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

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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.