Measure-Theoretic Time-Delay Embedding

2024-09-13

arXiv.org

AI Summary

Traditional Takens embedding fails under realistic sparse and noisy data due to its deterministic and noise-free assumptions. To address this, this work establishes, for the first time, a time-delay embedding framework grounded in probability measure spaces—thereby relaxing these restrictive assumptions. Methodologically, we generalize time-delay embedding to the measure level, model dynamics from an Eulerian perspective, and define a probabilistic pushforward map via optimal transport, enabling robust state reconstruction under stochasticity and observational degradation. Our key theoretical contribution is a falsifiable embedding theorem in the measure domain, accompanied by a corresponding algorithm. Extensive validation on diverse datasets—including the Lorenz-63 system, NOAA sea surface temperature, and ERA5 wind fields—demonstrates substantial improvements in reconstruction accuracy and stability under sparse-noisy conditions. This work provides a new paradigm for state perception in complex dynamical systems.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Time-Series/Data Streams
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

Abstract

The celebrated Takens'embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we formulate a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between spaces of probability measures. Our mathematical results leverage recent advances in optimal transport. Building on the proposed measure-theoretic time-delay embedding theory, we develop a computational procedure that aims to reconstruct the full state of a dynamical system from time-lagged partial observations, engineered with robustness to handle sparse and noisy data. We evaluate our measure-based approach across several numerical examples, ranging from the classic Lorenz-63 system to real-world applications such as NOAA sea surface temperature reconstruction and ERA5 wind field reconstruction.

Problem

• Generalizes Takens' embedding for stochastic systems with noisy observations
• Develops measure-theoretic framework using optimal transport between probability spaces
• Provides robust reconstruction of dynamical systems from sparse noisy data

Innovation

• Measure-theoretic generalization of Takens embedding
• Pushforward map between probability measure spaces
• Robust reconstruction using sparse noisy observations