# Correlation and causation

Linear correlation (Pearson's) is vastly applied in meteorology/climatology to assess the relationship between two variables, say precipitation and SST, for example.

However, we know that correlation does not necessarily imply causation, mainly for two factors: there may be external factors acting on both series or spurious coincidences may also happen.

What are possible and comprehensible ways, that have been used and are possible to reproduce in the Earth Sciences (Meteorology, Oceanography, Clmatology...), to go further and make a point to show that correlation does imply causation in some situations?

EDIT to make it more specific:

Imagine correlation is found between sea surface temperature (SST) in some region and rainfall in another. How to test if the variability of the two series are not being externally forced by a third party?

Thank you.

• If you're asking how do prove causation (which I believe is a blurry subject), that'd really be a Math SE question. Examples in Earth Science would likely use the same methods.... and asking for a list would be probably a little too open-ended? Jul 1 '17 at 16:50
• too broad... question needs to be about a specific Earth Science problem. I've had several of my own questions closed for the same reason.
– f.thorpe
Jul 1 '17 at 17:00
• In terms of meteorology related topics, I think a lot of statistical studies are more based on determining a relationship rather than proving explicit causation. Finding shear vs CAPE profiles favorable for tornadoes, setups that indicate heavy rain, etc... it's about being able to find such events, not the reasoning. Obviously that's not true of all topics, but it is a lot. Otherwise quite often observational correlation studies work in tandem with dynamical studies exploring the actual mechanisms to further the solidity and understanding (plus model studies that seek idealized evidence) Jul 1 '17 at 18:12
• But one topic that surely has a lot of attention in the correlation/causation topic more heavily is global warming, so perhaps narrowing this question down to just that would make it more viable. Jul 1 '17 at 18:14
• @BarryCarter thanks, yes, small physical models allow that, however a bit more difficult regarding the climate system. Jul 2 '17 at 7:47

Convergent cross mapping (CCM) is a recently developed tool to answer the question you've asked. It's based on tools developed in nonlinear time series analysis and dynamical systems theory. It allows you to:
1) determine if a causal relationship between two variables is present
2) establish the direction of causality
3) do so even in the presence of noise.

As for an interesting application, check out the paper Causal feedbacks in climate change [van Nes et al., 2015], where CCM is applied to co2 and temperature based on the Vostok data sets.

EDIT: Below I've added a more detailed explanation of CCM to show the original poster that this technique does indeed answer their question, as well as to show it has a rigorous mathematical underpinning.

The general idea of convergent cross mapping is based on phase space reconstruction [F. Takens, 1981],[H. Abarbanel, 1996]. Numbers 1 through 5 explain the idea behind phase space reconstruction, which is needed to understand CCM. Numbers 6 through 8 very briefly explain CCM. References are listed at the bottom for more depth.

1) A physical system that is described by a set of equations (e.g. conservation of mass, momentum, etc) has a phase space. The solution to the system of equations is a trajectory through (or subset of) the phase space.

2) An attractor is a subset of the phase space that the trajectories/solutions evolve toward.

3) If you know the governing attractor, then you have all solutions of the system for all time.

4) Taken’s theorem says that one can reconstruct the attractor of the system based on a single observable. For example, if temperature, pressure, and velocity are the three variables of the system, then you only need measurements from one of these variables to reconstruct the attractor of the system. State space reconstruction

5) The reconstructed attractor is not exactly the “true” attractor, but it has a direct 1:1 mapping to the true attractor. Taken's theorem

6) If two observables belong to the same system, then they each have a reconstructed attractor with a direct mapping to the true attractor. The reconstructed attractors also have a direct mapping to one another. Convergent cross mapping

7) It is then possible to make predictions on one observable, based on the reconstructed attractor of the other observable, if they are in fact from the same attractor (causally related). Time series and attractors

8) Last, a series of tests/predictions with the data help to establish the direction of, strength of, and linearity of the interactions between the two variables. This is detailed in the papers [sugihara et al., 2012] and [van Nes et al., 2015].

To answer your question "given two observed variables, how do you tell if an third variable is simply forcing the two observed variables, making them appear correlated? First, the process of phase space reconstruction would yield an estimate of the "embedding dimension", which is an estimate of the dimension of the phase space (how many variables there). In the CCM framework, a one-way forcing relationship between the two known variables (v1 and v2) would be attractor for v1 can make skillful predictions of v2, but attractor v2 can not make skillful predictions for v1. Contingent upon the situation where you have an idea of what the third variable is (v3), I think what you could do is the following, take the reconstructed attractor of v3 and make predictions on both v1 and v2, and show that v3 has more predictive power on v1 (compared to v2), and that v3 has more predictive power on v2 (compared to v1). I'm not sure about this though.

Also, if the forcing (v3) is thought to be linear, you could simply remove/detrend v3 from v1 and v2, as is done when you remove seasonality from temperature data.

Note: There is MATLAB code available to mess around with this technique. I believe you can find similar codes in R as well.

