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Statistical Methods in Data Mining of Brain Dynamics

Permanent Link: http://ufdc.ufl.edu/UFE0022141/00001

Material Information

Title: Statistical Methods in Data Mining of Brain Dynamics
Physical Description: 1 online resource (83 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: autoregression, cointegrated, eeg, electroencephalogram, phase, synchronization, var
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study discusses statistical approaches used for data mining of multichannel electroencephalogram recordings. Such recordings represent massive data sets that contain hidden patterns of complex dynamical processes in the brain. Formally, multichannel EEG can be viewed as a multiple time series, and therefore, a natural idea for summarizing such data is to utilize autoregressive modeling of multivariate stochastic processes. In particular, we thoroughly discuss various concepts and approaches related vector autoregressive processes, including stable stationary VAR models of order p and nonstationary systems with integrated and cointegrated variables, as well as procedures for estimating parameters of the systems (e.g., order, lag, or cointegration rank). The work highlights some stability issues that may arise in the application of vector autoregression to mining EEG data, and questions the applicability of Granger causality in the frequency domain to multichannel EEG. Synchronization has been found to be an important characteristic of the abnormal brain dynamics manifested by epilepsy and Parkinson's disease. We review two approaches for extracting the instantaneous phase from time series. In this study, we generalize the concept of the phase synchronization, and propose a novel approach based on multivariate analysis via modeling cointegrated VAR(p) processes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022141:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022141/00001

Material Information

Title: Statistical Methods in Data Mining of Brain Dynamics
Physical Description: 1 online resource (83 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: autoregression, cointegrated, eeg, electroencephalogram, phase, synchronization, var
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This study discusses statistical approaches used for data mining of multichannel electroencephalogram recordings. Such recordings represent massive data sets that contain hidden patterns of complex dynamical processes in the brain. Formally, multichannel EEG can be viewed as a multiple time series, and therefore, a natural idea for summarizing such data is to utilize autoregressive modeling of multivariate stochastic processes. In particular, we thoroughly discuss various concepts and approaches related vector autoregressive processes, including stable stationary VAR models of order p and nonstationary systems with integrated and cointegrated variables, as well as procedures for estimating parameters of the systems (e.g., order, lag, or cointegration rank). The work highlights some stability issues that may arise in the application of vector autoregression to mining EEG data, and questions the applicability of Granger causality in the frequency domain to multichannel EEG. Synchronization has been found to be an important characteristic of the abnormal brain dynamics manifested by epilepsy and Parkinson's disease. We review two approaches for extracting the instantaneous phase from time series. In this study, we generalize the concept of the phase synchronization, and propose a novel approach based on multivariate analysis via modeling cointegrated VAR(p) processes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022141:00001


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STATISTICAL METHODS IN DATA MINING OF BRAIN DYNAMICS


By

ALLA R. KAMMERDINER



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008


































2008 Alla R. Kammerdiner

































To my wonderful family.









ACKNOWLEDGMENTS

I would like to show my heartfelt appreciation to my advisor Dr. Panos M.

Pardalos for his support and mentoring. Working with him has helped me grow not

only professionally, but also as a person. I am also very grateful to other members

who served on my supervisory committee, J. Cole Smith, William W. Hager, Vladimir

L. Boginski, and H. Edwin Romeijn, for their valuable comments on my research for

this dissertation. Last but not least, I thank my husband Jason, my parents Olga and

Aleksandr, my brother Mikhail, and all the rest of my great family for their unconditional

love and support.









TABLE OF CONTENTS


page


ACKNOW LEDGMENTS .................................

LIST O F TABLES . . . . . . . . . .

LIST OF FIGURES . . . . . . . . .

LIST OF SYMBOLS ................... ................

A B ST R A C T . . . . . . . . . .

CHAPTER


1 INTRODUCTION ..................................

1.1 Statistical Methods for Data Mining .....................
1.2 Electroencephalographic Recordings ......................
1.3 Feature extraction . . . . . . . .
1.4 Contribution Summ ary .............................
1.4.1 Testing Applicability of Frequency Domain Estimates of Granger
Causality for EEG time series .. .................
1.4.2 Generalization of Phase Synchronization via Cointegrated VAR .

2 AUTOREGRESSIVE MODELING OF MULTIPLE TIME SERIES .......

2.1 Multivariate Autoregressive Modeling in EEG Data Mining .........
2.2 Tests of Granger Causality ........................
2.3 Vector Autoregressive Models (VAR) .. .................
2.3.1 Methods for VAR Parameter Estimation .. ............
2.3.2 VAR Order Selection Criteria .. .................
2.3.3 Stability Condition and Other Assumptions of VAR .........
2.4 Inegrated and Cointegrated VAR .. ...................
2.4.1 Augmented Dickey-Fuller Test for Testing the Null Hypothesis of
the Presence of a Unit Root .. ..................
2.4.2 Phillips-Ouliaris Cointegration Test ................
2.4.3 Estimation of Cointegrated VAR(p) Processes .. .........
2.4.4 Testing for the Rank of Cointegration .................

3 PHASE SYNCHRONY IN BRAIN DYNAMICS ..................


The Role of Phase Synchronization in Neural Dynamics .......
Phase Estimation using Hilbert Transform .. ...........
Phase Estimation via Wavelet Transform .. ............
Comparison between Two Approaches to Phase Extraction ......
Measures of Phase Synchrony .. .................









4 APPLICATION OF VECTOR AUTOREGRESSION TO MINING BRAIN DY-
N A M ICS . . . . . . . . . 60

4.1 Numerical Issues in Estimating Parameters of Vector Autoregression from
EEG ................... .. .................. 60
4.2 Multivariate Approach to Phase Synchrony via Cointegrated VAR . 62
4.2.1 Cointegration Rank as a Measure of Synchronization among Different
EEG C('!i ,. . ... .... ...... ..... .. .. 64
4.2.2 Absence Seizures ..... . . .... .. ....... 67
4.2.3 Numerical Study of Synchrony in Multichannel EEG Recordings
from Patients with Absence Epilepsy ................ .. 68

5 CONCLUSION .................. ................. 73

REFERENCES .............................. .. ..... 75

BIOGRAPHICAL SKETCH .................. ............ 83









LIST OF TABLES


Table page

2-1 ADF test: Critical values of the T(c- 1) and i statistics . ..... 49

2-2 Phillips-Ouliaris demeaned: Critical values of the Z, statistic . ... 49

2-3 Johansen test: Critical values of the ALR(r, K) statistic . . ..... 49

2-4 Johansen test: Critical values of the ALR(r, r + 1) statistic . .... 50

4-1 Seizure 1: Results of the ADF unit root tests .................. 70

4-2 Seizure 2: Results of the ADF unit root tests ......... . ....... 70

4-3 Seizure 3: Results of the ADF unit root tests .................. 70

4-4 Seizure 1: Results of the Johansen cointegration rank tests . .... 70

4-5 Seizure 2: Results of the Johansen cointegration rank tests . .... 70

4-6 Seizure 3: Results of the Johansen cointegration rank tests . .... 70









LIST OF FIGURES


Figure page

1-1 The International 10-20 system for placement of EEG electrodes . ... 27

4-1 Raw data: Numbers of unstable roots for different T and p . ..... 71

4-2 Filtered data: Numbers of unstable roots for different T and p . ... 71

4-3 EEG segment with absence seizure .................. ..... .. 72









LIST OF SYMBOLS, NOMENCLATURE, OR ABBREVIATIONS

Most of the notation is unambiguously defined in the text where it is introduced. To

provide some general guidelines, we include the following list of commonly used symbols.
ARIMA(p, d, q) autoregressive integrated moving average process

AR(p) autoregressive process of order p

CPV C 1 r!v principal value

X complex conjugate of X

converges to

Sconverges in distribution to

Sconvolution of functions

det determinant

e element of

equals

: equals by definition

Equivalent to

x estimator of x

3 exists

E expectation

exp exponential function

V for all

I identity matrix

IK (K x K) identity matrix

S imaginary part of a complex value

oo infinity

I(d) integrated process of order d

S is distributed as

lim limit










in natural logarithm

Pr probability of a random event

rank rank of a matrix

R real part of a complex value

Y sum of terms
o superposition

trace trace

X' transpose of X

Var variance

VAR(p) vector autoregressive process of order p









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

STATISTICAL METHODS IN DATA MINING OF BRAIN DYNAMICS

By

Alla R. Kammerdiner

May 2008

C'!I i': Panos M. Pardalos
Major: Industrial and Systems Engineering

This study discusses statistical approaches used for data mining of multichannel

electroencephalogram recordings. Such recordings represent massive data sets that contain

hidden patterns of complex dynamical processes in the brain. Formally, multichannel EEG

can be viewed as a multiple time series, and therefore, a natural idea for summarizing such

data is to utilize autoregressive modeling of multivariate stochastic processes.

In particular, we thoroughly discuss various concepts and approaches related

vector autoregressive processes, including stable stationary VAR models of order p and

nonstationary systems with integrated and cointegrated variables, as well as procedures for

estimating parameters of the systems (e.g. order, lag, or cointegration rank).

The work highlights some stability issues that may arise in the application of vector

autoregression to mining EEG data, and questions the applicability of Granger causality in

the frequency domain to multichannel EEG.

Synchronization has been found to be an important characteristic of the abnormal

brain dynamics manifested by epilepsy and Parkinson's disease. We review two approaches

for extracting the instantaneous phase from time series. In this study, we generalize the

concept of the phase synchronization, and propose a novel approach based on multivariate

analysis via modeling cointegrated VAR(p) processes.









CHAPTER 1
INTRODUCTION

1.1 Statistical Methods for Data Mining

With significant improvements in obtaining, processing and storing the information

electronically in the last decades, it became routine to accumulate large amounts of data

in various fields of research and business. However, any information is only useful if it

can be analyzed to draw some meaningful conclusions. A drastic increase in information

loads makes the task of interpreting collected data especially challenging. Not only

visual inspection and analysis of such massive data become extremely time-consuming

and often ineffective, but also the limitations of the traditional numerical data analysis

techniques in application to the massive collections of data necessitate the development

of new approaches. As a result of continuous attempts of scientific community to extract

useful information from large data sets, a multi-discipline field of data mining has been

developed.

Data mining together with data preprocessing constitute the central part of a more

general process of knowledge discovery in databases (KDD). The KKD process can

be described as a sequence of actions, which selects the raw data in data warehouses

and transforms the selected data in order to discover valid, understandable, novel and

potentially useful knowledge from the data. Data preprocessing that is applied to raw

data to improve the quality of the data often influences the selection and facilitates

the application of data mining techniques. Proper preprocessing of raw data leads to a

decrease in the time needed to mine the data, and boosts the overall mining efficiency.

The techniques used for data preprocessing can be roughly subdivided into data

(1. ,iii,: data integration and data reduction [64]. Data cleaning techniques handle the

problem of incomplete, inconsistent and erroneous data, remove the noise inherently

present in the raw data, minimize redundancy in the data, etc. Data integration is

concerned with combining heterogeneous data collected from different sources to form a









consistent data set. The process of data reduction amounts to identifying useful features,

which are capable of adequately representing the data, and it is usually performed using

dimensionality reduction and feature extraction methods.

The two fundamental tasks assigned to data mining are a descriptive task of

discovering hidden patterns and relationships in given data, and a predictive task

of forecasting or classifying the model's behavior from available data. Data mining

includes regression, classification, (1 l-1. i-i.- image restoration, learning association

rules and extracting functional dependencies, data summarization, etc. Data mining

is closely connected to other research areas such as statistics, machine learning and

artificial intelligence, optimization, visualization and databases. Data mining utilizes many

important results from the related fields, while keeping the main focus on the algorithms

and architectures, scalability of the number of features and instances, and automated

managing of massive quantities of diverse data.

t ii,: areas of data mining employ various approaches developed in the field of

optimization. In particular, it is shown in [7] that many basic problems in data mining,

including classification and (< -1. ii11 .- can be formulated as mathematical programming

problems and solved using optimization techniques.

In fact, Bradley at el. [7] demonstrated that a problem of minimizing the number

of misclassified points in two-class classification can be viewed as a linear program

with equilibrium constraints (LPEC). LPEC is a linear program (LP) with a single

complementarity constraint. Such constraint imposes a condition of orthogonality between

two linear functions. LPEC formulation arises in the instances when the constraints of the

problem include another LP problem.

In addition, the problem of feature selection in two-class classification by finding a

separating plane that utilizes minimum number of features can be given a mathematical

programming formulation as a parametric problem. Furthermore, the classification

via support vector machines (SVMs that find the separating plane maximizing the









margin between two classes while minimizing misclassification errors) can be stated

as a quadratic programming problem [6]. Moreover, as indicated in [7], the above

mathematical programming formulations have been extended to be effectively emploi,- .1

by other data mining approaches, including neural networks training, calculation of

nonlinear discriminants, and building decision trees. The clustering problem has a

complex formulation as a minimization problem with the objective given by a sum of

the minimums of a set of convex functions [5]. In general, this objective function is

neither convex nor concave. See review by Bradley at el. [7] for additional information

about mathematical programming formulations for various problems in data mining, the

application of optimization techniques, as well as the challenges that the field of data

mining offers to optimization.

i in!;: data mining approaches, such as classification,
segmentation, have been applied to time series a", -ii-- [45]. Many traditional statistical

approaches are also applied to mining time series. For instance, regression is one of the

most commonly used techniques for modeling and forecasting time series. Among the

statistical models applied to regression in time series are linear autoregression (AR),

autoregressive moving average process (ARMA), autoregressive integrated moving average

process (ARIMA), as well as their multivariate analogs (i.e. vector autoregression, etc.)

Time series arise in various applied areas, including economics and finance, meteorology,

biomedicine, etc. For instance, the study of seismic activity related to earthquakes

produces two-dimensional time series, where each measurement consists of the time

and the magnitude of a registered seismic event. Many biomedical signals, such as

electrocardiogram (ECG), electroencephalogram (EEG) and electrooculogram (EOG)

represent time series that can be interpreted via application of regression, segmentation,

neural networks, and other data mining methodologies. Sound signals are another example

of time series that are effectively analyzed using different data mining techniques.









Time series, which originate in different fields, are generated by diverse underlying

processes, and as a result they are often characterized by very distinct properties. Indeed,

as indicated in [14], although the normal time scale is a very natural choice of parameter

for the time series describing physical processes, the regular time looses its natural

meaning when dealing with many financial time series. Because most of financial time

series are irregularly spaced in physical time, the concept of "business time" or ,ll ,ii-,

time" is introduced to represent a new time parameter with respect to which time series

are regularly scaled. This procedure of time deformation allows the relabeled time series to

be viewed as stationary on a new time scale. Financial time series also often exhibit clear

seasonal trends, which obviously cannot be found when examining time series produced by

speech.

Statistical testing of several multivariate time series determined that the time series

from AUSLAN and BCI data sets can be considered stationary, whereas BCI MPI and

EEG contained non-stationary time series [108]. As a result of inherent differences in time

series data from diverse sources, some data mining methods that are successfully applied

to time series in one research area may not necessarily be applicable to analysis of time

series that stem from another applied field.

Time series obtained from electroencephalogram (EEG) recordings have several

interesting properties that distinguish them from other time series. Although some studies

apply one-dimensional modeling by considering one channel in EEG recording at a time,

in general, EEG data should be treated as multivariate time series. The multivariate

approach becomes especially important in view of its ability to investigate spatio-temporal

dependencies in the EEG data in contrast to being limited to only temporal relations in

a one-dimensional case. There is a disagreement among researchers studying EEG data

on whether the series should be modeled by a non-linear stochastic process or they can be

better described by a deterministic chaotic dynamical system.









Storage of continuous EEG recordings sampled from multiple channels at high

frequency during several hours (or even di -) from multiple subjects, wether it is for a

sleep study or diagnosis of neurological disorder, may take gigabytes of memory. The

application of data mining approaches to EEG time series allows automatic handling and

analysis of such large data sets. With new technological advances in collecting EEG data,

it becomes particularly important to develop new efficient data mining methods designed

specifically for mining EEG data.

1.2 Electroencephalographic Recordings

Electroencephalography is one of the most commonly applied methods of extracting

neurophysiological signals. It originated in 1875, when an English physician Richard Caton

measured the electrical activity from the exposed brains of monkeys and rabbits [100].

Generally -p1' i1:;i EEG represents a digital or a graphic record of the electrical

activity in the brain, and can be measured by either non-invasive or invasive methods.

EEG (obtained during a non-invasive procedure) is defined as a record of electrical activity

of an alternating type measured from the scalp surface after being picked up by metal

electrodes and conductive media [67]. There are two types of EEG produced by invasive

procedures, the electrocorticogram, which measures the brain's electrical activity directly

from the cortical surface, and the electrogram, which is an EEG obtained using deep

probes.

EEG estimates and records the relative change in electric potentials produced by

a large number of electric dipoles during a period of neural excitations. The activation

of neurons (brain cells) generates local current flows in the brain. EEG records mostly

the electrical currents that flow during synaptic excitations of the dendrites of numerous

pyramidal neurons in the cerebral cortex. EEG recorded from the scalp surface can only

detect the electrical activity produced by massive populations of active neurons. On the

other hand, EEG recorded using deep probe electrodes implanted into the brain can pick

up a signal from a small group of neurons, which can be further filtered out to obtain the









electric potentials generated by individual neurons. EEG has become an effective device

in the area of neurological research as well as clinical neurology, because of its capacity

to reveal both abnormal and normal brain activity. It is believed that by birth, human

brain has already developed the full number of neural cells, which is approximately 1011

neurons [68]. This gives an average density of about 104 neural cells per cubic millimeter

of the brain. Neural cells are interconnected through synaptic connections in the brain

into neural nets. The brain of an average adult contains approximately 500 trillion

synapses. The total number of neurons decreases with age. As a result the total number

of synaptic connections declines with .I-,: even though the number of synapses per one

neuron increases with age.

To ensure the consistency in referencing locations of electrodes in EEG experiments,

the International 10-20 system for EEG electrode placement was developed [37]. The 10-20

EEG system is used to describe the respective locations of scalp electrodes during EEG

recording in relation to the underlying area of cerebral cortex.

According to the 10-20 system, anatomical landmarks of a skull, such nasion, inion

and preauricular points, are identified for consecutive placement of the electrodes at fixed

distances from these points in steps of either 10 or 20 percent. This approach is devised

to take into account possible variations of head size. In addition, the method is easily

applicable in practical use. As a result, the 10-20 EEG system became very widely used

for positioning electrodes.

In the 10-20 system, the points are denoted with one or two letters, and can be also

followed by a number (as shown on Figure 1-1). The letters roughly represent the lobe

location (with exception of letters C and Z), whereas the numbers serve for identifying

the corresponding hemisphere. More specifically, the points located on the left hemisphere

of the brain are represented by odd numbers (1, 3, 5, and 7), and the sites on the right

hemisphere are marked with even numbers (2, 4, 6, and 8). The sites located on the

frontal, temporal, parietal and occipital lobes are denoted by the corresponding initials









F, T, P, and 0. The letters C and Z refer to the points placed in the central area. In

particular, Z represents a point on the midline, and C refers to the line parallel to the

midline. Note that the central area is not a lobe.

EEG signal resembles a collection of sinusoids of various amplitude and frequency.

Power spectrum is extracted from the raw EEG data using Fourier transform to obtain the

information about the contribution of sinusoidal waves of different frequency. The power

spectrum of EEG is continuous, ranging from 0 Hz up to a half of the sampling frequency.

Depending on the state of the brain, certain frequencies appear to be more prevalent.

There are four in i r frequency bands, alpha, beta, delta and theta, which presence in

EEG during various states of consciousness has been extensively studied. These bands

represent sine waves of relatively low frequency, with delta ranging from 0.5 to 4 Hz, theta

4 8 Hz, alpha 8 13Hz, and beta over 13 Hz. Alpha waves discovered by Adrian and

Matthews in 1934 are the best-known and the most studied among the four frequency

bands [100]. They are induced by closing eyes and by relaxation, and terminated with eyes

opening or due to thinking, calculating, and other analytical activities. In particular, in

most people, eye closing produces rapid changes in brain activity manifesting themselves

in EEG as an adjustment of the dominant frequency band from beta to alpha. EEG is

capable of discriminating between different states, such as i1 ii.- alertness, stress state,

various sleep stages, hypnosis, etc. Presence of beta band is dominant during the state

of alertness with eyes open. Drowsiness or the resting are usually characterized by the

rise in alpha activity. During the sleep, presence of lower frequency waves becomes more

apparent. A higher proportion of delta band frequencies is observed during stages III and

IV of the non-rapid eye movement sleep (NREM). EEG recorded from distinct regions in

the brain exhibits different spectrum of wave frequencies. In addition, the brain patterns

are unique for every individual.

Practical applications of EEG include epilepsy research and localization of the focus

of epileptic seizures, testing of epilepsy drug effects; determining areas of damage due









to stroke, head injury, etc.; monitoring alertness, coma and brain death; testing afferent

pathr- --, by evoked potentials; research in sleep physiology and sleep disorder; controlling

anaesthesia depth, etc. [3].

1.3 Feature extraction

Formally, EEG signal can be described as a deterministic multidimensional nonlinear

non-stationary time series [94]. In order to properly reflect the spatio-temporal properties

of brain dynamics, the i, i1 ,-i of EEG data must involve a simultaneous investigation of

the dependencies across channels with respect to time.

Different features have been proposed for analysis of EEG time series, including

Fourier transform, wavelets, cross-correlation, coherence, Granger causality and partial

directed coherence, mutual information and transfer entropy, global and phase synchronization,

Lyapunov exponents and correlation dimension, etc.

Since EEG can be viewed as a collection of sine waves, EEG series are often Jin 1v. 1

in a frequency domain. In addition, some frequency bands have shown to p1 iv specific

roles in various states of consciousness, and so the frequency information in EEG can be

particularly important. Subsequently, the Fourier transform with a running time window,

also known as short time Fourier transform (STFT), became one of the most widely used

methods for extracting features from EEG. STFT is obtained from Fourier transform by

applying a time window function g with a time shift r. Mathematically, STFT is given by

the following formula:


S(r, f) x(t)g(t -r) exp{-217r f t}dt, (1-1)
-oo

where S(r, f) denotes the STFT with time window g located at time 7, corresponding to

frequency f; and x(t) is a signal at time t. In other words, STFT S(r, f) represents the

power spectrum of the signal estimated around time 7. The drawback of STFT is that

there is a trade off between time accuracy and frequency precision. By making the window









g smaller, the resolution of the time parameter is improved, unfortunately, at the expense

of the resolution in frequency.

An alternative to Fourier transform is wavelet transform (WT), which is a transformation

of the signal based on a special function, called mother wavelet (M\W). The mother

wavelet is shifted in time by a location parameter 7, and then adjusted by a scale

parameter a. More precisely, the wavelet transform is defined by the following formula:

1 +0 t T
W(r,a) = x(t)( )dt, (1-2)


where ) is a mother wavelet, a is scale parameter, is a time location parameter, and x is

a signal.

The scale parameter a in WT is analogous to the frequency parameter f in STFT.

In particular, the large values of parameter a (a > 1) stretch the wavelet, and so they

represent low frequencies, whereas the small values of a (a < 1) shrink the wavelet

function, which corresponds to higher frequencies. An advantage of using wavelets is that

the high frequency components can be analyzed with a higher time accuracy than the

lower frequency components of the signal.

As follows from (1-2), W(r, a) can be interpreted as the projection of the signal onto

the appropriately shifted and scaled wavelet W, i.e. W(r, a) is a contribution of the wavelet

to the signal x(t).

While Fourier and wavelet transforms are usually applied to study each channel of

EEG signal individually, the cross-correlation, coherence and Granger causality measure

the interdependency between different channels. The cross-correlation function quantifies

the linear correlation between two processes. Given two normalized signals x(t) and

y(t) with zero means and unit variances, the cross-correlation between these signals is

estimated as:
N-T
C (r) xr+tt)y(t), (1-3)
t=i









where N denotes the total number of sampled points, and r is a time delay parameter

between two signals. The cross-correlation estimate ranges between -1 and 1. The positive

values of cross-correlation indicate the direct correlation between the signals (i.e. x and y

tend to be similar in both their absolute value and have the same sign), and the negative

values correspond to inversely correlated signals (i.e. signals have similar absolute values,

but different signs). Although zero cross-correlation value shows that the signals are not

linearly correlated, it does not necessarily imply that two signals are not interrelated in a

non-linear fashion.

