Delay Coordinate Embedding

In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space.

Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions.

Delay embedding theorems are simpler to state for
discrete-time dynamical systems.
The state space of the dynamical system is a $\nu$-dimensional manifold $M$. The dynamics is given by a smooth map

:$f: M \to M.$

Assume that the dynamics $f$ has a strange attractor $A\sub M$ with box counting dimension $d_A$. Using ideas from Whitney's embedding theorem, $A$ can be embedded in $k$-dimensional Euclidean space with

:$k > 2 d_A.$

That is, there is a diffeomorphism $\phi$ that maps $A$ into $\R^k$ such that the derivative of $\phi$ has full rank.

A delay embedding theorem uses an ''observation function'' to construct the embedding function. An observation function $\alpha : M \to \R$ must be twice-differentiable and associate a real number to any point of the attractor $A$. It must also be typical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function

:$\phi_T(x)=\left(\alpha(x), \alpha\left(f(x)\right), \dots, \alpha\left(f^(x)\right)\right)$

is an embedding of the strange attractor $A$ in $\R^k$.

Simplified, slightly inaccurate version

Suppose the $d$-dimensional
state vector $x_t$ evolves according to an unknown but continuous
and (crucially) deterministic dynamic. Suppose, too, that the
one-dimensional observable $y$ is a smooth function of $x$, and “coupled”
to all the components of $x$. Now at any time we can look not just at
the present measurement $y(t)$, but also at observations made at times
removed from us by multiples of some lag $\tau: y_, y_$, etc. If we use
$k$ lags, we have a $k$-dimensional vector. One might expect that, as the
number of lags is increased, the motion in the lagged space will become
more and more predictable, and perhaps in the limit $k \to \infty$ would become
deterministic. In fact, the dynamics of the lagged vectors become
deterministic at a finite dimension; not only that, but the deterministic
dynamics are completely equivalent to those of the original state space (More exactly, they are related by a smooth, invertible change of coordinates,
or diffeomorphism.) The magic embedding dimension $k$ is
at most $2d+1$, and often less.

See also

* Whitney embedding theorem
* Nonlinear dimensionality reduction

References

Further reading

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* {{cite journal
, journal = Remote Sensing of Environment
, year = 2015
, title = Estimating determinism rates to detect patterns in geospatial datasets
, pages = 11–20
, author = R. A. Rios, L. Parrott, H. Lange and R. F. de Mello
, volume = 156
, doi = 10.1016/j.rse.2014.09.019, bibcode = 2015RSEnv.156...11R

External links

Attractor Reconstruction (scholarpedia)


Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface.

Theorems in dynamical systems