# 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 lot of statisticians will argue exactly what the phrase non-parametric statistics means exactly (I know the definition that applies to my field...), but "lack of certainty in data distributions's parameters" can be easily misinterpreted as parametric statistical inference ("I have uncertainty about $\mu$ and $\sigma$ that define my normal distribution"). Commented Oct 15, 2015 at 20:02
• @CliffAB: Thank you for pointing that out. My wording indeed could be more precise. As you can guess, the implied meaning is not the uncertainty about, but the lack of any distributional assumptions (if I can use these terms; please correct me, as I am not a statistician). Commented Oct 15, 2015 at 20:07
• I'm going to get flack for this, I can see it, but...the definition of non-parametric estimator that I prefer is "an estimator whose solution cannot be characterized by a finite number of parameters". For example, the simplest NP estimator is the empirical distribution function, which places probability mass $1/n$ at each observed time point. All possible monotonic step functions cannot be described by a finite set up of parameters, unlike, say all possible normal distributions. Commented Oct 15, 2015 at 20:27
• Note that the tag stochastic-processes does not describe chaotic dynamics, which are deterministic. If you examine the evolution in time from two nearby points in state space, the difference between the two trajectories will be random for a stochastic system, but the trajectories will diverge exponentially in time for a chaotic system.
– EdM
Commented Oct 15, 2015 at 21:11
• @AleksandrBlekh, for me a good characterization of nonparametric is lack of distributional assumptions (as you wrote above). Suppose we have variables $y$ and $x$. In a parameteric case, we assume the shape of the conditional distribution of $y|x$ to be known but the specific parameters unknown (e.g. Normal with unknown mean, unknown variance); we then go after estimating the unknown parameters. In a nonparametric case we do not assume to know the shape. Commented Oct 16, 2015 at 6:46

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.

• Thank you so much for taking time to answer my question. I appreciate your detailed answer (+1). It will take some time for me to process the information, considering that I'm an aspiring complex systems enthusiast. Commented Oct 15, 2015 at 22:08
• To EdM's point in the quote he cites, distributed lags of a univariate time series amount to a Koyck model in econometrics and is nothing new. Where I think EDM models do diverge significantly from traditional time series analysis is in leveraging a non-reductionistic, non-mechanistic approach to model building. This is a definite rejection of the "Occam's Razor" assumptions underlying nearly all traditional analysis. Commented Oct 16, 2015 at 11:58
• +1. But I had a brief look at the PNAS paper (as well as at the popular article that @AleksandrBlekh linked to), and somehow it looks like a fairly simple exercise in time series analysis, but filled with buzzwords, complemented with appropriate strawmen and unbelievably oversold and over-hyped. That's to put it politely. I don't understand what it does in PNAS, but then it's not the first time I have this feeling about PNAS. Not surprising that people get confused about how all these fancy terms are related. Commented Oct 16, 2015 at 22:56