Timeline for Interpreting the covariance of Poincaré plots
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16 at 23:03 | answer | added | Augusto Santos | timeline score: 1 | |
| Aug 16 at 13:16 | comment | added | Augusto Santos | All-in-all, the information entailed in $C$ depends on the underlying dynamical law. For reference, I also explored the role of these lags for structure identification at arxiv.org/abs/2312.05974 and at ojs.aaai.org/index.php/AAAI/article/view/26085 | |
| Aug 16 at 13:12 | comment | added | Augusto Santos | @ManfredWeis: Assuming the dynamical system is initialized at the invariant distribution, then $C_{i,i+k} = R(k)$, where $R(k)$ is the $k$-lag covariance matrix. If the process abides by a linear dynamical law $x[n+1] = \rho x[n]+\xi_{n+1}$ (and under the proper assumptions), $R(k) = \rho^k R(0)$. If you "expand" $R(0)$, you get that $R(1)-R(3) = \rho$ -- a critical parametric information for the dynamics. This gets more interesting at the higher-dimensional setting where you can recover the "interacting network" of the dynamical system under the proper assumptions on the dynamics. | |
| Aug 16 at 0:50 | comment | added | Michael Hardy | Eigenvalues of the covariance matrix are variances of certain linear combinations of the $k$ components of the vector and those linear combinations are uncorrelated with each other. But I'm guessing that's not enough to address what you're asking. | |
| Aug 15 at 6:46 | history | asked | Manfred Weis | CC BY-SA 4.0 |