Instantaneous and lagged measurements of linear and nonlinear dependence between groups of multivariate time series: frequency decomposition

The terms “multivariate time series”, “multiple time series”, and “vector time series” have identical meaning in this paper.

For general notation and definitions, see e.g. Brillinger (1981) for stationary multivariate time series analysis, and see e.g. Mardia et al (1979) , and defined as:

Eq. 1 It will be assumed throughout that ω X and ω Y each have zero mean.

Let:

Eq. 3

The discrete Fourier transforms in Eq. 1 and Eq. 2 contain both phase and amplitude information, which carries over to the covariance matrices in Eq. 3,Eq. 4,Eq. 5,and Eq. 6. This means that for the analysis of phase information, the amplitudes must be factored out by an appropriate normalization method. This is achieved by using the following definition for the normalized complex-valued discrete Fourier transform vector: Eq. 7 ( )

X X X and:

Eq. 8

( )

Note that this normalization operation, although deceivingly simple, is a highly nonlinear transformation.

The corresponding covariance matrices containing phase information (without amplitude information) are:

Eq. 9 Eq. 11

Note that the normalization used in Eq. 7 and Eq. 8 will be the basis for the analysis of phase synchronization between the multivariate time series X and Y.

The instantaneous, zero-phase, zero-lag covariance matrix corresponding to a multivariate time series at frequency ω, is simply the real part of the Hermitian covariance matrix at frequency ω, i.e. ( )

To justify this, consider the multivariate time series

In a first step, filter the time series to leave exclusively the frequency ω component.

Filtered jt ω

. Note that, by construction, the spectral density of ( ) Filtered jt ω X is zero everywhere except at frequency ω.

In a second step, compute the instantaneous, zero-lag, zero phase shifted, time domain, symmetric covariance matrix for the filtered time series ( ) Filtered jt ω X at frequency ω:

Eq. 13

)( )

Finally, by making use of Parseval’s theorem for the filtered time series, the following relation holds: given by Eq. 3 above.

These arguments apply identically to the normalized time series, as in Eq. 7 to Eq. 12 above, when considering the phase-information cross-spectra. This means that the instantaneous, zero-phase, zero-lag covariance matrix corresponding to a normalized multivariate time series X at frequency ω, is simply the real part of the phase-information Hermitian covariance matrix at frequency ω, i.e. ( )

The section entitled “Appendix 1” gives a brief description of the problems that arise in neurophysiology, where any measure of dependence is highly contaminated with an instantaneous, non-physiological contribution due to volume conduction and low spatial resolution.

The definitions presented here are largely motivated by the seminal paper on linear feedback by Geweke (1982).

The measure of linear dependence between time series X and Y at frequency ω is defined as:

where M denotes the determinant of M. The matrix in the numerator of Eq. 15 is a blockdiagonal matrix, with 0 denoting a matrix of zeros, which in this case is of dimension q p × .

This measure of linear dependence is expressed as the sum of the lagged linear dependence

and instantaneous linear dependence

The measure of instantaneous linear dependence is defined as:

Eq. 17 Finally, the measure of lagged linear dependence is: All three measures are non-negative. They take the value zero only when there is independence of the pertinent type (lagged, instantaneous, or both).

Not that the measure of linear dependence ( )

in Eq. 15 can be interpreted as follows:

Eq. 19

was defined as the general coherence in Pascual-Marqui (2007a; see Eq. 7 therein):

Eq. 20

( )

Some relevant literature that motivated the definition of the general coherence

in the previous study (Pascual-Marqui 2007a) follows. In the case of real-valued stochastic variables, Mardia et al (1979) review several “measures of correlation between vectors”. In particular, Kent (1983) proposed a general measure of correlation that is closely related to the vector alienation coefficient (Hotelling 1936, Mardia et al 1979). This measure of general coherence is also equivalent to the coefficient of determination as defined by Pierce (1982). All these definitions can be straightforwardly generalized to the complex valued domain.

In order to illustrate and further motivate these measures of linear dependence, a detailed analysis for the simple case of two univariate time series is presented.

In the case that the two time series are univariate, the measure of linear dependence ( )

in Eq. 15 is:

where:

In Eq. 22, ρ is the ordinary squared coherence (see e.g. Equation 3in Nolte et al 2004).

The measure of instantaneous linear dependence is:

Re

Note that we can define the instantaneous

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