The coherence function is a frequency domain analog of the cross-correlation measure.

Coherence is obtained from cross-correlation by applying Fourier transform to (1-3). The

estimate of the coherence spectrum of two signals is called periodogram. The periodogram

is calculated by subdividing the signals into a number of epochs of the same length, and

then applying the following formula:

| 2S1 (f) 12
P)2 (f) = (1-4)

where Sx(f), Sxx(f), Syy(f) denote the average values of the cross-spectral density

between x and y, and the individual auto-spectral densities of x and y, respectively. The

periodogram values range between 0 and 1, where zero coherence indicates that the signals

are linearly independent at a given frequency, and the maximum periodogram value of 1

shows that two signals are completely linearly dependent at a chosen frequency.

While the cross-correlation and coherence are features, which reflect the linear

dependency between two channels in the data, the concept of Granger causality is capable

of not only establishing the linear dependency, but also specifying the direction of such

dependency. In other words, by applying Granger causality, it becomes possible to

identify causal relationship among the channels of EEG. Granger causality is based on

the multivariate autoregressive modeling of time series. It has also received an alternative

reformulation in the frequency domain via spectral decomposition for stochastic processes.









Several studies have shown that EEG can be effectively modeled using chaos

theory [35, 89]. In chaotic systems, trajectories originating from very close initial

conditions diverge exponentially. The system dynamics are characterized by the rate

of the divergence of the trajectories, which is measured by Lyapunov exponents and

dynamical phase.

Short term largest Lyapunov exponent (denoted STLmax), which is an estimate of

the maximum Lyapunov exponent for non-stationary data, is a dynamical measure of the

chaoticity in the brain. Next, the method for estimating STLmax is summarized.

First, using the method of d. 1 ,i' [69], the embedding phase space is constructed from

a data segment x(t) with t E [0, T], so that the vector Xi of the phase space is given by

Xi (x(t),x(t + 7),..., x(ti + (p- 1))), (1-5)

where ti E [1, T (p 1)r], p is a chosen dimension of the embedding phase space, and T

denotes the time d. 1 iv between the components of each phase space vector.

Next, the estimate L of the short term largest Lyapunov exponent STLmax is

computed as follows:

1 -2 X(t + At) X(t + At) (
L 0 log2 (- ( -' (1-6)
i-N X(ti) X(ti)
where Na is the total number of local maximum Lyapunov exponents that are estimated

during the time interval [0, T]; At is the evolution time for the displacement vector

X(ti) X(tj); X(ti) represents the point of the fiducial trajectory such that t = ti,

X(to) = (x(to),x(to + -),... ,x(to + (p 1)r)), and X(tj) is an appropriately selected

vector that is .,.i i,:ent to X(ti) in the embedding phase space. In [34], lasemedis at

el. si-- -1- .1 a method for estimating STLmax in the EEG data based on the Wolf's

algorithm for time series [107].

The short term largest Lyapunov exponent STLmax is proved to be an especially

useful EEG feature for studying the dynamics of the epileptic brain [35, 89]. In particular,









spatio-temporal transitions during interictal, preictal, ictal, and postictal states can be

characterized by the changes in STLmax profiles [90].

1.4 Contribution Summary

1.4.1 Testing Applicability of Frequency Domain Estimates of Granger
Causality for EEG time series

The rules of interaction between various parts of the brain are one of the key

problems in theoretical and applied neuroscience. One of the most commonly used

v-,- to record the neural information in multiple areas of the brain is via a multichannel

electroencephalogram recording. Quantitatively, such recording is a multivariate time

series. In the last decade, the investigation of causal relationships exhibited by the

electroencephalographic time series became a very active area of research in neuroscience

and related fields. Several approaches for studying causality in EEG are proposed, most

of which are based on the definition of Granger causality via the spectral representation of

time series (or processes), which was introduced by Geweke [25] for analysis of econometric

series. Because frequency domain contains valuable information about the brain processes,

Geweke's definition of causality seems to be particularly useful for analysis of EEG data.

Both definitions of Granger causality, the original one given by Granger and the

frequency domain definition, are introduced via vector autoregressive modeling of multiple

time series. Precisely, the linear vector autoregression is used to fit the data. Based on this

model, two competing hypothesis about the data are considered and tested statistically to

determine, which of these two assumptions is supported by the data. In other words, to

test the causality, the hypothesis of data being modeled as linear autoregression (that does

not include another series) is compared to the alternative of the data best described by

including information from the other series.

Since Granger causality is defined based on linear vector autoregressive model (VAR),

the applicability of Granger causality depends upon the underlying assumptions of the

model. The fundamental assumptions of the VAR are stability, stationarity, and gaussian









distribution of the error term. If such conditions are violated, the model is not suitable for

the data. In this case, one needs to find a model that is more appropriate for the given

data. For example, many financial time series data exhibit periodic or seasonal behavior.

To account for such cases, various modifications of vector autoregression (including those

relaxing the stability condition) are constructed for analysis of econometric time series.

Although Bernasconi and Konig [2] tested the stationarity of different EEG data, and

concluded that a time interval of 1 second is the interval on which the EEG time series

can be considered stationary, the underlying assumptions of the VAR modeling such as

stationarity and stability are rarely statistically tested in applications to EEG data.

In particular, the stability assumption means that the reverse characteristic

polynomial of the model does not have roots inside a unit circle. In order to highlight

the importance of the stability condition for VAR, it is necessary to point out that when

stability is violated, the model may simply follow a random walk, or it may even exhibit

explosive behavior.

To the best of our knowledge, the stability condition of the VAR estimated from the

EEG time series was not investigated before our study [43]. In many studies that utilize

the vector autoregression to examine Granger causality among series, the verification of

the conditions assumed by the VAR model is often omitted.

The results of our numerical experiments indicate that the stability condition of

vector autoregressive model is often violated in application to the EEG data. More

specifically, we found that the stability assumption imposed on the linear VAR models

may be violated even in the case when the sample size parameter T is much larger than

the lag parameter p of the estimated model. In addition, we showed that despite the

fact that it is common in practice to filter the EEG data within a certain frequency

band, filtering the EEG time series within some restricted frequency band often results

in significant reduction of the (T,p) domain, where the estimated VAR(p) models remain

stable.









Based on our numerical studies, we concluded that suitable extensions of multivariate

autoregression to unstable processes may fit the data better, and so, be more appropriate

for the EEG time series analysis than linear vector autoregressive modeling [43].

Comprehensive statistical testing is necessary in order to make conclusions on what

multivariate models are the most appropriate for extracting the directional dependencies

between channels in a frequency domain from multichannel EEG data.

1.4.2 Generalization of Phase Synchronization via Cointegrated VAR

The temporal integration of various functional areas in different parts of the brain is

believed to be essential for normal cognitive processes. This results in constant interaction

among the brain regions. Many studies highlight the importance of neural synchrony in

such large-scale integration [15, 101, 104, 105]. Actually, it was found that oscillation

of various neuronal groups in given frequency bands leads to temporary phase-locking

between such groups of neurons. This observation has stimulated the development of

robust approaches that allow one to measure the phase-synchrony in a given frequency

band from experimentally recorded biomedical signals such as EEG.

In particular, the importance of synchronization of neuronal discharges has been

shown by a variety of animal studies using microelectrode recordings of brain activity [83,

96], and even at coarser levels of resolution by other studies in animals and humans [21].

The phase synchronization in the brain extracted from EEG data using Hilbert or

wavelet transforms has recently been shown to be an especially promising tool in ain 1, -;

of EEG data recorded from patients with various types of epilepsy [86].

In our recent study [44], we introduce a novel concept of generalized phase synchronization,

which is based on vector autoregressive modeling. This new notion of phase synchronization

is constructed as an extension of the classical definition of phase synchronization between

two systems. In fact, the phase synchronization is usually defined as the condition that

some integer combination of the instantaneous phases of two signals is constant. Often this

condition is relaxed by allowing for a bounded linear combination of two phases, in order









to account for noise in the measurements. This classical approach is clearly bivariate. But

what if we are interested in studying a synchrony among several parts of a system? Is

there such a notion?

To construct a more general multivariate concept of phase synchronization, we

extended the classical definition by considering a linear combination of phases for a

finite number of signals that represents a stationary process. All the individual signals

together form a common system described by some multivariate process. We note that a

vector process, such that a linear combination of its individual components is a stationary

process, can be modeled as a cointegrated vector autoregressive time series.

Furthermore, it is easy to see (as shown in Section 4.2.1) that the cointegrated rank of

the regression determines how restricted the behavior of such system is. This means that

the rank r of cointegrated autoregressive model, estimated from the multiple time series

of the instantaneous phases, measures how large the vector subspace, which generates the

changes in the phase values, is.

This new measure of cointegration was applied to absence epilepsy EEG data in [44].

The data sets collected from the patients with other types of epilepsy are currently being

investigated.































A top view


B profile view

Figure 1-1. The International 10-20 system for placement of EEG electrodes









CHAPTER 2
AUTOREGRESSIVE MODELING OF MULTIPLE TIME SERIES

2.1 Multivariate Autoregressive Modeling in EEG Data Mining

Several methods for joint spatio-temporal analysis of multichannel EEG recordings

based on the idea of Granger causality were presented in the last decade. The concept

of Granger causality was first introduced by Clive Granger [26, 27] for measuring

linear dependence and feedback in economic time series. Later, this idea was further

extended by John Geweke [25], who proposed an equivalent measure based on the spectral

representation of time series. Both Granger's and Geweke's approaches employ the vector

autoregressive modeling to derive estimates of underlying causal relations in the data.

However, the latter approach is found particularly useful for analysis of EEG time series,

since it investigates the causal relation in the frequency domain instead of the time

domain as in the former approach. In particular, the spectral measure of Granger causality

proposed by John Geweke was employ, ,1 on intracortical local field potentials recorded

from 8 electrodes during go/no-go trials of cat's visual responses [2]. Another study [57]

utilized a similar method of directed transfer function (which is equivalent to the spectral

measure of Granger causality) to examine causal influences in the primate visual cortex

during the task of visual pattern recognition. The direct transfer function approach to

Granger causality was also applied to analyzing brain connectivity patterns on human

EEG data recorded during stage 2 sleep [42].

Michael Eichler proposed a graphical approach for modeling Granger-causal

relationships in multivariate time series [17] and later applied this method to studying

connectivity in neural systems [18, 19]. Luiz Baccala and Koichi Sameshima introduced

a concept of the partial directed coherence for inference of Granger causality in the

frequency domain based on the linear vector autoregressive modeling, and applied it to

investigating the functional interactions among different brain structures [1, 92].









Vector autoregressive modeling is a common basis in various approaches to estimating

Granger causality. Next, we review the details of the multivariate autoregressive modeling

and investigate some limitations in its application to the EEG data.

2.2 Tests of Granger Causality

w1 i: different tests of Granger Causality are developed. Some versions of the test are

based on a vector autoregressive model, others are based on a multivariate moving average

representation. For simplicity, we present the alternative definition based on the bivariate

regression. For a detailed review of Granger causality tests, see the book on time series by

Hamilton [30].

Let X(t) and Y(t), t E Z denote two time series (or discrete time stochastic processes)

with the corresponding realizations xt and yt, t E Z. Suppose that Qx,t and Qy,t denote

all the information about the realizations of processes X and Y, respectively, up to time t.

Then, the relationship of Granger causality between such series can be formally defined as

follows:

A time series X(t) is said to Gri,,n,. -cause Y(t) if there exists p = 1, 2,... such that

the mean squared error (lMSl) of the p-step forecast of Y(t) based on the information

Qx,t and QY,t is smaller than the MSE of the p-step forecast of Y(t) based on Qyt alone,
i.e.

p = 1,2,... : Ey(plx| t, y, t) < Ey(plQY,t), (2-1)

where Ey(p|fQ) is the MSE of the p-step forecast of Y(t) based on information f.

Using the above definition, we now present the test for Granger causality based on the

bivariate autoregressive model. Suppose that for some integer lag parameter p > 0, the

realizations of time series Y(t) are given by the model


yt = v + aiy-i + Oixt-i + Et, (2-2)

where Et is a standard white noise (or innovation process, i.e. Et has zero mean and zero

autocorrelation).









We test the hypothesis


Ho: i 0 Vi ,2,...,p (2-3)


against the alternative

H1 : 3j, 1 < j < p s.t. 0j 0 (2-4)

Note that if the null hypothesis is accepted, then a time series Y(t) is believed to be NOT

Granger-caused by X(t). Meanwhile, rejecting the null hypothesis (i.e. accepting the

alternative) means that X(t) is believed to cause Y(t) in Granger's sense.

Let T be the sample size parameter. The model parameters for the null hypothesis Ho

and the parameters for the alternative HI are estimated from the sample data using the

ordinary least squares method (or other methods) to obtain the estimates of the forecast

errors ot and Eit, respectively, t = 1, 2,..., T. Then the sum of squared residuals RSSo

under the assumption of null hypothesis Ho is

T
RSSo0 = t, (2-5)
t=1

and the sum of squared residuals RSSI under the alternative H1 is

T
RSS, = t. (2-6)
t=1

By conducting the F-test of the null hypothesis, one can find the test statistic

(RSSo RSSi)/p
1 RSS ~ Fp, T-2p-1. (2-7)
RSSl/(T 2p 1)

If the test statistic Si exceeds the specified critical value, then the null hypothesis that

X(t) does not Granger-causes Y(t) is rejected. Otherwise, Ho is accepted.

An .,-vmptotically equivalent test of Granger causality is given by the following

statistic
T(RSSo RSS)
2 (RSSp) (28)
RSS1









It is noteworthy to point out that the tests of Granger causality are very sensitive to

the choice of the lag length parameter p, and to the methods utilized for handling any

non-stationarity in the time series.

The bivariate approach to testing Granger causality can be naturally extended to

the multivariate case by partitioning the vector autoregressive process Z(t) into two

components X(t) and Y(t), so that Z(t) = (X(t),Y(t)), and then testing the suitable

zero constraints on the coefficients of vector autoregression. For the derivation of the Wald

statistic and the F-statistic for testing Granger causality in the multivariate case, see the

book on multiple time series by Liitkepohl [59].

2.3 Vector Autoregressive Models (VAR)

2.3.1 Methods for VAR Parameter Estimation

The vector autoregressive (VAR) model of finite order serves as a foundation for

establishing Granger causal relations in multidimensional time series.

Let p denote a positive integer, and let yt denote the K-variate time series (i.e.

realizations of K-dimensional process Y(t)). A vector autoregressive model of order p,

denoted VAR(p), is formally defined as follows:


yt =v+Ayt+...+A tp t, t 1, ... yt.., (2-9)

where yt = (lt, ... Kt)' is a (K x 1) random vector, v (vl,..., VK)' is a fixed

(K x 1) vector representing a non-zero mean EY(t), the Aj, i = ,... ,p are fixed

(K x K)-dimensional coefficient matrices, and Et = ( Et,..., Kt)' is a K-dimensional white

noise process (i.e. E [Et] = 0, E [E:'] = 0, for s / t, and E [FsE] = EY). It is assumed

that the covariance matrix Es is nonsingular. In addition, three important conditions are

usually imposed on the time series in the VAR model. The first condition is stability of

the process Y(t), the second is stationarity of Y(t), while the third one supposes that the

underlying white noise process Et is Gaussian.









Suppose that the the lag length parameter p is specified. Although, in the above

definition of a VAR(p) model, the process mean v, the coefficient matrices Ai, and the

covariance matrix Es are assumed to be known, in practice, these parameters must

be derived from the sample data. There are three main approaches to estimating the

parameters of a VAR(p) time series, namely, the multivariate least squares method,

the Yule-Walker estimator, and the maximum likelihood estimation [59]. Under the

assumptions of stability and Gaussian distribution, these approaches lead to estimators

with the same .,-vmptotic properties. However, the .,-vmptotic results should be used

cautiously in inference from small samples. As a result, different approaches may

sometimes lead to different results when estimating the model parameters using small

samples.

Let us now briefly present the multivariate least squares estimation, which is a higher

dimensional extension of the well-known method of ordinary least squares. For more

detailed discussion, refer to [59].

Suppose that the available data include (T + p) successive realizations of estimated

multiple time series represented by K-dimensional vectors


Y-p+ ... ,o, Y 1, l .. YT

where p is the fixed lag length, and T is the sample size parameter. For convenience, we

partition the data into the pre-sample y-p+,... yo and the sample yl,..., yr values. In

addition, the following notation is introduced:


Y:= (y,,...,yr) (K x T),

1

Z:= yt ((Kp+ 1) x 1), (2-10)


Ut-p+1
Z:= (Zo,...,ZT-) ((Kp + ) x T),









B:= (v,A,,...,Ap) (K x (Kp+ 1)),

E:= (1E,..., T) (K xT),


where t = 1,..., T. Then using this notation in (2-9), the vector autoregressive model of

order p can be represented in the compact form:


Y = BZ + E, (2-11)


and the coefficients B of the model are given by the least squares estimator:


B = YZ'(ZZ')-\. (2-12)


The covariance matrix can be estimated in various v--iv. Since ZE = E [EtE'], the

estimator
T
E f E ti't = (Y BZ)(Y BZ)' (2-13)
t=1
is consistent. However, this estimator of the covariance matrix Es is not unbiased.

Therefore, it is often replaced by the following unbiased estimator


E~ t' (Y BZ)(Y BZ)' (214)
T- Kp- 1 t T-Kp 1

Obviously, both estimators are consistent estimators of the covariance matrix, and they

are .,- mptotically equivalent.

When estimating the coefficients of the vector autoregressive model from data, we

assumed the order p of the VAR(p) to be known. In practice, however, it is unknown, and

therefore, needs to be derived from the data. Since zero coefficient matrices are allowed,

one could simply set p to some upper bound on the VAR order. On the other hand,

selecting an unnecessary large p would affect the forecast precision of the estimated model.

Therefore, it is advantageous to apply some suitable criteria for optimal selection of the

lag length parameter p.









2.3.2 VAR Order Selection Criteria

Various criteria for choosing the optimal model order are developed. Some of the

most commonly used are the final prediction error (FPE) criterion, Akaike's information

criterion (AIC), Hannan-Quinn criterion (HQ), and Schwarz or B ,-i ,- information

criterion (SC) [59].

Let E,(m) denote the maximum likelihood estimator of ~, computed by fitting the

VAR model of order m. The FPE criterion proposed by Akaike in 1969 is based on the

idea that minimizing the mean square error improves the forecast of the model. For a

VAR(p) time series, the FPE criterion is defined as


FPE(m) ( ) K det (1 (m)) (2-15)

Using the FPE criterion, the estimate PFPE of the model order p is selected so that

FPE (PFPE) min FPE(m), (2-16)
m=1,...,M

where M denotes some upper boundary on the model order. In other words, first, for each

m = 1,... M, the vector autoregressive model of order m is estimated from the data, and

the respective values of the FPE(m) are calculated using (2-15); then the order producing

the smallest value of FPE(m) is chosen among the possible orders m 1,..., M.

AIC is another popular order selection criteria that was also introduced by Akaike.

Given a VAR(m) model, the Akaike information criteria is defined as follows:

2m inK2 I
AIC(m) =2 +n (I1,(m) (2-17)

Similarly to the FPE criterion, the VAR(m) models are estimated for different m =

1,..., M to obtain the corresponding AIC(m) values for each order. Then the estimate

PAIC of the model order p with the smallest AIC(m) is selected.









The Hannan-Quinn order selection criterion, HQ, is given by:


HQ(m) mK2 nT + In (m)) (2-18)
T

As before, among the model parameters m = 1,..., M, the parameter m having the

smallest value of HQ(m) is chosen as the estimator PHQ of the true model order p.

Last, but not least, we present Schwarz criterion, which was derived using B li-, -i i

arguments. The SC is formulated as:

mK21n T 1
SC(m) = nT +ln (,(m ) (2-19)

and the order minimizing SC(m) is chosen among m = 1,..., M as the estimator psc of

the model order p.

Some interesting statistical properties of the above criteria are proved in [59].

In particular, it is shown that AIC and FPE criteria for VAR order selection are

.,-i-,,il! 11 ically equivalent, although these estimators of the model order are not consistent.

On the other hand, the other two criteria provide consistent estimators of the order

parameter p. More precisely, in the univariate case (K = 1), the Hannan-Quinn criterion

is consistent (i.e. limTr,+ Pr{p = p} = 1). In addition, the HQ criterion is strongly

consistent for K > 2 (i.e. Pr {limT,+,- p } = 1). The SC is shown to be strongly

consistent for any dimension K.

It is important to keep in mind that even though FPE and AIC do not provide

consistent estimators, they are not necessarily inferior to HQ and SC. Actually, in small

samples, and even in larger samples, FPE and AIC may produce better forecast, although

they may not estimate the model order correctly.

2.3.3 Stability Condition and Other Assumptions of VAR

As mentioned above, the conditions of stability, stationarity and Gaussian distribution

are usually imposed on time series when dealing with the VAR models. Below we define

these conditions, and discuss their role. A K-dimensional VAR(p) time series (2-9) are









called stable, if


det(IK Alz ... Apz) / 0 for complex z : zl < 1. (2-20)


In other words, the VAR(p) process (2-9) satisfies the -Il,.1ii:,1 condition when its reverse

characteristic p ..1;,,;. ..;;,,.l (given by det(IK Alz ... ApzP)) has no roots on and inside

the complex unit circle.

The stability condition guarantees that there exists a moving average (1 A)

representation for the VAR(p) process. Also stability ensures that the process is a

well-defined stochastic process with the distributions of its univariate components and

joint distribution of the process yt uniquely determined by the innovation process Et. For

a stable VAR(p)process, both the process mean and the autocovariance are time-invariant

(which, according to the definition below, implies stationarity).

When the stability condition is violated, the process variance is increasing with time

and unbounded. Specifically, if the reverse characteristic polynomial of the time series has

a single unit root, and all the other roots are outside the complex unit circle, then the

time series behavior is similar to a random walk. In this special case, the variance increases

linearly with time, the correlation between yt and yt+h approaches 1, and the process mean

E [Y(t)] exhibits a linear trend for v / 0. In addition, if one of the roots of the reverse

characteristic polynomial lies strictly inside the complex unit circle, then such process is

explosive, i.e. the process variance grows exponentially. Various approaches are developed

in the time series literature to address the time series with the unit roots. For example,

the unit roots can be removed by taking differences. However, the explosive time series are

not as well-studied, because it is believed that an exponential increase in the variance of

the economic time series is not well founded. As one can see the stability assumption pi' "-

an important role in VAR(p).

A wide-sense stationarity for stochastic processes is imposed on the VAR time series

as follows. A stochastic process Y(t) is considered stat.:..., *i ;, if









1. E [Y(t)] = v for all t;

2. E [(Y(t + h) v) (Y(t) -)'] = F(h) = F(-h) for all t and = 0,1,....

In other words, the stationarity condition supposes that the first and the second moments

are time invariant. Also note that the process mean v and the autocovariance matrix

Fy(h) are finite. It is shown (see Proposition 2.1 in [59]) that

A stable VAR(p) time series yt, t 0, 1, 2,... is stationary.

Since stability of a time series implies that the series is stationary, the stability condition (2-20)

is sometimes cited in the literature as the stationarity condition. However, it is important

to remember that these two conditions are not equivalent. In fact, although a stable vector

autoregressive series is alv-- stationary, the converse is not true, i.e. an unstable time

series is not necessary non-stationary.

The Gaussian distribution assumption is introduced into the VAR(p) model through

Et. Specifically, given representation (2-9) of the VAR (p), the innovation process Et is

assumed to be Gaussian white noise. This condition implies that yt is a Gaussian process,

i.e. any subcollection t,... t+h follows a multivariate normal distribution for all possible

values of t and h.

2.4 Inegrated and Cointegrated VAR

In previous sections of this chapter, we considered VAR processes, for which the

stationarity and stability assumptions are satisfied. However, in practice, many time series

data are fit better by unstable non-stationary processes. In this chapter, we introduce

integrated and cointegrated processes, which are found especially useful in econometric

studies, and for which the stability and stationarity conditions are violated.

