Selecting optimal embedding dimension

[kkhader]

I understand the general approach for estimating the optimal embedding dimension by selecting the value of E that corresponds with the highest 'forecast skill'. What is suggested in the case where what is optimal is not obvious? For example when forecast skill is identical (or nearly identical) across multiple values of E? Are there specific conclusions one can draw from this situation? Should a 'sensitivity analysis' be done for each of those values? As an example, included is a figure showing forecast skill across a range of values for E that are obtained from real-world time series data. As you can see, there are some time series where the choice is clear but others where there may be 2-3 choices that produce similar forecast performance. Thanks for any guidance you have!

[SoftwareLiteracy]

Hi kkhader,

This is imo part of the "art" of applying EDM. A few general principles I try to keep in-mind:

EDM is predicated on representing low-dimensional, deterministic dynamics. It performs remarkably well on real world data in the presence of noise, but is founded on low-dimensional, deterministic dynamics. Takens theorem presumes a univariate time series adequately encapsulates the entire dynamical behavior of interest, and, that the behavior arises from a single dynamical system. These two presumptions are often violated; evenso, EDM works remarkably well. But these principles are at the core of estimating dynamical system dimension using predictability.

The literature is replete with studies relating embedding dimensions to manifold trajectories/states toward the ideal of best unfolding or disentangling trajectories/states.

With this in mind one generally prefers the lowest dimension that represent the dynamics.

Multiple peaks is perhaps a sign that one or more of the foregoing presumptions have been violated.

How well the dynamics can be represented (predicted) is important. I often reject results below some reasonable threshold of rho. Looking at simplex predictions at the suggested E can help inform your level of comfort.

Those are my thoughts... I expect those whom are better versed in EDM fundamentals & application can offer more accurate/insightful perspective.

[kkhader]

Thank you! I hope it's okay to ask a follow up on the 'art of applying EDM' related to CCM.

In an example from an rEDM tutorial, it says "In the following example, we reproduce the analysis from (Sugihara et al. 2012) to identify causality between anchovy landings in California and Newport Pier sea-surface temperature. For this example, a previously identified value of 3 for the embedding dimension will be used." The implementation of CCM then evaluates CCM of anchovy:np_sst as well as np_sst:anchovy using E = 3. But presumably, both np_sst and anchovy could both have their own (and different) optimal embedding dimension. In that case, is there any theory behind the use of a single value of E versus two different (optimal) values of E for both np_sst and anchovy to evaluate np_sst:anchovy and anchovy:np_sst?

And lastly, while I understand that the performance of forecasts with EDM methods is influenced by degree of nonlinearity (as measured by S-map). My understanding is that EDM embeddings are not optimal when there is nonlinearity is not present. I assume the same is true about CCM? What can we say in a setting where we see convergence in the CCM but one or both of the underlying time series do not exhibit nonlinear dynamics (measured by S-map)?

[SoftwareLiteracy]

Hi kkhader,

Re:

presumably, both np_sst and anchovy could both have their own (and different) optimal
embedding dimension. In that case, is there any theory behind the use of a single value
of E versus two different (optimal) values of E for both np_sst and anchovy to evaluate
np_sst:anchovy and anchovy:np_sst?

You may wish to review Convergent cross mapping, Detecting Causality in Complex Ecosystems & State Space Reconstruction: Convergent Cross Mapping reflecting that CCM presumes the causal variables are part of the same dynamical system. The EDM packages carry this presumption into using the same time delay embedding dimension for "forward" and "reverse" mapping as a convenience.

Regarding the "art" ; ) one could perform each of the CCM's with different optimal E's for each target variable. If the two variables are part of the same system one may have difficulty explaining why they have different embeddings, If they are not part of the same system it seems reasonable to me one could use different embedding dimensions. So it seems a bit circular that one would need to first establish two variables are not part of the same system before justifying different embeddings.

What can we say in a setting where we see convergence in the CCM but one or 
both of the underlying time series do not exhibit nonlinear dynamics (measured
by S-map)?

Indeed, All models are wrong.

My opinion is if the system is not a low-dimensional, deterministic dynamical system, one does not need EDM. I therefore have not looked closely at applicability of EDM/CCM to linear or stochastic systems. S-map assesses the scale of state-dependence in the dynamics, and thus the nonlinearity of the dynamics. If the system is low-dimensional, state-dependent and not stochastic, then I think EDM is a viable and most useful analytical approach.