# Does convergent cross-mapping require you to control for other variables?

There's a really cool method called convergent cross-mapping (Tsonis et al. (2018)) that's used to see if two time-series are causally linked within a dynamic system. It seems really powerful and like it might have a wide range of applications.

I'm wondering if it's sufficient to just "plug in" two variables you're interested in and go, or if you need to control for potential omitted variables like you do in multivariate regression studies.

Thanks!

Tsonis, Anastasios A., Ethan R. Deyle, Hao Ye, and George Sugihara. Convergent cross mapping: theory and an example. In Advances in nonlinear geosciences, pp. 587-600. Springer, Cham, 2018.

• It is important to bear in mind that Sugihara's empirical dynamic modeling methods (of which convergent cross-mapping is one) are predicated on Takens' Theorem, which says if you have a bunch of causally linked variables (and I disagree about what cure says about Granger causality with respect to Takens) then the behavior of all causes is encoded in a time series of their effects; if variables are mutually-causing, then the behavior of all variables in the system is encoded in every variable in the system. Commented Jun 2, 2022 at 5:02
• Visual representation of Taken's Theorem relevant to convergent cross-mapping. Commented Jun 2, 2022 at 5:03

TLDR: They test for causality in Granger sense. It is not causality in the interventional meaning as defined in Pearl et al. (2016). If you seek for Granger causality - simply plug in any two variables. If you wish to perform causal inference - things are never so simple.

This is indeed very cool method and interesting question. However, as many other authors, Tsonis et al. (2018) call causality in Granger sense "just causality", which is in my opinion very misleading attitude. There are many definitions of causality and Granger causality is not one of them. It is something different.

To show an example how it is misleading let me first cite Tsonis et al. (2018):

"if past sea surface temperatures can be estimated from time series of sardine abundance, temperature had a measurable and recoverable influence on the population dynamics of sardines"

Ok. But what if we used something different than the temperature measures around the particular sea? What if we measured the number of sunburns got by the population of people sunbathing by the sea? Let me paraphrase their sentence:

"if past numbers of sunburns can be estimated from time series of sardine abundance, sunburns had a measurable and recoverable influence on the population dynamics of sardines"

Now, this looks bad. But it is how causality in Granger sense works. It seeks for predictors, and has purely observational meaning. If we observe unnaturally high number of sunburns this summer, will we observe changes in sardines population in the future? Very likely. But if we ban suntan cream from use, will population of sardines change? Highly unlikely.

It is important to note, that the variables Tsonis et al. (2018) say that they influence something, are arguably "very exogenous". It is extremely unlikely to find variables that influence temperature in short term, it is much more unlikely to find variables that influence cosmic radiation.

Such situation helps causal inference (and make this article even more misleading, because it makes sense) but it it is an external knowledge (assumption about DGP), which can not be derived directly from the data. And causal inference always requires such external knowledge.

Pearl, J., Glymour, M. and Jewell, N.P., 2016. Causal inference in statistics: A primer. John Wiley & Sons.

Tsonis, Anastasios A., Ethan R. Deyle, Hao Ye, and George Sugihara. Convergent cross mapping: theory and an example. In Advances in nonlinear geosciences, pp. 587-600. Springer, Cham, 2018.

• Thanks for the thorough reply! In the sunburns example, the standard way to address endogeneity like this would be adding the temperature as a variable to test for; this would eliminate the relationship between sunburns and sardines. Here, is it possible to do something like that within the context of a Granger test? Commented Dec 29, 2020 at 1:21
• Also, for Convergent Cross-Mapping in particular, Sugihara shows some results in this video (youtube.com/watch?v=uhONGgfx8Do&t=1153s) that seems to address this endogeneity problem. Here, the population of sardines and anchovies appear to be correlated (like sunburns and anchovies), but somehow CCM is able to "see" that they aren't causally linked. Meanwhile, temperatures and anchovies, and temperatures and sardines are each shown as causally linked. Do you have an idea of how they were able to accomplish this? Commented Dec 29, 2020 at 1:29
• I think that sardines and anchovies are different example than sunburns and sardines. I intended the variable 'sunburns' to be reacting immediately on temperature. Anchovies and sardines probably both react after some time. Therefore it is possible, that they do not predict each other, but temperature predicts both of them. If you want to make more drastic example, think that instead of sunburns we put the exact measurement of the thermometer floating on the sea. If thermometer breaks, will population of sardines change?
– cure
Commented Dec 29, 2020 at 2:01
• Makes sense! Thanks for the explanation. So when applying a technique like CCM, do you know of any ways to "close forks" or separate the influence of temperature on anchovies, from the influence of sunburns on anchovies? Wondering how to apply the idea from multivariate regression to this context. Commented Dec 29, 2020 at 4:21
• Unfortunately I do not know methods to control for any variables in CCM. If you want to search for them, or explore the method further, I think that the best way would be to take a look at rEDM package in R. Some tutorials are in the internet.
– cure
Commented Dec 30, 2020 at 17:25