EAGER: Real-Time: Search for dynamical dependencies and natural time-scales of physical processes

物理科学和工程中的一个基本问题是识别相互作用过程之间的依赖关系和动态关系，以理解因果关系和推进预测建模。事实上，从历史上看，这种依赖关系常常构成物理定律的基础。在现代技术世界中，关于复杂的自然或工程系统的大量大型数据集带来了一种紧迫感，需要可靠和通用的机器学习工具来检测过程之间的关系。该项目的出发点是认识到通常收集数据的观测时间尺度可能不是发生相互作用的本地时间尺度，因此采样可能混淆这种关系的性质。事实上，连续时间过程之间的线性动态关系可能不容易从以任何有限采样率收集的数据中检测出来。该项目将开发各种方法，以便从（通常较粗糙的）观测时间尺度的数据中完全恢复自然时间尺度上所寻求的过程之间的关系。为此目的将开发的理论和统计学习工具将应用于地球物理过程，例如利用地面和多卫星传感器在不同时空尺度上的一套观测资料确定气候变量之间的关系。当变量之间的关系是动态的（即，交互依赖于系统中的内存），采样隐藏了动态依赖的本质。具体来说，在线性随机过程中，当在过程的自然时间尺度上估计功率谱密度矩阵时，向量值过程之间的动态依赖关系反映在功率谱密度矩阵的零值中。在观测采样率下，相应的零值不再与动力学关系的结构有关。然而，通过适当的分析，可以通过将样本模型投射到所涉及过程的自然时间尺度来恢复动力学关系。事实上，对于线性随机过程，存在一个与粗尺度观测数据一致的最快过程时间尺度，正是在这个精细尺度上，模型可以通过求解合适的代数方程来预测。在任何较粗的尺度上，动态依赖关系都不容易被检测到。该项目将侧重于如何学习和恢复必须寻求动态依赖的自然时空尺度。将开发统计学习工具，并将其应用于地球物理数据，作为概念的证明。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。

A fundamental problem in physical sciences and engineering is to identify dependencies and dynamical relations between interacting processes for understanding causal relationships and advancing predictive modeling. Indeed, historically, such dependences often formed the basis of physical laws. The abundance of large data-sets about complex natural or engineered systems in our modern technological world, has brought a sense of urgency to the need for reliable and versatile machine learning tools to detect relations between processes. The starting point of the project is the realization that often the observational time scale at which data is collected may not be the native time scale at which interactions occur and thus sampling might obfuscate the nature of such relations. Indeed, linear dynamical relations between continuous-time processes may not be readily detectable from data collected at any finite sampling rate. This project will develop methodologies that will allow to fully recover sought relations between processes at the natural time-scale from data at a (typically coarser) observational time-scale. Theory and statistical learning tools that will be developed for that purpose will be applied to geophysical processes, such as identifying relationships between climate variables using a suite of observations from ground and multi-satellite sensors at different spatio-temporal scales. When relations between variables are dynamic (i.e., the interaction relies on memory in the system), sampling hides the nature of dynamical dependencies. Specifically, in linear stochastic processes, dynamical dependencies between vector-valued processes are reflected in the nullity of the power spectral density matrix when this is estimated at the natural time-scale of the process. At the observational sampling rate, the corresponding nullity no longer relates to the structure of the dynamical relations. Yet, with proper analysis the dynamical relations can be recovered by projecting sample-models to the natural time-scale of the processes involved. In fact, for linear stochastic processes there is a fastest process time-scale that is consistent with the coarser scale observational data, and it is at that fine scale that models can be projected via solving suitable algebraic equations. At any coarser scale, dynamical dependencies cannot be readily detected. The project will focus on how to learn and recover the natural time-space scale at which dynamical dependencies must be sought. Statistical learning tools will be developed and applied to geophysical data as proof of concept.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.