# State space reconstruction

October 14, 2014 — August 30, 2022

Disclaimer: I know next to nothing about this.

But I think it’s something like: Looking at the data from a, possibly stochastic, dynamical system. and hoping to infer cool things about the kinds of hidden states it has, in some general sense, such as some measure of statistical of computational complexity, or how complicated or “large” the underlying state space, in some convenient representation, is.

TBH I don’t understand this framing, but possibly because I don’t come from a dynamical systems group; I just dabble in special cases thereof. Surely you either do *physics*, and work out the dynamics of your system from experiment, or you do *statistics* and select an appropriate model to minimise some estimated predictive loss trading off data set, model complexity and algorithmic complexity. I need to read more to understand the rationale here, clearly.

Anyway, tools seem to include inventing large spaces of hidden states (Takens embedding); does this get us some nice algebraic properties? Also, how does delay embedding relate? Is that the same? Sample complexity results seem to be scanty, possibly because they usually want their chaos to be deterministic and admitting noise would be fiddly.

OTOH, from a statistical perspective there are lots of useful techniques to infer special classes of dynamical systems state-space. It is especially interesting inmgrammatical inference of formal syntax where there are many lovely and faintly depressing computational complexity results.

## 2 Stuff that I might actually use

Hirata’s reconstruction looks like good clean decorative fun — you can represent graphs by an equivalent dynamical system.

## 3 Incoming

## 4 References

*Complexity: Hierarchical Structures and Scaling in Physics*. Cambridge Nonlinear Science Series.

*Proceedings of the National Academy of Sciences*.

*IEEE Transactions on Signal Processing*.

*Nature Computational Science*.

*Physical Review Letters*.

*Pattern Recognition*.

*Proceedings of the Seventh ACM International Conference on Multimedia (Part 1)*. MULTIMEDIA ’99.

*International Journal of Bifurcation and Chaos*.

*arXiv:1611.05414 [Physics, Stat]*.

*The European Physical Journal Special Topics*.

*Physical Review E*.

*Nonlinear Time Series Analysis*.

*arXiv:1703.08596 [Cs, Math, Stat]*.

*The European Physical Journal Special Topics*.

*Advances in Complex Systems*.

*Science*.