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→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.

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 → 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).

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).

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).

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).

TACI applied to coupled non-stationary Hénon Maps. A) A plot of the TACI inference when applied to the coupled Hénon Maps system where the coupling from X → Y is set to either Cxy = 0.6 (blue bar above the plot) or Cxy = 0 (no bar above the plot). B) Same as A but with a toggle from Cxy = 0.6 to Cyx = 0.6 (where the blue and red bars above the plot flip). C) Same as A but with multiple pulses of Cxy = 0.6 of varying sizes. Error bars are generated using a bootstrapping procedure (see Materials and Methods).

TACI applied to coupled non-stationary Hénon Maps with ramped couplings. A) Inferred causal coupling as a function of time during the simulation. B Time series of how the coupling from X to Y was stepped up and then down. Error bars are generated using a bootstrapping procedure (see Materials and Methods).

Time series of Temperature, Dew Point, Relative Humidity, and Vapor Pressure Deficit from the Jena Climate Dataset.

Summary of Jena Climate Dataset Features

Causal interactions with relative humidity from the Jena Climate Dataset. A) Empirical relationship between relative humidity and air temperature (assumes Tdew = 10). Note the large negative partial derivative at low values of T. B) TACI predictions for causal interactions for how the other 13 variables in Table I affect relative humidity as a function of time across the eight years of the dataset (gray lines, mean trajectory is the black line). Note how causal influence peaks consistently when the temperature (C) is at its nadir, just as predicted by the plot in A.

Parameters used in the TACI model training and prediction phases (ranges indicate the parameter range used across the examples in this chapter)

Interactions between brain regions in ECoG data. Each plot here shows the average interaction between all electrodes within each of the 8 coarse-grained regions described in the text. The left matrices are from before the anesthesia was administered, the middle matrices are from when the monkey was anesthetized, and the right plots are from the recovery period. A is the Pearson correlation between the signals, B is the TACI-derived inference of causal interaction, and C displays the TACI Directionality – the difference between the CSGI score in one direction minus the CSGI score in the other direction.

Causal interactions across time between Parietal and medial Prefrontal Cortices. Plot of the average TACI-derived interactions between PC and mPFC over the course of the anesthesia experiment. Error bars are the standard errors of the mean across all electrode interactions, and the dashed lines represent change points in the experimental protocol (labeled above the axes).