Chaos theory, equation-free modeling and non-parametric statistics Being interested in complex systems and trying to get a beginner's understanding of the field, today I ran across this interesting article in Quanta Magazine on chaos theory and equation-free modeling. I realize that non-parametric statistics implies lack of certainty in data distributions' parameters (please correct me, if I'm wrong). I'm not sure about chaos theory, but but it seems to me that, at least, equation-free modeling is a term, closely related to non-parametric statistics.
Question: what are relations, if any, between the emphasized topics and non-parametric statistics (I am not interested in details, but rather sources and nature of relations; references are welcome)?
 A: In the limit, this becomes a question of whether you consider time-series analysis to be non-parametric statistics.
The approach of the PNAS paper by Ye at al., cited in the Quanta Magazine article, might be considered a generalization of standard time-series analysis. As stated in the Supporting Information Appendix to the article: "reconstructions of a dynamic system can be made using successive lags of a single time series...if enough lags are taken, this form of reconstruction...preserves essential mathematical properties of the original system." This Appendix has (for me, at least) the clearest explanation of the approach.
What this approach adds is a weighting procedure that can deal better with the problems posed by underlying non-linear dynamics. (I will leave aside the question of whether there really is a chaos theory, as raised in comments above.) A "tuning parameter," $\theta$, is added to the model, which gives nearby points "stronger weighting, allowing the model to be adaptive to local influences and therefore, nonlinear." 
For $\theta=0$, "the model reduces to an autoregressive model"; hence the first sentence of this answer. Forecast skill is then assessed as a function of $\theta$. In the data analyzed in that paper, "forecast skill peaks when $\theta$ is ~ 2, which is evidence for nonlinearity in the aggregate time series." Information from multiple time series is then used to try "to identify informative environmental variables and elucidate potential mechanisms."
I suppose that one could find an analogy between this approach and non-parametric tests that make no assumptions about underlying distributions, in that the approach of Ye et al. makes no assumptions about the form of the equations of the underlying dynamic model. But I think that drawing the analogy would be a disservice both to "non-parametric statistics," whatever that means, and to the work presented by Ye et al.
