Disclaimer: I know basically 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 here 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? (Further bonus question – did SiJR04 rediscover delay embedding, or have they extended it?) Sample complexity results here 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, even with nonlinear dynamics. e.g. in plain old model-based count time series such as branching processes, and grammatical inference of formal syntax, and nonlinear system identification.
I would be interested to see a compelling new insight from the dynamical system perspective on these problems. New estimators; models outside the ken of Kalman filters?
Some stuff I saw that’s maybe related
- TISEAN is a collection of state reconstruction algorithms
- CSSR ?
- symbolic dynamics
- learning automata
Stuff that I might actually use
Hirata’s reconstruction looks like good clean decorative fun – you can represent graphs by an equivalent dynamical system.
- Marw08: N. Marwan (2008) A historical review of recurrence plots. The European Physical Journal Special Topics, 164(1), 3–12. DOI
- ShSh04: Cosma Rohilla Shalizi, Kristina Lisa Shalizi (2004) Blind construction of optimal nonlinear recursive predictors for discrete sequences. (pp. 504–511).
- BaPo99: Remo Badii, Antonio Politi (1999) Complexity: Hierarchical Structures and Scaling in Physics. Cambridge University Press
- ShCr00: Cosma Rohilla Shalizi, James P Crutchfield (2000) Computational Mechanics: Pattern and Prediction, Structure and Simplicity.
- SMYH12: George Sugihara, Robert May, Hao Ye, Chih-hao Hsieh, Ethan Deyle, Michael Fogarty, Stephan Munch (2012) Detecting Causality in Complex Ecosystems. Science, 338(6106), 496–500. DOI
- ChBR16: Adam Charles, Aurele Balavoine, Christopher Rozell (2016) Dynamic Filtering of Time-Varying Sparse Signals via l1 Minimization. IEEE Transactions on Signal Processing, 64(21), 5644–5656. DOI
- CrYo89: James P. Crutchfield, Karl Young (1989) Inferring statistical complexity. Physical Review Letters, 63(2), 105–108. DOI
- ShCr02: Cosma Rohilla Shalizi, James P. Crutchfield (2002) Information bottlenecks, causal states, and statistical relevance bases: how to represent relevant information in memoryless transduction. Advances in Complex Systems, 05(01), 91–95. DOI
- HaBS16: Franz Hamilton, Tyrus Berry, Timothy Sauer (2016) Kalman-Takens filtering in the presence of dynamical noise. ArXiv:1611.05414 [Physics, Stat].
- DuDE05: P Dupont, François Denis, Y Esposito (2005) Links between probabilistic automata and hidden Markov models: probability distributions, learning models and induction algorithms. Pattern Recognition, 38, 1349–1371. DOI
- GrSS91: Peter Grassberger, Thomas Schreiber, C Schaffrath (1991) Nonlinear time sequence analysis. International Journal of Bifurcation and Chaos, 1(3), 521–547. DOI
- KaSc04: Holger Kantz, Thomas Schreiber (2004) Nonlinear time series analysis. Cambridge, UK ; New York: Cambridge University Press
- HiSA06: Yoshito Hirata, Hideyuki Suzuki, Kazuyuki Aihara (2006) Reconstructing state spaces from multivariate data using variable delays. Physical Review E, 74(2), 026202. DOI
- HiHA08: Yoshito Hirata, Shunsuke Horai, Kazuyuki Aihara (2008) Reproduction of distance matrices and original time series from recurrence plots and their applications. The European Physical Journal Special Topics, 164(1), 13–22. DOI
- Levi17: David N. Levin (2017) The Inner Structure of Time-Dependent Signals. ArXiv:1703.08596 [Cs, Math, Stat].
- Foot99: Jonathan Foote (1999) Visualizing music and audio using self-similarity. In Proceedings of the seventh ACM international conference on Multimedia (Part 1) (pp. 77–80). New York, NY, USA: ACM DOI