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[Trinidad Baltzell]

In the previous chapters we talked about Simplex Projection, a forecasting technique that looks for similar trends in the past to forecast the future by computing for nearest neighbors on an embedding. In this chapter, we discuss Convergent Cross Mapping (CCM) also formulated by Sugihara et al., 2012 as a methodology that uses ideas from Simplex Projection to identify causality between variables in a complex dynamical system (e.g. ecosystem) using just time series data.

We will go through the key ideas of CCM, how it addresses the limitations of Granger causality, and the algorithm behind it. We will then test the CCM framework on simulated data where we will deliberately adjust the influence of one variable over the other. Finally, we will apply CCM on some real world data to infer the relationships between variables in a system.

This chapter explains the CCM methodology in detail. If you wish to apply this in your own projects, install the framework using pip install causal-ccm. See using_causal_ccm_package.ipynb notebook for details how to use.

Granger causality framework does not apply to cases where information about variables are not separable from the rest of the system especially for those whose causalities are weak to moderate (like in our test systems in the following sections). Finally, Granger Causality is dependent on the performance of a linear predictive model that assumes linear combination of variables. This model may not perform well for some systems that need nonlinear models to improve predictions.

Convergence in CCM means that for variables with causalities, the longer our observation period (or more data we gather), the better we can predict one variable using the other. In the attractor example above, we can imagine the attractor will get denser as time goes on since our system will eventually fill in some of the gaps in the manifold. Stated another way, more defined manifolds imply that for variables with causalities, we can expect the accuracy of \(\hatYM_x\) to improve. If two variables have no causal link, then improving their manifolds will not translate to improvement in predictions.

Convergence means we are able to improve cross mapping accuracy the longer the period \(L\) we consider. To recap, this happens only when we are able to enhance our reconstruction of the shared attractor between two variables using more data.

In the plot above, we find that both cross mappings converge, however one converges more than the other. We have satisfied the two criteria for CCM: convergence and cross mapping. We can thus say causality exists. However, the magnitudes of causality going from one direction or the other are different.

Although for time period up to 7, it might look like there is convergence as shown by improvement in crossmapping correlation, proceeding further results in drop in convergence. We conclude that these two variables are not causally linked.

Based on the results above, we can infer a network of causal relationships. Relationships with unclear convergence are shown in broken lines while relationships with clear convergence are shown in solid lines.

Based on the results above, we can infer a network of causal relationships. There seems to be a strong feedback loops between all three elements in the system. Upon checking past researches on the topic, we find that there are papers that support the seemingly unintuitive interactions in the inferred causal network:

In this chapter we discussed the theory and algorithm behind Convergent Cross Mapping methodology to infer causality between variables using time series data. We then applied it on simulated and real world data. In the next chapter we will discuss techniques for finding pattens in time series data namely cross correlation, Fourier transform, and Wavelet transform.

Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two variables that, like the Granger causality test, seeks to resolve the problem that correlation does not imply causation.[1] While Granger causality is best suited for purely stochastic systems where the influences of the causal variables are separable (independent of each other), CCM is based on the theory of dynamical systems and can be applied to systems where causal variables have synergistic effects. As such, CCM is specifically aimed to identify linkage between variables that can appear uncorrelated with each other.

In the event one has access to system variables as time series observations, Takens' embedding theorem can be applied. Takens' theorem generically proves that the state space of a dynamical system can be reconstructed from a single observed time series of the system, X \displaystyle X . This reconstructed or shadow manifold M X \displaystyle M_X is diffeomorphic to the true manifold, M \displaystyle M , preserving instrinsic state space properties of M \displaystyle M in M X \displaystyle M_X .

Convergent Cross Mapping (CCM) leverages a corollary to the Generalized Takens Theorem[2] that it should be possible to cross predict or cross map between variables observed from the same system. Suppose that in some dynamical system involving variables X \displaystyle X and Y \displaystyle Y , X \displaystyle X causes Y \displaystyle Y . Since X \displaystyle X and Y \displaystyle Y belong to the same dynamical system, their reconstructions via embeddings M X \displaystyle M_X and M Y \displaystyle M_Y , also map to the same system.

The causal variable X \displaystyle X leaves a signature on the affected variable Y \displaystyle Y , and consequently, the reconstructed states based on Y \displaystyle Y can be used to cross predict values of X \displaystyle X . CCM leverages this property to infer causality by predicting X \displaystyle X using the M Y \displaystyle M_Y library of points (or vice-versa for the other direction of causality), while assessing improvements in cross map predictability as larger and larger random samplings of M Y \displaystyle M_Y are used. If the prediction skill of X \displaystyle X increases and saturates as the entire M Y \displaystyle M_Y is used, this provides evidence that X \displaystyle X is causally influencing Y \displaystyle Y .

CCM is used to detect if two variables belong to the same dynamical system, for example, can past ocean surface temperatures be estimated from the population data over time of sardines or if there is a causal relationship between cosmic rays and global temperatures. As for the latter it was hypothesised that cosmic rays may impact cloud formation, therefore cloudiness, therefore global temperatures. [3]

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An important problem across many scientific fields is the identification of causal effects from observational data alone. Recent methods (convergent cross mapping, CCM) have made substantial progress on this problem by applying the idea of nonlinear attractor reconstruction to time series data. Here, we expand upon the technique of CCM by explicitly considering time lags. Applying this extended method to representative examples (model simulations, a laboratory predator-prey experiment, temperature and greenhouse gas reconstructions from the Vostok ice core and long-term ecological time series collected in the Southern California Bight), we demonstrate the ability to identify different time-delayed interactions, distinguish between synchrony induced by strong unidirectional-forcing and true bidirectional causality and resolve transitive causal chains.

A fundamental question in science is identifying the causal relationships between variables. The conventional approach to this problem is to observe the outcomes of controlled experiments; however, this is not always possible due to moral, legal, or feasibility reasons. Consequently, the ability to infer causality using only observational data is a highly valuable tool with applications in many fields of study [e.g., financial systems, ecosystems, neuroscience1,2,3,4].

This extension of CCM has several additional applications: the identification of time delays in causation can be informative, for instance in understanding delays in interventions or manipulations. It can also be used to identify the causal effects of stochastic drivers that have no dynamics (for which general cross mapping may not succeed) and can even correctly determine the order of variables in a transitive causal chain.

In Figure 4C, we show the results of extended CCM applied to long-term time series of chlorophyll-a and sea surface temperature measured at the Scripps Institution of Oceanography pier. As expected, there is no effect of chlorophyll-a on SST (red line). However, we do identify a causal influence of SST on chlorophyll-a, suggesting that the physical environment plays a role in determining phytoplankton abundances (which are proxied by concentrations of chlorophyll-a). Moreover, optimal cross mapping occurs with a lag of 3 weeks, suggesting that the physical drivers of algae populations act with a lag of several weeks. Ideally, if other causal drivers show similar time delays in their effects, then it may be possible to produce models that can forecast events such as algal blooms several weeks in advance!

We note that in certain systems, especially those with stochastic drivers that contain unique information, Granger causality may correctly identify causal interactions. Indeed, Granger causality has been successful when applied to system consisting solely of stochastic components. However, in situations where both cause and effect have deterministic dynamics, causal information cannot be isolated from amongst the affected variables and alternative methods, such as CCM must therefore be used.

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