References

Sugihara et al., 2012, Detecting causality in complex ecosystems.
van Nes et al., 2015, Causal feedbacks in climate change.
Abarbanel, Analysis of Observed Chaotic Data,1996, Springer publishing.
Takens, 1981, Detecting strange attractors in turbulence

• What is making you think this? The area of mathematics called dynamical systems theory suggests you can. I’ll add an edit to the post explaining the procedure for clarity.
– Z W
Jul 2 '17 at 19:46
• @BarryCarter I've referenced the mathematical underpinnings as per your comment.
– Z W
Jul 2 '17 at 20:58
• @ouranos I've added in more detail a rigorous method that can be used two determine causality between two observables beyond mere statistical correlation measures, i.e. a method based on the systems dynamics (but without knowledge of the governing equations). As for your specific example of SST and rainfall, you could easily use this method - see the reference to causal feedbacks in climate change, you can follow their method exactly by replacing temperature with SST, and CO2 with rainfall.
– Z W
Jul 2 '17 at 20:59
• @BarryCarter so actually the CCM algorithm was partly a response to the limitations of Granger causality (which is designed for stochastic variables).
– Z W
Jul 3 '17 at 5:49
• @ouranos deep diving for a deep question! Here's a link to some matlab code mathworks.com/matlabcentral/fileexchange/… also there is a nice discussion of the technique in a less formal/more digestible style with links to more matlab codes askhamwhat.github.io/blog/2016/03/09/detecting-causality Lastly, some of the applications you probably notice are ecological, but that doesn't really matter, all that matters is that the system under question has an associated set of differential equations that describe it (i.e. a dynamical system)
– Z W
Jul 5 '17 at 20:43

I can give some examples from atmospheric science:

1. Wind and temperature in the vertical direction: as you increase in height, the temperature decreases due to the conservation of geopotential energy. Also the wind speed increases, due to the lack of friction.

2. In data assimilation, spurious correlations are quite common, especially for large distances. Localization is one way to reduce the influence of spurious correlations.

3. It is not an uncommon rule of thumb that precipitation is associated with low pressure. However, that rule of thumb does not always apply to thunderstorms, where pressure may increase in the downdraft.

4. It is a known fact that pressure and temperature are proportional ($P=\rho R T$). However, if you look at hurricane data, warm air can strengthen a hurricane, according to the wind-induced surface heat exchange. Also, warm advection can also lower the pressure of an extratropical cyclone.

5. Spurious Correlations has a list of examples of correlations that are not many times inexplicable, though not all are relegated to earth science. For example, precipitation in South Carolina is negatively correlated with precipitation in Richland County, Wisconsin (-0.77).

• in #2, there are telleconnections in the atmosphere, like the impact of ENSO in virtually all around the globe, PNA, PSA... in #4, that rule applies to a closed system I believe, imagine a sealed box where P and T are proportional. In open atmosphere, I can intuitively think of increase in temperature leading to lowering density, expanding air, therefore lowering pressure. Jul 2 '17 at 7:33
• @BaroclinicCplusplus El Nino lags IOD by two seasons. Does that mean IOD causes El Nino ? Positive IOD is usually followed by La Nina and negative IOD is usually followed by El Nino.Correlation leads to causation ? Jul 2 '17 at 14:34
• @Ouranos While some correlations do exist because of teleconnections, often those are not used in data assimilation. Otherwise you could infer that an observation in Argentina could tell you about the weather in Greenland. In addition, many teleconnections are not straight-forward, such as ENSO, the NAO, etc. They are normalized by climatology. Jul 2 '17 at 18:10
• @gansub I never actually talked about teleconnections until ouranos brought it up. I don't know much of the IOD, but I'll answer this in regards to the EnKF. The EnKF do not use a temporal lag to compute observations, but instead use an ensemble spread and observation locality to assimilate data. That is not the case for 4D hybrids, but the modeling studies are usually sub-seasonal. Consider how representative an observation 3 miles away is to 500 miles away. That is effectively what is considered. Jul 2 '17 at 18:29
• @BarocliniCplusplus I don't know much about data assimilation, sorry for not noticing the specific context. I just tried to highlight that causal correlations may exist despite of long distances. Jul 2 '17 at 21:13

As an alternative to Convergent Cross Mapping (CCM), the recent Tigramite is a fast python library for causal discovery that promises to ...

... outperforms current approaches in detection power and scales up to high-dimensional datasets. It overcomes detection biases, especially when strong autocorrelations are present, and allows ranking associations in large-scale analyses by their causal strength.

The paper introducing the method contains a real data application on an atmospheric and climatic dataset:

Runge, Jakob, Dino Sejdinovic, and Seth Flaxman. “Detecting Causal Associations in Large Nonlinear Time Series Datasets.” arXiv:1702.07007 [Physics, Stat], February 22, 2017. http://arxiv.org/abs/1702.07007.

At first sight the difference with CCM would be:

• CCM is a deterministic approach as it test if to time series are related to the same hypothetical attractor while Tigramite test for statistical dependence between delays.
• CCM is a pairwise approach while Tigramite is faster and easily applied to larger database.

Those might be complementary.