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设数据点是彼此独立采样的。然而，在许多问题中，这种假设是不正确的。这个项目关注的是一个观测影响其他观测的数据，要么是强化（可能具有相似的值），要么是排斥（可能具有非常不同的值）。这种相互作用可能发生在静态测量之间，在空间/网络中发展的轨迹之间，或者可能是算法中促进鲁棒性，多样性或公平性等目标的理想偏差。这可能是由于对有限资源的竞争，由于传播社会影响产生的富得更富的动态，由于物理和生物系统中的相互作用过程，或者出于学习复杂系统的紧凑表示的愿望。例子包括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、用户在社交媒体上的导航、神经元的峰值或疾病的传播。这项研究汇集了机器学习、统计学、物理学和计算机科学等领域的应用问题和理论思想。这些工具为数据总结、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。本项目的应用方面为本科生的研究提供了机会，并通过随机过程和模拟的本科课程将研究与教学相结合。在技术层面上，该项目开发了有原则的统计模型和有效的算法，放宽了共享空间中观测数据之间的独立性假设。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：用于强化的Hawkes过程和用于排斥的Matern iii型过程。这两个过程都共享一个潜在泊松过程的直观和机械生成方案，其速率由事件历史调节。这允许在静止和轨迹数据中联合建模更丰富的排斥和强化相互作用的框架。与泊松过程的联系允许新的模型和机制的加强和排斥，以及新的，可扩展的算法，允许调查在实际应用中的非泊松性的基本作用。将排斥先验纳入层次模型的潜在变量也允许新的排斥潜在变量模型倾向于简约和可解释性。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。

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.