A First Analysis of the Stability of Takens' Embedding

TL;DR

This paper investigates the stability of Takens' Embedding in noisy conditions, using compressed sensing techniques to provide theoretical guidance on selecting the number of delays for stable attractor reconstruction.

Contribution

It introduces a theoretical framework applying compressed sensing to determine the number of delays needed for stable embedding under noise.

Findings

Delay-coordinate maps can be stable embeddings with proper delays and sampling.

Theoretical conditions for stability depend on system and measurement properties.

Provides guidelines for empirical delay selection in noisy time-series analysis.

Abstract

Takens' Embedding Theorem asserts that when the states of a hidden dynamical system are confined to a low-dimensional attractor, complete information about the states can be preserved in the observed time-series output through the delay coordinate map. However, the conditions for the theorem to hold ignore the effects of noise and time-series analysis in practice requires a careful empirical determination of the sampling time and number of delays resulting in a number of delay coordinates larger than the minimum prescribed by Takens' theorem. In this paper, we use tools and ideas in Compressed Sensing to provide a first theoretical justification for the choice of the number of delays in noisy conditions. In particular, we show that under certain conditions on the dynamical system, measurement function, number of delays and sampling time, the delay-coordinate map can be a stable embedding of the dynamical system's attractor.