Recall that the VAR(p) process (2-9) satisfies the -/.i..:1/; condition when its reverse

characteristic 1y. Jl;;.i,,.:/.l det(IK A1z ... ApzP) has no roots on and inside a complex

unit circle. If an unstable process has a single unit root and all the other roots outside

of the complex unit circle, then such process exhibits a behavior similar to that of a

random walk. In other words, the variance of such process increases linearly to infinity,









and the correlation between the variables Y(t) and Y(t h) tends to 1 as t o0. On

the other hand, when the root of reverse characteristic polynomial lies inside the unit

circle, the process becomes explosive, i.e. its variance increases exponentially. In real-life

applications, the former case is of the most practical interest.

This renders the following definition of an integrated process.

A one-dimensional process with d roots on the unit circle is said to be integrated of

order d (denoted as I(d)).

It can be shown [59] that the integrated I(d) process Y(t) of order d with all roots of

its reverse characteristic polynomial being equal to 1 can be made stable by differencing

the original process d times. For example, the integrated I(1) process Y(t) becomes stable

after taking the first differences (1 L)Y(t) = Y(t) Y(t 1), where L represents the

lag operator. More generally, for the I(d) process Y(t), its transformation (1 L)dY(t) is

stable.

An example of an integrated I(d) process in the univariate case is an autoregressive

integrated moving average process ARIMA(p, d, q), which is sometimes called fractionally

difference autoregressive moving average process for d c (-0.5, 0.5). The one-dimensional

process Y(t) is said to be ARIMA(p, d, q), if Z(t) := (1 L)dY(t) is a stationary

autoregressive moving average ARMA(p,q) process, i.e.
P q
aZ(t i) = bjt-j, (2-21)
i=0 j=0

where Et-js are independent normally distributed random variables with mean 0 and

variance a2, and L is the differencing operator introduced above.

It is noteworthy to point out that taking differences may distort the relationship

among the variables (i.e. one-dimensional components) in some VAR(p) models. In

particular, this is the case for systems with cointegrated variables. It turns out that fitting

VAR(p) model after differencing the original cointegrated process produces inadequate

results. Next, we discuss such processes.









Cointegrated processes were first introduced by Clive Granger in 1981, and gained

a great deal of popularity in both theoretical and applied econometrics. Indeed, many

economic variables are expected to be in equilibrium relationship, for example, household

income and expenditures, or prices of a given commodity in different markets.

Suppose that sampled values yit of K different variables of interest i(t) are combined

into the K-dimensional vectors yt = lt,..., YKt)'. In addition, suppose that the variables

are in a long-run equilibrium relation


c Y(t) := ci Yi(t) +... + CK YK(t) = 0, (2-22)

where c = (c1,..., CK) is a K-dimensional real vector. During any given time interval, the

relation (2-22) may not necessarily be satisfied precisely by the sample yt, instead we may

have:

c yt := ci yt + C. + CK YKt = Et, (2-23)

where Et is a stochastic process that denotes the deviation from the equilibrium relation at

time t. If our assumption about the long-run equilibrium among individual variables I(t),

i = 1,..., K is valid then it is reasonable to expect that the variables Y(t) move together,

i.e. the stochastic process Et is stable. On the other hand, this does not contradict the

possibility that the variables deviate substantially as a group. Therefore, it is possible that

although each individual component i(t) is integrated, there is a linear combination of

Y(t), i = 1,..., K, which is stationary. Integrated processes with such property are called

cointegrated.

Without loss of generality, we assume that all individual one-dimensional components

Yi(t) (i = 1,..., K) are either I(1) or I(0) processes. Then the combined K-dimensional

VAR(p) process

Y(t) = v+ AIY(t- 1) +... + ApY(t- p) + Et (2-24)









is said to be cointegrated of rank r, when the correspondent matrix


I IK A1 ... Ap (2-25)


has rank r.

Since some one-dimensional components of the cointegrated VAR(p) process are

integrated processes, one may be interested in testing the presence of a unit root in the

univariate series. In the following section, we present a commonly used unit root test,

which was derived by Dickey and Fuller [16].

2.4.1 Augmented Dickey-Fuller Test for Testing the Null Hypothesis of the
Presence of a Unit Root

The augmented Dickey-Fuller (or ADF) test is a widely used statistical test for

detecting the existence of a unit-root of the reverse characteristic polynomial in a

univariate time series. By fitting an autoregressive AR(k) model, this test investigates

the null hypothesis of an autoregressive integrated moving average ARIMA(p, 1, 0)

process against the alternative of a stationary ARIMA(p + 1, 0, 0) process. The limiting

distribution of the ADF test for p < k 1 was derived by Dickey and Fuller [16], and it

can be shown that this distribution is the same for k > 1 and for k = 1. Fuller tabulated

the approximate critical values for the ADF test with k > 1 and p < k 1 for -/ .. '.:

sample sizes.

Finite-sample critical values for the ADF test for i,1, sample size were obtained

by means of response surface analysis by MacKinnon [60], who also showed that an

approximate .,-,iii!,ll ic distribution function for the test can be derived via response

surface estimation of quantiles [61].

Although the ..-, ,.ii!ll ic distribution of the ADF test statistic does not depend on

the lag order, it is noted by C'!. uii et al. [13] that empirical applications must deal with

finite samples, in which case the distribution of the ADF test statistic can be sensitive to









the lag order. Taking this into account, they closely examined the roles of the sample size

and the lag order in finding the finite-sample critical values of the ADF test.

As we noted above, the limiting distribution of the ADF test statistic is the same for

k > 1 and k = 1. Hence, for simplicity, we consider the case of k = 1. In fact, let Y denote

the autoregressive AR(1) model


Y(t) c Y(t- 1)+ Et, t 1,2,... (2-26)


where Y(0) = 0, c is a real number, and Et ~ N(0, a2) (i.e. Et is normally distributed with

zero mean and variance a2 for all t = 1, 2,...).

Note that when |c| < 1, the process Y(t) converges to a stationary process as t oo;

whereas, in the case of c| = 1, the process Y(t) is not stationary with variance ta2.

Furthermore, when |cl > 1, not only the process is not stationary, but the variance of Y(t)

grows exponentially with time t.

From the AR(1) model (2-26), one can see that in the case when c = 1, in order to

make the process stationary, the series can be appropriately transformed by differencing.

Furthermore, notice that the condition c = 1 in (2-26) is clearly equivalent to the

requirement that the reverse characteristic polynomial det(1 cz) = 1 z of AR(1) has a

unit root. In other words, to determine whether an autoregressive time series AR(1) has a

unit root, we must test the null hypothesis Ho : c = 1.

Let y y2, .. yT denote a sample of T consecutive observations of the AR(1) process

Y(t), then the maximum likelihood estimator of c is the least squares estimator

C t= tt- (2 27)
T y2 1

Note that 'c is a consistent estimator of the regression coefficient c.

Since each yt, t 1,..., T is a realization of an AR(1) process, it follows from (2-26)

that yt = c yt-1 + Et holds, and so by plugging this last condition into Equation (2-27), the









estimator c of the regression coefficient can also be written as:

(zc =tt I 1+t) 1 (2-28)
T
S z2T 1
c t 1 -1 i+ t1 t-1it
zT 12
Et 1 2t t
Yt 1 Lt- Et
c+
Y, E I-itL
3t 1u/ 1

Subtracting c from both sides of Equation 2-28 and multiplying each side by T lead

to the ADF statistic

T(Z- c) = i t-I t (2-29)
2 t2 1 t[1t- 1
Dickey and Fuller [16] derived the following representation of the limiting distribution for

statistic T(c- c):

T(c-c)= i- (W2 -1), as T oo (2-30)
2

where
OO
r F= df Xf, (2-31)
i=1

W = v2 dXj, (2-32)
i=1
2(-1)i+1
d = )' (2-33)


and random variables X, i = 1, 2,..., are independent and identically distributed

according to the normal distribution with zero mean and variance a2. Note that a symbol

= denotes convergence in distribution.

In [16], Dickey and Fuller considered the following "Stu. li ii. i i statistic based on

the likelihood ratio test of the hypothesis Ho : c = 1.



^ S y) (2-34)
t=2-2









where

^S2 7 _1,- i12 (2-35)
T 2

and is computed from (2-27)

Tables of the critical values for the .i-i1!,l .1' ic distributions of the ADF test statistic

T(c 1) and the statistic ? can be found in Fuller [22]. We summarize some of the

information in Table 2-1, which lists the p-values for .,-vmptotic distributions of T(c- 1)

and ? corresponding to percentiles of 90, 95, and 99 percent.

2.4.2 Phillips-Ouliaris Cointegration Test

The unit root tests based on analysis of residuals were introduced by Phillips [75]. In

particular, in his study Phillips first considered two statistics Z, and Zt for testing the

null of no cointegration in time series.

Because many unit root tests, constructed before 1987, were founded on the

assumption that the errors in the regression are independent with common variance

(which is rarely met in practice), Phillips wanted to relax the rather strict condition that

the time series are driven by independent identically distributed innovations. In other

words, he wanted to develop the testing procedures based on the least squares regression

estimation and the associated regression t statistic, which would allow for rather general

weakly dependent and heterogeneously distributed sequence of error terms.

The properties of .-i- i!! i i.. i distributions of residual based tests for the presence

of cointegration in multiple time series were thoroughly investigated by Phillips and

Ouliaris [76]. The characteristic feature of these tests is that they utilize the residuals

computed from regressions among the univariate components of multivariate series. The

residual based procedures developed by Phillips and Ouliaris are designed to test the null

of no cointegration by means of testing the null hypothesis of the unit root presence in

the residuals against the alternative of a root that lies inside the complex unit circle. The

hypothesis Ho of the absence of cointegration is rejected, if the null of a unit root in the









residuals is rejected. In the nutshell, the procedures are simply residual based unit root

tests.

As noted in [76], the residual based unit root tests are .,-vi! ,tically similar, and can

be represented via the standard Brownian motion. Moreover, the ADF and Zt tests are

proved to be .,-imptotically equivalent. However, these two tests are not as powerful as

the test based on statistic Zo, because it was shown by Phillips and Ouliaris [76] that the

rate of divergence under cointegration assumption is slower for the ADF and Zt than other

tests, such as the Z,-statistic test. The later test (i.e. the cointegration test based on Z,)

is also widely known as the Phillips-Ouliaris cointegration test.

It is noteworthy that the null hypothesis for the Phillips-Ouliaris test is that of no

cointegration (instead of cointegration). This formulation is chosen because of some 1n i i"

pitfalls found in procedures that are designed to test the null of cointegration in multiple

time series. These defects (discussed in more detail in [76]) are significant enough to be a

strong argument against the indiscriminate use of the test formulations based on the null

of cointegration, and to support the continuing use of residual based unit root tests.

Consider the K-dimensional vector autoregressive process Y(t). Let us partition

Y(t) = (Ut, V )' into the univariate component Ut = Y1(t) and the (K 1)-dimensional

Vt Y( t),..., YK(t))'.

The residuals are determined by fitting linear cointegrating regression


U(t)= cV(t) + t, t 1,2,... (2-36)


Residual based tests are formulated to test the null hypothesis that the multiple time

series Y(t) are not cointegrated using the scalar unit root tests, such as the ADF test,

which are applied to the residuals t, t = 1, 2,... in (2-36).

In [76], the ADF test as well as two additional tests Z, and Zt, developed earlier by

Phillips [75], were applied to check for the presence of a unit root in the residuals t. In









order to perform the unit root test, we fit an AR(1) model to t, t = 1, 2,... according to


at = _- + Qt, t 1,2,... (2-37)

Then the statistic Z, in Phillips-Ouliaris test is defined as follows:

1 2 2
Z = T( 1 1) (2-38)
T2 t=-2 2t-I

whereas the Zt statistic is given by the following formula:

(t 2 1) 1 Si -S 2
Z Y ) = _2 l 1 (2 39)
where I1 T

where


s 2 2 Q (2-40)

T T T
4Si -T Yj Qt +r7"; T L s, (241)
t=1 s=1 t=s+1
,.. 1- (2-42)
1+1

Note that s2 and STi are consistent estimators for the variance a2 of Qt and the partial sum

variance a2 =limT, E (E 52), where ST z= t1 is the partial sum of the error terms

in (2-37).

The critical values for Z, and Zt statistics can be found in [76] (Tables I and II).

Phillips and Ouliaris tabulated the values for cointegrating regressions with at most 5

explanatory variables. Some estimates of the critical values for the Phillips-Ouliaris test

(Zn) are listed in Table 2-2.

2.4.3 Estimation of Cointegrated VAR(p) Processes

Several methods can be employ, ,1 to estimate the parameters of a Cointegrated

VAR(p) model, including modifications of the approaches used for estimation of the

standard VAR(p) processes.









In this section we present the maximum likelihood approach to estimating a Gaussian

cointegrated VAR(p) process. Suppose yt is a realization of a K-dimensional VAR(p)

process with cointegration rank r, such that 0 < r < K. Without loss of generality, we

assume that Y(t) has zero mean, i.e. the intercept v = 0 in (2-24).

Given a realization yt, t = 2,..., of Y(t), one seeks to determine the coefficients of

the following model:


Yt= Alyt-1 + ... + Apyt+p+ Et, t= 1,2,..., (2-43)

subject to the constraint


rank(I) = rank(IK A -...- A) = r. (2-44)


Note that Et is assumed to be a Gaussian white noise with a nonsingular covariance matrix

Es. Furthermore, the initial conditions y-p+l,... ,yo are supposed to be fixed.

In order to impose the cointegration constraint, the model (2-43) is reparameterized

in the following fashion [59]:


Ayt DAyt-_ +... + Dp_,Ayt-p+l + Fyt-p + t, t 1,2,... (2-45)


where Ay = yt yt-1, and matrix II can be represented as a product II = HC of matrices

of rank r, i.e. H is (K x r) and C is (r x K).

Consider


AY :- [AY1,...,Ayr],

Ayt

AXt : (2-46)

AXYt-p+2
AX := [AXo,..., AX,_-] ,

D := [D1,...,Dp-1],

Y-p := [y-p, ...-, YT-p]









Then the log-likelihood function for a sample of size T can be written as:

KT T
In 1 ln[2r] TIn [det S,]
2 2
trace ((AY DAX + HCYp)' E (AY DAX + HCYp)) (2 47)
2

The proof of the following theorem on the maximum likelihood estimators of a

cointegrated VAR process can be found in [59] (Proposition 11.1).

Theorem 1. (reproduced from [59])



M : I AX'(AXAX')-AX,

Ro := AYM,

R1: Y_, -,
1
Sj RiR j, =0, 1.

Let G be the lower ', :,,,,,il,1ir matrix with positive .':,i..'.1i such that GS G' = IK

Denote A1 > ... > AK to be the .i; ,; ,;,,,. of GSloSoo So1G',

and

vi,..., v2 be the corresponding orthonormal eigenvectors.

Then the 1 A-1.:. O 7, ,.od function in (2-47) is maximized for

C := [v ,... ,]'G,

H -AYMY'IC' (CY, _IY'C')1

-So1C' (CS11C')-

D : (AY + HCYp)AX (AXAX')-,

Z := (AY DAX + HCY_) (AY DAX + HCY_)'.
T









The maximum is


KT T + ln KT(24
max[lnl] K2 ln[27]- 2 In [det oo] + ( A) (2-48)
i= 1

2.4.4 Testing for the Rank of Cointegration

Based on Theorem 1, one can easily derive the likelihood ratio statistic for testing

a candidate value ro of the cointegration rank r of a VAR(p) process against a larger

cointegration rank rl.

Given a VAR(p) process y(t) defined by (2-24), suppose we wish to test a hypothesis

Ho against an alternative H1, where


Ho : r= ro against H1 : ro < r < rl. (2-49)


Under assumption that the noise Et is a Gaussian process, the maximum of the likelihood

function for a cointegrated VAR(p) model with cointegration rank r is computed in

Theorem 1. From that result, the value of the LR statistic for testing (2-49) can be

determined in the following manner:


ALR(ro,r1) 2[lnLmax(ri)- lnLmax(ro)] (2-50)
r1 TO



-T ln( A),
i=ro+1

where Lmax(ri), i = 0, 1, denotes the maximum of the Gaussian likelihood function

for cointegration rank ri. The advantage of this test is in the simplicity with which the

LR statistic can be computed. On the other hand, the .,-i-,iii!l' tic distribution of the

LR statistic (2-50) is nonstandard. Specifically, the LR statistic is not ..i-','i!iil1 l ly

distributed according to x2-distribution. Nevertheless, the ..i-mptotic distribution of the

cointegration rank test statistic ALR depends only on two factors:









the difference K r between the process dimensionality and the cointegration rank;

and

the alternative hypothesis.

As a result, the selected percentage points of the .i-:-.1,l' I '1ic distribution of the test

statistic ALR were tabulated by Johansen and Juselius in [38]. The percentage points of

.I-i-,iIl ,-l ic distribution of ALR are given in Tables 2-3 and 2-4.

Table 2-1. Critical values of the .,-imptotic distributions of the T(c- 1) and r for
performing unit root check by the ADF test (reproduced from [22])
Statistic 911 95' 9' '.
c 0.93 1.28 2.03
S0.89 1.28 2.01


Table 2-2.


Critical values of the ..i-mptotic distributions of the Z, statistic for testing the
null of no cointegration (Phillips-Ouliaris demeaned, reproduced from [76]).
Parameter n (n = K 1) represents the number of explanatory variables


911' .
-17.0390
-22.1948
-27.5846
-32.7382
-37.0074


Table 2-3.


95',
-20.4935
-26.0943
-32.0615
-37.1508
-41.9388


9-r
-28.3218
-34.1686
-41.1348
-47.5118
-52.1723


Percentage points of the .,-i-ii! l 'i ic distributions of the ALR(r, K) for testing
the cointegration rank (reproduced from [38])
91 r' 95' 9' '
6.69 8.08 11.58
15.58 17.84 21.96
28.44 31.26 37.29































Table 2-4. Percentage points of the .,-i~,ll, '1 ic distributions of the ALR(r, r + 1) for testing
the cointegration rank (reproduced from [38])
K -r 91 r', 95' I
1 6.69 8.08 11.58
2 12.78 14.60 18.78
3 18.96 21.28 26.15









CHAPTER 3
PHASE SYNCHRONY IN BRAIN DYNAMICS

In this chapter we introduce a concept of phase synchronization, and consider two

methods for estimating the phase of a signal, specifically using the Hilbert transform and

via the wavelet transform.

3.1 The Role of Phase Synchronization in Neural Dynamics

The word "synchrony" originates from a combination of two Greek words cva (syn,

meaning common) and Xpovoc (chronos, meaning time), and it can be translated as

!' iplening at the same time". A concept of synchronization can be defined as a process

of active adjustment between the rhythms of different oscillating systems due to some

kind of interaction or coupling between them [78]. Synchronization phenomena were

discovered in the seventeenth century by C. Huygens who first observed synchronization

between two pendulum clocks hanging from a common support [33]. Since then, the

study of synchronization between dynamical systems became an active field of research

in many scientific and technical disciplines, including solid state physics [74], plasma

physics [84], communication [11, 48], electronics [72, 77], laser dynamics [20, 87, 98], and

control [80, 88].

Synchronization phenomena can also be found in physiological systems, such as heart

and brain. Synchronization processes in physiological systems were discovered by B. van

der Pol in the beginning of the twentieth century. In particular, van der Pol was the first

to apply oscillation theory to the human heart [103].

One of the important research areas in neuroscience explores the role of synchronization

in neural dynamics. Much effort is given to investigation of synchronization phenomena on

all different levels of organization of brain tissue, starting with pairs of individual neurons

to larger scales, such as within a given area of the brain or between distinct parts of the

brain.









M ini: studies emphasize that normal cognitive processes call for the transient

integration of numerous functional areas in various regions of the brain, and as a result,

the brain regions are in constant interaction with each other [15, 101, 104, 105]. Neural

synchrony p1 ii- a vital role in such large-scale integration. In fact, various neuronal

groups oscillate in specific frequency bands and become phase-locked over a limited period

of time. This observation has stimulated the development of robust approaches that

allow one to measure the phase-synchrony in a given frequency band from experimentally

recorded biomedical signals, such as EEG.

In particular, the importance of synchronization of neuronal discharges has been

shown by a variety of animal studies using microelectrode recordings of brain activity [83,

96]. The findings in the microelectrode-recording studies are also supported at coarser

levels of resolution by other studies in animals and humans [21].

An electrophysiological signal is recorded via a low impedance extracellular

microelectrode by placing the microelectrode sufficiently far from individual local neurons

in order to prevent any particular cell from dominating the signal. Next, to obtain the

local field potential (LFP), the signal is low-pass filtered, with a cut off at approximately

300 Hz. Due to the low impedance and positioning of the micro electrode, the recorded

signal is predominantly induced by the activity of a large number of neurons. The

unfiltered signal reflects the sum of action potentials from cells within approximately

50-350 micrometers from the tip of the electrode [53] and slower ionic events from within

0.5-3 millimeters from the tip of the electrode [39]. The spike component of the signal is

removed by low-pass filter, whereas the lower frequency signal, the LFP, is preserved in

the signal. It is assumed that the local field potential characterizes the synchronized input

into the observed area, in the contrast to the spike data, which represents the output from

the area.

Local field potentials (LFPs) of various degrees of spatial resolution can be recorded

by scalp EEG or MEG. In fact, studies have shown that the presence of gamma and beta









band responses can be detected during visual discrimination tasks on the human scalp [97]

and in subdural electrocorticograms [50, 54]. In addition, some recent findings -~i--. -1

that long-range synchronization analogous to the one found in microelectrode studies in

animals can also be detected between surface recordings [82].

It has been shown that synchronization is a significant attribute of the signal recorded

from the patients affected by several neurological disorders. In particular, researchers have

found that epilepsy [65] and Parkinsons disease [99] manifest as a pathological form of the

synchronization process.

It is noted in [56] that although the cross-correlograms between spike discharges may

be adequate for microelectrode studies, the quantification of phase synchrony between

meso- or macro-electrodes (i.e. EEG/\! I G, intracranial recordings) calls for entirely

different methods. Therefore, they emphasize an importance of clearly distinguishing

between synchrony as an appropriate estimate of phase relation, and the classical measures

of coherence or spectral covariance that have been extensively used in neuroscience [8, 10,

62]. Le Van Quyen et al. discuss two important limitations of coherence [56].

The first limitation arises because the standard approaches for measuring coherence [12]

based on Fourier analysis are known to be highly dependent on the stationarity of the

measured signal, whereas the signals recorded from the brain, such as EEG, appear

to be clearly non-stationary. Applying the timefrequency estimation method, which is

not founded on the assumption of stationarity, could improve this limitation towards

estimating a stable, instantaneous coherence as well as synchrony between two concurrent

brain signals.

The second limitation stems from the fact that classical coherence is a measure of

spectral covariance. Hence, it is not able to separate the effects of amplitude and phase

in the relations between two signals. Because we are concerned with examining the

specific hypothesis that phase-locking synchrony is the pertinent biological mechanism of

transient integration in the brain, coherence serves only as an indirect measure. In order









to investigate the phase relations between different areas in the brain directly, the phase

component should be extracted separately from the amplitude component for a given

frequency or frequency band, which can be quite unstable or even chaotic. In a nutshell,

coherence gives only an indirect and approximate indication of phase synchrony. There has

been a general increase of interest in understanding bivariate data by studying their phase

synchronization over time not only in neuroscience, but also in other research fields [86].

In other words, even though our discussion of the phase synchrony is focused on EEG

data, its applications can also be extended to the fields other than neuroscience.

Classical concept of the synchronization of two oscillators is described as an active

adjustment of their rhythmicity that manifests in phase-locking between the synchronized

oscillators. Specifically, given two signals X1(t) and X2(t), and their corresponding

instantaneous phases 01(t) and 02(t), the basic definition of the phase locking states that


ni(t) m2(t) = C cost, (3-1)

where integers n and m specify the phase locking ratio.

When investigating phase synchrony in neurophysiological signals, one must assume

that the constant phase locking ratio is valid within a limited time interval T, which

usually means a few hundreds of milliseconds. When examining neural signals, one

must keep in mind that discovering the presence of the phase locking between EEG

recordings from two distant parts of the brain is not straightforward. The detection of

phase synchrony in neural signals is problematic because of several factors particularly

when working not on the level of a single neuron, but rather with large neuronal

populations, whose activity is picked up by macroscopic or meso- electrodes. As noted

in [56], as a consequence of volume conduction effects in brain tissues, the activity of a

single neuronal population can be recorded by two distant electrodes, which results in

spurious phase-locking between their signals. Furthermore, in non-invasive EEG, the

true synchronies are hidden in a significant background noise, and so, in the synchronous









state, the phase shifts back and forth around some constant value. Hence, the signals

can be viewed as synchronous or not synchronous only in a statistical sense. Therefore,

this necessitates the development of novel approaches that are capable of extracting the

true phase synchronies from noisy data, and so, the condition (3-1) must be adjusted to

account for the noise as follows:


|ni(t)- m0r2(t)|
where C denotes a positive constant.

