Collaborative research: Leveraging low-dimensional structure for time series analysis and prediction

题目：合作研究：利用低维结构进行时间序列分析和预测pi: Christopher J. Rozell，佐治亚理工学院co- pi: Michael B. Wakin，密歇根大学安娜堡分校预测复杂系统的行为对于许多具有重大科学和国家重要性的任务至关重要，包括气象、金融市场和全球冲突等领域。现代科学有一个根深蒂固的前提，即对一种动态现象的反复观察有助于理解其驱动机制并预测其未来行为。研究人员研究的方法，以提高我们的能力，以表征和预测这样的系统，即使他们是非常大的（即，有许多相互作用的因素）或出现高度无序（即，混沌系统）。这项研究利用了新的数学结果，使分析人员能够有效地捕捉到即使在看起来非常复杂的系统中也经常出现的简单结构。这些结果导致了基于人工神经网络的启发式预测方法的改进和性能保证，这些方法在实践中经常使用，但有时会莫名其妙地失败。时间序列预测通常是通过假设一个隐藏系统驱动数据生成的结构化模型来实现的。该项目借鉴了低维信号建模的最新进展，在类似的低维结构存在时，推进时间序列分析和预测工具的最新状态。对于线性系统，本研究开发了有效的估计策略，通过鼓励稀疏解来改进经典技术。对于非线性模型，本项目建立在Takens嵌入定理的基础上，该定理指出，可以使用时间序列观测序列重建吸引流形的图像，以保证吸引流形的定量稳定嵌入。此外，本研究旨在改进并保证油藏计算方法的性能，其中随机连接的神经网络已被确定为预测混沌时间序列的有效机制。

Title: Collaborative Research: Leveraging Low-dimensional Structure for Time Series Analysis and PredictionPI: Christopher J. Rozell, Georgia Institute of Technologyco-PI: Michael B. Wakin, University of Michigan, Ann ArborPredicting the behavior of complex systems is central to many tasks of great scientific and national importance, including arenas such as meteorology, financial markets and global conflict. Modern science is ingrained with the premise that repeated observations of a dynamic phenomenon can help in understanding its driving mechanisms and predicting its future behavior. The investigators study methods for improving our ability to characterize and predict such systems even when they are very large (i.e., with many interacting factors) or appear highly unordered (i.e., chaotic systems). This research leverages new mathematical results that enable analysts to efficiently capture the simple structure that is often present even in systems that appear very complex. These results lead to improvements and performance guarantees for heuristic prediction methods based on artificial neural networks, which are often used in practice but can sometimes fail inexplicably.Time series prediction is often approached by postulating a structured model for a hidden system driving data generation. This project borrows from recent advances in low-dimensional signal modeling to advance the state of the art in time series analysis and prediction tools when similar low-dimensional structure is present. For linear systems, this research develops efficient estimation strategies that improve upon classical techniques by encouraging sparse solutions. For nonlinear models, this project builds upon Takens' Embedding Theorem, which states that the image of an attractor manifold can be reconstructed using a sequence of time series observations, to guarantee a quantifiably stable embedding of the attractor manifold. Furthermore, this research aims to improve upon and make performance guarantees for reservoir computing methods, where randomly-connected neural networks have been identified as effective mechanisms for predicting chaotic time series.