Multivariate approaches to extract neural
interrelations between EEG channels
Amir Omidvarnia
22 Oct. 2010
Outline
 Introduction to multivariate AR models
 Multivariate connectivity based on time-invariant methods
 Non-parametric approaches
 Parametric approaches
 Multivariate connectivity based on time-varying methods
 Non-iterative approaches
 Iterative approaches
Introduction
 Methods based on the estimation of coherence/cross-
correlation functions are widely used to extract mutual and
synchronized activities between EEG channels.
 Most of these methods use multivariate AR models to define
proper criteria.
 Detecting the direction of the information flow between
EEG channel pairs is one of most important objectives of the
newly suggested methods.
 As the EEG signal is non-stationary, time-varying MVAR
based solutions should be taken into consideration.
Multivariate AR models
 The MVAR model with N variables is defined by the
equations [1]:
Multivariate AR models
 x1(n), . . ., xN(n) are the current values of each time series.
 a11(i) . . . aNN(i) are predictor coefficients at delay i.
 M is the model order, indicating the number of previous data
points used for modelling.
 e1(n) . . . eN(n) are one-step prediction errors [1]
Multivariate connectivity based on
time-invariant methods
Multivariate connectivity based on
time-invariant methods
 The input signal is considered as stationary and statistically
time-invariant.
 These methods can be divided into two main groups;
 Non-parametric measures
 Extract multivariate Cross-Power Spectral Density matrix using Fourier
transforms of the signals directly.
 Parametric measures
 Extract multivariate Cross-Power Spectral Density matrix using the
fitted MVAR model on the multichannel data.
Multivariate connectivity based on
time-invariant methods (cont.)
 Non-parametric measures
 Ordinary Coherence: Reflects the correlation (linear relationship) between
channels k and j in the frequency domain [2].
 Partial Coherence: Removes linear influences from all other channels in order to
detect directly interaction between channels i and j [2,3].
 Multiple Coherence: Describes the proportion of the power of the i’th channel at
a certain frequency which is explained by the influences of all other channels (the
rest) [4,5].
Multivariate connectivity based on
time-invariant methods (cont.)
 Corresponding multichannel matrices of the previously
indicated criteria are symmetric.
 There is no difference between the measures of channel i-
channel j and channel j-channel i pairs.
 In other words, none of the ordinary, partial and multiple
coherence measures show the direction of the information
flow between channels.
Multivariate connectivity based on
time-invariant methods (cont.)
 Parametric approach
 MVAR coefficient matrices need to be transferred into the frequency domain:
Multivariate connectivity based on
time-invariant methods (cont.)
 Parametric approach
 Cross-Power Spectral Density and Transfer Function matrices can be estimated based
on a fitted MVAR model on the multichannel data [6]:
∑: Noise covariance matrix of
the fitted MVAR model
Multivariate connectivity based on
time-invariant methods (cont.)
 Granger causality: The main idea originates from this fact
that a cause must precede its effect [12,13].
 A dynamical process X is said to Granger-cause a dynamical
process Y, if information of the past of process X enhances
the prediction of the process Y compared to the knowledge of
the past of process Y alone.
 Granger causality can be investigated by using MVAR
models.
Multivariate connectivity based on
time-invariant methods (cont.)
 Parametric measures
 Granger Causality Index (GCI): A time-domain criterion which
investigates directed influences from channel i to channel j in a
multichannel dynamical system [13].
 In an AR(2) model including two channels, if channel X causes channel Y,
the variance of the prediction error decreases for two-dimensional
modelling, because the past of channel X improves the prediction of channel
Y [14,15].
 If X Granger-causes Y, F will be positive, otherwise F is negative.
Multivariate connectivity based on
time-invariant methods (cont.)
 Parametric measures
All parametric measures are defined in the frequency domain based on S, A and H matrices.
 Directed Coherence: A unique decomposition of the ordinary coherence function which
represents the feedback aspects of the interaction between channels [6,7].
 Directed Transfer Function (DTF): The same as Directed Coherence when the effect of the
noise is ignored (σjj=1) [6,8].
Multivariate connectivity based on
time-invariant methods (cont.)
 Parametric measures

direct Directed Transfer Function (dDTF): DTF shows all direct and cascade flows together.
For example, both propagation 1→2→3 and propagation 1→3 are reflected in the DTF results.
dDTF can separate direct flows from indirect flows [9,10].