The following two key steps are instrumental in investigating the phase synchrony:

1. estimate instantaneous phase of each signal;

2. provide a statistical criteria to quantify the degree of phase-locking.

Two methods for detecting phase-locking applied to neuronal signals have recently

been considered in the literature. Tass and colleagues [99] extracted the instantaneous

phases from original signals by means of the Hilbert transform, and then applied to

magnetoencephalographic (\ll;G) motor signals in patients affected by Parkinson's

disease [99], and also to the synchronization between cardiovascular and respiratory

rhythms [93]. On the other hand, Lachaux et al. [50] estimated the phases from the

original signals by means of convolution with a complex wavelet, and then applied it to

EEG and intracranial data during cognitive tasks [51, 82].

The first step in quantifying phase synchronization between two time series X and Y

is the determination of their instantaneous phases Qx(t) and Qy(t). This is achieved either

via the Hilbert transform or via the wavelet transform. These approaches are presented in

the next two sections.

3.2 Phase Estimation using Hilbert Transform

The first method used to extract the instantaneous phase from the time series is

based on the ,1:.~i/.:l .-..1..,l approach, which was first introduced by D. Gabor [23] and

later extended for model systems and experimental data [70, 86].









The Hilbert ',. .", irm of a given real-valued function f(t) with domain T is defined as

a real-valued function f(t) on T as follows:


f(t) = CPV f(T)g(t r)dTr CPV g(T)h(t T)dr, (3-3)

where
1
g(t) := t eT,
7tT
and symbol CPV signifies that the integral is taken in the sense of C',,. 1I,; principal value.

Notice that f(t) can be viewed as a convolution g(t) f(t) of the original function

f(t) with the function g(t). This means that the Hilbert transform can be performed by

applying an ideal filter, whose amplitude response equals to 1, and phase response is a

constant r/2 lag at all frequencies.

Given an arbitrary continuous real-valued time series X(t), the corresponding I,.,l.l.'

/.::,,l is defined as the following complex-valued function:


x(t) = X(t) + 2 X(t) = ax(t) exp {z. x(t)}, (3-4)

where t denotes time, z is a unit on the complex axis, X(t) denotes the Hilbert transform

of the time series X(t), ax(t) is the corresponding instantaneous amplitude, and Ox(t)

represents the instantaneous phase of the signal via Hilbert convolution.

It follows from Equation 3-4 that the instantaneous phase Ox(t) of X(t) can be

computed as:

x (t) arctan () (3-5)
X (t)
A key advantage of the analytic approach is that the phase can be easily computed

for an arbitrary broad-band signal. On the other hand, instantaneous amplitude and phase

have a clear physical meaning only if X(t) is a narrow-band signal. Therefore, filtration

is required in order to separate the frequency band of interest from the background brain

activity.









3.3 Phase Estimation via Wavelet Transform

An alternative approach to determining the instantaneous phase of the signal is

based on the wavelet transform. This method of phase estimation was proposed by

Lachaux and colleagues [50, 52], and is similar to the Hilbert transform method presented

above. In their approach, Lachaux et al. extract the instantaneous phase by applying the

convolution of the original signal with a complex Morlet wavelet. They consider the Morlet

wavelet (also known as Gabor function) at time t and frequency w given by the following

formula:

t,' (e) = exp {z 27 ( t)} exp- -2t} (3-6)

Notice that t,,(-r) is the product of a sinusoidal wave at frequency w, and a Gaussian

function centered at time t, with a standard deviation a proportional to the inverse of w.

It depends solely on o, which sets the number of cycles of the wavelet to 6wJa.

According to [56], given the time series X(t), the coefficient of the Morlet transform

as a function of time t and frequency w is defined as follows:

Wx(t,) Xr)(-).r,,,)dr, (37)

where t,, (r) denotes the complex conjugate of the Morlet wavelet t,(-(r).

The following slight modification of the Morlet wavelet is introduced in [81]:


(t) exp 1- } exp {z. ,t} exp -2 ) (3-8)

where parameters wo and a represent the center frequency and the rate of decay of the

wavelet function, respectively. This is proportional to the number of cycles and related to

the frequency span by the uncertainty principle.

Similarly to the above, a complex time series of wavelet coefficients is obtained via the

convolution of X(t) with Q(t):


Wx() = o X)(t) = J r)X(t 7)dr iix(t) exp I () (3-9)
J-+









where ax(t) and Qx(t), respectively, are the instantaneous amplitude and the phase of the

signal X(t) extracted via the Morlet wavelet.

As in the case of the Hilbert transform, the phases can be determined from

Equation (3-9) as

x(t) arctan { [Wx( }, (3 10)

where Z [Wx(t)] and S [Wx(t)] denote the real and imaginary parts of the complex

transformed time series Wx(t), respectively.

3.4 Comparison between Two Approaches to Phase Extraction.

The above two definitions of the instantaneous phase are closely related, despite

the fact that they are based on very different transformations. In particular, the

connection between the phases obtained via the Hilbert and wavelet transforms was

demonstrated experimentally in [56] and also explained theoretically by Quian Quiroga

and colleagues [81].

In a nutshell, the phase Qx(t) extracted from the signal using the wavelet transform

corresponds approximately to the phase Qx(t) determined via the Hilbert convolution,

which would be performed after band pass filtering the time series. Furthermore, if the

phase estimation based on wavelet transform were performed by a convolution with an

analytic wavelet, and if this wavelet were applied to do the band pass filtering in the

Hilbert approach, then such approaches would, in fact, be equivalent.

It is easy to see that in the method based on the wavelet convolution, the center

frequency u and the rate of decay a of the wavelet can serve as parameters that allow us

to modify the frequency range of interest. On the other hand, the actual phase extraction

via the Hilbert transform is free of parameters, and so the correspondent phase preserves

information from the entire power spectrum and not just the main frequency band as in

the case of the wavelet convolution. As a result, it is possible to achieve a comparison

of narrow band and broad band synchronization simply by using both methods of phase

extraction without performing any additional filtering.









3.5 Measures of Phase Synchrony

Various measures of phase synchrony between two signal are proposed based on the

phases extracted via the Hilbert and the wavelet transforms, including standard deviation,

mutual information and Shannon entropy [49, 56]. However, most of the currently used

measures of phase synchronization are based on bivariate indexes.

In C'! lpter 4, we propose a novel multivariate approach to detecting phase synchronization

in the phases extracted from multiple time series, such as multichannel EEG.









CHAPTER 4
APPLICATION OF VECTOR AUTOREGRESSION TO MINING BRAIN DYNAMICS

4.1 Numerical Issues in Estimating Parameters of Vector Autoregression
from EEG

In the applications of Granger causality and related measures to the EEG, the

directional dependencies in neural data are analyzed based on autoregressive modeling.

Although, Bernasconi and Konig [2] applied statistical testing to verify the stationarity

of the data, and established the duration of the stationary interval for EEG to be

approximately 1 second. The statistical testing of underlying assumptions of the VAR

(which was thoroughly discussed in C'! lpter 4) is often omitted in the later studies.

As shown above, the stability condition is a very important assumption of vector

autoregression. In this study, we examine how the parameters of the model order and

sample size influence the stability of the derived VAR model.

In order to estimate VAR(p) model from data and investigate the properties of the

derived model, the rodent intracranial EEG data were used. The data set consisted of

the electroencephalographic recordings from 6 electrodes (implanted in left frontal, right

frontal, two left hippocampal and two right hippocampal parts of the rodent's brain)

sampled at 200 Hz.

To examine the applicability of the vector autoregressive modeling to EEG data, we

estimated the VAR(p) model parameters for different values of lag order p and different

sample sizes T. The sample sizes T E [1, 300] were used, and the lag length parameter

p varied between 1 and 30. In addition, we filtered data using a Rectangular band pass

Hamming window with 100 coefficients into the frequency bands of 0 -30 Hz, 30 -60 Hz,

60 -90 Hz, 90 120 Hz, and 120 150 Hz. The raw data and the five differently filtered

data represented separate data sets. For each data set, we ran the model estimation

procedure with T = 1,2,..., 300 and p = 1,2,..., 30. The procedure for estimating

coefficients of the model was implemented in the MATLAB environment based on the

multivariate least squares method presented above. For every VAR model derived from









the data, the stability condition was checked, and the number of the roots of the reverse

characteristic polynomial (RCP) on and inside the complex unit circle was stored. For

each data set, a 3-dimensional surface plot was produced by graphing (T, p, n), where

n denotes the number of the RCP roots on and inside the unit circle, whereas T and p

represent the sample size and the model order (or lag length), respectively. The obtained

surface plots are di- l i' '1 in Figures 4-1 and 4-2.

The surface plot in Figure 4-1 supports the fact [59] that the sample size parameter

should significantly exceed the lag length p, i.e. T > p. On the other hand, it can be

seen from Figure 4-1 that even for T > p, the stability assumptions may still be violated.

Indeed, for T = 132 considerably larger than p = 2, the estimated VAR has two RCP roots

on or inside the unit circle (n = 2).

Figure 4-2 shows that for the filtered data, the (T,p) region, where the stability

condition of the estimated VAR(p) model is violated, covers almost the whole domain.

Whereas the (T,p) region that corresponds to stable VAR(p) models is much smaller

than the stable region in Figure 4-1, and characterized by large T and very small p. Very

similar results were obtained for the number of the RCP roots inside the unit circle, when

estimating parameters of VAR(p) with different p using filters in the 30-60 Hz, 60-90 Hz,

90-120 Hz, and 120-150 Hz bands.

The experiment was repeated with consistent results on various samples from the six

data sets. The results of our experiments clearly show that the stability condition imposed

on the linear VAR model is often violated even for the parameters T > p. Furthermore,

filtering the data within some restricted frequency band often leads to reduction of the

(T,p) domain where the estimated VAR(p) models remain stable. In practice, the EEG

data are often filtered within a certain frequency band. In many studies that utilize the

vector autoregression to investigate Granger causality in the biomedical time series, the

verification of the conditions assumed by the VAR model is often omitted. From our point









of view, the relevant statistical tests ju- I iVii:-; the suitability of the model should alvb-7

be performed when estimating the model from data.

Various modifications of vector autoregression (which relax the stability condition of

no roots inside unit circle) are developed for analysis of economic time series. Although

Bernasconi and Konig [2] examined the stationarity of different EEG data, and concluded

that a time interval of approximately one second is the interval during which the EEG

time series can be considered stationary, the underlying assumptions of the VAR modeling

such as stationarity and stability are rarely statistically tested in applications to EEG

data. The results of our experiments indicate that the stability assumptions of vector

autoregressive model are often violated in application to the EEG data. This observation

-i --.- -i; that suitable extensions of multivariate autoregression to unstable processes may

fit the data better, and as a result, such extensions of VAR may be more appropriate for

the EEG time series analysis than the standard linear vector autoregressive modeling.

Additional statistical testing is required in order to make conclusions on what

multivariate models are the most suitable for extracting the directional dependencies

between channels in a frequency domain from multichannel EEG data.

4.2 Multivariate Approach to Phase Synchrony via Cointegrated VAR

We propose a new approach to measuring the synchrony among the instantaneous

phases extracted from multivariate time series. This approach is based on the Cointegrated

VAR modeling of time series.

Given the signal represented formally as a multiple time series X(t), one can extract

the instantaneous phases Qx(t) from each one-dimensional component Xi(t) of the signal

as shown in C'! lpter 3 (either via a convolution with the Morlet wavelet or by applying

the Hilbert transform). The phase extraction procedure produces a new multiple time

series Qx(t) of the correspondent phases.

Next, we derive new measures of phase synchrony of the signal based on the concepts

introduced in C'!i pter 3. Let us observe that the left-hand side of Equation (3-1)









represents the linear combination of the respective phases Ox, (t) and x,2 (t) with integer

coefficients. Also recall that condition (3-1), which defines phase-locking between two

signals XI(t) and X2(t), needs to be modified in practice to account for the noise in the

signal. Taking into account presence of the stochastic noise in the phase series, let us

introduce a i,. ',/fi. ,1 concept of the phase synchrony between two signals by rI' 1, i.,:. the

'i, ,i,,d:l;/ condition on the coefficients in the linear combination as follows.

Two signals XI(t) and X2(t) are considered to be gener /ll/; phase-synchronized, if the

correspondent instantaneous phases Ox, (t) and x,2 (t) satisfy the condition below:


3 c1, C2 : C1, () + 2x (t) = Zt, (4 1)

where zt ~ N(C, a') is a stochastic variable that represents the deviation from the

constant level C as a result of the noise. Notice that in the contrast to condition (3-1)

in the classic definition of phase synchronization, the coefficients cl and c2 in the

definition (4-1) of generalized phase-synchrony do not need to be integer.

Furthermore, it is straightforward that the new condition (4-1) means that a

two-dimensional process X(t) = (XI(t),X2(t))' is cointegrated. Based on this observation,

we can extend our modified concept of phase synchronization between two signals to the

multivariate case in the following manner.

The multichannel signal X(t) = (Xi(t),... ,XK(t)) is considered to be phase-

synchronized of rank r, if the process Ox(t) composed of the correspondent instantaneous

phases Ox,(t), i 1,..., K is cointegrated of rank r.

In the subsequent subsections, we first discuss the role of the cointegration rank in

the framework of multivariate phase-synchronization, and then apply this approach to

multichannel EEG data collected from the patients with absence epilepsy.









4.2.1 Cointegration Rank as a Measure of Synchronization among Different
EEG Channels

Note that integrated autoregressive processes I(d) are shown to exhibit behavior

similar to that of a random walk. In a short paper [66], Michael Murray used an example

of drunkard and her dog to illustrate the concept of the cointegration. To explain

our reasoning behind the rank of cointegration as a measure of synchrony, we briefly

summarize and then further extend his analogy.

The nonstationary processes (such as a random walk) are often introduced by teachers

of statistics by comparing them (it) with the drunkard's walk. The drunkard wonders

aimlessly, so that the direction of each step is random and completely independent of her

previous steps. In other words, the meandering of the drunkard is described by a random

walk:

Xt Xt-1= Et, t= 1,2,..., (4-2)

where Xt represents the position of the drunk at time t, and Et is a stationary white-noise,

which models the drunk's step at time t).

As Murray noticed [66], an unleashed puppy is another creature, whose behavior

reminds a random walk. Indeed, each new scent that puppy's nose comes upon dictates

a direction for the pup's next step so strongly that the last scent along with its direction

is forgotten as soon as the new scent appears. Having shown that the puppies follow the

random walk yt, t = 1, 2,..., let us represent the puppy's walk as:


yt Yt-1 = t, t = 2,..., (4-3)

where ct is a stationary white-noise (i.e. puppy's step at time t).

The well-known feature of a random walk is that the best predictor of the future

value is the most recently observed one. In other words, the longer it has been since we

had seen the drunk, or the dog, the further away from the initial place, on average, they









are at the moment. As a result, even if the drunk and the dog crossed their walks at some

location, as the time goes on, they tend to wander further away from each other.

However, if the puppy belongs to the drunkard, then they will remain relatively close

to each other at all the time, similarly to the individual integrated processes that together

form a cointegrated process. Indeed, the drunk would still wonder aimlessly in a random

walk fashion, as would her puppy. However, from time to time she would remember about

her dog and call for it, the puppy would recognize her voice, and bark. They would hear

each other and make their next step in each other's direction.

The paths of the drunk and her dog are still nonstationary, but they are no longer

independent from each other. As a matter of fact, at each time, the puppy and its master

are likely to be found not far from each other. If this is true, then the distance between

two paths is stationary, and the walks of the drunk xt and her dog yt are said to be

cointegrated, i.e. xt and yt are integrated 1(1), and there is a linear combination of xt and

yt (with non-zero weights) that is I(0), i.e. stationary.
Mathematically, the cointegrating relationship between a lady and her puppy can be

written as:


Xt- Xt_- =t + C(Yt-1 Xt), (4-4)

yt yt-1 = t + d(xt-1 yt-1), (4-5)

at time t = 1, 2,.... Note that Et and et, as before, represent the stationary white-noise

steps of the drunk and her dog.

Since Equation 4-4 can be easily rewritten in form of (2-45) as follows:


Xt Et C -C Xt-1
A (4-6)
t t -d d y/t-1









then


c -c
-d d


and so, rank(II) = 1. This shows that the cointegrating relationship between the drunk

lady and her puppy has the cointegration rank 1.

Note that rank(I) = 0, if and only if c = d = 0. In such case, (4-4) becomes simply a

system of equations (4-2) and (4-3), which models two independent random walks driven

by independent white noise processes E and e. On the other hand, when at least one of the

coefficients c and d is non-zero, then by multiplying system (4-6) by a vector [d, c]', we

have:

dAxt + cAyt = dEt + et, t = t, 2,..., (4-7)

which means that the model is driven by a single common stochastic trend dEt + ct.

Although the example described by Murray is clearly a bivariate cointegrated

VAR(1), it can be extended to an illustration of the multivariate cointegrated process.

Consider, for example, a heard of sheep guarded by two dogs, where the sheep wonder

aimlessly in the field, while the dogs run around and bring the sheep that have -11 i, l- too

far back into the flock. f -, for example, a faster dog guards sheep from the east, south,

and west, whereas a slower dog from the north, then the cointegrated process appear to

have the cointegration rank of 2. Clearly, two dogs are able to keep a flock of sheep closer

together, than a single dog can. In other words, the higher cointegration rank the more

restrictive it is.

In fact, let us consider a K-dimensional cointegrated vector autoregressive process,

and let r denote the cointegration rank of the process. Similarly to the bivariate example

above, we can see that when the rank is zero (r = 0), the univariate components of the

process are independent, and the model is driven by K independent white noise processes

(i.e. there is no cointegration). In the case of r = 1, we can decompose the multivariate









process onto K 2 independent components, and two dependent components that form a

common stochastic trend. Hence, in the case r = 1, the cointegrated model is driven by

(K 2) + 1 = K 1 independent stochastic processes. By induction, we can show that for

a cointegrated VAR process with the cointegration rank r, 0 < r < K 1, the VAR model

is generated by K r independent stochastic trends.

Therefore, the smaller is the cointegration rank r, the larger is the number K r

of the underlying independent stochastic trends, and so (the larger) is the vector space

in which our cointegrated model can travel. And the other way around, increasing the

cointegration rank of the model shrinks the underlying domain of the process, i.e. makes it

bounded to a smaller hyperplane. For r = K, the VAR(p) is a stable process, which clearly

has the most constrained domain. For r = 0, the VAR process is not cointegrated and

unrestricted.

Thus, in the framework of generalized phase-synchronization introduced above, the

cointegration rank represents a fundamental measure of synchrony in the multi-channel

signal, such as EEG. In particular, we z v that the signal is ../,ipl. /. ;/ r;,. ;. ronous,

if the cointegration rank r is zero. On the other hand, when the multivariate process is

stable (i.e. the rank coincides with the dimension of the process, r = K), the signal is said

to be perf,. I/,/ synchronous.

4.2.2 Absence Seizures

Absence seizures (or petit mal seizures) are known to occur in several forms of

epilepsy, whereas absence epilepsy refers to a type of epilepsy in which only the absence

seizures occur. Absence epilepsy is usually characterized by age of onset, and often affects

teenage population. Absence seizures usually begin in childhood or adolescence, and often

run in families, which may -ir--.- -1 a genetic predisposition. Absence seizures are marked

by momentary lapses of consciousness. Absence seizures often have no visible symptoms,

although some patients may have purposeless movements during a seizure, such as rapidly

blinking eyes. Absence seizures often have a brief duration, and a person may resume









the previous activity immediately after the seizure [73]. These brief seizures can happen

several times during a d,,-, but in some patients, the frequency of absence seizures can

be as high as hundred of times a di-, which interferes with the daily activities of a child

such as school. In some cases of childhood absence epilepsy, the seizures stop when a child

reaches puberty. Absence seizures exhibit a characteristic spike-and-wave EEG pattern at

a 3 Hz frequency [73].

Figure 4-3 di-pl'i-1 a multichannel EEG recording that includes an absence seizure.

The duration of the seizure is approximately 4 seconds. The figure vividly illustrates a

characteristic spike-and-wave activity during the seizure.

4.2.3 Numerical Study of Synchrony in Multichannel EEG Recordings from
Patients with Absence Epilepsy

The proposed approach to studying synchronization among multiple channels was

applied to analysis of EEG data recorded from the patience with absence epilepsy.

First, the multiple time series of the instantaneous phases were extracted from the

raw EEG data using the Hilbert transform approach as described in Section 3.2. In

particular, we took advantage of the functions hilbert and angle readily available in the

MATLAB R 2006a environment.

The VAR modeling and testing were implemented using the R 2.6.1 statistical

software. In our analysis of the instantaneous phases, we incorporated ar, adf. test,

po.test, caj olst and other functions found in packages series and urea.

Next, we illustrate our approach on the example of the EEG data file that includes

three seizure intervals. The file contains a 16-channel recording of scalp EEG sampled

at the 200 Hz frequency as well as two auxiliary channels, which were discarded. The

instantaneous phase values were estimated from the EEG time series by means of Hilbert

transform, and the resulting phase series were tested using the ADF test introduced in

Section 2.4.1. Specifically, we applied the Augmented Dickey-Fuller procedure to test the









presence of a unit root in the individual univariate components of the multiple time series

of estimated phases.

The results of our experiments for seizures 1, 2, and 3 are presented in Tables 4-1,

4-2, and 4-3, respectively. The channels, for which the ADF unit root test has detected a

presence of a unit root at the significance level a = 0.01, are listed as integrated. Whereas

the channels, for which the null hypothesis of a unit root has been rejected by the ADF

at the 1 percent level, are denoted by stat.:. ; : The channels for which the p-values of

the ADF test exceed 2.5' are marked with *. Notice that all three seizure segments are

considered stable, when the ADF is applied at a 0.025 significance level.

Next, we fit vector autoregression to the multiple time series of phase estimates, for

each of three different segments (before, during, and after a seizure) in order to determine

appropriate lag length parameter p. To find appropriate lags p, the Akaike Information

Criteria (AIC) was used. This led us to choose several lag length for each segment and

each seizure. Finally, Johansen cointegration rank procedure was applied to determine the

values of cointegration rank r for each case. The results are summarized in Tables 4-4, 4-5,

and 4-6.

Notice that during the seizure the system becomes stable, especially when modeled

using a short estimate of the lag parameter. Since the durations of the seizure 1 and

seizure 2 are rather short, and only include 440-500 sample points, the models estimated

under a long lag parameter may not adequately represent the underlying processes in

seizure 1 and 2. On the other hand, seizure 3 is estimated based on almost 1200 sample

values, and therefore the long lag model of a longer seizure 3 may be more realistic, than

the long lag models for shorter seizures 1 and 2. Overall, the models based on a short

lag p for all three seizures provide an evidence of absolute synchronization among the

channels. Whereas, the the pre-seizure and post-seizure models are more likely to be less

restricted, and seem to exhibit a cointegration rank between 9 and 16.










Results of the ADF unit root tests for each channel during three segments (2
seconds immediately before seizure, during seizure, and 2 seconds after seizure)
for seizure 1. Note that the significance at 2.5'. level is denoted by *


pre-seizure
3,4,7,9,15
1*,2*,5,6*,8* ,10,11,12*, 13* ,14* ,16*


seizure
1-3,5-14
4,15,16


post-seizure
1,3,5,6,8,10,12,15
2*,4*,7,9,13*, 14,16*


Table 4-1.



