This article explains the extended dynamic mode decomposition (EDMD) algorithm. It is about the spectral decomposition of nonlinear dynamics based on the Koopman operator theory.
Particularly, it is about the finite approximation $\mathbf{K}$ of the Koopman operator $\mathcal{K}$
$$ (\mathcal{K}\psi)(\mathbf{x}) = (\mathbf{\Psi} \; \circ \; \mathbf{F})(\mathbf{x})\mathbf{a} = \mathbf{\Psi}(\mathbf{x})(\mathbf{K}\mathbf{a}) + r(\mathbf{x}), $$
where $\mathbf{\Psi}$ denotes a tuple of so called observables which are basically base functions of the spectral decomposition, $\mathbf{F}$ is a mapping from manifold to manifold, $\mathbf{a}$ are weighting coefficients, and $r$ is an error that needs to be minimized.
In the example given for a Koopman-based approximation of a linear time-invariant (LTI) dynamical system with two states $(x_1, x_2)$, permutations of products of hermite polynomials $H_i(x)H_j(y), i,j=0...4$ are used as base functions, as in
$$ \begin{eqnarray} \mathbf{\Psi}(\mathbf{x}) & = & \left( \psi_{00}(\mathbf{x}), \psi_{01}(\mathbf{x}), \psi_{02}(\mathbf{x}), ..., \psi_{44}(\mathbf{x}) \right) \\ & = & \left( H_0(x_1)H_0(x_2), H_0(x_1)H_1(x_2), ..., H_4(x_1)H_4(x_2) \right). \end{eqnarray} $$
While this attempt works well for the given LTI case, it fails for data which exhibits periodic patterns, like this one:
My questions are:
- What base functions $(\psi_{00}, \psi_{01}, ...)$ are promising candidates for the periodic signal?
- What makes this particular choice of hermite polynomials suitable for the example system?