Amir Omidvarnia 3 June 2011 Synchrony measures for newborn EEG analysis.

Amir Omidvarnia 3 June 2011 Synchrony measures for newborn EEG analysis

Outline The concept of cointegration Johansen test Bivariate phase synchronization Multivariate phase synchronization Surrogate data method for phase synchronization Results White noise Combination of white noise and random walk Asymm./Asynch. data

The concept of cointegration Suppose two drunkards are wandering aimlessly, while they dont know each other. Their movement can be considered as two independent random walks. Mathematically, random walk refers to a trajectory that consists of taking successive random steps.

The concept of cointegration Random walk

The concept of cointegration Bivariate cointegration: A drunk and her dog [2] Now, imagine a drunk walking with her dog. Each of the two trajectories is still a random walk by itself. But, the distance between two paths is fairly predictable, as the location of the one can roughly tell us the location of the other one.

The concept of cointegration Bivariate cointegration: A drunk and her dog

The concept of cointegration Bivariate cointegration A long-run equilibrium relationship between the drunk and her dog causes a stationary distance between their random walks. The co-movement between two random walks is called a bivariate cointegrating relationship.

The concept of cointegration Bivariate cointegration x 1 (t), x 2 (t): Random walk -d < x 1 (t) + x 2 (t) < d x x 1 (t) ( / )x 2 (t) ~ white noise [ ] : cointegrating vector

The concept of cointegration multivariate cointegration: A sheep herd and Shepherd dogs [6] Imagine a herd of sheep wandering aimlessly in the field (multiple random walks). Consider that a herding dog guards the flock by running around and returning the sheep that have gone too far back to the herd.

The concept of cointegration multivariate cointegration: A sheep herd and Shepherd dogs The dog makes a co-movement among the sheep. Thus, we can say that there is a cointegrating relationship (rank of one) within the sheep trajectories. It is obvious that two dogs (rank of two) are able to restrict the movements within the flock more than one dog. The higher cointegration rank,The higher cointegration rank, the more restricted movementsthe more restricted movements

Johansen test A test for estimation of the cointegrating vectors and co- movements within a K-channel signal. Two types of hypothesis testing are applied on a mathematical feature extracted from the signal: Trace test H 0 : There are r cointegrating relations H 1 : There are K cointegrating relations Maximum eigenvalue test H 0 : There are r cointegrating relations H 1 : There are r+1 cointegrating relations.

Johansen test Johansen test gives all linear combinations between the dimensions which represent a stationary random process. Let x(t) be K-channel signal (t = 1,...,T). A cointegrating relationship among the channels (out of r, r

Amir Omidvarnia 3 June 2011 Synchrony measures for newborn EEG analysis.

Documents

dog slide

higher cointegration

cointegrating vector

data slide

random walk d x

t d x x

random walk asymm

johansen test johansen