Seizure #
stationar'
integrate

Table 4-2.


pre-seizure
3,4,7,9-11,13,16
1*,2*,5*,6,8*,12*,14*,15


seizure
1-16
none


post-seizure
3,7,11-16
1*,2,4*,5,6*,8,9*,10


Results of the ADF unit root tests for each channel during three segments (2
seconds immediately before seizure, during seizure, and 2 seconds after seizure)
for seizure 3. Note that the significance at 2.5'. level is denoted by *


pre-seizure
7,11,13,14
1*,2*,3,4*,5*,6*,8*,9,10*,12,15,16*


seizure
1-16
none


post-seizure
2,4,5,11,13,15
1,3*,6*,7,8,9*,10*,12,14*,16


Results of the Johansen cointegration rank procedure for the multiple series
during three segments (2 seconds immediately before seizure, during seizure,
and 2 seconds after seizure) for seizure 1. Significance level is 1 Full rank is
denoted by t


Segment
long lag
short lag

Table 4-5.




Seizure #
long lag
short lag

Table 4-6.




Seizure #
long lag
short lag


pre-seizure
p = 22,r = 12
p 2,r 13


p 23, r
p


seizure
11, p= 20, r = 13
S2,r 16t


post-seizure
p 20, r = 12
p 2, r = 10


Results of the Johansen cointegration rank procedure for the multiple series
during three segments (2 seconds immediately before seizure, during seizure,
and 2 seconds after seizure) for seizure 2. Significance level is 1 full rank is
denoted by t


pre-seizure
S21, r 14
2, r 16t


seizure
p =26, r = 12, p= 20, r 9
p =3, r = 16t


post-seizure
p 20, r = 10
p 2, r 13


Results of the Johansen cointegration rank procedure for the multiple series
during three segments (2 seconds immediately before seizure, during seizure,
and 2 seconds after seizure) for seizure 3. Significance level is 1 full rank is
denoted by t


pre-seizure
p 24, r 13
p 2, r 9


seizure
p = 26, r = 16t, p = 20,r = 16t
p 2, r = 16t


post-seizure
p 20, r = 13
p 2,r 16t


Results of the ADF unit root tests for each channel during three segments (2
seconds immediately before seizure, during seizure, and 2 seconds after seizure)
for seizure 2. Note that the significance at 2.5'. level is denoted by *


y
d


Seizure #
stationary
integrated


Table 4-3.


Seizure #
stationary
integrated


Table 4-4.





















100


0 0


Figure 4-1. Numbers n of roots of the reverse characteristic polynomial (RCP) for
VAR(p), which lie on and inside the complex unit circle, computed for different
sample sizes T and for different model orders p using the raw data


0 0


Figure 4-2.


Numbers n of roots of the reverse characteristic polynomial (RCP) for
VAR(p), which lie on and inside the complex unit circle, computed for different
sample sizes T and for different model orders p using the 0-30 Hz band filtered
data


























-Jl~


=r



-J1






L~ i'


- -7 I


s
N
m

0



-o

0
0;



0
ccO





0



0
r^
Cs








0b




0
0
0
U


0



o







0
0;
Cs







0




V




0
o
E
5D
r^j

--)


(i~l i I iiii iiiiii


C


C









CHAPTER 5
CONCLUSION

Investigation of spatio-temporal properties of the EEG data by data mining and

optimization approaches posts various challenges. Numerous features and methods have

been proposed for studying the multivariate series that is EEG. The analysis of EEG time

series is often approached from two different points of view, the one that treats EEG data

as produced by a deterministic chaotic dynamical system, and the other more traditional

approach of linear autoregressive modeling.

In this work, we investigated several statistical approaches that are recently

introduced for data mining brain dynamics. In particular, we examined the application of

vector autoregressive modeling and linear Granger causality to raw and filtered EEG data.

Motivated by recent success in application of phase synchronization to analysis of

dynamic processes in epileptic brain, we developed a concept of generalized ;.'. I,.-

nization based on the novel idea of extending the classical synchronization condition

of a bounded linear combination of two phases. This simple bivariate condition in the

multivariate case is analogous to a cointegrating relationship in the multiple time series.

Thus, we can analyze the synchrony among different parts of the common interrelated

system (such as a human brain), by modeling the phases extracted from a finite number

of signals in the systems by means of cointegrated vector autoregression. Moreover, we

showed that the cointegration rank in the cointegrated VAR model of the phase time series

can be viewed as a measure of synchrony among the phases of different components of the

EEG signal.

Not only this new measure of multivariate phase synchrony can be tested on various

biomedical data, such as multichannel EEG recorded from an epileptic brain, but also

the new multiple phase synchronization can be employ, 1 in different areas of applied and

theoretic research (including physics, communication, electronics, laser dynamics, and









control) for studying synchronization among several dynamical systems or a system that

consists of several parts.









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BIOGRAPHICAL SKETCH

Alla Revenko Kammerdiner was born in Kiev, Ukraine. An older of two children,

she grew up in Kiev, Ukraine, graduating from School #32 in 1994. She earned her B.S.

in Probability Theory and Mathematical Statistics from the National Taras Shevchenko

University of Kyiv in 1998.

In January 2001, Alla joined a graduate program in the Mathematics Department at

the University of Florida. Upon graduating in May 2004 with her M.S. in mathematics,

Alla entered the Ph.D. program in industrial and systems engineering at the University of

Florida.

Alla has been happily married to Jason R. Kammerdiner for the last 3 years. On

February 24, 2008 she completed her first marathon in 3:53:09.





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IwouldliketoshowmyheartfeltappreciationtomyadvisorDr.PanosM.Pardalosforhissupportandmentoring.Workingwithhimhashelpedmegrownotonlyprofessionally,butalsoasaperson.Iamalsoverygratefultoothermemberswhoservedonmysupervisorycommittee,J.ColeSmith,WilliamW.Hager,VladimirL.Boginski,andH.EdwinRomeijn,fortheirvaluablecommentsonmyresearchforthisdissertation.Lastbutnotleast,IthankmyhusbandJason,myparentsOlgaandAleksandr,mybrotherMikhail,andalltherestofmygreatfamilyfortheirunconditionalloveandsupport. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 LISTOFSYMBOLS .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 12 1.1StatisticalMethodsforDataMining ..................... 12 1.2ElectroencephalographicRecordings ...................... 16 1.3Featureextraction ............................... 19 1.4ContributionSummary ............................. 23 1.4.1TestingApplicabilityofFrequencyDomainEstimatesofGrangerCausalityforEEGtimeseries ..................... 23 1.4.2GeneralizationofPhaseSynchronizationviaCointegratedVAR ... 25 2AUTOREGRESSIVEMODELINGOFMULTIPLETIMESERIES ....... 28 2.1MultivariateAutoregressiveModelinginEEGDataMining ......... 28 2.2TestsofGrangerCausality ........................... 29 2.3VectorAutoregressiveModels(VAR) ..................... 31 2.3.1MethodsforVARParameterEstimation ............... 31 2.3.2VAROrderSelectionCriteria ..................... 34 2.3.3StabilityConditionandOtherAssumptionsofVAR ......... 35 2.4InegratedandCointegratedVAR ....................... 37 2.4.1AugmentedDickey-FullerTestforTestingtheNullHypothesisofthePresenceofaUnitRoot ...................... 40 2.4.2Phillips-OuliarisCointegrationTest .................. 43 2.4.3EstimationofCointegratedVAR(p)Processes ............ 45 2.4.4TestingfortheRankofCointegration ................. 48 3PHASESYNCHRONYINBRAINDYNAMICS .................. 51 3.1TheRoleofPhaseSynchronizationinNeuralDynamics ........... 51 3.2PhaseEstimationusingHilbertTransform .................. 55 3.3PhaseEstimationviaWaveletTransform ................... 57 3.4ComparisonbetweenTwoApproachestoPhaseExtraction. ......... 58 3.5MeasuresofPhaseSynchrony ......................... 59 5

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....................................... 60 4.1NumericalIssuesinEstimatingParametersofVectorAutoregressionfromEEG ....................................... 60 4.2MultivariateApproachtoPhaseSynchronyviaCointegratedVAR ..... 62 4.2.1CointegrationRankasaMeasureofSynchronizationamongDierentEEGChannels .............................. 64 4.2.2AbsenceSeizures ............................ 67 4.2.3NumericalStudyofSynchronyinMultichannelEEGRecordingsfromPatientswithAbsenceEpilepsy ................. 68 5CONCLUSION .................................... 73 REFERENCES ....................................... 75 BIOGRAPHICALSKETCH ................................ 83 6

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Table page 2-1ADFtest:CriticalvaluesoftheT(bc1)andbstatistics ............. 49 2-2Phillips-Ouliarisdemeaned:CriticalvaluesoftheZstatistic .......... 49 2-3Johansentest:CriticalvaluesoftheLR(r;K)statistic .............. 49 2-4Johansentest:CriticalvaluesoftheLR(r;r+1)statistic ............ 50 4-1Seizure1:ResultsoftheADFunitroottests ................... 70 4-2Seizure2:ResultsoftheADFunitroottests ................... 70 4-3Seizure3:ResultsoftheADFunitroottests ................... 70 4-4Seizure1:ResultsoftheJohansencointegrationranktests ............ 70 4-5Seizure2:ResultsoftheJohansencointegrationranktests ............ 70 4-6Seizure3:ResultsoftheJohansencointegrationranktests ............ 70 7

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Figure page 1-1TheInternational10-20systemforplacementofEEGelectrodes ......... 27 4-1Rawdata:NumbersofunstablerootsfordierentTandp 71 4-2Filtereddata:NumbersofunstablerootsfordierentTandp 71 4-3EEGsegmentwithabsenceseizure ......................... 72 8

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Mostofthenotationisunambiguouslydenedinthetextwhereitisintroduced.Toprovidesomegeneralguidelines,weincludethefollowinglistofcommonlyusedsymbols. ARIMA(p;d;q)autoregressiveintegratedmovingaverageprocessAR(p)autoregressiveprocessoforderpCPVCauchyprincipalvalue 9

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Thisstudydiscussesstatisticalapproachesusedfordataminingofmultichannelelectroencephalogramrecordings.Suchrecordingsrepresentmassivedatasetsthatcontainhiddenpatternsofcomplexdynamicalprocessesinthebrain.Formally,multichannelEEGcanbeviewedasamultipletimeseries,andtherefore,anaturalideaforsummarizingsuchdataistoutilizeautoregressivemodelingofmultivariatestochasticprocesses. Inparticular,wethoroughlydiscussvariousconceptsandapproachesrelatedvectorautoregressiveprocesses,includingstablestationaryVARmodelsoforderpandnonstationarysystemswithintegratedandcointegratedvariables,aswellasproceduresforestimatingparametersofthesystems(e.g.order,lag,orcointegrationrank). TheworkhighlightssomestabilityissuesthatmayariseintheapplicationofvectorautoregressiontominingEEGdata,andquestionstheapplicabilityofGrangercausalityinthefrequencydomaintomultichannelEEG. SynchronizationhasbeenfoundtobeanimportantcharacteristicoftheabnormalbraindynamicsmanifestedbyepilepsyandParkinson'sdisease.Wereviewtwoapproachesforextractingtheinstantaneousphasefromtimeseries.Inthisstudy,wegeneralizetheconceptofthephasesynchronization,andproposeanovelapproachbasedonmultivariateanalysisviamodelingcointegratedVAR(p)processes. 11

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Dataminingtogetherwithdatapreprocessingconstitutethecentralpartofamoregeneralprocessofknowledgediscoveryindatabases(KDD).TheKKDprocesscanbedescribedasasequenceofactions,whichselectstherawdataindatawarehousesandtransformstheselecteddatainordertodiscovervalid,understandable,novelandpotentiallyusefulknowledgefromthedata.Datapreprocessingthatisappliedtorawdatatoimprovethequalityofthedataofteninuencestheselectionandfacilitatestheapplicationofdataminingtechniques.Properpreprocessingofrawdataleadstoadecreaseinthetimeneededtominethedata,andbooststheoverallminingeciency. Thetechniquesusedfordatapreprocessingcanberoughlysubdividedintodatacleaning,dataintegrationanddatareduction[ 64 ].Datacleaningtechniqueshandletheproblemofincomplete,inconsistentanderroneousdata,removethenoiseinherentlypresentintherawdata,minimizeredundancyinthedata,etc.Dataintegrationisconcernedwithcombiningheterogeneousdatacollectedfromdierentsourcestoforma 12

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Thetwofundamentaltasksassignedtodataminingareadescriptivetaskofdiscoveringhiddenpatternsandrelationshipsingivendata,andapredictivetaskofforecastingorclassifyingthemodel'sbehaviorfromavailabledata.Dataminingincludesregression,classication,clustering,imagerestoration,learningassociationrulesandextractingfunctionaldependencies,datasummarization,etc.Dataminingiscloselyconnectedtootherresearchareassuchasstatistics,machinelearningandarticialintelligence,optimization,visualizationanddatabases.Dataminingutilizesmanyimportantresultsfromtherelatedelds,whilekeepingthemainfocusonthealgorithmsandarchitectures,scalabilityofthenumberoffeaturesandinstances,andautomatedmanagingofmassivequantitiesofdiversedata. Manyareasofdataminingemployvariousapproachesdevelopedintheeldofoptimization.Inparticular,itisshownin[ 7 ]thatmanybasicproblemsindatamining,includingclassicationandclustering,canbeformulatedasmathematicalprogrammingproblemsandsolvedusingoptimizationtechniques. Infact,Bradleyatel.[ 7 ]demonstratedthataproblemofminimizingthenumberofmisclassiedpointsintwo-classclassicationcanbeviewedasalinearprogramwithequilibriumconstraints(LPEC).LPECisalinearprogram(LP)withasinglecomplementarityconstraint.Suchconstraintimposesaconditionoforthogonalitybetweentwolinearfunctions.LPECformulationarisesintheinstanceswhentheconstraintsoftheproblemincludeanotherLPproblem. Inaddition,theproblemoffeatureselectionintwo-classclassicationbyndingaseparatingplanethatutilizesminimumnumberoffeaturescanbegivenamathematicalprogrammingformulationasaparametricproblem.Furthermore,theclassicationviasupportvectormachines(SVMsthatndtheseparatingplanemaximizingthe 13

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6 ].Moreover,asindicatedin[ 7 ],theabovemathematicalprogrammingformulationshavebeenextendedtobeeectivelyemployedbyotherdataminingapproaches,includingneuralnetworkstraining,calculationofnonlineardiscriminants,andbuildingdecisiontrees.Theclusteringproblemhasacomplexformulationasaminimizationproblemwiththeobjectivegivenbyasumoftheminimumsofasetofconvexfunctions[ 5 ].Ingeneral,thisobjectivefunctionisneitherconvexnorconcave.SeereviewbyBradleyatel.[ 7 ]foradditionalinformationaboutmathematicalprogrammingformulationsforvariousproblemsindatamining,theapplicationofoptimizationtechniques,aswellasthechallengesthattheeldofdataminingoerstooptimization. Manydataminingapproaches,suchasclassication,clustering,indexingandsegmentation,havebeenappliedtotimeseriesanalysis[ 45 ].Manytraditionalstatisticalapproachesarealsoappliedtominingtimeseries.Forinstance,regressionisoneofthemostcommonlyusedtechniquesformodelingandforecastingtimeseries.Amongthestatisticalmodelsappliedtoregressionintimeseriesarelinearautoregression(AR),autoregressivemovingaverageprocess(ARMA),autoregressiveintegratedmovingaverageprocess(ARIMA),aswellastheirmultivariateanalogs(i.e.vectorautoregression,etc.) Timeseriesariseinvariousappliedareas,includingeconomicsandnance,meteorology,biomedicine,etc.Forinstance,thestudyofseismicactivityrelatedtoearthquakesproducestwo-dimensionaltimeseries,whereeachmeasurementconsistsofthetimeandthemagnitudeofaregisteredseismicevent.Manybiomedicalsignals,suchaselectrocardiogram(ECG),electroencephalogram(EEG)andelectrooculogram(EOG)representtimeseriesthatcanbeinterpretedviaapplicationofregression,segmentation,neuralnetworks,andotherdataminingmethodologies.Soundsignalsareanotherexampleoftimeseriesthatareeectivelyanalyzedusingdierentdataminingtechniques. 14

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14 ],althoughthenormaltimescaleisaverynaturalchoiceofparameterforthetimeseriesdescribingphysicalprocesses,theregulartimeloosesitsnaturalmeaningwhendealingwithmanynancialtimeseries.Becausemostofnancialtimeseriesareirregularlyspacedinphysicaltime,theconceptof\businesstime"or\intrinsictime"isintroducedtorepresentanewtimeparameterwithrespecttowhichtimeseriesareregularlyscaled.Thisprocedureoftimedeformationallowstherelabeledtimeseriestobeviewedasstationaryonanewtimescale.Financialtimeseriesalsooftenexhibitclearseasonaltrends,whichobviouslycannotbefoundwhenexaminingtimeseriesproducedbyspeech. StatisticaltestingofseveralmultivariatetimeseriesdeterminedthatthetimeseriesfromAUSLANandBCIdatasetscanbeconsideredstationary,whereasBCIMPIandEEGcontainednon-stationarytimeseries[ 108 ].Asaresultofinherentdierencesintimeseriesdatafromdiversesources,somedataminingmethodsthataresuccessfullyappliedtotimeseriesinoneresearchareamaynotnecessarilybeapplicabletoanalysisoftimeseriesthatstemfromanotherappliedeld. Timeseriesobtainedfromelectroencephalogram(EEG)recordingshaveseveralinterestingpropertiesthatdistinguishthemfromothertimeseries.Althoughsomestudiesapplyone-dimensionalmodelingbyconsideringonechannelinEEGrecordingatatime,ingeneral,EEGdatashouldbetreatedasmultivariatetimeseries.Themultivariateapproachbecomesespeciallyimportantinviewofitsabilitytoinvestigatespatio-temporaldependenciesintheEEGdataincontrasttobeinglimitedtoonlytemporalrelationsinaone-dimensionalcase.ThereisadisagreementamongresearchersstudyingEEGdataonwhethertheseriesshouldbemodeledbyanon-linearstochasticprocessortheycanbebetterdescribedbyadeterministicchaoticdynamicalsystem. 15

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100 ]. Generallyspeaking,EEGrepresentsadigitaloragraphicrecordoftheelectricalactivityinthebrain,andcanbemeasuredbyeithernon-invasiveorinvasivemethods.EEG(obtainedduringanon-invasiveprocedure)isdenedasarecordofelectricalactivityofanalternatingtypemeasuredfromthescalpsurfaceafterbeingpickedupbymetalelectrodesandconductivemedia[ 67 ].TherearetwotypesofEEGproducedbyinvasiveprocedures,theelectrocorticogram,whichmeasuresthebrain'selectricalactivitydirectlyfromthecorticalsurface,andtheelectrogram,whichisanEEGobtainedusingdeepprobes. EEGestimatesandrecordstherelativechangeinelectricpotentialsproducedbyalargenumberofelectricdipolesduringaperiodofneuralexcitations.Theactivationofneurons(braincells)generateslocalcurrentowsinthebrain.EEGrecordsmostlytheelectricalcurrentsthatowduringsynapticexcitationsofthedendritesofnumerouspyramidalneuronsinthecerebralcortex.EEGrecordedfromthescalpsurfacecanonlydetecttheelectricalactivityproducedbymassivepopulationsofactiveneurons.Ontheotherhand,EEGrecordedusingdeepprobeelectrodesimplantedintothebraincanpickupasignalfromasmallgroupofneurons,whichcanbefurtherlteredouttoobtainthe 16

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68 ].Thisgivesanaveragedensityofabout104neuralcellspercubicmillimeterofthebrain.Neuralcellsareinterconnectedthroughsynapticconnectionsinthebrainintoneuralnets.Thebrainofanaverageadultcontainsapproximately500trillionsynapses.Thetotalnumberofneuronsdecreaseswithage.Asaresultthetotalnumberofsynapticconnectionsdeclineswithaging,eventhoughthenumberofsynapsesperoneneuronincreaseswithage. ToensuretheconsistencyinreferencinglocationsofelectrodesinEEGexperiments,theInternational10-20systemforEEGelectrodeplacementwasdeveloped[ 37 ].The10-20EEGsystemisusedtodescribetherespectivelocationsofscalpelectrodesduringEEGrecordinginrelationtotheunderlyingareaofcerebralcortex. Accordingtothe10-20system,anatomicallandmarksofaskull,suchnasion,inionandpreauricularpoints,areidentiedforconsecutiveplacementoftheelectrodesatxeddistancesfromthesepointsinstepsofeither10or20percent.Thisapproachisdevisedtotakeintoaccountpossiblevariationsofheadsize.Inaddition,themethodiseasilyapplicableinpracticaluse.Asaresult,the10-20EEGsystembecameverywidelyusedforpositioningelectrodes. Inthe10-20system,thepointsaredenotedwithoneortwoletters,andcanbealsofollowedbyanumber(asshownonFigure 1-1 ).Thelettersroughlyrepresentthelobelocation(withexceptionoflettersCandZ),whereasthenumbersserveforidentifyingthecorrespondinghemisphere.Morespecically,thepointslocatedonthelefthemisphereofthebrainarerepresentedbyoddnumbers(1,3,5,and7),andthesitesontherighthemispherearemarkedwithevennumbers(2,4,6,and8).Thesiteslocatedonthefrontal,temporal,parietalandoccipitallobesaredenotedbythecorrespondinginitials 17

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EEGsignalresemblesacollectionofsinusoidsofvariousamplitudeandfrequency.PowerspectrumisextractedfromtherawEEGdatausingFouriertransformtoobtaintheinformationaboutthecontributionofsinusoidalwavesofdierentfrequency.ThepowerspectrumofEEGiscontinuous,rangingfrom0Hzuptoahalfofthesamplingfrequency.Dependingonthestateofthebrain,certainfrequenciesappeartobemoreprevalent.Therearefourmajorfrequencybands,alpha,beta,deltaandtheta,whichpresenceinEEGduringvariousstatesofconsciousnesshasbeenextensivelystudied.Thesebandsrepresentsinewavesofrelativelylowfrequency,withdeltarangingfrom0.5to4Hz,theta4{8Hz,alpha8{13Hz,andbetaover13Hz.AlphawavesdiscoveredbyAdrianandMatthewsin1934arethebest-knownandthemoststudiedamongthefourfrequencybands[ 100 ].Theyareinducedbyclosingeyesandbyrelaxation,andterminatedwitheyesopeningorduetothinking,calculating,andotheranalyticalactivities.Inparticular,inmostpeople,eyeclosingproducesrapidchangesinbrainactivitymanifestingthemselvesinEEGasanadjustmentofthedominantfrequencybandfrombetatoalpha.EEGiscapableofdiscriminatingbetweendierentstates,suchasresting,alertness,stressstate,varioussleepstages,hypnosis,etc.Presenceofbetabandisdominantduringthestateofalertnesswitheyesopen.Drowsinessortherestingareusuallycharacterizedbytheriseinalphaactivity.Duringthesleep,presenceoflowerfrequencywavesbecomesmoreapparent.AhigherproportionofdeltabandfrequenciesisobservedduringstagesIIIandIVofthenon-rapideyemovementsleep(NREM).EEGrecordedfromdistinctregionsinthebrainexhibitsdierentspectrumofwavefrequencies.Inaddition,thebrainpatternsareuniqueforeveryindividual. PracticalapplicationsofEEGincludeepilepsyresearchandlocalizationofthefocusofepilepticseizures,testingofepilepsydrugeects;determiningareasofdamagedue 18

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3 ]. 94 ].Inordertoproperlyreectthespatio-temporalpropertiesofbraindynamics,theanalysisofEEGdatamustinvolveasimultaneousinvestigationofthedependenciesacrosschannelswithrespecttotime. DierentfeatureshavebeenproposedforanalysisofEEGtimeseries,includingFouriertransform,wavelets,cross-correlation,coherence,Grangercausalityandpartialdirectedcoherence,mutualinformationandtransferentropy,globalandphasesynchronization,Lyapunovexponentsandcorrelationdimension,etc. SinceEEGcanbeviewedasacollectionofsinewaves,EEGseriesareoftenanalyzedinafrequencydomain.Inaddition,somefrequencybandshaveshowntoplayspecicrolesinvariousstatesofconsciousness,andsothefrequencyinformationinEEGcanbeparticularlyimportant.Subsequently,theFouriertransformwitharunningtimewindow,alsoknownasshorttimeFouriertransform(STFT),becameoneofthemostwidelyusedmethodsforextractingfeaturesfromEEG.STFTisobtainedfromFouriertransformbyapplyingatimewindowfunctiongwithatimeshift.Mathematically,STFTisgivenbythefollowingformula: whereS(;f)denotestheSTFTwithtimewindowglocatedattime,correspondingtofrequencyf;andx(t)isasignalattimet.Inotherwords,STFTS(;f)representsthepowerspectrumofthesignalestimatedaroundtime.ThedrawbackofSTFTisthatthereisatradeobetweentimeaccuracyandfrequencyprecision.Bymakingthewindow 19

