Granger causality with signal-dependent noise

Abstract

It is generally believed that the noise variance in in vivo neuronal data exhibits time-varying volatility, particularly signal-dependent noise. Despite a widely used and powerful tool to detect causal influences in various data sources, Granger causality has not been well tailored for time-varying volatility models. In this technical note, a unified treatment of the causal influences in both mean and variance is naturally proposed on models with signal-dependent noise in both time and frequency domains. The approach is first systematically validated on toy models, and then applied to the physiological data collected from Parkinson patients, where a clear advantage over the classical Granger causality is demonstrated.© 2011 Elsevier Inc. All rights reserved. IntroductionThe past few years have witnessed a significant growth in the application of Granger causality to various research areas, especially to physiological recordings and functional MRI data. Proposed by Granger (1969) and further extended to the frequency domain by Geweke (1982), Granger causality has become a powerful tool to detect causal influences and functional connectivity from the huge amount of temporal data easily accessible nowadays.The classical Granger causal analysis is based on an autoregressive (AR) model (Friston, 2009a), which assumes that the current state of a process can be predicted by a linear function of its previous states plus a white noise or innovation process. A process is said to be the cause of another process if the inclusion of its past information can improve the prediction of the second process, i.e., reduce the variance of the prediction error. In spite of its easy implementation, wide justification and successful applications, some of the working assumptions might be an over-simplification when performing time series analysis in some situations. One particular scenario is the violation of time-invariant noise, that is, the volatility changes over time. As a common phenomenon in financial market series, similar characteristics have been observed in many physiological recordings, such as the data from epilepsy patients, Parkinson patients and so on. Because of the close relationship of many series, it is obvious to conjecture that changes in the volatility of one series may have an impact on the mean activity or volatility of another series and vice versa, which means that causal influences may happen on the second order statistics or there may be feedback between the first and second moments of the series. In these scenarios, a simple AR model obviously cannot capture these effects and a more detailed modeling of the volatility of the time series as well as the corresponding causality inference techniques are desired.The development of specific models for changing volatility, or say, conditionally heteroskedastic data, has long been launched, primarily motivated by the research in the literature of economics. A natural development was boosted by Engle's invention of the autoregressive conditional heteroskedasticity (...

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