Takens' theorem
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In mathematics, Takens' delay embedding theorem is a result of Floris Takens on the embedding dimension of nonlinear (chaotic) systems. The theorem states that a dynamical system can be reconstructed from a sequence of observations of the state of the dynamical system.
Takens' result is simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a ν-dimensional manifold M. The dynamics is given by a smooth map
- f : M → M.
Assume that the dynamics f has an invariant manifold A with Hausdorff dimension dA (it could be a strange attractor). Using Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with
- k > 2 dA.
That is, there is a diffeomorphism φ that maps A into R k such that the derivative of φ has full rank.
Takens' result states that an observation function can be used to construct the embedding function. An observation function α must be twice-differentiable and associate a real number to any point of the manifold. It must also be generic, so its derivative is of full rank and has no special symmetries in its components. Takens' delay embedding theorem states that the function
- <math>\varphi_T(x)=( \alpha(x), \alpha(f(x)), \dots, \alpha(f^{k-1}(x)) )<math>
is an embedding of the invariant submanifold A.
The embedding dimension of time series gives a hint about the underlying model, and is required to apply most of the machine learning algorithms (artificial neural networks, support vector machines).
External link
- Takens Theorem (http://www.maths.ox.ac.uk/~hardenbe/defs/node2.html)