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AnalternativetoFouriertransformiswavelettransform(WT),whichisatransformationofthesignalbasedonaspecialfunction,calledmotherwavelet(MW).Themotherwaveletisshiftedintimebyalocationparameter,andthenadjustedbyascaleparametera.Moreprecisely,thewavelettransformisdenedbythefollowingformula: a)dt;(1{2) whereisamotherwavelet,aisscaleparameter,isatimelocationparameter,andxisasignal. ThescaleparameterainWTisanalogoustothefrequencyparameterfinSTFT.Inparticular,thelargevaluesofparametera(a1)stretchthewavelet,andsotheyrepresentlowfrequencies,whereasthesmallvaluesofa(a<1)shrinkthewaveletfunction,whichcorrespondstohigherfrequencies.Anadvantageofusingwaveletsisthatthehighfrequencycomponentscanbeanalyzedwithahighertimeaccuracythanthelowerfrequencycomponentsofthesignal. Asfollowsfrom( 1{2 ),W(;a)canbeinterpretedastheprojectionofthesignalontotheappropriatelyshiftedandscaledwavelet,i.e.W(;a)isacontributionofthewavelettothesignalx(t). WhileFourierandwavelettransformsareusuallyappliedtostudyeachchannelofEEGsignalindividually,thecross-correlation,coherenceandGrangercausalitymeasuretheinterdependencybetweendierentchannels.Thecross-correlationfunctionquantiesthelinearcorrelationbetweentwoprocesses.Giventwonormalizedsignalsx(t)andy(t)withzeromeansandunitvariances,thecross-correlationbetweenthesesignalsisestimatedas: 20

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Thecoherencefunctionisafrequencydomainanalogofthecross-correlationmeasure.Coherenceisobtainedfromcross-correlationbyapplyingFouriertransformto( 1{3 ).Theestimateofthecoherencespectrumoftwosignalsiscalledperiodogram.Theperiodogramiscalculatedbysubdividingthesignalsintoanumberofepochsofthesamelength,andthenapplyingthefollowingformula: where Whilethecross-correlationandcoherencearefeatures,whichreectthelineardependencybetweentwochannelsinthedata,theconceptofGrangercausalityiscapableofnotonlyestablishingthelineardependency,butalsospecifyingthedirectionofsuchdependency.Inotherwords,byapplyingGrangercausality,itbecomespossibletoidentifycausalrelationshipamongthechannelsofEEG.Grangercausalityisbasedonthemultivariateautoregressivemodelingoftimeseries.Ithasalsoreceivedanalternativereformulationinthefrequencydomainviaspectraldecompositionforstochasticprocesses. 21

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35 89 ].Inchaoticsystems,trajectoriesoriginatingfromverycloseinitialconditionsdivergeexponentially.Thesystemdynamicsarecharacterizedbytherateofthedivergenceofthetrajectories,whichismeasuredbyLyapunovexponentsanddynamicalphase. ShorttermlargestLyapunovexponent(denotedSTLmax),whichisanestimateofthemaximumLyapunovexponentfornon-stationarydata,isadynamicalmeasureofthechaoticityinthebrain.Next,themethodforestimatingSTLmaxissummarized. First,usingthemethodofdelays[ 69 ],theembeddingphasespaceisconstructedfromadatasegmentx(t)witht2[0;T],sothatthevectorXiofthephasespaceisgivenby whereti2[1;T(p1)],pisachosendimensionoftheembeddingphasespace,anddenotesthetimedelaybetweenthecomponentsofeachphasespacevector. Next,theestimateLoftheshorttermlargestLyapunovexponentSTLmaxiscomputedasfollows: whereNaisthetotalnumberoflocalmaximumLyapunovexponentsthatareestimatedduringthetimeinterval[0;T];4tistheevolutiontimeforthedisplacementvectorX(ti)X(tj);X(ti)representsthepointoftheducialtrajectorysuchthatt=ti,X(t0)=(x(t0);x(t0+);:::;x(t0+(p1))),andX(tj)isanappropriatelyselectedvectorthatisadjacenttoX(ti)intheembeddingphasespace.In[ 34 ],Iasemedisatel.suggestedamethodforestimatingSTLmaxintheEEGdatabasedontheWolf'salgorithmfortimeseries[ 107 ]. TheshorttermlargestLyapunovexponentSTLmaxisprovedtobeanespeciallyusefulEEGfeatureforstudyingthedynamicsoftheepilepticbrain[ 35 89 ].Inparticular, 22

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90 ]. 1.4.1TestingApplicabilityofFrequencyDomainEstimatesofGrangerCausalityforEEGtimeseries 25 ]foranalysisofeconometricseries.Becausefrequencydomaincontainsvaluableinformationaboutthebrainprocesses,Geweke'sdenitionofcausalityseemstobeparticularlyusefulforanalysisofEEGdata. BothdenitionsofGrangercausality,theoriginalonegivenbyGrangerandthefrequencydomaindenition,areintroducedviavectorautoregressivemodelingofmultipletimeseries.Precisely,thelinearvectorautoregressionisusedtotthedata.Basedonthismodel,twocompetinghypothesisaboutthedataareconsideredandtestedstatisticallytodetermine,whichofthesetwoassumptionsissupportedbythedata.Inotherwords,totestthecausality,thehypothesisofdatabeingmodeledaslinearautoregression(thatdoesnotincludeanotherseries)iscomparedtothealternativeofthedatabestdescribedbyincludinginformationfromtheotherseries. SinceGrangercausalityisdenedbasedonlinearvectorautoregressivemodel(VAR),theapplicabilityofGrangercausalitydependsupontheunderlyingassumptionsofthemodel.ThefundamentalassumptionsoftheVARarestability,stationarity,andgaussian 23

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AlthoughBernasconiandKonig[ 2 ]testedthestationarityofdierentEEGdata,andconcludedthatatimeintervalof1secondistheintervalonwhichtheEEGtimeseriescanbeconsideredstationary,theunderlyingassumptionsoftheVARmodelingsuchasstationarityandstabilityarerarelystatisticallytestedinapplicationstoEEGdata. Inparticular,thestabilityassumptionmeansthatthereversecharacteristicpolynomialofthemodeldoesnothaverootsinsideaunitcircle.InordertohighlighttheimportanceofthestabilityconditionforVAR,itisnecessarytopointoutthatwhenstabilityisviolated,themodelmaysimplyfollowarandomwalk,oritmayevenexhibitexplosivebehavior. Tothebestofourknowledge,thestabilityconditionoftheVARestimatedfromtheEEGtimeserieswasnotinvestigatedbeforeourstudy[ 43 ].InmanystudiesthatutilizethevectorautoregressiontoexamineGrangercausalityamongseries,thevericationoftheconditionsassumedbytheVARmodelisoftenomitted. TheresultsofournumericalexperimentsindicatethatthestabilityconditionofvectorautoregressivemodelisoftenviolatedinapplicationtotheEEGdata.Morespecically,wefoundthatthestabilityassumptionimposedonthelinearVARmodelsmaybeviolatedeveninthecasewhenthesamplesizeparameterTismuchlargerthanthelagparameterpoftheestimatedmodel.Inaddition,weshowedthatdespitethefactthatitiscommoninpracticetoltertheEEGdatawithinacertainfrequencyband,lteringtheEEGtimeserieswithinsomerestrictedfrequencybandoftenresultsinsignicantreductionofthe(T;p)domain,wheretheestimatedVAR(p)modelsremainstable. 24

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43 ]. ComprehensivestatisticaltestingisnecessaryinordertomakeconclusionsonwhatmultivariatemodelsarethemostappropriateforextractingthedirectionaldependenciesbetweenchannelsinafrequencydomainfrommultichannelEEGdata. 15 101 104 105 ].Actually,itwasfoundthatoscillationofvariousneuronalgroupsingivenfrequencybandsleadstotemporaryphase-lockingbetweensuchgroupsofneurons.Thisobservationhasstimulatedthedevelopmentofrobustapproachesthatallowonetomeasurethephase-synchronyinagivenfrequencybandfromexperimentallyrecordedbiomedicalsignalssuchasEEG. Inparticular,theimportanceofsynchronizationofneuronaldischargeshasbeenshownbyavarietyofanimalstudiesusingmicroelectroderecordingsofbrainactivity[ 83 96 ],andevenatcoarserlevelsofresolutionbyotherstudiesinanimalsandhumans[ 21 ]. ThephasesynchronizationinthebrainextractedfromEEGdatausingHilbertorwavelettransformshasrecentlybeenshowntobeanespeciallypromisingtoolinanalysisofEEGdatarecordedfrompatientswithvarioustypesofepilepsy[ 86 ]. Inourrecentstudy[ 44 ],weintroduceanovelconceptofgeneralizedphasesynchronization,whichisbasedonvectorautoregressivemodeling.Thisnewnotionofphasesynchronizationisconstructedasanextensionoftheclassicaldenitionofphasesynchronizationbetweentwosystems.Infact,thephasesynchronizationisusuallydenedastheconditionthatsomeintegercombinationoftheinstantaneousphasesoftwosignalsisconstant.Oftenthisconditionisrelaxedbyallowingforaboundedlinearcombinationoftwophases,inorder 25

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Toconstructamoregeneralmultivariateconceptofphasesynchronization,weextendedtheclassicaldenitionbyconsideringalinearcombinationofphasesforanitenumberofsignalsthatrepresentsastationaryprocess.Alltheindividualsignalstogetherformacommonsystemdescribedbysomemultivariateprocess.Wenotethatavectorprocess,suchthatalinearcombinationofitsindividualcomponentsisastationaryprocess,canbemodeledasacointegratedvectorautoregressivetimeseries. Furthermore,itiseasytosee(asshowninSection 4.2.1 )thatthecointegratedrankoftheregressiondetermineshowrestrictedthebehaviorofsuchsystemis.Thismeansthattherankrofcointegratedautoregressivemodel,estimatedfromthemultipletimeseriesoftheinstantaneousphases,measureshowlargethevectorsubspace,whichgeneratesthechangesinthephasevalues,is. ThisnewmeasureofcointegrationwasappliedtoabsenceepilepsyEEGdatain[ 44 ].Thedatasetscollectedfromthepatientswithothertypesofepilepsyarecurrentlybeinginvestigated. 26

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Bproleview TheInternational10-20systemforplacementofEEGelectrodes 27

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26 27 ]formeasuringlineardependenceandfeedbackineconomictimeseries.Later,thisideawasfurtherextendedbyJohnGeweke[ 25 ],whoproposedanequivalentmeasurebasedonthespectralrepresentationoftimeseries.BothGranger'sandGeweke'sapproachesemploythevectorautoregressivemodelingtoderiveestimatesofunderlyingcausalrelationsinthedata.However,thelatterapproachisfoundparticularlyusefulforanalysisofEEGtimeseries,sinceitinvestigatesthecausalrelationinthefrequencydomaininsteadofthetimedomainasintheformerapproach.Inparticular,thespectralmeasureofGrangercausalityproposedbyJohnGewekewasemployedonintracorticallocaleldpotentialsrecordedfrom8electrodesduringgo/no-gotrialsofcat'svisualresponses[ 2 ].Anotherstudy[ 57 ]utilizedasimilarmethodofdirectedtransferfunction(whichisequivalenttothespectralmeasureofGrangercausality)toexaminecausalinuencesintheprimatevisualcortexduringthetaskofvisualpatternrecognition.ThedirecttransferfunctionapproachtoGrangercausalitywasalsoappliedtoanalyzingbrainconnectivitypatternsonhumanEEGdatarecordedduringstage2sleep[ 42 ]. MichaelEichlerproposedagraphicalapproachformodelingGranger-causalrelationshipsinmultivariatetimeseries[ 17 ]andlaterappliedthismethodtostudyingconnectivityinneuralsystems[ 18 19 ].LuizBaccalaandKoichiSameshimaintroducedaconceptofthepartialdirectedcoherenceforinferenceofGrangercausalityinthefrequencydomainbasedonthelinearvectorautoregressivemodeling,andappliedittoinvestigatingthefunctionalinteractionsamongdierentbrainstructures[ 1 92 ]. 28

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30 ]. LetX(t)andY(t),t2Zdenotetwotimeseries(ordiscretetimestochasticprocesses)withthecorrespondingrealizationsxtandyt,t2Z.SupposethatX;tandY;tdenotealltheinformationabouttherealizationsofprocessesXandY,respectively,uptotimet.Then,therelationshipofGrangercausalitybetweensuchseriescanbeformallydenedasfollows: AtimeseriesX(t)issaidtoGranger-causeY(t)ifthereexistsp=1;2;:::suchthatthemeansquarederror(MSE)ofthep-stepforecastofY(t)basedontheinformationX;tandY;tissmallerthantheMSEofthep-stepforecastofY(t)basedonY;talone,i.e. whereY(pj)istheMSEofthep-stepforecastofY(t)basedoninformation. Usingtheabovedenition,wenowpresentthetestforGrangercausalitybasedonthebivariateautoregressivemodel.Supposethatforsomeintegerlagparameterp>0,therealizationsoftimeseriesY(t)aregivenbythemodel where"tisastandardwhitenoise(orinnovationprocess,i.e."thaszeromeanandzeroautocorrelation). 29

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againstthealternative Notethatifthenullhypothesisisaccepted,thenatimeseriesY(t)isbelievedtobeNOTGranger-causedbyX(t).Meanwhile,rejectingthenullhypothesis(i.e.acceptingthealternative)meansthatX(t)isbelievedtocauseY(t)inGranger'ssense. LetTbethesamplesizeparameter.ThemodelparametersforthenullhypothesisH0andtheparametersforthealternativeH1areestimatedfromthesampledatausingtheordinaryleastsquaresmethod(orothermethods)toobtaintheestimatesoftheforecasterrors^"0tand^"1t,respectively,t=1;2;:::;T.ThenthesumofsquaredresidualsRSS0undertheassumptionofnullhypothesisH0is andthesumofsquaredresidualsRSS1underthealternativeH1is ByconductingtheF-testofthenullhypothesis,onecanndtheteststatistic RSS1=(T2p1)Fp;T2p1:(2{7) IftheteststatisticS1exceedsthespeciedcriticalvalue,thenthenullhypothesisthatX(t)doesnotGranger-causesY(t)isrejected.Otherwise,H0isaccepted. AnasymptoticallyequivalenttestofGrangercausalityisgivenbythefollowingstatistic 30

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ThebivariateapproachtotestingGrangercausalitycanbenaturallyextendedtothemultivariatecasebypartitioningthevectorautoregressiveprocessZ(t)intotwocomponentsX(t)andY(t),sothatZ(t)=(X(t);Y(t)),andthentestingthesuitablezeroconstraintsonthecoecientsofvectorautoregression.ForthederivationoftheWaldstatisticandtheF-statisticfortestingGrangercausalityinthemultivariatecase,seethebookonmultipletimeseriesbyLutkepohl[ 59 ]. 2.3.1MethodsforVARParameterEstimation Letpdenoteapositiveinteger,andletytdenotetheK-variatetimeseries(i.e.realizationsofK-dimensionalprocessY(t)).Avectorautoregressivemodeloforderp,denotedVAR(p),isformallydenedasfollows: whereyt=(y1t;:::;yKt)0isa(K1)randomvector,=(1;:::;K)0isaxed(K1)vectorrepresentinganon-zeromeanEY(t),theAi,i=1;:::;parexed(KK)-dimensionalcoecientmatrices,and"t=("1t;:::;"Kt)0isaK-dimensionalwhitenoiseprocess(i.e.E["t]=0,E["s"0t]=0,fors6=t,andE["s"0t]=").Itisassumedthatthecovariancematrix"isnonsingular.Inaddition,threeimportantconditionsareusuallyimposedonthetimeseriesintheVARmodel.TherstconditionisstabilityoftheprocessY(t),thesecondisstationarityofY(t),whilethethirdonesupposesthattheunderlyingwhitenoiseprocess"tisGaussian. 31

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59 ].UndertheassumptionsofstabilityandGaussiandistribution,theseapproachesleadtoestimatorswiththesameasymptoticproperties.However,theasymptoticresultsshouldbeusedcautiouslyininferencefromsmallsamples.Asaresult,dierentapproachesmaysometimesleadtodierentresultswhenestimatingthemodelparametersusingsmallsamples. Letusnowbrieypresentthemultivariateleastsquaresestimation,whichisahigherdimensionalextensionofthewell-knownmethodofordinaryleastsquares.Formoredetaileddiscussion,referto[ 59 ]. Supposethattheavailabledatainclude(T+p)successiverealizationsofestimatedmultipletimeseriesrepresentedbyK-dimensionalvectorsyp+1;:::;y0;y1;:::;yT

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2{9 ),thevectorautoregressivemodeloforderpcanberepresentedinthecompactform: andthecoecientsBofthemodelaregivenbytheleastsquaresestimator: ^B=YZ0(ZZ0)1:(2{12) Thecovariancematrixcanbeestimatedinvariousways.Since"=E["t"0t],theestimator ~"=1 isconsistent.However,thisestimatorofthecovariancematrix"isnotunbiased.Therefore,itisoftenreplacedbythefollowingunbiasedestimator ^"=1 Obviously,bothestimatorsareconsistentestimatorsofthecovariancematrix,andtheyareasymptoticallyequivalent. Whenestimatingthecoecientsofthevectorautoregressivemodelfromdata,weassumedtheorderpoftheVAR(p)tobeknown.Inpractice,however,itisunknown,andtherefore,needstobederivedfromthedata.Sincezerocoecientmatricesareallowed,onecouldsimplysetptosomeupperboundontheVARorder.Ontheotherhand,selectinganunnecessarylargepwouldaecttheforecastprecisionoftheestimatedmodel.Therefore,itisadvantageoustoapplysomesuitablecriteriaforoptimalselectionofthelaglengthparameterp. 33

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59 ]. Let~u(m)denotethemaximumlikelihoodestimatorofucomputedbyttingtheVARmodeloforderm.TheFPEcriterionproposedbyAkaikein1969isbasedontheideathatminimizingthemeansquareerrorimprovestheforecastofthemodel.ForaVAR(p)timeseries,theFPEcriterionisdenedas UsingtheFPEcriterion,theestimate^pFPEofthemodelorderpisselectedsothat whereMdenotessomeupperboundaryonthemodelorder.Inotherwords,rst,foreachm=1;:::;M,thevectorautoregressivemodelofordermisestimatedfromthedata,andtherespectivevaluesoftheFPE(m)arecalculatedusing( 2{15 );thentheorderproducingthesmallestvalueofFPE(m)ischosenamongthepossibleordersm=1;:::;M. AICisanotherpopularorderselectioncriteriathatwasalsointroducedbyAkaike.GivenaVAR(m)model,theAkaikeinformationcriteriaisdenedasfollows: SimilarlytotheFPEcriterion,theVAR(m)modelsareestimatedfordierentm=1;:::;MtoobtainthecorrespondingAIC(m)valuesforeachorder.Thentheestimate^pAICofthemodelorderpwiththesmallestAIC(m)isselected. 34

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T+ln~u(m)(2{18) Asbefore,amongthemodelparametersm=1;:::;M,theparametermhavingthesmallestvalueofHQ(m)ischosenastheestimator^pHQofthetruemodelorderp. Last,butnotleast,wepresentSchwarzcriterion,whichwasderivedusingBayesianarguments.TheSCisformulatedas: T+ln~u(m);(2{19) andtheorderminimizingSC(m)ischosenamongm=1;:::;Mastheestimator^pSCofthemodelorderp. Someinterestingstatisticalpropertiesoftheabovecriteriaareprovedin[ 59 ].Inparticular,itisshownthatAICandFPEcriteriaforVARorderselectionareasymptoticallyequivalent,althoughtheseestimatorsofthemodelorderarenotconsistent.Ontheotherhand,theothertwocriteriaprovideconsistentestimatorsoftheorderparameterp.Moreprecisely,intheunivariatecase(K=1),theHannan-Quinncriterionisconsistent(i.e.limT!+1Prf^p=pg=1).Inaddition,theHQcriterionisstronglyconsistentforK2(i.e.PrflimT!+1^p=pg=1).TheSCisshowntobestronglyconsistentforanydimensionK. ItisimportanttokeepinmindthateventhoughFPEandAICdonotprovideconsistentestimators,theyarenotnecessarilyinferiortoHQandSC.Actually,insmallsamples,andeveninlargersamples,FPEandAICmayproducebetterforecast,althoughtheymaynotestimatethemodelordercorrectly. 2{9 )are 35

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det(IKA1z:::Apzp)6=0forcomplexz:jzj1:(2{20) Inotherwords,theVAR(p)process( 2{9 )satisesthestabilityconditionwhenitsreversecharacteristicpolynomial(givenbydet(IKA1z:::Apzp))hasnorootsonandinsidethecomplexunitcircle. Thestabilityconditionguaranteesthatthereexistsamovingaverage(MA)representationfortheVAR(p)process.Alsostabilityensuresthattheprocessisawell-denedstochasticprocesswiththedistributionsofitsunivariatecomponentsandjointdistributionoftheprocessytuniquelydeterminedbytheinnovationprocess"t.ForastableVAR(p)process,boththeprocessmeanandtheautocovariancearetime-invariant(which,accordingtothedenitionbelow,impliesstationarity). Whenthestabilityconditionisviolated,theprocessvarianceisincreasingwithtimeandunbounded.Specically,ifthereversecharacteristicpolynomialofthetimeserieshasasingleunitroot,andalltheotherrootsareoutsidethecomplexunitcircle,thenthetimeseriesbehaviorissimilartoarandomwalk.Inthisspecialcase,thevarianceincreaseslinearlywithtime,thecorrelationbetweenytandyt+happroaches1,andtheprocessmeanE[Y(t)]exhibitsalineartrendfor6=0.Inaddition,ifoneoftherootsofthereversecharacteristicpolynomialliesstrictlyinsidethecomplexunitcircle,thensuchprocessisexplosive,i.e.theprocessvariancegrowsexponentially.Variousapproachesaredevelopedinthetimeseriesliteraturetoaddressthetimeserieswiththeunitroots.Forexample,theunitrootscanberemovedbytakingdierences.However,theexplosivetimeseriesarenotaswell-studied,becauseitisbelievedthatanexponentialincreaseinthevarianceoftheeconomictimeseriesisnotwellfounded.AsonecanseethestabilityassumptionplaysanimportantroleinVAR(p). Awide-sensestationarityforstochasticprocessesisimposedontheVARtimeseriesasfollows.AstochasticprocessY(t)isconsideredstationaryif 36

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2. Inotherwords,thestationarityconditionsupposesthattherstandthesecondmomentsaretimeinvariant.Alsonotethattheprocessmeanandtheautocovariancematrixy(h)arenite.Itisshown(seeProposition2.1in[ 59 ])that AstableVAR(p)timeseriesyt,t=0;1;2;:::isstationary. Sincestabilityofatimeseriesimpliesthattheseriesisstationary,thestabilitycondition( 2{20 )issometimescitedintheliteratureasthestationaritycondition.However,itisimportanttorememberthatthesetwoconditionsarenotequivalent.Infact,althoughastablevectorautoregressiveseriesisalwaysstationary,theconverseisnottrue,i.e.anunstabletimeseriesisnotnecessarynon-stationary. TheGaussiandistributionassumptionisintroducedintotheVAR(p)modelthrough"t.Specically,givenrepresentation( 2{9 )oftheVAR(p),theinnovationprocess"tisassumedtobeGaussianwhitenoise.ThisconditionimpliesthatytisaGaussianprocess,i.e.anysubcollectionyt;:::;yt+hfollowsamultivariatenormaldistributionforallpossiblevaluesoftandh. RecallthattheVAR(p)process( 2{9 )satisesthestabilityconditionwhenitsreversecharacteristicpolynomialdet(IKA1z:::Apzp)hasnorootsonandinsideacomplexunitcircle.Ifanunstableprocesshasasingleunitrootandalltheotherrootsoutsideofthecomplexunitcircle,thensuchprocessexhibitsabehaviorsimilartothatofarandomwalk.Inotherwords,thevarianceofsuchprocessincreaseslinearlytoinnity, 37

