Inferring the time-varying coupling of dynamical systems with temporal convolutional autoencoders

TL;DR

Time-varying causality in nonlinear, nonstationary dynamical systems is addressed by introducing Temporal Autoencoders for Causal Inference (TACI), which couples a two-headed Temporal Convolutional Network autoencoder with a surrogate-data-based Comparative Surrogate Granger Index (CSGI) to quantify directional influence over time. Key contributions include the CSGI metric, defined as $\chi_{x\to y} = \frac{R^2_{xy}-R^2_{x^{(s)} y}}{\frac{1}{2}(R^2_{xy}+R^2_{x^{(s)} y})}$, and a training/prediction pipeline that evaluates causal influence on moving windows across four network configurations. Validation on synthetic benchmarks (Rössler-Lorenz, bidirectional Henon, coupled autoregressive, and non-stationary Henon) and real-world data (Jena climate and non-human primate ECoG) shows that TACI outperforms SLGC, CCM, and TE in discerning direction, strength, and time-varying changes, and that a single model trained on the full time series can infer dynamics without retraining. Potential applications include neuroscience and climate dynamics; limitations include computational cost and risk of overfitting, which the authors discuss.

Abstract

Most approaches for assessing causality in complex dynamical systems fail when the interactions between variables are inherently non-linear and non-stationary. Here we introduce Temporal Autoencoders for Causal Inference (TACI), a methodology that combines a new surrogate data metric for assessing causal interactions with a novel two-headed machine learning architecture to identify and measure the direction and strength of time-varying causal interactions. Through tests on both synthetic and real-world datasets, we demonstrate TACI's ability to accurately quantify dynamic causal interactions across a variety of systems. Our findings display the method's effectiveness compared to existing approaches and also highlight our approach's potential to build a deeper understanding of the mechanisms that underlie time-varying interactions in physical and biological systems.

Figures (11)

• Figure 1: Schematic of the Temporal Autoencoders for Causal Inference (TACI) Networks. We use a two-headed network consisting of Temporal Convolutional Networks that interact through a shared latent space to predict a time-shifted version of one of the two input time series. For each pair of variables we wish to examine (here, $X$ and $Y$), we train two networks for each causal direction: one using $X$ and $Y$ as inputs and another using $X$ and a randomized version of $Y$. We consider an interaction from $Y\to X$ to be causal if the network using the actual value of $Y$ predicts the future of $X$ better than the network using the surrogate version of $Y$. In this particular case, we show the approach applied to two different variables from the Lorenz system.
• Figure 2: Causal inference in the Rössler-Lorenz System. A) 2-dimensional projections of the Rössler attractor (left) and the Lorenz system (right three plots) as $C$ increases. Mathematically, there is only coupling from $X\to Y$, but starting near $C=2.14$, the two systems become synchronized, making finding the causal interactions an ill-posed problem. B-E) Results from applying the four methods to the system. Note that only TACI accurately predicts the unidirectional coupling in the regime above $C>0$ and before synchronization occurs. Error bars are generated using a bootstrapping procedure (see Materials and Methods).
• Figure 3: Causal inference in the bidirectional species system. A-D) Results from applying the four methods to the bidirectional species system. Error bars are generated using a bootstrapping procedure (see Materials and Methods).
• Figure 4: Causal inference in the coupled autoregressive models system. A-D) Results from applying the four methods to the coupled autoregressive models system. Error bars are generated using a bootstrapping procedure (see Materials and Methods).
• Figure 5: Causal inference in the coupled Hénon Maps system. A-D) Results from applying the four methods to the coupled Hénon Maps system. Here, only TACI accurately predicts univariate coupling across all values of $C$ prior to synchronization. Error bars are generated using a bootstrapping procedure (see Materials and Methods).