Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data

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Abstract

It is often useful in multivariate time series analysis to determine statistical causal relations between different time series. Granger causality is a fundamental measure for this purpose. Yet the traditional pairwise approach to Granger causality analysis may not clearly distinguish between direct causal influences from one time series to another and indirect ones acting through a third time series. In order to differentiate direct from indirect Granger causality, a conditional Granger causality measure in the frequency domain is derived based on a partition matrix technique. Simulations and an application to neural field potential time series are demonstrated to validate the method.

Introduction

The concept of causality introduced by Wiener (1956) and formulated by Granger (1969) has played a considerable role in investigating the relations among stationary time series. The original definition of Granger (1969), which is well named as Granger causality, refers to the improvement in predictability of a series that derives from incorporation of the past of a second series, above the predictability based solely on the past of the first series. This definition only involves the relation between two time series. As pointed out by Granger, 1969, Granger, 1980, if a third series is taken into account, a spurious or indirect causality due to the third series may be detected. Then he defined a prima facie cause (Granger, 1980): Yis said to be a prima facie cause of X if the observations of Y up to time t (Y(τ):τt) help one predict X(t+1) when the corresponding observations of X and Z are available (X(τ),Z(τ):τt). We refer to this idea as conditional Granger causality since it gives a measure of causality between two time series, X and Y, conditional on a third, Z. Evaluation of this conditional Granger causality in the time domain is fairly straightforward through comparison of two predictions of X(t+1), one when (X(τ),Z(τ):τt) are given, the other when (X(τ),Y(τ),Z(τ):τt) are given. However, evaluating causality by frequency decomposition may allow more meaningful interpretations in cases where oscillations are involved.

After giving clear measurements of linear dependence and feedback between two blocks of time series (Geweke, 1982), he also presented a measure of conditional linear dependence and feedback (Geweke, 1984). Both a time domain measure, consistent with that of Granger, and its frequency decomposition were given. Although Hosoya presented some improvements on Geweke’s methods (both bivariate (Hosoya, 1991) and conditional versions (Hosoya, 2001)), they have not been widely accepted because his time domain implementation departs from Granger’s original idea, and its physical interpretation is less clear.

We point out that Geweke’s use of the term “feedback” is equivalent to “causality” in the present discussion. In applying Geweke’s frequency-domain conditional Granger causality measure to neural field potential data, we have found that negative values, which have no meaning in terms of causality, may occur at some frequencies. This finding casts doubt on the applicability of Geweke’s method for neural time series analysis. We believe that the negative values result from the lack of identity of estimates of the same spectrum when different autoregressive (AR) models are used. This non-identity of different estimates of the same spectrum is a general practical problem in numerical analysis that causes errors in Geweke’s implementation because it requires the estimates to be identical. In this paper, we employ a partition matrix method to overcome this problem. Comparison of the results from our procedure with Geweke’s original procedure, clearly shows the validity of the current procedure. In the following sections: we first provide an introduction to Granger causality; then present an overview of Geweke’s procedure on conditional causality, pointing out the importance of obtaining a correct measure; and then derive our procedure. Finally, results of simulations and application to neural field potential time series data are provided.

Section snippets

Background

Consider a multiple stationary time series of dimension n,W={wt}. The series has the following moving average representation with use of the lag operator L:wt=A(L)εt,where E(εt)=0, var(εt)=Σ and A0=In, the n×n identity matrix. Assume there exists the autoregressive representation:B(L)wt=εt,where B0=In.

Suppose that wt has been decomposed into two vectors xt and yt with k and l dimensions, respectively: wt=(xt,yt), where the prime denotes matrix transposition. Denote Wt1 as the subspace

Geweke’s measure of conditional feedback causality

Now suppose that wt has been decomposed into three vectors xt, yt and zt with k, l and m dimensions, respectively: wt=(xt,yt,zt). The measure given by Geweke for the linear dependence of X on Y, conditional on Z, in the time domain (Geweke, 1984) is:FYX|Z=lnvar(xt|Xt1,Zt1)var(xt|Xt1,Yt1,Zt1),which is consistent with Granger’s definition of a prima facie cause (Granger, 1980).

Time series prediction is achieved by the fitting of MVAR models. In order to implement Eq. (12), two MVAR

Partition matrix improvement

For three blocks of time series xt,yt,zt, we can fit a three-variable MVAR model as in Eq.(14) and we can also derive its frequency domain expression as in Eq. (18). From Eq. (18), writing an expression only for X(λ)and Z(λ) (making partitions) we have:X(λ)Z(λ)=Hxx(λ)Hxz(λ)Hzx(λ)Hzz(λ)E¯x(λ)E¯z(λ),where E¯x(λ) and E¯z(λ) have the following moving average expression:E¯x(λ)E¯z(λ)=Ex(λ)Ez(λ)+Hxx(λ)Hxz(λ)Hzx(λ)Hzz(λ)1×Hxy(λ)Hzy(λ)Ey(λ).We realize that Eq. (23) is actually a summation of multiple

Application to simulated data

We performed conditional Granger causality analysis on the delay driving and sequential driving systems presented above in Section 2. For the delay driving case (Section 2.1 and Fig. 1), the Granger causality spectrum from yto z, conditional on x, is presented in Fig. 3. It is obvious from Fig. 3 that the conditional Granger causality measure eliminated the indirect causal influence of y on z which appeared in Fig. 1(b). For the sequential driving case (Section 2.2 and Fig. 2), the Granger

Acknowledgements

The work was supported by US NIMH grants MH64204, MH070498 and MH71620, and NSF grant 0090717.

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