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Thisrendersthefollowingdenitionofanintegratedprocess. Aone-dimensionalprocesswithdrootsontheunitcircleissaidtobeintegratedoforderd(denotedasI(d)). Itcanbeshown[ 59 ]thattheintegratedI(d)processY(t)oforderdwithallrootsofitsreversecharacteristicpolynomialbeingequalto1canbemadestablebydierencingtheoriginalprocessdtimes.Forexample,theintegratedI(1)processY(t)becomesstableaftertakingtherstdierences(1L)Y(t)=Y(t)Y(t1),whereLrepresentsthelagoperator.Moregenerally,fortheI(d)processY(t),itstransformation(1L)dY(t)isstable. AnexampleofanintegratedI(d)processintheunivariatecaseisanautoregressiveintegratedmovingaverageprocessARIMA(p,d,q),whichissometimescalledfractionallydierencedautoregressivemovingaverageprocessford2(0:5;0:5).Theone-dimensionalprocessY(t)issaidtobeARIMA(p,d,q),ifZ(t):=(1L)dY(t)isastationaryautoregressivemovingaverageARMA(p,q)process,i.e. where"tj'sareindependentnormallydistributedrandomvariableswithmean0andvariance2,andListhedierencingoperatorintroducedabove. Itisnoteworthytopointoutthattakingdierencesmaydistorttherelationshipamongthevariables(i.e.one-dimensionalcomponents)insomeVAR(p)models.Inparticular,thisisthecaseforsystemswithcointegratedvariables.ItturnsoutthatttingVAR(p)modelafterdierencingtheoriginalcointegratedprocessproducesinadequateresults.Next,wediscusssuchprocesses. 38

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SupposethatsampledvaluesyitofKdierentvariablesofinterestYi(t)arecombinedintotheK-dimensionalvectorsyt=(y1t;:::;yKt)0.Inaddition,supposethatthevariablesareinalong-runequilibriumrelation wherec=(c1;:::;cK)0isaK-dimensionalrealvector.Duringanygiventimeinterval,therelation( 2{22 )maynotnecessarilybesatisedpreciselybythesampleyt,insteadwemayhave: where"tisastochasticprocessthatdenotesthedeviationfromtheequilibriumrelationattimet.Ifourassumptionaboutthelong-runequilibriumamongindividualvariablesYi(t),i=1;:::;KisvalidthenitisreasonabletoexpectthatthevariablesYi(t)movetogether,i.e.thestochasticprocess"tisstable.Ontheotherhand,thisdoesnotcontradictthepossibilitythatthevariablesdeviatesubstantiallyasagroup.Therefore,itispossiblethatalthougheachindividualcomponentYi(t)isintegrated,thereisalinearcombinationofYi(t),i=1;:::;K,whichisstationary.Integratedprocesseswithsuchpropertyarecalledcointegrated. Withoutlossofgenerality,weassumethatallindividualone-dimensionalcomponentsYi(t)(i=1;:::;K)areeitherI(1)orI(0)processes.ThenthecombinedK-dimensionalVAR(p)process 39

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=IKA1:::Ap(2{25) hasrankr. Sincesomeone-dimensionalcomponentsofthecointegratedVAR(p)processareintegratedprocesses,onemaybeinterestedintestingthepresenceofaunitrootintheunivariateseries.Inthefollowingsection,wepresentacommonlyusedunitroottest,whichwasderivedbyDickeyandFuller[ 16 ]. 16 ],anditcanbeshownthatthisdistributionisthesamefork>1andfork=1.FullertabulatedtheapproximatecriticalvaluesfortheADFtestwithk1andpk1forspecicsamplesizes. Finite-samplecriticalvaluesfortheADFtestforanysamplesizewereobtainedbymeansofresponsesurfaceanalysisbyMacKinnon[ 60 ],whoalsoshowedthatanapproximateasymptoticdistributionfunctionforthetestcanbederivedviaresponsesurfaceestimationofquantiles[ 61 ]. AlthoughtheasymptoticdistributionoftheADFteststatisticdoesnotdependonthelagorder,itisnotedbyCheungetal.[ 13 ]thatempiricalapplicationsmustdealwithnitesamples,inwhichcasethedistributionoftheADFteststatisticcanbesensitiveto 40

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Aswenotedabove,thelimitingdistributionoftheADFteststatisticisthesamefork>1andk=1.Hence,forsimplicity,weconsiderthecaseofk=1.Infact,letYdenotetheautoregressiveAR(1)model whereY(0)=0,cisarealnumber,and"tN(0;2)(i.e."tisnormallydistributedwithzeromeanandvariance2forallt=1;2;:::). Notethatwhenjcj<1,theprocessY(t)convergestoastationaryprocessast!1;whereas,inthecaseofjcj=1,theprocessY(t)isnotstationarywithvariancet2.Furthermore,whenjcj>1,notonlytheprocessisnotstationary,butthevarianceofY(t)growsexponentiallywithtimet. FromtheAR(1)model( 2{26 ),onecanseethatinthecasewhenc=1,inordertomaketheprocessstationary,theseriescanbeappropriatelytransformedbydierencing.Furthermore,noticethattheconditionc=1in( 2{26 )isclearlyequivalenttotherequirementthatthereversecharacteristicpolynomialdet(1cz)=1zofAR(1)hasaunitroot.Inotherwords,todeterminewhetheranautoregressivetimeseriesAR(1)hasaunitroot,wemusttestthenullhypothesisH0:c=1. Lety1;y2;:::;yTdenoteasampleofTconsecutiveobservationsoftheAR(1)processY(t),thenthemaximumlikelihoodestimatorofcistheleastsquaresestimator Notethatbcisaconsistentestimatoroftheregressioncoecientc. Sinceeachyt,t=1;:::;TisarealizationofanAR(1)process,itfollowsfrom( 2{26 )thatyt=cyt1+"tholds,andsobypluggingthislastconditionintoEquation( 2{27 ),the 41

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=cPTt=1y2t1+PTt=1yt1"t 2{28 andmultiplyingeachsidebyTleadtotheADFstatistic DickeyandFuller[ 16 ]derivedthefollowingrepresentationofthelimitingdistributionforstatisticT(bcc): 21(W21);asT!1(2{30) where =1Xi=1d2iX2i; andrandomvariablesXi,i=1;2;:::,areindependentandidenticallydistributedaccordingtothenormaldistributionwithzeromeanandvariance2.Notethatasymbol)denotesconvergenceindistribution. In[ 16 ],DickeyandFullerconsideredthefollowing\Studentized"statisticbasedonthelikelihoodratiotestofthehypothesisH0:c=1. 2;(2{34) 42

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andbciscomputedfrom( 2{27 ) TablesofthecriticalvaluesfortheasymptoticdistributionsoftheADFteststatisticT(bc1)andthestatisticbcanbefoundinFuller[ 22 ].WesummarizesomeoftheinformationinTable 2-1 ,whichliststhep-valuesforasymptoticdistributionsofT(bc1)andbcorrespondingtopercentilesof90,95,and99percent. 75 ].Inparticular,inhisstudyPhillipsrstconsideredtwostatisticsZandZtfortestingthenullofnocointegrationintimeseries. Becausemanyunitroottests,constructedbefore1987,werefoundedontheassumptionthattheerrorsintheregressionareindependentwithcommonvariance(whichisrarelymetinpractice),Phillipswantedtorelaxtheratherstrictconditionthatthetimeseriesaredrivenbyindependentidenticallydistributedinnovations.Inotherwords,hewantedtodevelopthetestingproceduresbasedontheleastsquaresregressionestimationandtheassociatedregressiontstatistic,whichwouldallowforrathergeneralweaklydependentandheterogeneouslydistributedsequenceoferrorterms. ThepropertiesofasymptoticdistributionsofresidualbasedtestsforthepresenceofcointegrationinmultipletimeserieswerethoroughlyinvestigatedbyPhillipsandOuliaris[ 76 ].Thecharacteristicfeatureofthesetestsisthattheyutilizetheresidualscomputedfromregressionsamongtheunivariatecomponentsofmultivariateseries.TheresidualbasedproceduresdevelopedbyPhillipsandOuliarisaredesignedtotestthenullofnocointegrationbymeansoftestingthenullhypothesisoftheunitrootpresenceintheresidualsagainstthealternativeofarootthatliesinsidethecomplexunitcircle.ThehypothesisH0oftheabsenceofcointegrationisrejected,ifthenullofaunitrootinthe 43

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Asnotedin[ 76 ],theresidualbasedunitroottestsareasymptoticallysimilar,andcanberepresentedviathestandardBrownianmotion.Moreover,theADFandZttestsareprovedtobeasymptoticallyequivalent.However,thesetwotestsarenotaspowerfulasthetestbasedonstatisticZ,becauseitwasshownbyPhillipsandOuliaris[ 76 ]thattherateofdivergenceundercointegrationassumptionisslowerfortheADFandZtthanothertests,suchastheZ-statistictest.Thelatertest(i.e.thecointegrationtestbasedonZ)isalsowidelyknownasthePhillips-Ouliariscointegrationtest. ItisnoteworthythatthenullhypothesisforthePhillips-Ouliaristestisthatofnocointegration(insteadofcointegration).Thisformulationischosenbecauseofsomemajorpitfallsfoundinproceduresthataredesignedtotestthenullofcointegrationinmultipletimeseries.Thesedefects(discussedinmoredetailin[ 76 ])aresignicantenoughtobeastrongargumentagainsttheindiscriminateuseofthetestformulationsbasedonthenullofcointegration,andtosupportthecontinuinguseofresidualbasedunitroottests. ConsidertheK-dimensionalvectorautoregressiveprocessY(t).LetuspartitionY(t)=(Ut;V0t)0intotheunivariatecomponentUt=Y1(t)andthe(K1)-dimensionalVt=(Y2(t);:::;YK(t))0. Theresidualsaredeterminedbyttinglinearcointegratingregression ResidualbasedtestsareformulatedtotestthenullhypothesisthatthemultipletimeseriesY(t)arenotcointegratedusingthescalarunitroottests,suchastheADFtest,whichareappliedtotheresidualst,t=1;2;:::in( 2{36 ). In[ 76 ],theADFtestaswellastwoadditionaltestsZandZt,developedearlierbyPhillips[ 75 ],wereappliedtocheckforthepresenceofaunitrootintheresidualst.In 44

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ThenthestatisticZinPhillips-Ouliaristestisdenedasfollows: 2s2Tls2% whereastheZtstatisticisgivenbythefollowingformula: 2(b1) 2s2Tls2% 2;(2{39) where l+1: Notethats2%andsTlareconsistentestimatorsforthevariance2%of%tandthepartialsumvariance2=limT!1E1 2{37 ). ThecriticalvaluesforZandZtstatisticscanbefoundin[ 76 ](TablesIandII).PhillipsandOuliaristabulatedthevaluesforcointegratingregressionswithatmost5explanatoryvariables.SomeestimatesofthecriticalvaluesforthePhillips-Ouliaristest(Z)arelistedinTable 2-2 45

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2{24 ). Givenarealizationyt,t=1;2;:::,ofY(t),oneseekstodeterminethecoecientsofthefollowingmodel: subjecttotheconstraint rank()=rank(IKA1:::Ap)=r:(2{44) Notethat"tisassumedtobeaGaussianwhitenoisewithanonsingularcovariancematrix".Furthermore,theinitialconditionsyp+1;:::;y0aresupposedtobexed. Inordertoimposethecointegrationconstraint,themodel( 2{43 )isreparameterizedinthefollowingfashion[ 59 ]: yt=D1yt1+:::+Dp1ytp+1+ytp+"t;t=1;2;:::;(2{45) whereyt=ytyt1,andmatrixcanberepresentedasaproduct=HCofmatricesofrankr,i.e.His(Kr)andCis(rK). Consider Y:=[y1;:::;yT];Xt:=266664yt...ytp+2377775; X:=[X0;:::;XT1];D:=[D1;:::;Dp1];Yp:=[y1p;:::;yTp]:

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lnl=KT 2trace(YDX+HCYp)01"(YDX+HCYp): TheproofofthefollowingtheoremonthemaximumlikelihoodestimatorsofacointegratedVARprocesscanbefoundin[ 59 ](Proposition11.1). 59 ]) Dene Denote1:::KtobetheeigenvaluesofGS10S100S01G0, and Thenthelog-likelihoodfunctionin( 2{47 )ismaximizedfor

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1 ,onecaneasilyderivethelikelihoodratiostatisticfortestingacandidatevaluer0ofthecointegrationrankrofaVAR(p)processagainstalargercointegrationrankr1. GivenaVAR(p)processy(t)denedby( 2{24 ),supposewewishtotestahypothesisH0againstanalternativeH1,where Underassumptionthatthenoise"tisaGaussianprocess,themaximumofthelikelihoodfunctionforacointegratedVAR(p)modelwithcointegrationrankriscomputedinTheorem 1 .Fromthatresult,thevalueoftheLRstatisticfortesting( 2{49 )canbedeterminedinthefollowingmanner: (2{50) =T"r1Xi=1ln(1i)+r0Xi=1ln(1i)#=Tr1Xi=r0+1ln(1i); 2{50 )isnonstandard.Specically,theLRstatisticisnotasymptoticallydistributedaccordingto2-distribution.Nevertheless,theasymptoticdistributionofthecointegrationrankteststatisticLRdependsonlyontwofactors: 48

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Asaresult,theselectedpercentagepointsoftheasymptoticdistributionoftheteststatisticLRweretabulatedbyJohansenandJuseliusin[ 38 ].ThepercentagepointsofasymptoticdistributionofLRaregiveninTables 2-3 and 2-4 Table2-1. CriticalvaluesoftheasymptoticdistributionsoftheT(bc1)andbforperformingunitrootcheckbytheADFtest(reproducedfrom[ 22 ]) Statistic90%95%99% Table2-2. CriticalvaluesoftheasymptoticdistributionsoftheZstatisticfortestingthenullofnocointegration(Phillips-Ouliarisdemeaned,reproducedfrom[ 76 ]).Parametern(n=K1)representsthenumberofexplanatoryvariables 1-17.0390-20.4935-28.32182-22.1948-26.0943-34.16863-27.5846-32.0615-41.13484-32.7382-37.1508-47.51185-37.0074-41.9388-52.1723 Table2-3. PercentagepointsoftheasymptoticdistributionsoftheLR(r;K)fortestingthecointegrationrank(reproducedfrom[ 38 ]) 16.698.0811.58215.5817.8421.96328.4431.2637.29 49

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PercentagepointsoftheasymptoticdistributionsoftheLR(r;r+1)fortestingthecointegrationrank(reproducedfrom[ 38 ]) 16.698.0811.58212.7814.6018.78318.9621.2826.15 50

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Inthischapterweintroduceaconceptofphasesynchronization,andconsidertwomethodsforestimatingthephaseofasignal,specicallyusingtheHilberttransformandviathewavelettransform. 78 ].SynchronizationphenomenawerediscoveredintheseventeenthcenturybyC.Huygenswhorstobservedsynchronizationbetweentwopendulumclockshangingfromacommonsupport[ 33 ].Sincethen,thestudyofsynchronizationbetweendynamicalsystemsbecameanactiveeldofresearchinmanyscienticandtechnicaldisciplines,includingsolidstatephysics[ 74 ],plasmaphysics[ 84 ],communication[ 11 48 ],electronics[ 72 77 ],laserdynamics[ 20 87 98 ],andcontrol[ 80 88 ]. Synchronizationphenomenacanalsobefoundinphysiologicalsystems,suchasheartandbrain.SynchronizationprocessesinphysiologicalsystemswerediscoveredbyB.vanderPolinthebeginningofthetwentiethcentury.Inparticular,vanderPolwasthersttoapplyoscillationtheorytothehumanheart[ 103 ]. Oneoftheimportantresearchareasinneuroscienceexplorestheroleofsynchronizationinneuraldynamics.Mucheortisgiventoinvestigationofsynchronizationphenomenaonalldierentlevelsoforganizationofbraintissue,startingwithpairsofindividualneuronstolargerscales,suchaswithinagivenareaofthebrainorbetweendistinctpartsofthebrain. 51

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15 101 104 105 ].Neuralsynchronyplaysavitalroleinsuchlarge-scaleintegration.Infact,variousneuronalgroupsoscillateinspecicfrequencybandsandbecomephase-lockedoveralimitedperiodoftime.Thisobservationhasstimulatedthedevelopmentofrobustapproachesthatallowonetomeasurethephase-synchronyinagivenfrequencybandfromexperimentallyrecordedbiomedicalsignals,suchasEEG. Inparticular,theimportanceofsynchronizationofneuronaldischargeshasbeenshownbyavarietyofanimalstudiesusingmicroelectroderecordingsofbrainactivity[ 83 96 ].Thendingsinthemicroelectrode-recordingstudiesarealsosupportedatcoarserlevelsofresolutionbyotherstudiesinanimalsandhumans[ 21 ]. Anelectrophysiologicalsignalisrecordedviaalowimpedanceextracellularmicroelectrodebyplacingthemicroelectrodesucientlyfarfromindividuallocalneuronsinordertopreventanyparticularcellfromdominatingthesignal.Next,toobtainthelocaleldpotential(LFP),thesignalislow-passltered,withacutoatapproximately300Hz.Duetothelowimpedanceandpositioningofthemicroelectrode,therecordedsignalispredominantlyinducedbytheactivityofalargenumberofneurons.Theunlteredsignalreectsthesumofactionpotentialsfromcellswithinapproximately50{350micrometersfromthetipoftheelectrode[ 53 ]andslowerioniceventsfromwithin0.5{3millimetersfromthetipoftheelectrode[ 39 ].Thespikecomponentofthesignalisremovedbylow-passlter,whereasthelowerfrequencysignal,theLFP,ispreservedinthesignal.Itisassumedthatthelocaleldpotentialcharacterizesthesynchronizedinputintotheobservedarea,inthecontrasttothespikedata,whichrepresentstheoutputfromthearea. Localeldpotentials(LFPs)ofvariousdegreesofspatialresolutioncanberecordedbyscalpEEGorMEG.Infact,studieshaveshownthatthepresenceofgammaandbeta 52

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97 ]andinsubduralelectrocorticograms[ 50 54 ].Inaddition,somerecentndingssuggestthatlong-rangesynchronizationanalogoustotheonefoundinmicroelectrodestudiesinanimalscanalsobedetectedbetweensurfacerecordings[ 82 ]. Ithasbeenshownthatsynchronizationisasignicantattributeofthesignalrecordedfromthepatientsaectedbyseveralneurologicaldisorders.Inparticular,researchershavefoundthatepilepsy[ 65 ]andParkinsonsdisease[ 99 ]manifestasapathologicalformofthesynchronizationprocess. Itisnotedin[ 56 ]thatalthoughthecross-correlogramsbetweenspikedischargesmaybeadequateformicroelectrodestudies,thequanticationofphasesynchronybetweenmeso-ormacro-electrodes(i.e.EEG/MEG,intracranialrecordings)callsforentirelydierentmethods.Therefore,theyemphasizeanimportanceofclearlydistinguishingbetweensynchronyasanappropriateestimateofphaserelation,andtheclassicalmeasuresofcoherenceorspectralcovariancethathavebeenextensivelyusedinneuroscience[ 8 10 62 ].LeVanQuyenetal.discusstwoimportantlimitationsofcoherence[ 56 ]. Therstlimitationarisesbecausethestandardapproachesformeasuringcoherence[ 12 ]basedonFourieranalysisareknowntobehighlydependentonthestationarityofthemeasuredsignal,whereasthesignalsrecordedfromthebrain,suchasEEG,appeartobeclearlynon-stationary.Applyingthetimefrequencyestimationmethod,whichisnotfoundedontheassumptionofstationarity,couldimprovethislimitationtowardsestimatingastable,instantaneouscoherenceaswellassynchronybetweentwoconcurrentbrainsignals. Thesecondlimitationstemsfromthefactthatclassicalcoherenceisameasureofspectralcovariance.Hence,itisnotabletoseparatetheeectsofamplitudeandphaseintherelationsbetweentwosignals.Becauseweareconcernedwithexaminingthespecichypothesisthatphase-lockingsynchronyisthepertinentbiologicalmechanismoftransientintegrationinthebrain,coherenceservesonlyasanindirectmeasure.Inorder 53

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86 ].Inotherwords,eventhoughourdiscussionofthephasesynchronyisfocusedonEEGdata,itsapplicationscanalsobeextendedtotheeldsotherthanneuroscience. Classicalconceptofthesynchronizationoftwooscillatorsisdescribedasanactiveadjustmentoftheirrhythmicitythatmanifestsinphase-lockingbetweenthesynchronizedoscillators.Specically,giventwosignalsX1(t)andX2(t),andtheircorrespondinginstantaneousphases1(t)and2(t),thebasicdenitionofthephaselockingstatesthat whereintegersnandmspecifythephaselockingratio. Wheninvestigatingphasesynchronyinneurophysiologicalsignals,onemustassumethattheconstantphaselockingratioisvalidwithinalimitedtimeintervalT,whichusuallymeansafewhundredsofmilliseconds.Whenexaminingneuralsignals,onemustkeepinmindthatdiscoveringthepresenceofthephaselockingbetweenEEGrecordingsfromtwodistantpartsofthebrainisnotstraightforward.Thedetectionofphasesynchronyinneuralsignalsisproblematicbecauseofseveralfactorsparticularlywhenworkingnotonthelevelofasingleneuron,butratherwithlargeneuronalpopulations,whoseactivityispickedupbymacroscopicormeso-electrodes.Asnotedin[ 56 ],asaconsequenceofvolumeconductioneectsinbraintissues,theactivityofasingleneuronalpopulationcanberecordedbytwodistantelectrodes,whichresultsinspuriousphase-lockingbetweentheirsignals.Furthermore,innon-invasiveEEG,thetruesynchroniesarehiddeninasignicantbackgroundnoise,andso,inthesynchronous 54

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3{1 )mustbeadjustedtoaccountforthenoiseasfollows: whereCdenotesapositiveconstant. Thefollowingtwokeystepsareinstrumentalininvestigatingthephasesynchrony: 1. estimateinstantaneousphaseofeachsignal; 2. provideastatisticalcriteriatoquantifythedegreeofphase-locking. Twomethodsfordetectingphase-lockingappliedtoneuronalsignalshaverecentlybeenconsideredintheliterature.Tassandcolleagues[ 99 ]extractedtheinstantaneousphasesfromoriginalsignalsbymeansoftheHilberttransform,andthenappliedtomagnetoencephalographic(MEG)motorsignalsinpatientsaectedbyParkinson'sdesease[ 99 ],andalsotothesynchronizationbetweencardiovascularandrespiratoryrhythms[ 93 ].Ontheotherhand,Lachauxetal.[ 50 ]estimatedthephasesfromtheoriginalsignalsbymeansofconvolutionwithacomplexwavelet,andthenappliedittoEEGandintracranialdataduringcognitivetasks[ 51 82 ]. TherststepinquantifyingphasesynchronizationbetweentwotimeseriesXandYisthedeterminationoftheirinstantaneousphasesX(t)andY(t).ThisisachievedeitherviatheHilberttransformorviathewavelettransform.Theseapproachesarepresentedinthenexttwosections. 23 ]andlaterextendedformodelsystemsandexperimentaldata[ 70 86 ]. 55

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^f(t)=CPVZ+1f()g(t)d=CPVZ+1g()h(t)d;(3{3) whereg(t):=1 Noticethat^f(t)canbeviewedasaconvolutiong(t)f(t)oftheoriginalfunctionf(t)withthefunctiong(t).ThismeansthattheHilberttransformcanbeperformedbyapplyinganideallter,whoseamplituderesponseequalsto1,andphaseresponseisaconstant=2lagatallfrequencies. Givenanarbitrarycontinuousreal-valuedtimeseriesX(t),thecorrespondinganalyticsignalisdenedasthefollowingcomplex-valuedfunction: wheretdenotestime,{isaunitonthecomplexaxis,^X(t)denotestheHilberttransformofthetimeseriesX(t),aX(t)isthecorrespondinginstantaneousamplitude,andX(t)representstheinstantaneousphaseofthesignalviaHilbertconvolution. ItfollowsfromEquation 3{4 thattheinstantaneousphaseX(t)ofX(t)canbecomputedas: Akeyadvantageoftheanalyticapproachisthatthephasecanbeeasilycomputedforanarbitrarybroad-bandsignal.Ontheotherhand,instantaneousamplitudeandphasehaveaclearphysicalmeaningonlyifX(t)isanarrow-bandsignal.Therefore,ltrationisrequiredinordertoseparatethefrequencybandofinterestfromthebackgroundbrainactivity. 56

