A first analysis of the stability of Takens' embedding

Abstract

Takens' Embedding Theorem asserts that when the states of a hidden dynamical system are confined to a lowdimensional 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, linear measurement function, number of delays and sampling time, the delay-coordinate map can be a stable embedding of the dynamical systems attractor.

Tangle: A metric for quantifying complexity and erratic behavior in short time series.

“…That is, an attractor can be reconstructed from discrete samples of a continuous process such that the reconstructed attractor is a smooth functional transformation of the attractor underlying the associated continuous process. This smooth transform preserves many qualities of the continuous attractor and it is generally accepted that many qualitative inferences obtained from a reconstructed attractor also hold true for its associated continuous time attractor, even in the presence of noise (Yap, Eftekhari, Wakin, & Rozell, 2014). In practice, D may be estimated by methods such as those proposed by Cao (1997) and Kennel, Brown,and Abarbanel (1992), and τ may be estimated through minimization of average mutual information or autocorrelation between X i and X i+τ or through the C-C method (Cao, 1997;Kennel, Brown, & Abarbanel, 1992;Kim, Eykholt, & Salas, 1999).…”

Stochastic embeddings of dynamical phenomena through variational autoencoders

“…The term involving tr A i (x t )Σ i A T i (x t ) d i=1 cannot be interpreted in isolation, since it would yield Σ i → 0. This does not occur due to the Kullback-Leibler divergence from Equation (15), which tries to keep each of the Σ i as close as possible to its prior value, say Σ (0) i . Therefore, the maximization of L requires a compromise, which (usually) results in values of Σ i comprised between 0 and Σ (0) i .…”

“…Therefore close points on the original attractor may end up far in the reconstructed state space. This makes Takens' theorem sensible to noise, since small fluctuations could have large effects on the delay reconstruction [15].…”

Permutation Entropy Based on Non-Uniform Embedding

“…Theoretical foundations by Takens [ 21 ], and expansions of his ideas by Sauer et al [ 22 ] indicate that the embedding dimension of (where m is the fractal dimension of the attractor) almost always ensures the reconstruction of the topology of the original attractor [ 20 ]. This surprising result states that time series output is sufficient to obtain complete information about hidden states of the dynamical system [ 23 ].…”