Here the concept of Granger causality is employed to propose a new criterion for unsupervised learning that is appropriate in the case of temporally-dependent source signals. The basic idea is to identify two projections of a multivariate time series such that the Granger causality among the resulting pair of components is maximized.

A new technique for unsupervised learning of time series data based on the notion of Granger causality is presented. The technique learns pairs of projections of a multivariate data set such that the resulting components -- driving and driven -- maximize the strength of the Granger causality between the latent time series (how strongly the past of the driving signal predicts the present of the driven signal). A coordinate descent algorithm that learns pairs of coefficient vectors in an alternating fashion is developed and shown to blindly identify the underlying sources (up to scale) on simulated vector autoregressive (VAR) data. The technique is tested on scalp electroencephalography (EEG) data from a motor imagery experiment where the resulting components lateralize with the side of the cued hand, and also on functional magnetic resonance imaging (fMRI) data, where the recovered components express previously reported resting-state networks.

A new technique for unsupervised learning of time series data based on the notion of Granger causality is presented. The technique learns pairs of projections of a multivariate data set such that the resulting components -- "driving" and "driven" -- maximize the strength of the Granger causality between the latent time series (how strongly the past of the driving signal predicts the present of the driven signal). A coordinate descent algorithm that learns pairs of coefficient vectors in an alternating fashion is developed and shown to blindly identify the underlying sources (up to scale) on simulated vector autoregressive (VAR) data. The technique is tested on scalp electroencephalography (EEG) data from a motor imagery experiment where the resulting components lateralize with the side of the cued hand, and also on functional magnetic resonance imaging (fMRI) data, where the recovered components express previously reported resting-state networks.

Identifying the causal structure of systems with multiple dynamic elements is critical to several scientific disciplines. The conventional approach is to conduct statistical tests of causality, for example with Granger Causality, between observed signals that are selected a priori. Here it is posited that, in many systems, the causal relations are embedded in a latent space that is expressed in the observed data as a linear mixture. A technique for blindly identifying the latent sources is presented: the observations are projected into pairs of components -- driving and driven -- to maximize the strength of causality between the pairs. This leads to an optimization problem with closed form expressions for the objective function and gradient that can be solved with off-the-shelf techniques. After demonstrating proof-of-concept on synthetic data with known latent structure, the technique is applied to recordings from the human brain and historical cryptocurrency prices. In both cases, the approach recovers multiple strong causal relationships that are not evident in the observed data. The proposed technique is unsupervised and can be readily applied to any multiple time series to shed light on the causal relationships underlying the data.