Bayesian Nonparametric Modeling and Inference Methods for Point Processes

该研究项目将针对不同类别的点过程开发灵活的统计模型和相应的推理方法。点过程理论的发展是为了研究在随机时间和/或空间位置发生的事件的分布。该项目将改进统计方法，以解决涉及点状过程产生的数据的一系列问题。一个关键目标将是通过开发统计模型来放宽最先进方法的限制性假设，从而扩大几类点数过程的范围。点过程方法在统计学领域很重要，在气候学、犯罪学、金融学和地震学等多个领域都有应用。新的模型将应用于重要的社会问题，包括研究不同类型犯罪的聚集模式和分析地震发生。为便利来自其他领域的从业人员和研究人员使用这些方法，将开发公开可用的软件来实施若干统计模型。该项目还将为研究生创造教育机会，并寻求增加妇女和代表性不足群体在研究中的参与。该研究项目将为时间或空间上的点过程开发一个通用的基于模型的框架，包括泊松过程和霍克斯过程。该建模框架包括点过程强度的结构化混合表示，在实现统计推断和预测时实现了模型灵活性和计算效率之间的平衡。该项目将为在不规则区域观察到的空间点过程开发易于处理的推理方法；例如，城市、州或国家边界。它还将扩大标记霍克斯过程的推理范围，这是一类多功能的随机点过程模型，已应用于不同领域，包括地震建模、金融、犯罪学和社会网络分析。尽管霍克斯过程的理论已经得到了很好的研究，但统计推断方法的开发相对较少，特别是在允许非标准数据特征和完全不确定量化的一般设置下。该研究项目在各种点过程模型的方法学发展方面有大量的分析部分，在实现可根据大量数据进行有效的模型拟合方面也有重要的计算部分。新方法的实用性将通过几项模拟研究和涉及地震和犯罪数据分析的实质性应用进行调查。这一奖项反映了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。

This research project will develop flexible statistical models and corresponding inference methods for different classes of point processes. The theory of point processes was developed to study the distribution of events that occur at random times and/or spatial locations. This project will advance statistical methodology for a range of problems involving data that arise from point processes. A key objective will be to increase the scope of several classes of point processes by developing statistical models that relax the restrictive assumptions of state-of-the-art methods. Point process methods are important for the field of statistics and have applications in a variety of areas, including climatology, criminology, finance, and seismology. The new models will be applied to important societal problems including the study of clustering patterns for different types of crime and the analysis of earthquake occurrences. To facilitate use of the methods by practitioners and researchers from other fields, publicly available software will be developed for implementing several of the statistical models. The project also will create educational opportunities for graduate students and seek to increase the participation of women and underrepresented groups in the research.This research project will develop a general model-based framework for point processes over time or space, including Poisson processes and Hawkes processes. The modeling framework involves structured mixture representations for point process intensities that achieve a balance between model flexibility and computational efficiency in implementation of statistical inference and prediction. The project will develop tractable inference methods for spatial point processes observed over irregular domains; for instance, city, state, or country boundaries. It also will increase the inferential scope for marked Hawkes processes, a versatile class of stochastic point process models that have been applied in diverse areas, including earthquake modeling, finance, criminology, and analysis of social networks. Even though the theory for Hawkes processes is well studied, statistical inference methods are relatively less developed, especially under general settings that allow for non-standard data features and for full uncertainty quantification. The research project has a substantial analytic component with regards to methodological development for the various point process models, as well as a significant computational component with regards to achieving efficient model fitting that can be scaled to large amounts of data. The practical utility of the new methods will be investigated with several simulation studies and through substantive applications involving analysis of earthquake and crime data.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.