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State-Space Reconstruction

Last update: 13 Dec 2024 21:11
First version: 20 August 2007; major expansion, 16 July 2016

An aspect of time series analysis: given that the time series came from a dynamical system, figure out the state space of that system from observation alone.

Here's the basic set-up. Suppose we have a deterministic dynamical system with state \( z(t) \) on a smooth manifold of dimension \( m \), evolving according to a nice system of differential equations, \( \dot{z}(t) = f(z(t)) \). What we observe is not the state \( z(t) \) but rather a smooth, instantaneous function of the state, \( x(t) = g(z(t)) \). Now, it should be obvious that in this set-up \( z \) is only going to be identified up to a smooth change of coordinates --- basically because we can use any coordinate system we like on the hidden manifold, without changing anything at all. What is surprising is that the system can, in fact, be identified up to a smooth, invertible change of coordinates (i.e., a diffeomorphism).

Fix a finite length of time \( \tau \) and a whole number \( k \), and set \[ s(t) = \left(x(t), x(t-\tau), x(t-2\tau), \ldots x(t-(k-1)\tau)\right) \]

For generic choices of \( f, g \) and \( \tau \), if \( k \geq 2m+1 \) , then \( z(t) = \phi(s(t)) \). This \( \phi \) is smooth and invertible (a diffeomorphism), and commutes with time-evolution, \( \frac{d}{dt}\phi(s(t)) = f(\phi(s(t))) \). Indeed, regressing \( \dot{s}(t) \) on \( s(t) \) will give \( \phi^{-1} \circ f \).

The first publication this subject was that by Packard et al. The first proof that this can work was that of Takens, which remains the standard reference. Note 8 in Packard et al. leads me to believe that the idea may actually have originated with David Ruelle.

I am especially interested in ways of making this idea work for stochastic systems.

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