Chunfeng Yang
email: chunfeng.yang@univ-rennes1.fr

Régine Le

Bouquin Jeannès

Gérard Faucon
email: faucon.gerard@gmail.com

H Shu

Extracting Information on Flow Direction in Multivariate Time Series

Keywords: Directed transfer function, ordinary coherence, partial coherence, phase slope index. I

Phase slope index is a measure which aims at detecting causal relation of interdependence in multivariate time series. One drawback of this approach relies in its incapability to distinguish the direct and indirect relations. So, in order to identify only direct relations, we propose to replace the ordinary coherence function used in the phase slope index with the partial coherence. Furthermore, we consider and compare two estimators of the coherence functions, the first one based on Fourier transform and the second one on an autoregressive model. In order to cope with the difficult issue of bidirectional flow, which cannot be addressed by the coherence based phase slope index, we propose another index based on the directed transfer function. Experimental results support the relevance of the new indices, both based on autoregressive modeling, in multivariate time series.

structures get involved during this FOA, in particular, in which way some structure can "drive" other ones. In order to detect direct causal relations and distinguish patterns of connectivity, we propose a new phase slope index based on partial coherence (PC) function instead of ordinary coherence function. If ordinary coherence function can be obtained using Fourier transform as in [START_REF] Nolte | Robustly estimating the flow direction of information in complex physical systems[END_REF], another approach consists in computing ordinary and partial coherences using autoregressive (AR) modeling of signals as proposed hereafter. In a second step, to face with the symmetric character of the coherence function and be able to detect bidirectional relations between signals, we introduce another phase slope index based on directed transfer function (DTF) [START_REF] Kaminski | A new method of the description of the information flow in the brain structures[END_REF] instead of ordinary coherence function.

II. METHODS PSI is a method to evaluate the information flow direction in multivariate time series [START_REF] Nolte | Robustly estimating the flow direction of information in complex physical systems[END_REF]. After recalling the principle of this method, we derive the expression of a new PSI based on the partial coherence function, and then the expression of another new index based on the DTF.

A. PSI Principle

The basic hypothesis relies on the phase linearity between signals. PSI is based on the slope of the phase of the crossspectrum between two time series and . The idea is to define an average measure in such a way that this quantity properly represents relative time delays between signals. This quantity, noted , is defined by [START_REF] Veeramani | Measuring the direction and the strength of coupling in nonlinear systems-a modeling approach in the state space[END_REF] where is the coherence function between signals and , is the frequency resolution, Im denotes the imaginary part and the asterisk denotes conjugate value. is the set of frequencies over which the index is computed. In this equation, the coherence function used by Nolte in [START_REF] Nolte | Robustly estimating the flow direction of information in complex physical systems[END_REF] is the ordinary coherence between signals and , noted as hereafter, and defined as follows: [START_REF] Carter | Coherence and time delay estimation[END_REF] where and are the auto-spectral density functions of signals and respectively, and is the cross-spectral density function. They are given by [START_REF] Sabesan | Information flow and application to epileptogenic focus localization from intracranial EEG[END_REF] where is the Fourier transform of the signal and denotes the expectation. The magnitudes of the coherences allow weighting the phase difference between two consecutive frequencies and, consequently, decrease its impact when they are low. The sign of PSI indicates the flow direction and its magnitude increases along with the delay. Given (1)-(3), when the information flow is from to , is positive. In the following, PSI using the OC is named PSI-OC: -

Given (3), the auto-spectral and cross-spectral density functions may be obtained by two different techniques, either from direct Fourier transforms of signals and as in [START_REF] Nolte | Robustly estimating the flow direction of information in complex physical systems[END_REF], or from AR modeling. In the first one, the expectation required to get the spectral density functions is obtained by averaging and overlap. In the second one, the methodology in the multivariate case can be derived as follows.

where each signal depends not only on its own past but also on the past of the other signals [START_REF] Ding | Granger causality: Basic theory and application to neuroscience[END_REF]

The coefficient evaluates the linear interaction of on , whatever , . These coefficients are estimated by least squares method [START_REF] Biscainho | AR model estimation from quantized signals[END_REF]. Using the lag operator , we rewrite (5) in the following form: . . . . . . [START_REF] Wendling | Interictal to ictal transition in human temporal lobe epilepsy: Insights from a computational model of intracerebral EEG[END_REF] where (see the equation at the bottom of the page).

Applying the Fourier transform to both sides of [START_REF] Wendling | Interictal to ictal transition in human temporal lobe epilepsy: Insights from a computational model of intracerebral EEG[END_REF] [START_REF] Ding | Granger causality: Basic theory and application to neuroscience[END_REF], and the symbol denotes Hermitian transpose. Finally, the corresponding PSI-OC can be calculated using (1), [START_REF] Carter | Coherence and time delay estimation[END_REF], and [START_REF] Bendat | Random Data-Analysis and Measurement Procedures[END_REF].

B. PSI Based on Partial Coherence (PC)

The partial coherence function gives the level of coupling between two signals and when the influence of the other signals is removed [START_REF] Bendat | Random Data-Analysis and Measurement Procedures[END_REF]. It is defined by [START_REF] Chen | Analyzing multiple nonlinear time series with extended Granger causality[END_REF] .

