Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data
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): is said to be a prima facie cause of if the observations of up to time t help one predict when the corresponding observations of X and Z are available . 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 , one when are given, the other when 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 . The series has the following moving average representation with use of the lag operator L:where , var and , the identity matrix. Assume there exists the autoregressive representation:where .
Suppose that has been decomposed into two vectors and with k and l dimensions, respectively: , where the prime denotes matrix transposition. Denote as the subspace
Geweke’s measure of conditional feedback causality
Now suppose that has been decomposed into three vectors , and with k, l and m dimensions, respectively: . The measure given by Geweke for the linear dependence of on , conditional on , in the time domain (Geweke, 1984) is: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 , 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 and (making partitions) we have:where and have the following moving average expression: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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