RI: Small: Dynamics of repulsion and reinforcement in point process, latent variable, and trajectory models

大多数传统的数据分析方法都假定数据点是相互独立地采样的。然而，在许多问题上，这种假设是不正确的。这个项目关注的是一个观察结果影响其他观察结果的数据，要么是强化的(可能有相似的值)，要么是排斥的(可能有很大不同的值)。这种相互作用可能发生在静态测量之间、在空间/网络上演变的轨迹之间，或者可能是算法中的理想偏差，以促进稳健性、多样性或公平性等目标。这些可能是竞争有限资源的结果，可能是因为传播社会影响而变得越来越丰富的动力，可能是因为物理和生物系统中的相互作用过程，或者是出于学习复杂系统的紧凑表示的愿望。例如细胞或服务站的位置、粒子或种群之间的相互作用、交通轨迹、用户浏览社交媒体、神经元的尖峰或疾病的传播。这项研究汇集了来自机器学习、统计学、物理学和计算机科学等领域的应用问题和理论想法。这些工具为数据汇总、探索和可视化开辟了新的途径，并允许从业者探索可解释性和预测准确性之间的权衡。这个项目的应用方面为本科生的研究提供了机会，并通过一门关于随机过程和模拟的本科课程为研究和教学的整合提供了机会。在技术层面上，这个项目开发了原则性的统计模型和有效的算法，放松了位于共享空间的观测之间相互独立的假设。它考虑了三类问题的相互作用：1)点过程模型，2)潜变量模型和3)轨迹模型。这项工作的核心是两种随机过程模型：用于增援的霍克斯过程和用于排斥的马特恩III型过程。这两个过程都共享来自潜在泊松过程的直观和机械的生成方案，泊松过程的速度受到事件历史的调节。这允许建立一个框架，对静止和轨迹数据中更丰富的排斥和强化相互作用进行联合建模。与泊松过程的联系允许新的强化和排斥的模型和机制，以及新的、可扩展的算法，使得研究非泊松在实际应用中的基本作用成为可能。在层级模型的潜在变量中加入排斥性先验，也允许新的排斥性潜变量模型偏向简约和可解释性。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。

Most traditional ways of analyzing data assume that data points are sampled independently of one another. In many problems, however, this assumption is incorrect. This project focuses on data where one observation influences others, either as reinforcing (likely to have a similar value) or repulsing (likely to have a greatly different value). Such interactions might arise between static measurements, between trajectories evolving in space/on a network, or may be desirable biases in algorithms to promote goals like robustness, diversity or fairness. These might arise as a consequence of competition for finite resources, because of rich-get-richer dynamics from propagating social influence, because of interacting processes in physical and biological systems, or out of a desire to learn compact representations of complex systems. Examples include the locations of cells or service stations, interactions among particles or populations, traffic trajectories, users navigating social media, the spiking of neurons or the spread of disease. The research brings together applied problems and theoretical ideas from fields like machine learning, statistics, physics and computer science. Such tools open new avenues to data-summarization, exploration and visualization, and allow practitioners to explore trade-offs between interpretability and predictive accuracy. The applied aspects of this project provide an opportunity for undergraduate research and for the integration of research and teaching through an undergraduate course on stochastic processes and simulation.At a technical level, this project develops principled statistical models and efficient algorithms that relax assumptions of independence among observations lying on a shared space. It considers interactions for three classes of problems: 1) point process models, 2) latent variable models and 3) trajectory models. Central to the work are two kinds of stochastic process models: the Hawkes process for reinforcement, and the Matern type-III process for repulsion. Both processes share intuitive and mechanistic generative schemes from an underlying Poisson process, whose rate is modulated by event history. This allows a framework that jointly models richer repulsive and reinforcing interactions in stationary and trajectory data. The connection with the Poisson process allows novel models and mechanisms of reinforcement and repulsion, as well as new, scalable algorithms, allowing investigations into the fundamental role of non-Poissonness in real applications. Incorporating repulsive priors into latent variables of hierarchical models also allow novel repulsive latent variable models with biases towards parsimony and interpretability.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.