Estimating Granger causality from Fourier and wavelet transforms of time series data

Extracting information flow in networks of coupled dynamical systems from the time series measurements of their activity is of great interest in physical, biological and social sciences. Such knowledge holds the key to the understanding of phenomena ranging from turbulent fluids to interacting genes and proteins to networks of neural ensembles. Granger causality [1] has emerged in recent years as a leading statistical technique for accomplishing this goal. The definition of Granger causality [1] is based on the theory of linear prediction [2] and its original estimation framework requires autoregressive (AR) modeling of time series data [1,3]. Such parametric Granger causality and associated spectral decompositions have been applied in a wide variety of fields including condensed matter physics [4], neuroscience [5,6,7,8], genetics [9], climate science [10,11], and economics [1,12]. However, the parametric modeling methods often encounter difficulties such as uncertainty in model parameters and inability to fit data with complex spectral contents [13]. On the other hand, the Fourier and wavelet transform-based nonparametric spectral methods are known to be free from such difficulties [13] and have been used extensively in the analysis of univariate and multivariate experimental time series [14,15]. A weakness of the current nonparametric framework is that it lacks the ability for estimating Granger causality. In this Letter, we overcome this weakness by proposing a nonparametric approach to estimate Granger causality directly from Fourier and wavelet transforms of data, eliminating the need of explicit AR modeling. Time-domain Granger causality can be obtained by integrating the corresponding spectral representation over frequency [3]. Below, we present the theory and apply it to simulated time series.

Granger causality: the parametric estimation approach. Granger causality [1] is a measure of causal or directional influence from one time series to another and is based on linear predictions of time series. Consider two simultaneously recorded time series: X 1 : x 1 (1), x 1 (2), …, x 1 (t), …; X 2 : x 2 (1), x 2 (2), …, x 2 (t), … from two stationary stochastic processes (X 1 , X 2 ). Now, using AR representations, we construct bivariate linear prediction models for x 1 (t) and x 2 (t):

along with the univariate models:

Here, ǫ’s are the prediction errors. If var(ǫ 1|2 (t)) < var(ǫ 1 (t)) in some suitable statistical sense, then X 2 is said to have a causal influence on X 1 . Similarly, if var(ǫ 2|1 (t)) < var(ǫ 2 (t)), then there is a causal influence from X 1 to X 2 . These causal influences are quantified in time domain [3] by F j→i = ln var(ǫ i (t)) var(ǫ i|j (t))

, where i = 1, 2 and j = 2, 1.

Experimental processes are often rich in oscillatory content, lending themselves naturally to spectral analysis. The spectral decomposition of Granger’s time-domain causality was proposed by Geweke in 1982 [3]. To derive the frequency-domain Granger causality, we start with Eq. (1-2). We rewrite these equations in a matrix form with a lag operator L:

where

where the components of the coefficient matrix

Then, the spectral density matrix S(f ) is given by

where * denotes matrix adjoint. To examine the causal influence from X 2 to X 1 , one needs to look at the auto-spectrum of x 1 (t)-series, which is

Here, because of the cross-terms in this expression for S 11 , the causal power contribution is not obvious. Geweke [3] introduced a transformation that eliminates the cross terms and makes an intrinsic power term and a causal power term identifiable. For X 1 -process, this transformation is achieved by left-multiplying Eq. ( 4) on both sides with 1 0 -Σ 12 /Σ 11 1

, which yields:

where

, where the first term accounts for the intrinsic power of x 1 (t) and the second term for causal power due to the influence from X 2 to X 1 . Since Granger causality is the natural logarithm of the ratio of total power to intrinsic power [3], causality from X 2 to X 1 (or, 2 to 1) at frequency f is

using the expressions for S 11 and Σ22 obtained after the transformation. Next, by taking the transformation matrix as 1 -Σ 12 /Σ 22 0 1 and performing the same analysis, one can get Granger causality I 1→2 (f ) from X 1 to X 2 , the expression for which can be obtained just by exchanging subscripts 1 and 2 in Eq. ( 8). Geweke [3] showed that the time-domain measure is theoretically related to the frequency-domain measure as

all processes of practical interest, the equality holds.

From the above discussion, it is clear that the estimation of frequency-domain Granger causality requires noise covariance and transfer function which are obtained as part of the AR data modeling. The mathematics behind this parametric approach to obtain these quantities is well-established. However, for nonparametric methods the current estimation framework does not contain provisions for computing these quantities. Moreover, the parametri

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