Correlation and causation

Question

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.

Answer 1

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

Answer 2

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).

Answer 3

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.