Emergent Granger Causality in Neural Networks: Can Prediction Alone Reveal Structure?

Granger Causality (GC) offers an elegant statistical framework to study the association between multivariate time series data. Vector autoregressive models (VAR) are simple and easy to fit, but have limited application because of their inherent inability to capture more complex (e.g., non-linear) associations. Numerous attempts have already been made in the literature that exploit the functional approximation power of deep neural networks (DNNs) for GC. However, these methods treat GC as a variable selection problem. We present a novel paradigm for investigating the learned GC from a single neural network used for joint modeling of all components of multivariate time series data, which is essentially linked with prediction and assessing the distribution shift in residuals. A deep learning model, with proper regularization, may learn the true GC structure when jointly used for all components of the time series when there is sufficient training data. We propose to uncover the learned GC structure by comparing the model uncertainty or distribution of the residuals when the past of everything is used as compared to the one where a specific time series component is dropped from the model. We also compare the effect of input layer dropout on the ability of a neural network to learn GC. We show that a well-regularized model can learn the true GC structure from the data without explicitly adding terms in the loss function that guide the model to select variables or perform sparse regression under specific settings. We also provide a comparison of deep learning architectures such as CNN, LSTM and transformer models on their ability to discover Granger Causality. The numerical experiments demonstrate that, compared to sparse regression models, a simple joint model is a strong baseline for learning the true GC which has the advantage that it does not require tuning of many extra hyper-parameters.