Takens' Embedding Theorem remarkably established that concatenating M previous outputs of a dynamical system into a vector (called a delay coordinate map) can be a one-to-one mapping of a low-dimensional attractor from the system state space. However, Takens' theorem is fragile in the sense that even small imperfections can induce arbitrarily large errors in this attractor representation. We extend Takens' result to establish deterministic, explicit and non-asymptotic sufficient conditions for a delay coordinate map to form a stable embedding in the restricted case of linear dynamical systems and observation functions. Our work is inspired by the field of Compressive Sensing (CS), where results guarantee that low-dimensional signal families can be robustly reconstructed if they are stably embedded by a measurement operator. However, in contrast to typical CS results, i) our sufficient conditions are independent of the size of the ambient state space, and ii) some system and measurement pairs have fundamental limits on the conditioning of the embedding (i.e., how close it is to an isometry), meaning that further measurements beyond some point add no further significant value. We use several simple simulations to explore the conditions of the main results, including the tightness of the bounds and the convergence speed of the stable embedding. We also present an example task of estimating the attractor dimension from time-series data to highlight the value of stable embeddings over traditional Takens' embeddings.
We present a new embedding theorem for time series, in the spirit of Takens's theorem, but requiring multivariate signals. Our result is part of a growing body of work that extends the domain of geometric time series analysis to some genuinely stochastic systems-including such natural examples where φ is some fixed map and the ηi are i.i.d. random displacements Takens' theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series. This theorem gives little guidance, however, about practical considerations for reconstructing a good state space. We extend Takens' treatment, applying statistical methods to incorporate the effects of observational noise and estimation error. We define the distortion matrix, which is proportional to the conditional covariance of a state, given a series of noisy measurements, and the noise amplification, which is proportional to root-mean-square time series prediction errors with an ideal model. We derive explicit formulae for these quantities, and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification. We identify several different scaling regimes for distortion and noise amplification, and derive asymptotic scaling laws. When the dimension and Lyapunov exponents are sufficiently large these scaling laws show that, no matter how the state space is reconstructed, there is an explosion in the noise amplification- from a practical point of view determinism is lost, and the time series is effectively a random process. In the low noise, large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation, in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension. However, in numerical experiments we find that estimation error complicates this issue. For local approximation methods, we analyze the effect of reconstruction on estimation error, derive a scaling law, and suggest an algorithm for reducing estimation errors.