C. PSI Based on Directed Transfer Function (DTF)

Directed transfer function was proposed by Kaminski and Blinowska in [START_REF] Kaminski | A new method of the description of the information flow in the brain structures[END_REF]. From [START_REF] Biscainho | AR model estimation from quantized signals[END_REF], the definition of DTF from to is defined by

(15)
where is the element of the matrix , corresponding to the contribution of signal onto signal . is a multichannel causality measure based on the AR model and measures the flow from channel to channel . It is constructed using elements of the transfer matrix of the AR model. This matrix is not symmetrical and, the value of the DTF, at a given frequency, represents a ratio between the inflow to channel from channel to all the inflows to channel . So, contrary to ordinary coherence, it can deal with bidirectional flow and answer to the question of effective connectivity. DTF is able to quantify the strength of such connectivity but the supplementary interest of the PSI based indicator relies on the phase information carried by each element of the transfer function used in the DTF. The relevance of PSI-DTF consists in using on the one hand the amplitude of the directed transfer function and on the other hand the variation of its phase between two adjacent frequencies. So, averaging over the frequency band allows assessing not only the strength of the coupling but also the importance of the delay. In the same way as previously, we define a phase slope index based on DTF, noted PSI-DTF, as follows: -(16)

III. RESULTS

We tested our approaches on models simulating practical situations. In our simulations, the model order is given by Akaike's criterion [START_REF] Ding | Granger causality: Basic theory and application to neuroscience[END_REF].

A. First Model

For the first linear stochastic system we considered, the three following signals were generated:

(17) where , , 2, 3, were independent white Gaussian noises with zero means and unit variances; the parameter was introduced to consider two patterns of causal interactions, either direct or indirect relationships. The sampling frequency was 200 Hz.

In Section II, ordinary and partial coherence functions are used to obtain two phase slope indices, respectively PSI-OC and PSI-PC. As indicated above, spectra can be obtained either by Fourier transform or using multivariate AR modeling. In the following, we denote by PSI-OC(FFT) (resp. PSI-PC(FFT)) the case where the ordinary (resp. partial) coherence is estimated by fast Fourier transform. In the same way, PSI-OC(AR) (resp. PSI-PC(AR)) denotes the case where the ordinary (resp. partial) coherence is estimated by AR modeling.

When using FFT, spectra were obtained using a sliding window of 64-point length and a 50% overlap. As for AR modeling, it was realized on the whole signal length. Simulation was carried out 100 times on 1024-point signals; the means and standard deviations (sd) were computed and reported in Table I. From this table, two main results can be drawn: i) both PSI-OC and PSI-PC perfectly point out the flow direction of information among the three signals. However, PSI-PC allows distinguishing the direct and indirect relations whatever the computation mode (FFT or AR). For example, in the case , when , corresponding to indirect relation, -is close to zero, and when , corresponding to direct relation, -increases significantly, whereas systematically presents important values whatever ; ii) the results obtained with AR modeling are preferred for two reasons: when there is some direct relation between two signals, the mean values are generally higher with AR modeling and the corresponding sd are lower.

The robustness of these methods against additive disturbing independent noises has also been tested. Choosing signal-tonoise ratios in the range did not affect the performance of the indicators and the same conclusions hold.

We also considered the case of nonlinearities as that described in [START_REF] Chen | Analyzing multiple nonlinear time series with extended Granger causality[END_REF] in this context of unidirectional flow. In this situation, it came out that the proposed indicators remained robust when

B. Second Model

For the second linear stochastic system, the three following signals were generated:

(18) where , , 2, 3, were independent white Gaussian noises with zero means and unit variances. In this situation, bidirectional causalities existed between signals and . The sampling frequency was 200 Hz.

Results on PSI-OC and PSI-PC

Since we already found that the indices based on AR modeling outperform those based on FFT, we only present results on AR modeling. Simulations were carried out 500 times on 1024-point signals, the means and sd were derived and reported in Table II. From this table, we can see that both PSI-OC and PSI-PC can point out the correct information flow when there is only unidirectional causality, e.g., between and . On the other hand, both PSI-OC and PSI-PC fail when bidirectional causality exists, e.g., between and .

Results on PSI-DTF

Simulation was also carried out 500 times on 1024-point signals, the means and sd were derived and reported in Table III. When there is no information flow, e.g., from to , or from to , -and -remain close to zero. When some information flow exists, e.g., from to , or from to in the case of unidirectional flows, or from to but also from to in the case of bidirectional flow, PSI-DTF is able to identify the different propagation graphs. For the bidirectional case, the difference obtained between and can be explained by the values of the delays used to generate signals and . Considering a symmetric interaction between two of these signals, and , the respective values of the PSI-DTF indicator from signal to signal and from signal to signal would be comparable. So, PSI-DTF appears as a relevant indicator to detect unidirectional and bidirectional causalities.

IV. CONCLUSION

In this letter, we focused on information propagation between multi-site observations using a phase slope index based approach. The interest of the technique we proposed can be summarized as follows: i) the introduction of partial coherence instead of ordinary coherence allows dealing with direct causal relations; ii) AR modeling reduces estimator variance. Combining both improvements allows us to distinguish direct and indirect causal relations in multivariable time series with the lowest error; iii) the introduction of directed transfer function instead of ordinary coherence reveals pertinent to deal with bidirectional causal relations. In a future work, we plan to test these approaches to reveal the underlying network organization of an epileptic seizure, in some difficult situation where time shifts between signals strongly vary in time.

TABLE I RESULTS

 I ON PSI-OC AND PSI-PC. THE FIRST VALUE IS THE MEAN, THE VALUE IN PARENTHESES IS THE SD

TABLE II RESULTS

 II ON PSI-OC AND PSI-PC. THE FIRST VALUE IS THE MEAN, THE VALUE IN PARENTHESES IS THE SD considering the particular case of nonlinear signals with linear coupling.

TABLE III RESULTS

 III ON PSI-DTF. THE FIRST VALUE IS THE MEAN, THE VALUE IN PARENTHESES IS THE SD