A copula approach to assessing Granger causality

Highlights

• Define a novel measure: a model-free, copula-based Granger causality.

• The new measure is able to reveal both nonlinear and higher-order moment causality.

• Copula-based Granger causality outperforms other available methods.

• Top-down GC difference is only observed in V2 → V1 in a visual illusion task.

Abstract

In neuroscience, as in many other fields of science and engineering, it is crucial to assess the causal interactions among multivariate time series. Granger causality has been increasingly used to identify causal influence between time series based on multivariate autoregressive models. Such an approach is based on linear regression framework with implicit Gaussian assumption of model noise residuals having constant variance. As a consequence, this measure cannot detect the cause-effect relationship in high-order moments and nonlinear causality. Here, we propose an effective model-free, copula-based Granger causality measure that can be used to reveal nonlinear and high-order moment causality. We first formulate Granger causality as the log-likelihood ratio in terms of conditional distribution, and then derive an efficient estimation procedure using conditional copula. We use resampling techniques to build a baseline null-hypothesis distribution from which statistical significance can be derived. We perform a series of simulations to investigate the performance of our copula-based Granger causality, and compare its performance against other state-of-the-art techniques. Our method is finally applied to neural field potential time series recorded from visual cortex of a monkey while performing a visual illusion task.

Introduction

Assessing causal relationship between time series is an important problem with a wide range of applications in science and engineering. The last decade in neuroscience has witnessed considerable interest in a class of techniques called Granger causality (GC) to provide a statistically principled way to measure directional influences of neural interactions. The basic idea of this method was introduced by Wiener (Wiener, 1956), and later formalized by Granger (Granger, 1969) in the context of linear regression models of stochastic processes. Specifically, if the past of one time series can be used to facilitate the prediction of the future of another time series, then we say there is a Granger causal influence from the former to the latter (Granger, 1969). A spectral formulation of GC (Ding et al., 2006) can be readily applied to neurophysiological data because causal influences between neuronal populations often depend on frequency-specific oscillatory synchrony (Bressler and Seth, 2011). Recent works applying this method to neural data have generated new insights into functional organization of brain networks (Brovelli et al., 2004, Gregoriou et al., 2009, Hu et al., 2011, Saalmann et al., 2012).

Typically Granger causality is inferred parametrically through autoregressive models of time series data (Ding et al., 2006). This approach is based on linear regression framework with implicit Gaussian assumption of model noise residuals having constant variance. As such, any cause-effect relationship due to high-order statistics is not captured by the model. It therefore merely measures the Granger causality in mean, does not quantify high-order moment causality. In addition, linear regression modeling framework is only limited to detecting linear cause-effect relationship, thus cannot be used to reveal nonlinear causality. These limitations inevitably hinder wide applications of the method in which the nonlinearity may occur and/or the causal relations go beyond simple mean dependence. It has been shown in financial market series (Asai et al., 2006, Cheung and Ng, 1996) that noise variance often exhibits time-varying volatility. Such signal-dependent noise has also been observed in both in vivo neuronal recordings (Luo et al., 2013) and neuroimaging data (Kruger et al., 2001). Indeed, there exists extensive evidence for skewed distributions in neuroscience, typically with heavy tails and asymmetric variations (Buzsáki and Mizuseki, 2014). Conventional Granger causality is not able to capture these high-order causal effects. Although a few attempts have been made to tackle the problems (Chen et al., 2004, Ancona et al., 2004, Marinazzo et al., 2008, Hafer and Herwartz, 2008), methods to account for both nonlinear and high-order causality are still lacking. As such, we propose a novel model-free, copula-based Granger causality measure by virtue of the log-likelihood ratio statistic (Geweke, 1984, Luo et al., 2013). We show that our method is able to detect not only nonlinear causality, but also high-order causality in the time series.

Central to our method is the predictability of time series, which is represented by conditional probability distribution. In our method, the Granger causality from a time series X to Y is assessed by comparing the predictability (likelihood) of time series Y with and without information from the history of the other time series X. If the likelihood ratio is less than one, there is causal influence from X to Y, and if the likelihood ratio is one, there is no causality. The causal influence in the opposite direction (from Y to X) can be analyzed in a similar way. Such definition is essentially to assess Granger causality in distribution, which is in stark contrast to conventional Granger causality in mean. Practical issues in estimating Granger causality in distribution are addressed by using conditional copula. Noting that, under normal assumption, our log-likelihood ratio based method is reduced to the model-based linear GC measure. The major advantage of our method is that it is model-free, thus can be used to reveal nonlinear, high-order causality. The use of the likelihood ratio statistics allows us to perform statistical significance test based on the chi-squared test. Additionally, the resampling techniques can also be used to build a baseline null-hypothesis distribution from which statistical significance is derived.

The main contributions of the paper are as follows. First, we introduce a model-free Granger causality measure based on copula that can be used to reveal nonlinear, high-order causality. Second, we derive an efficient estimation procedure using conditional copula. Third, we devise a variability assessment strategy based on the ideas of resampling that allows statistical testing of significance to be carried out on the derived Granger causality.

We performed extensive simulations to study the strengths and weaknesses of the proposed method compared to other available methods, and we showed that our method is able to correctly identify the causal relationship of time series generated by both linear and nonlinear regression models with different forms of heteroskedasticity (time-varying variance). In addition, we found our method is robust to short data lengths, varying noise levels and even outliers in the time series. Subsequently, we applied our approach to the neural field potential data from monkey while performing a visual illusion task (Wilke et al., 2003) to demonstrate the usefulness and advantage of the proposed method in revealing bottom-up and top-down influence in the cortex over conventional Granger causality measure.