VariableLag Granger Causality for Time Series Analysis Granger causality is a fundamental technique for causal inference in time series data, commonly used in the social and biological sciences. Typical operationalizations of Granger causality make a strong assumption that every time point of the effect time series is influenced by a combination of other time series with a fixed time delay. However, the assumption of the fixed time delay does not hold in many applications, such as collective behavior, financial markets, and many natural phenomena. To address this issue, we develop variablelag Granger causality, a generalization of Granger causality that relaxes the assumption of the fixed time delay and allows causes to influence effects with arbitrary time delays. In addition, we propose a method for inferring variablelag Granger causality relations. We demonstrate our approach on an application for studying coordinated collective behavior and show that it performs better than several existing methods in both simulated and realworld datasets. Our approach can be applied in any domain of time series analysis.

Robust brain causality network construction based on Bayesian multivariate autoregression Abstract   Background  Cognitive processes involve information integration among multiple encephalic regions, which can be measured by causal networks. However, the estimation of causal networks by means of some traditional methods with the least square will lead to distorted networks because of the unexpected outlier noise and the small number of signal samples in real applications.    New method  In this work, we adopted Bayesian inference to estimate parameters in a multivariate autoregression model (MVAR), to restrain the influence of outliers.    Results  Through the simulation study, we observed that our proposed method can efficiently suppress outlier influence and shows stable performance when sample sizes become small. Application to real motor imagery functional magnetic resonance imaging (fMRI) also revealed that the proposed approach can capture the inherent hemispheric lateralization of motor imagery even with a small number of fMRI samples.    Comparison with existing methods  We compared our proposed Bayesianbased Granger analysis with traditional Granger causality analysis.    Conclusions  The analyses conducted in the current work demonstrate the robustness of Bayesianbased Granger analysis to outlier conditions or physiological signals with small sample sizes.