Multivariable Causal Analysis of Nonlinear Dynamical Systems using Convergent Cross Mapping

Abstract

Convergent Cross Mapping (CCM) was introduced as a data-driven technique to identify causal links in weakly coupled deterministic non-linear systems where other causal definitions, like the celebrated Granger Causality, fail due to their limited applicability to stochastic systems only. CCM is based on the idea of quantifying the extent to which a potential causal signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x[k]$</tex> is recoverable from another effect signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$y[k]$</tex> with increasing data length. A major drawback of the CCM is its inability to distinguish between direct and indirect causal links that is necessary for reconstructing the direct causal network from observed time series. In this work, we propose a multivariable approach to solve this issue. First, we perform the pair-wise CCM analysis and identify all the effects (both direct and indirect) linked to a cause. Next, we perform a multivariable state-space reconstruction using the identified effect variables and use it to recover the cause variable. We then evaluate the incremental improvement in the recovery as compared to the univariable case. A significant improvement indicates that the effect is an indirect one, while the converse indicates a direct effect. We also address a second shortcoming of CCM by proposing an improved metric for quantifying cross mapping of variables. Case studies on simulated and real data sets are presented to demonstrate the success of proposed developments.