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50 52 ],andissimilartotheHilberttransformmethodpresentedabove.Intheirapproach,Lachauxetal.extracttheinstantaneousphasebyapplyingtheconvolutionoftheoriginalsignalwithacomplexMorletwavelet.TheyconsidertheMorletwavelet(alsoknownasGaborfunction)attimetandfrequency!givenbythefollowingformula: Noticethatt;!()istheproductofasinusoidalwaveatfrequency!,andaGaussianfunctioncenteredattimet,withastandarddeviationproportionaltotheinverseof!.Itdependssolelyon,whichsetsthenumberofcyclesofthewaveletto6!. Accordingto[ 56 ],giventhetimeseriesX(t),thecoecientoftheMorlettransformasafunctionoftimetandfrequency!isdenedasfollows: where ThefollowingslightmodicationoftheMorletwaveletisintroducedin[ 81 ]: whereparameters!0andrepresentthecenterfrequencyandtherateofdecayofthewaveletfunction,respectively.Thisisproportionaltothenumberofcyclesandrelatedtothefrequencyspanbytheuncertaintyprinciple. Similarlytotheabove,acomplextimeseriesofwaveletcoecientsisobtainedviatheconvolutionofX(t)with(t): 57

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AsinthecaseoftheHilberttransform,thephasescanbedeterminedfromEquation( 3{9 )as where<[WX(t)]and=[WX(t)]denotetherealandimaginarypartsofthecomplextransformedtimeseriesWX(t),respectively. 56 ]andalsoexplainedtheoreticallybyQuianQuirogaandcolleagues[ 81 ]. Inanutshell,thephaseeX(t)extractedfromthesignalusingthewavelettransformcorrespondsapproximatelytothephaseX(t)determinedviatheHilbertconvolution,whichwouldbeperformedafterbandpasslteringthetimeseries.Furthermore,ifthephaseestimationbasedonwavelettransformwereperformedbyaconvolutionwithananalyticwavelet,andifthiswaveletwereappliedtodothebandpasslteringintheHilbertapproach,thensuchapproacheswould,infact,beequivalent. Itiseasytoseethatinthemethodbasedonthewaveletconvolution,thecenterfrequency!andtherateofdecayofthewaveletcanserveasparametersthatallowustomodifythefrequencyrangeofinterest.Ontheotherhand,theactualphaseextractionviatheHilberttransformisfreeofparameters,andsothecorrespondentphasepreservesinformationfromtheentirepowerspectrumandnotjustthemainfrequencybandasinthecaseofthewaveletconvolution.Asaresult,itispossibletoachieveacomparisonofnarrowbandandbroadbandsynchronizationsimplybyusingbothmethodsofphaseextractionwithoutperforminganyadditionalltering. 58

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49 56 ].However,mostofthecurrentlyusedmeasuresofphasesynchronizationarebasedonbivariateindexes. InChapter 4 ,weproposeanovelmultivariateapproachtodetectingphasesynchronizationinthephasesextractedfrommultipletimeseries,suchasmultichannelEEG. 59

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2 ]appliedstatisticaltestingtoverifythestationarityofthedata,andestablishedthedurationofthestationaryintervalforEEGtobeapproximately1second.ThestatisticaltestingofunderlyingassumptionsoftheVAR(whichwasthoroughlydiscussedinChapter 4 )isoftenomittedinthelaterstudies.Asshownabove,thestabilityconditionisaveryimportantassumptionofvectorautoregression.Inthisstudy,weexaminehowtheparametersofthemodelorderandsamplesizeinuencethestabilityofthederivedVARmodel. InordertoestimateVAR(p)modelfromdataandinvestigatethepropertiesofthederivedmodel,therodentintracranialEEGdatawereused.Thedatasetconsistedoftheelectroencephalographicrecordingsfrom6electrodes(implantedinleftfrontal,rightfrontal,twolefthippocampalandtworighthippocampalpartsoftherodent'sbrain)sampledat200Hz. ToexaminetheapplicabilityofthevectorautoregressivemodelingtoEEGdata,weestimatedtheVAR(p)modelparametersfordierentvaluesoflagorderpanddierentsamplesizesT.ThesamplesizesT2[1;300]wereused,andthelaglengthparameterpvariedbetween1and30.Inaddition,weltereddatausingaRectangularbandpassHammingwindowwith100coecientsintothefrequencybandsof0{30Hz,30{60Hz,60{90Hz,90{120Hz,and120{150Hz.Therawdataandthevedierentlyltereddatarepresentedseparatedatasets.Foreachdataset,weranthemodelestimationprocedurewithT=1;2;:::;300andp=1;2;:::;30.TheprocedureforestimatingcoecientsofthemodelwasimplementedintheMATLABenvironmentbasedonthemultivariateleastsquaresmethodpresentedabove.ForeveryVARmodelderivedfrom 60

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4-1 and 4-2 ThesurfaceplotinFigure 4-1 supportsthefact[ 59 ]thatthesamplesizeparametershouldsignicantlyexceedthelaglengthp,i.e.Tp.Ontheotherhand,itcanbeseenfromFigure 4-1 thatevenforTp,thestabilityassumptionsmaystillbeviolated.Indeed,forT=132considerablylargerthanp=2,theestimatedVARhastwoRCProotsonorinsidetheunitcircle(n=2). Figure 4-2 showsthatfortheltereddata,the(T;p)region,wherethestabilityconditionoftheestimatedVAR(p)modelisviolated,coversalmostthewholedomain.Whereasthe(T;p)regionthatcorrespondstostableVAR(p)modelsismuchsmallerthanthestableregioninFigure 4-1 ,andcharacterizedbylargeTandverysmallp.VerysimilarresultswereobtainedforthenumberoftheRCProotsinsidetheunitcircle,whenestimatingparametersofVAR(p)withdierentpusingltersinthe30{60Hz,60{90Hz,90{120Hz,and120{150Hzbands. Theexperimentwasrepeatedwithconsistentresultsonvarioussamplesfromthesixdatasets.TheresultsofourexperimentsclearlyshowthatthestabilityconditionimposedonthelinearVARmodelisoftenviolatedevenfortheparametersTp.Furthermore,lteringthedatawithinsomerestrictedfrequencybandoftenleadstoreductionofthe(T;p)domainwheretheestimatedVAR(p)modelsremainstable.Inpractice,theEEGdataareoftenlteredwithinacertainfrequencyband.InmanystudiesthatutilizethevectorautoregressiontoinvestigateGrangercausalityinthebiomedicaltimeseries,thevericationoftheconditionsassumedbytheVARmodelisoftenomitted.Fromourpoint 61

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Variousmodicationsofvectorautoregression(whichrelaxthestabilityconditionofnorootsinsideunitcircle)aredevelopedforanalysisofeconomictimeseries.AlthoughBernasconiandKonig[ 2 ]examinedthestationarityofdierentEEGdata,andconcludedthatatimeintervalofapproximatelyonesecondistheintervalduringwhichtheEEGtimeseriescanbeconsideredstationary,theunderlyingassumptionsoftheVARmodelingsuchasstationarityandstabilityarerarelystatisticallytestedinapplicationstoEEGdata.TheresultsofourexperimentsindicatethatthestabilityassumptionsofvectorautoregressivemodelareoftenviolatedinapplicationtotheEEGdata.Thisobservationsuggeststhatsuitableextensionsofmultivariateautoregressiontounstableprocessesmaytthedatabetter,andasaresult,suchextensionsofVARmaybemoreappropriatefortheEEGtimeseriesanalysisthanthestandardlinearvectorautoregressivemodeling. AdditionalstatisticaltestingisrequiredinordertomakeconclusionsonwhatmultivariatemodelsarethemostsuitableforextractingthedirectionaldependenciesbetweenchannelsinafrequencydomainfrommultichannelEEGdata. GiventhesignalrepresentedformallyasamultipletimeseriesX(t),onecanextracttheinstantaneousphasesXi(t)fromeachone-dimensionalcomponentXi(t)ofthesignalasshowninChapter3(eitherviaaconvolutionwiththeMorletwaveletorbyapplyingtheHilberttransform).ThephaseextractionprocedureproducesanewmultipletimeseriesX(t)ofthecorrespondentphases. Next,wederivenewmeasuresofphasesynchronyofthesignalbasedontheconceptsintroducedinChapter 3 .Letusobservethattheleft-handsideofEquation( 3{1 ) 62

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3{1 ),whichdenesphase-lockingbetweentwosignalsX1(t)andX2(t),needstobemodiedinpracticetoaccountforthenoiseinthesignal.Takingintoaccountpresenceofthestochasticnoiseinthephaseseries,letusintroduceamodiedconceptofthephasesynchronybetweentwosignalsbyrelaxingtheintegralityconditiononthecoecientsinthelinearcombinationasfollows. TwosignalsX1(t)andX2(t)areconsideredtobegenerallyphase-synchronized,ifthecorrespondentinstantaneousphasesX1(t)andX2(t)satisfytheconditionbelow: whereztN(C;0)isastochasticvariablethatrepresentsthedeviationfromtheconstantlevelCasaresultofthenoise.Noticethatinthecontrasttocondition( 3{1 )intheclassicdenitionofphasesynchronization,thecoecientsc1andc2inthedenition( 4{1 )ofgeneralizedphase-synchronydonotneedtobeinteger. Furthermore,itisstraightforwardthatthenewcondition( 4{1 )meansthatatwo-dimensionalprocessX(t)=(X1(t);X2(t))0iscointegrated.Basedonthisobservation,wecanextendourmodiedconceptofphasesynchronizationbetweentwosignalstothemultivariatecaseinthefollowingmanner. ThemultichannelsignalX(t)=(X1(t);:::;XK(t))isconsideredtobephase-synchronizedofrankr,iftheprocessX(t)composedofthecorrespondentinstantaneousphasesXi(t),i=1;:::;Kiscointegratedofrankr. Inthesubsequentsubsections,werstdiscusstheroleofthecointegrationrankintheframeworkofmultivariatephase-synchronization,andthenapplythisapproachtomultichannelEEGdatacollectedfromthepatientswithabsenceepilepsy. 63

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66 ],MichaelMurrayusedanexampleofdrunkardandherdogtoillustratetheconceptofthecointegration.Toexplainourreasoningbehindtherankofcointegrationasameasureofsynchrony,webrieysummarizeandthenfurtherextendhisanalogy. Thenonstationaryprocesses(suchasarandomwalk)areoftenintroducedbyteachersofstatisticsbycomparingthem(it)withthedrunkard'swalk.Thedrunkardwondersaimlessly,sothatthedirectionofeachstepisrandomandcompletelyindependentofherprevioussteps.Inotherwords,themeanderingofthedrunkardisdescribedbyarandomwalk: wherextrepresentsthepositionofthedrunkattimet,and"tisastationarywhite-noise,whichmodelsthedrunk'sstepattimet). AsMurraynoticed[ 66 ],anunleashedpuppyisanothercreature,whosebehaviorremindsarandomwalk.Indeed,eachnewscentthatpuppy'snosecomesupondictatesadirectionforthepup'snextstepsostronglythatthelastscentalongwithitsdirectionisforgottenassoonasthenewscentappears.Havingshownthatthepuppiesfollowtherandomwalkyt,t=1;2;:::,letusrepresentthepuppy'swalkas: wheretisastationarywhite-noise(i.e.puppy'sstepattimet). Thewell-knownfeatureofarandomwalkisthatthebestpredictorofthefuturevalueisthemostrecentlyobservedone.Inotherwords,thelongerithasbeensincewehadseenthedrunk,orthedog,thefurtherawayfromtheinitialplace,onaverage,they 64

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However,ifthepuppybelongstothedrunkard,thentheywillremainrelativelyclosetoeachotheratallthetime,similarlytotheindividualintegratedprocessesthattogetherformacointegratedprocess.Indeed,thedrunkwouldstillwonderaimlesslyinarandomwalkfashion,aswouldherpuppy.However,fromtimetotimeshewouldrememberaboutherdogandcallforit,thepuppywouldrecognizehervoice,andbark.Theywouldheareachotherandmaketheirnextstepineachother'sdirection. Thepathsofthedrunkandherdogarestillnonstationary,buttheyarenolongerindependentfromeachother.Asamatteroffact,ateachtime,thepuppyanditsmasterarelikelytobefoundnotfarfromeachother.Ifthisistrue,thenthedistancebetweentwopathsisstationary,andthewalksofthedrunkxtandherdogytaresaidtobecointegrated,i.e.xtandytareintegratedI(1),andthereisalinearcombinationofxtandyt(withnon-zeroweights)thatisI(0),i.e.stationary. Mathematically,thecointegratingrelationshipbetweenaladyandherpuppycanbewrittenas: attimet=1;2;:::.Notethat"tandt,asbefore,representthestationarywhite-noisestepsofthedrunkandherdog. SinceEquation 4{4 canbeeasilyrewritteninformof( 2{45 )asfollows: 264xtyt375=264"tt375264ccdd375264xt1yt1375; 65

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=264ccdd375; Notethatrank()=0,ifandonlyifc=d=0.Insuchcase,( 4{4 )becomessimplyasystemofequations( 4{2 )and( 4{3 ),whichmodelstwoindependentrandomwalksdrivenbyindependentwhitenoiseprocesses"and.Ontheotherhand,whenatleastoneofthecoecientscanddisnon-zero,thenbymultiplyingsystem( 4{6 )byavector[d;c]0,wehave: whichmeansthatthemodelisdrivenbyasinglecommonstochastictrendd"t+ct. AlthoughtheexampledescribedbyMurrayisclearlyabivariatecointegratedVAR(1),itcanbeextendedtoanillustrationofthemultivariatecointegratedprocess.Consider,forexample,aheardofsheepguardedbytwodogs,wherethesheepwonderaimlesslyintheeld,whilethedogsrunaroundandbringthesheepthathavestrayedtoofarbackintotheock.Say,forexample,afasterdogguardssheepfromtheeast,south,andwest,whereasaslowerdog-fromthenorth,thenthecointegratedprocessappeartohavethecointegrationrankof2.Clearly,twodogsareabletokeepaockofsheepclosertogether,thanasingledogcan.Inotherwords,thehighercointegrationrankthemorerestrictiveitis. Infact,letusconsideraK-dimensionalcointegratedvectorautoregressiveprocess,andletrdenotethecointegrationrankoftheprocess.Similarlytothebivariateexampleabove,wecanseethatwhentherankiszero(r=0),theunivariatecomponentsoftheprocessareindependent,andthemodelisdrivenbyKindependentwhitenoiseprocesses(i.e.thereisnocointegration).Inthecaseofr=1,wecandecomposethemultivariate 66

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Therefore,thesmalleristhecointegrationrankr,thelargeristhenumberKroftheunderlyingindependentstochastictrends,andso(thelarger)isthevectorspaceinwhichourcointegratedmodelcantravel.Andtheotherwayaround,increasingthecointegrationrankofthemodelshrinkstheunderlyingdomainoftheprocess,i.e.makesitboundedtoasmallerhyperplane.Forr=K,theVAR(p)isastableprocess,whichclearlyhasthemostconstraineddomain.Forr=0,theVARprocessisnotcointegratedandunrestricted. Thus,intheframeworkofgeneralizedphase-synchronizationintroducedabove,thecointegrationrankrepresentsafundamentalmeasureofsynchronyinthemulti-channelsignal,suchasEEG.Inparticular,wesaythatthesignaliscompletelyasynchronous,ifthecointegrationrankriszero.Ontheotherhand,whenthemultivariateprocessisstable(i.e.therankcoincideswiththedimensionoftheprocess,r=K),thesignalissaidtobeperfectlysynchronous. 67

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73 ].Thesebriefseizurescanhappenseveraltimesduringaday,butinsomepatients,thefrequencyofabsenceseizurescanbeashighashundredoftimesaday,whichinterfereswiththedailyactivitiesofachildsuchasschool.Insomecasesofchildhoodabsenceepilepsy,theseizuresstopwhenachildreachespuberty.Absenceseizuresexhibitacharacteristicspike-and-waveEEGpatternata3Hzfrequency[ 73 ]. Figure 4-3 displaysamultichannelEEGrecordingthatincludesanabsenceseizure.Thedurationoftheseizureisapproximately4seconds.Thegurevividlyillustratesacharacteristicspike-and-waveactivityduringtheseizure. First,themultipletimeseriesoftheinstantaneousphaseswereextractedfromtherawEEGdatausingtheHilberttransformapproachasdescribedinSection 3.2 .Inparticular,wetookadvantageofthefunctionshilbertandanglereadilyavailableintheMATLABR2006aenvironment. TheVARmodelingandtestingwereimplementedusingtheR2.6.1statisticalsoftware.Inouranalysisoftheinstantaneousphases,weincorporatedar,adf.test,po.test,cajolstandotherfunctionsfoundinpackagestseriesandurca. Next,weillustrateourapproachontheexampleoftheEEGdatalethatincludesthreeseizureintervals.Thelecontainsa16-channelrecordingofscalpEEGsampledatthe200Hzfrequencyaswellastwoauxiliarychannels,whichwerediscarded.TheinstantaneousphasevalueswereestimatedfromtheEEGtimeseriesbymeansofHilberttransform,andtheresultingphaseseriesweretestedusingtheADFtestintroducedinSection 2.4.1 .Specically,weappliedtheAugmentedDickey-Fullerproceduretotestthe 68

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Theresultsofourexperimentsforseizures1,2,and3arepresentedinTables 4-1 4-2 ,and 4-3 ,respectively.Thechannels,forwhichtheADFunitroottesthasdetectedapresenceofaunitrootatthesignicancelevel=0:01,arelistedasintegrated.Whereasthechannels,forwhichthenullhypothesisofaunitroothasbeenrejectedbytheADFatthe1percentlevel,aredenotedbystationary.Thechannelsforwhichthep-valuesoftheADFtestexceed2.5%aremarkedwith*.Noticethatallthreeseizuresegmentsareconsideredstable,whentheADFisappliedata0.025signicancelevel. Next,wetvectorautoregressiontothemultipletimeseriesofphaseestimates,foreachofthreedierentsegments(before,during,andafteraseizure)inordertodetermineappropriatelaglengthparameterp.Tondappropriatelagsp,theAkaikeInformationCriteria(AIC)wasused.Thisledustochooseseverallaglengthforeachsegmentandeachseizure.Finally,Johansencointegrationrankprocedurewasappliedtodeterminethevaluesofcointegrationrankrforeachcase.TheresultsaresummarizedinTables 4-4 4-5 ,and 4-6 Noticethatduringtheseizurethesystembecomesstable,especiallywhenmodeledusingashortestimateofthelagparameter.Sincethedurationsoftheseizure1andseizure2arerathershort,andonlyinclude440-500samplepoints,themodelsestimatedunderalonglagparametermaynotadequatelyrepresenttheunderlyingprocessesinseizure1and2.Ontheotherhand,seizure3isestimatedbasedonalmost1200samplevalues,andthereforethelonglagmodelofalongerseizure3maybemorerealistic,thanthelonglagmodelsforshorterseizures1and2.Overall,themodelsbasedonashortlagpforallthreeseizuresprovideanevidenceofabsolutesynchronizationamongthechannels.Whereas,thethepre-seizureandpost-seizuremodelsaremorelikelytobelessrestricted,andseemtoexhibitacointegrationrankbetween9and16. 69

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ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure1.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#pre-seizureseizurepost-seizure stationary3,4,7,9,151-3,5-141,3,5,6,8,10,12,15integrated1*,2*,5,6*,8*,10,11,12*,13*,14*,16*4,15,162*,4*,7,9,13*,14,16* Table4-2. ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure2.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#pre-seizureseizurepost-seizure stationary3,4,7,9-11,13,161-163,7,11-16integrated1*,2*,5*,6,8*,12*,14*,15none1*,2,4*,5,6*,8,9*,10 Table4-3. ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure3.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#pre-seizureseizurepost-seizure stationary7,11,13,141-162,4,5,11,13,15integrated1*,2*,3,4*,5*,6*,8*,9,10*,12,15,16*none1,3*,6*,7,8,9*,10*,12,14*,16 Table4-4. ResultsoftheJohansencointegrationrankprocedureforthemultipleseriesduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure1.Signicancelevelis1%.Fullrankisdenotedbyy longlagp=22;r=12p=23;r=11,p=20;r=13p=20;r=12shortlagp=2;r=13p=2;r=16yp=2;r=10 Table4-5. ResultsoftheJohansencointegrationrankprocedureforthemultipleseriesduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure2.Signicancelevelis1%,fullrankisdenotedbyy longlagp=21;r=14p=26;r=12,p=20;r=9p=20;r=10shortlagp=2;r=16yp=3;r=16yp=2;r=13 Table4-6. ResultsoftheJohansencointegrationrankprocedureforthemultipleseriesduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure3.Signicancelevelis1%,fullrankisdenotedbyy longlagp=24;r=13p=26;r=16y,p=20;r=16yp=20;r=13shortlagp=2;r=9p=2;r=16yp=2;r=16y

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Numbersnofrootsofthereversecharacteristicpolynomial(RCP)forVAR(p),whichlieonandinsidethecomplexunitcircle,computedfordierentsamplesizesTandfordierentmodelorderspusingtherawdata Numbersnofrootsofthereversecharacteristicpolynomial(RCP)forVAR(p),whichlieonandinsidethecomplexunitcircle,computedfordierentsamplesizesTandfordierentmodelorderspusingthe0{30Hzbandltereddata 71

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SegmentofmultichannelEEGrecordingthatcontainsafoursecondlongabsenceseizure

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Investigationofspatio-temporalpropertiesoftheEEGdatabydataminingandoptimizationapproachespostsvariouschallenges.NumerousfeaturesandmethodshavebeenproposedforstudyingthemultivariateseriesthatisEEG.TheanalysisofEEGtimeseriesisoftenapproachedfromtwodierentpointsofview,theonethattreatsEEGdataasproducedbyadeterministicchaoticdynamicalsystem,andtheothermoretraditionalapproachoflinearautoregressivemodeling. Inthiswork,weinvestigatedseveralstatisticalapproachesthatarerecentlyintroducedfordataminingbraindynamics.Inparticular,weexaminedtheapplicationofvectorautoregressivemodelingandlinearGrangercausalitytorawandlteredEEGdata. Motivatedbyrecentsuccessinapplicationofphasesynchronizationtoanalysisofdynamicprocessesinepilepticbrain,wedevelopedaconceptofgeneralizedsynchro-nizationbasedonthenovelideaofextendingtheclassicalsynchronizationconditionofaboundedlinearcombinationoftwophases.Thissimplebivariateconditioninthemultivariatecaseisanalogoustoacointegratingrelationshipinthemultipletimeseries.Thus,wecananalyzethesynchronyamongdierentpartsofthecommoninterrelatedsystem(suchasahumanbrain),bymodelingthephasesextractedfromanitenumberofsignalsinthesystemsbymeansofcointegratedvectorautoregression.Moreover,weshowedthatthecointegrationrankinthecointegratedVARmodelofthephasetimeseriescanbeviewedasameasureofsynchronyamongthephasesofdierentcomponentsoftheEEGsignal. Notonlythisnewmeasureofmultivariatephasesynchronycanbetestedonvariousbiomedicaldata,suchasmultichannelEEGrecordedfromanepilepticbrain,butalsothenewmultiplephasesynchronizationcanbeemployedindierentareasofappliedandtheoreticresearch(includingphysics,communication,electronics,laserdynamics,and 73

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74

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AllaRevenkoKammerdinerwasborninKiev,Ukraine.Anolderoftwochildren,shegrewupinKiev,Ukraine,graduatingfromSchool#32in1994.SheearnedherB.S.inProbabilityTheoryandMathematicalStatisticsfromtheNationalTarasShevchenkoUniversityofKyivin1998.InJanuary2001,AllajoinedagraduateprogramintheMathematicsDepartmentattheUniversityofFlorida.UpongraduatinginMay2004withherM.S.inmathematics,AllaenteredthePh.D.programinindustrialandsystemsengineeringattheUniversityofFlorida.AllahasbeenhappilymarriedtoJasonR.Kammerdinerforthelast3years.OnFebruary24,2008shecompletedherrstmarathonin3:53:09. 83