Nonparametric Prediction and Structure Discovery for Spatial Dynamics

该项目开发了非参数预测、过滤和结构发现的新方法，主要用于时空数据，但也适用于其他一系列具有高维观测的环境，如网络。调查方法有两个新颖的方面。首先，它不是试图对时空数据进行全局预测，也不是根据固定的模式进行预测，而是利用系统的动力学来开发一种新的局部预测形式，这种预测仍然可以捕捉到长尺度结构。其次，虽然传统的非参数平滑是基于预测变量空间的通常几何，但这是对具有类似预测结果的输入点进行补充平滑，实际上发现了新的几何。这种方法借鉴了早期在非线性动力学信息论方面的工作，允许以计算高效的方式准确地预测大型时空系统的演化。它还允许自动发现此类数据中复杂的更高级别的结构。科学数据越来越多地随着复杂的测量结果在空间和时间上传播。科学家们需要一些方法来预测这类系统将如何进化，并自动将系统的重要(但可能是微妙的)模式与无关紧要的“背景”区分开来，因为这些结构通常对理解动力学至关重要。这个项目共同解决了这两个具有挑战性的统计问题。它将信息论和非线性物理的思想与灵活的统计建模的现代工具相结合，从数据本身发现系统的内在动力学，并将这些结构用于预测和过滤。潜在的应用领域包括神经科学、流体动力学和生态学，在这些领域，它将有助于预测复杂系统的行为，并帮助找到控制这种行为的关键的有组织的结构。

This project develops new methods for non-parametric prediction, filtering, and structure discovery, primarily for spatio-temporal data but also in a range of other settings with high-dimensional observations, such as networks. There are two novel aspects to the investigation's approach. First, rather than trying to predict spatio-temporal data globally, or according to a fixed pattern, it exploits the dynamics of the system to develop a novel form of local prediction which still captures long-scale structure. Second, while conventional non-parametric smoothing is based on the usual geometry of the space of predictor variables, this is supplemented smoothing together input points which have similar predictive consequences, in effect discovering a new geometry. This approach, which draws on earlier work on information theory in nonlinear dynamics, allows for accurate forecasting of the evolution of large spatio-temporal systems in a computationally efficient manner. It also allows for the automatic discovery of complex higher-level structures in such data. Scientific data increasingly comes as complex measurements spread over space and time. Scientists need ways to forecast how such systems will evolve, and to automatically separate important (but perhaps subtle) patterns from inconsequential "background" of the system, since the structures are often crucial to understanding the dynamics. This project tackles both of these challenging statistical problems together. It combines idea from information theory and nonlinear physics with modern tools of flexible statistical modeling to discover the intrinsic dynamics of the system from the data itself, and uses these structures for both prediction and filtering. Areas of potential application include neuroscience, fluid dynamics, and ecology, where it would help forecast the behavior of complex systems, and help to find the organized structures which are keys to controlling that behavior.