dDTF is the product of the non-normalized DTF and partial coherence over frequency [3]:

Partial Directed Coherence (PDC): Provides a frequency description of Granger causality.
This criterion is defined using the MVAR –derived form of the partial coherence function [6].
Partial Coherence
Partial Directed Coherence
Multivariate connectivity based on
time-invariant methods (cont.)
 Example of DTF and PDC functions [6]:
Multivariate connectivity based on
time-invariant methods (cont.)
 Difference of the DTF and PDC [2]:
 Directed Transfer Function is normalized by the sum of the
influencing processes (i’th row of the Transfer Function
matrix H).
 Partial Directed Coherence is normalized by the sum of the
influenced processes (j’th column of the MVAR matrix A).
Multivariate connectivity based on
time-invariant methods (cont.)
 Generalized Partial Directed Coherence (GPDC)
 This criterion combines the idea of DTF (to show the
influencing effects) and PDC (to reflect influenced effects)
between channel i and channel j [10,11].
Time-frequency representations of the
coherence measures
 Time-Frequency Coherence Estimate (TFCE)
 Ordinary coherence measure can be extended to the time-
frequency domain for the class of positive TFDs [18].
Time-frequency representations of the
coherence measures (cont.)
 Short-time DFT and PDC
 The whole data is divided into short overlapping time windows.
 Then either the DFT function or the PDC function is extracted
in each window.
 Finally, a time-frequency representation of the information flow
can be obtained for each pair combination of channels.
 Bootstrap or surrogate data approaches can be used to obtain
statistical significance of the results [19,20].
Multivariate connectivity based on
time-varying methods
Multivariate connectivity based on
time-varying methods
 Time-varying MVAR model estimation
 Least-Squared based algorithms have been suggested to estimate
time-varying MVAR coefficient matrices for several realizations
of the multichannel signal (e.g., ERP and VEP signal analysis)
[16].
 If there is only one realization of the signal in each step (e.g.,
spontaneous EEG), both Least-square approaches and Kalman
filtering based algorithms have been proposed [17].
Multivariate connectivity based on
time-varying methods (cont.)
 Instantaneous EEG coherence [16]
 Similar to the previous study [14], time-varying MVAR matrix is
updated in each step for a batch of ERP signals using a RLS-based
approach. In each step, ordinary coherence and multiple coherence
measures are extracted from the MVAR model parameters. Finally,
time-frequency representations of the coherence values can be
plotted.
Multivariate connectivity based on
time-varying methods (cont.)
 Instantaneous EEG coherence [16]
K’th epoch of the M-channel system
Wn = (Yn1,…,Ynp)
All MVAR parameters in time n
Multivariate connectivity based on
time-varying methods (cont.)
 Time-varying Granger Causality [14]
 In a recursive method based on RLS algorithm and for a batch
of multichannel signals (ERP data), noise covariance matrix of
the MVAR model is updated and Granger causality index is
computed using the time-varying covariance matrix ∑.
 This algorithm is not applicable for spontaneous EEG, as there
is only one realization of the signal in each step.
Multivariate connectivity based on
time-varying methods (cont.)
 Time-varying PDC based on Extended Kalman Filter [21]
 MVAR(M,p) is re-written as M*p AR(1) models.
 State space equations are extracted using the equivalent AR(1)
models.
 Another state space is considered for AR coefficients (the coefficients
are considered as time-varying processes).
 Two Kalman filters are applied on two state spaces to estimate time-
varying AR(1) coefficients and states.
Multivariate connectivity based on
time-varying methods (cont.)
 Time-varying PDC based on Extended Kalman Filter
[21]
1
2
3
Multivariate connectivity based on
time-varying methods (cont.)
 Time-varying PDC based on Extended Kalman Filter
[21]
 General form of the Kalman filter
Multivariate connectivity based on
time-varying methods (cont.)
 Time-varying PDC based on Extended Kalman Filter
[21]
Conclusion
 Time-invariant coherence measures based on the time-
invariant MVAR models are not sufficient to investigate the
interrelations of the brain.
 Least-Square based algorithms as well as Kalman filtering
tools have been suggested for adaptive estimation of timevarying MVAR coefficients in spontaneous EEG signals.
 Extended Kalman filtering seems to be a good candidate for
the problem, as it will consider both non-stationarity and
non-linearity.
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