Among approaches to quantify the coupling between multivariate time series, we can distinguish between linear and non-linear methods as well as approaches based on bivariate and multivariate estimators. For a long time the standard methods of establishing relations between signals have been cross-correlation and coherence. However, in case of these bivariate measures we don’t know if the two channels are really coupled or if there is another channel that drives them.
This point may be elucidated by simulation illustrating the situation where the ac- tivity is measured in different distances from the source (Figure 3.4).The signals in channels 2-5 were constructed by delaying the signal from channel one (the source) by 1, 2, 3, 4 samples, respectively, and adding the noise in each step. The coherences
FIGURE 3.4: Propagations estimated for simulation scheme shown in a) by means of: b) bivariate coherences, c) bivariate DTF model, d) multivariate DTF. In panel b) above the diagonal, the magnitudes of coherences, below diagonal, phases of the coherences are shown. From phases, the delays (values printed in each box) were calculated. For c) and d) in each box DTF as a function of frequency, on the diagonal power spectra; propagation from channels labeled above to the channels labeled at the left of the picture. Resulting propagation schemes shown in the right part of each
were calculated between all signals pairwise. For the peak frequency the phases were found and assuming the linear phase relations, the delays were calculated. The re- sulting scheme of propagation presented inFigure 3.4 b)shows many false flows. Similar results were obtained for bivariate AR model (Figure 3.4 c). This situation is common to all bivariate methods; namely the propagation is found in each case where the phase difference is present [Blinowska et al., 2004a]. Sometimes, for bi- variate measures even the reverse propagation may be found [Kus et al., 2004]. The results obtained by means of the multivariate method (DTF) show the correct scheme of propagation (Figure 3.4 d).
Another problem is that, from correlation or coherence (also partial coherence), it is not possible to detect reciprocal flows (interaction in both directions). Such pat- terns of propagation may be found by means of estimators based on the causality principle.
All non-linear methods described above are bivariate, so they suffer from the dis- advantages pointed out above. There were some attempts to apply non-linear mea- sures for number of channels bigger than two. Chen et al. [Chen et al., 2004] pro- posed conditional extended Granger causality and applied it to 3 channel simulated non-linear time series, but the advantages over the linear approach were not clearly demonstrated and the influence of noise was not studied. The method requires deter- mination of embedding dimension and neighborhood size, which are difficult to find in an objective and optimal way. The same objections concern another methods aim- ing to extend non-linear estimators for a number of channels higher than two. The problems concerning state space reconstruction and noise sensitivity become even more serious for a higher number of channels.
In the study devoted to comparison of linear and non-linear methods of cou- pling [Netoff et al., 2006], non-linear estimators: mutual information, phase cor- relation, continuity measure2were compared with correlation in case of non-linear signals in the presence of noise. The authors found that any method that relies on an accurate state space reconstruction will be inherently at a disadvantage over measures which do not rely on such assumptions. Another finding was the high sensitivity to noise of non-linear estimators. The authors conclude: “We have been as guilty as
any of our colleagues in being fascinated by the theory and methods of nonlinear dynamics. Hence we have continually been surprised by robust capabilities of lin- ear CC (correlation) to detect weak coupling in nonlinear systems, especially in the presence of noise.”
We can conclude that non-linear methods are not recommended in most cases. They might be used only when there is clear evidence that there is a good reason to think that there is non-linear structure either in data themselves or in the interde- pendence between them [Pereda et al., 2005]. We have to bear in mind that many systems composed of highly non-linear components exhibit an overall linear type of behavior; the examples may be some electronic devices, as well as brain signals
2Measure similar to GS testing for continuity of mapping between neighboring points in one data set to
such as electroencephalograms or local field potentials (LFP). This problem will be further discussed in Sect. 4.1. In this context it is worth mentioning that some linear methods, e.g., DTF, work quite well also for non-linear time series [Winterhalder et al., 2005].
In the above quoted reference comparison of linear techniques inferring directed interactions in multivariate systems was carried out. Partial phase spectrum (PC) and three methods based on Granger causality: GCI, DTF and PDC, were tested in re- spect to specificity in absence of influences, correct estimation of direction, direct versus indirect transmissions, non-linearity of data, and influences varying in time. All methods performed mostly well with some exceptions. PC failed in the most important aspect—estimation of direction. It was due to the large errors in phase es- timation for particular frequencies and the inability to detect reciprocal interactions. GCI failed in the presence of non-linearities. DTF did not distinguish direct from cascade flows, but when such distinction is important dDTF may be used. PDC for signals with a non-linear component gave correct results only for very high model orders, which makes the method hardly applicable for strongly non-linear signals be- cause of limitations connected with the number of parameters of MVAR mentioned inSect. 3.3.2.3.2.
More important drawbacks of PDC were pointed out by [Schelter et al., 2009], namely:
i) PDC is decreased when multiple signals are emitted from a given source, ii) PDC is not scale-invariant, since it depends on the units of measurement of the source and target processes,
iii) PDC does not allow conclusions on the absolute strength of the coupling. The authors proposed the renormalization of PDC similar to the one used in the definition of DTF, which helped in alleviation of the above problems.
Point iii) is illustrated inFigure 3.5.In a situation when activity is emitted in sev- eral directions PDC shows weaker flows than in the situation when the same activity is emitted in one direction only. Another feature of PDC is a weak dependence on frequency. The PDC spectrum is practically flat, whereas DTF spectrum (especially for non-normalized DTF) reflects the spectral characteristics of the signal. An exam- ple of application of DTF and PDC to experimental data (EEG) will be given in Sect. 4.1.6.2.2.
We can conclude that the methods estimating direction which perform best are the linear multivariate methods based on the Granger causality concept, and in the frequency domain the methods or choice are DTF (or dDTF) and renormalized PDC.
FIGURE 3.5: Comparison of DTF (panel B) and PDC (panel D) for simulation scheme A. Resulting schemes of propagation for DTF and PDC below corresponding panels. Thickness of arrows proportional to the flow intensities. The convention of presenting DTF/PDC is as inFigure 3.4.Note that the weak flow from channel 5 is enhanced by PDC.