Extracting the reconstruction of state matrices (inferring coupling of multivariate signals using PyKoopman)

[snjie209]

Hi authors,

I am trying to extract the degree of coupling between signals in a dynamical system using Pykoopman. For example, if we leverage Lotka Volterra as an example, the coupling terms may be associated with the $\beta$ or $\delta$ terms assuming

$$\dot{x} = \alpha x - \beta xy$$

$$\dot{y} = \delta xy - \gamma y$$

where $x, y > 0$.

One conjecture of how to do this is by assuming that the coupling can be approximated by the eigenvalues and eigenvectors of the original time series $x$, however we can only extract coupling from the eigenvectors in the lifted space $z$. Hence, I was curious if there is a way to extract the $C$ or $W$ matrix as listed below in the diagram, which is the reconstruction of state $x$ which may represent such coupling characteristics? Thanks!

[snjie209]

Actually, I am curious if the $C$ and $W$ terms can be extracted from functions C and W in lines 422 and 438 respectively, perhaps answered my own question there but feel free to double check haha 😆