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Concept of cointegration

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Concept of cointegration

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Generalized Mean Phase Coherence for Asynchrony Abnormality Detection in Multichannel Newborn EEG

Amir Omidvarnia, Sampsa Vanhatalo, Mostefa Mesbah, Ghasem Azemi, Paul Colditz, Boualem Boashash

The University of Queensland, Australia

Centre for Clinical Research (UQCCR)

RESULTS – NEWBORN EEG DATA

INTRODUCTION

METHOD

- The very low variation of the thresholds with time suggests a consistent probability distribution of the null hypothesis (the state of asynchrony) for all time segments.
- The findings suggest that a consistent time shift may exist between the EEG bursts recorded from the left and right hemispheres of an abnormal newborn brain.

- Synchrony of spatially segregated brain activities is an important feature that mediates their functional connections. Inter-hemispheric asynchrony of EEG bursts in full-term neonates is an indication of brain abnormality.
- A bivariate measure is unlikely to capture the key feature of asynchrony seen in the sick neonatal brain, which is characterized by a global disruption of synchrony.
- Different approaches including Evolution Map Approach , Instantaneous Period Approach, Mutual Prediction Approach, General Field Synchronization and Stochastic Event Synchrony have also been proposed for phase synchrony computation within bivariate and multivariate data.
- The concept of cointegration, introduced in econometrics, has been used to develop a parametric multivariate phase synchrony measure for EEG data.
- The present study uses the concept of cointegration to generalize the bivariate Mean Phase Coherence (MPC) to the multivariate case which is suitable for analysis of newborn EEG asynchrony.

Concept of cointegration

- A one-dimensional stochastic process is said to be integrated of order d (I(d)) if the reverse characteristic polynomial of its fitted multivariate autoregressive (MVAR) model has d roots on the unit circle in the complex plane. Two or more integrated signals can be in a long-term relation with each other if there is linear combination of these signals that results in a stationary process. In such case, these signals are referred to as cointegrated signals.
- Generalized Phase Synchrony (GePS)
- Two signals x1(t) and x2(t) are said to be in a generalized phase synchrony if the following equation is satisfied:
- where z(t)~N(m,σ) is a normal stationary stochastic process. Such relationship is generalized to a multivariate cointegrating relationship among K phase signals extracted from a multivariate signal x(t) as
- x(t) is said to be phase synchronized of rank r, if there are r cointegrating relationships within the phases extracted from the signals variables. A higher rank implies a larger number of interconnected phase signals and therefore, higher synchrony within channels.
- Generalized Mean Phase Coherence (GMPC)
- The signals are divided into short duration segments using a sliding window with an optional overlap.
- The multivariate phase signal is computed from each segment using the analytic signal approach.
- The multivariate Johansen test is applied on the extracted phase signals to obtain all coefficients of r cointegrating relationships.
- The Generalized Mean Phase Coherence (GMPC) is defined as:
- where the residual noise zi(t) represents the fluctuation around zero and is related to the strength of the i'th cointegrating relationship (i=1,…,r).
- E. A surrogate data method (here, shuffling over the segments) is then applied to determine the significance threshold for the measure.

Figure 3: A 19-channel cap adjusted for the newborn EEG recordings according to the 10-20 standard. Dashed areas represent the selected left and right groups of electrodes.

Figure 4: The GMPC measures and their corresponding binary masks extracted from the left and right groups of electrodes. The lowest panel illustrates the point-wise AND operator of the left and right hemisphere binary masks. Circles and arrows demonstrate a correspondence between the left and right binary masks.

BACKGROUND

Mean Phase Coherence (MPC)

Let zx(t) and zy(t) be the analytic signals associated with two one-dimensional stochastic real-valued signals x(t) and y(t), respectively; that is:

where and are the Hilbert transforms of x(t) and y(t), respectively. The two signals x(t) and y(t) are said to be phase-locked of order (m,n) if

The ratio m/n is assumed to be rational. In the case of discrete signals and for the case m = n = 1, the MPC is used as a measure of phase synchrony:

where ∆t is the sampling period and N the length of the two signals in samples. R can be re-written in a simpler form as:

CONCLUSIONS

RESULTS – SIMULATED DATA

- The results obtained from simulated signals and newborn EEG show the performance of the GMPC measure in detecting phase synchrony.
- This approach is able to extract phase interdependencies of a multivariate signal independent of the signal amplitude.
- Ratio of the synchronous periods detected by the GMPC measure comparing to the asynchronous ones in the multichannel EEG can be used as a quantitative measure of EEG inter-hemispheric asynchrony.

- GMPC has clearly marked synchronous time periods within the simulated bivariate/multivariate signal. Considering the multivariate binary masks, the majority of variables present a trend-stationary phase during the marked intervals.

Figure 2: Binary phase masks for the simulated multivariate data (left). The GMPC measure extracted from the simulated multivariate data (right). Shadowed areas emphasis on the synchronous intervals detected by the GMPC.

Figure 1: Binary phase masks for the simulated bivariate data (left). MPC versus GMPC extracted from the simulated bivariate data (right).