In the context of extreme value analysis, the methodology will focus on appropriate quantification of uncertainty for rare but catastrophic events such as torrential rains, severe droughts, or stock index crashes. For these and related applications, improved prediction of the probability of occurrence of extreme events and understanding of associated factors can have an important impact on effective decision making.