  While most classical approaches to Granger causality detection assume linear
dynamics, many interactions in real-world applications, like neuroscience and
genomics, are inherently nonlinear. In these cases, using linear models may
lead to inconsistent estimation of Granger causal interactions. We propose a
class of nonlinear methods by applying structured multilayer perceptrons (MLPs)
or recurrent neural networks (RNNs) combined with sparsity-inducing penalties
on the weights. By encouraging specific sets of weights to be zero--in
particular, through the use of convex group-lasso penalties--we can extract the
Granger causal structure. To further contrast with traditional approaches, our
framework naturally enables us to efficiently capture long-range dependencies
between series either via our RNNs or through an automatic lag selection in the
MLP. We show that our neural Granger causality methods outperform
state-of-the-art nonlinear Granger causality methods on the DREAM3 challenge
data. This data consists of nonlinear gene expression and regulation time
courses with only a limited number of time points. The successes we show in
this challenging dataset provide a powerful example of how deep learning can be
useful in cases that go beyond prediction on large datasets. We likewise
illustrate our methods in detecting nonlinear interactions in a human motion
capture dataset.
