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# Robust Statistics for Describing Causality in Multivariate Time Series.

*Proceedings of the Neural Information Processing Systems Mini-Symposium on Causality in Time Series*, PMLR 12:30-64, 2011.

#### Abstract

A widely agreed upon definition of time series causality inference, established in the seminal 1969 article of Clive Granger (1969), is based on the relative ability of the history of one time series to predict the current state of another, conditional on all other past information. While the Granger Causality (GC) principle remains uncontested, its literal application is challenged by practical and physical limitations of the process of discretely sampling continuous dynamic systems. Advances in methodology for time-series causality subsequently evolved mainly in econometrics and brain imaging: while each domain has specific data and noise characteristics the basic aims and challenges are similar. Dynamic interactions may occur at higher temporal or spatial resolution than our ability to measure them, which leads to the potentially false inference of causation where only correlation is present. Causality assignment can be seen as the principled partition of spectral coherence among interacting signals using both auto-regressive (AR) modeling and spectral decomposition. While both approaches are theoretically equivalent, interchangeably describing linear dynamic processes, the purely spectral approach currently differs in its somewhat higher ability to accurately deal with mixed additive noise. Two new methods are introduced 1) a purely auto-regressive method named Causal Structural Information is introduced which unlike current AR-based methods is robust to mixed additive noise and 2) a novel means of calculating multivariate spectra for unevenly sampled data based on cardinal trigonometric functions is incorporated into the recently introduced phase slope index (PSI) spectral causal inference method (Nolte et al. 2008). In addition to these, PSI, partial coherence-based PSI and existing AR-based causality measures were tested on a specially constructed data-set simulating possible confounding effects of mixed noise and another additionally testing the influence of common, background driving signals. Tabulated statistics are provided in which true causality influence is subjected to an acceptable level of false inference probability.