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27e30003cf2c5ceecd5d8db09b46db6e7d09d02b 58356 F20101109_AABJDE kammerdiner_a_Page_54.pro 800fa431c9e9747283e7d70f674d970a 3732bcb2d7ae530ac84dc6fca2f27e2fba6083ab 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 DickeyFuller Test for Testing the Null Hypothesis of the Presence of a Unit Root .. .................. 2.4.2 PhillipsOuliaris 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 21 ADF test: Critical values of the T(c 1) and i statistics . ..... 49 22 PhillipsOuliaris demeaned: Critical values of the Z, statistic . ... 49 23 Johansen test: Critical values of the ALR(r, K) statistic . . ..... 49 24 Johansen test: Critical values of the ALR(r, r + 1) statistic . .... 50 41 Seizure 1: Results of the ADF unit root tests .................. 70 42 Seizure 2: Results of the ADF unit root tests ......... . ....... 70 43 Seizure 3: Results of the ADF unit root tests .................. 70 44 Seizure 1: Results of the Johansen cointegration rank tests . .... 70 45 Seizure 2: Results of the Johansen cointegration rank tests . .... 70 46 Seizure 3: Results of the Johansen cointegration rank tests . .... 70 LIST OF FIGURES Figure page 11 The International 1020 system for placement of EEG electrodes . ... 27 41 Raw data: Numbers of unstable roots for different T and p . ..... 71 42 Filtered data: Numbers of unstable roots for different T and p . ... 71 43 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 timeconsuming 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 multidiscipline 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 l1. ii. 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 twoclass 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 twoclass 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 twodimensional 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 nonstationary 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 onedimensional 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 spatiotemporal dependencies in the EEG data in contrast to being limited to only temporal relations in a onedimensional case. There is a disagreement among researchers studying EEG data on whether the series should be modeled by a nonlinear 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 noninvasive or invasive methods. EEG (obtained during a noninvasive 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 1020 system for EEG electrode placement was developed [37]. The 1020 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 1020 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 1020 EEG system became very widely used for positioning electrodes. In the 1020 system, the points are denoted with one or two letters, and can be also followed by a number (as shown on Figure 11). 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 bestknown 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 nonrapid 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 nonstationary time series [94]. In order to properly reflect the spatiotemporal 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, crosscorrelation, 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, (11) 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, (12) 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 (12), 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 crosscorrelation, coherence and Granger causality measure the interdependency between different channels. The crosscorrelation function quantifies the linear correlation between two processes. Given two normalized signals x(t) and y(t) with zero means and unit variances, the crosscorrelation between these signals is estimated as: NT C (r) xr+tt)y(t), (13) t=i where N denotes the total number of sampled points, and r is a time delay parameter between two signals. The crosscorrelation estimate ranges between 1 and 1. The positive values of crosscorrelation 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 crosscorrelation value shows that the signals are not linearly correlated, it does not necessarily imply that two signals are not interrelated in a nonlinear fashion. The coherence function is a frequency domain analog of the crosscorrelation measure. Coherence is obtained from crosscorrelation by applying Fourier transform to (13). 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) = (14) where Sx(f), Sxx(f), Syy(f) denote the average values of the crossspectral density between x and y, and the individual autospectral 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 crosscorrelation 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 nonstationary 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))), (15) 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 ( ( ' (16) iN 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, spatiotemporal 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 largescale integration [15, 101, 104, 105]. Actually, it was found that oscillation of various neuronal groups in given frequency bands leads to temporary phaselocking between such groups of neurons. This observation has stimulated the development of robust approaches that allow one to measure the phasesynchrony 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 11. The International 1020 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 spatiotemporal 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/nogo 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 Grangercausal 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 pstep forecast of Y(t) based on the information Qx,t and QY,t is smaller than the MSE of the pstep forecast of Y(t) based on Qyt alone, i.e. p = 1,2,... : Ey(plx t, y, t) < Ey(plQY,t), (21) where Ey(pfQ) is the MSE of the pstep 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 + aiyi + Oixti + Et, (22) 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 (23) against the alternative H1 : 3j, 1 < j < p s.t. 0j 0 (24) Note that if the null hypothesis is accepted, then a time series Y(t) is believed to be NOT Grangercaused 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, (25) t=1 and the sum of squared residuals RSSI under the alternative H1 is T RSS, = t. (26) t=1 By conducting the Ftest of the null hypothesis, one can find the test statistic (RSSo RSSi)/p 1 RSS ~ Fp, T2p1. (27) RSSl/(T 2p 1) If the test statistic Si exceeds the specified critical value, then the null hypothesis that X(t) does not Grangercauses 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 nonstationarity 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 Fstatistic 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 Kvariate time series (i.e. realizations of Kdimensional 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.., (29) where yt = (lt, ... Kt)' is a (K x 1) random vector, v (vl,..., VK)' is a fixed (K x 1) vector representing a nonzero mean EY(t), the Aj, i = ,... ,p are fixed (K x K)dimensional coefficient matrices, and Et = ( Et,..., Kt)' is a Kdimensional 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 YuleWalker 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 wellknown 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 Kdimensional vectors Yp+ ... ,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 presample yp+,... 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), (210) Utp+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 (29), the vector autoregressive model of order p can be represented in the compact form: Y = BZ + E, (211) and the coefficients B of the model are given by the least squares estimator: B = YZ'(ZZ')\. (212) The covariance matrix can be estimated in various viv. Since ZE = E [EtE'], the estimator T E f E ti't = (Y BZ)(Y BZ)' (213) 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 TKp 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), HannanQuinn 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)) (215) Using the FPE criterion, the estimate PFPE of the model order p is selected so that FPE (PFPE) min FPE(m), (216) 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 (215); 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) (217) 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 HannanQuinn order selection criterion, HQ, is given by: HQ(m) mK2 nT + In (m)) (218) 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 ) (219) 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 HannanQuinn 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 Kdimensional VAR(p) time series (29) are called stable, if det(IK Alz ... Apz) / 0 for complex z : zl < 1. (220) In other words, the VAR(p) process (29) 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 welldefined 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 timeinvariant (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 wellstudied, 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 widesense 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 (220) 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 nonstationary. The Gaussian distribution assumption is introduced into the VAR(p) model through Et. Specifically, given representation (29) 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 nonstationary 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 (29) 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 reallife applications, the former case is of the most practical interest. This renders the following definition of an integrated process. A onedimensional 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 onedimensional 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) = bjtj, (221) i=0 j=0 where Etjs 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. onedimensional 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 Kdimensional vectors yt = lt,..., YKt)'. In addition, suppose that the variables are in a longrun equilibrium relation c Y(t) := ci Yi(t) +... + CK YK(t) = 0, (222) where c = (c1,..., CK) is a Kdimensional real vector. During any given time interval, the relation (222) may not necessarily be satisfied precisely by the sample yt, instead we may have: c yt := ci yt + C. + CK YKt = Et, (223) where Et is a stochastic process that denotes the deviation from the equilibrium relation at time t. If our assumption about the longrun 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 onedimensional components Yi(t) (i = 1,..., K) are either I(1) or I(0) processes. Then the combined Kdimensional VAR(p) process Y(t) = v+ AIY(t 1) +... + ApY(t p) + Et (224) is said to be cointegrated of rank r, when the correspondent matrix I IK A1 ... Ap (225) has rank r. Since some onedimensional 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 DickeyFuller Test for Testing the Null Hypothesis of the Presence of a Unit Root The augmented DickeyFuller (or ADF) test is a widely used statistical test for detecting the existence of a unitroot 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. Finitesample 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 finitesample 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,... (226) 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 (226), 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 (226) 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 (226) that yt = c yt1 + Et holds, and so by plugging this last condition into Equation (227), the estimator c of the regression coefficient can also be written as: (zc =tt I 1+t) 1 (228) T S z2T 1 c t 1 1 i+ t1 t1it zT 12 Et 1 2t t Yt 1 Lt Et c+ Y, E IitL 3t 1u/ 1 Subtracting c from both sides of Equation 228 and multiplying each side by T lead to the ADF statistic T(Z c) = i tI t (229) 2 t2 1 t[1t 1 Dickey and Fuller [16] derived the following representation of the limiting distribution for statistic T(c c): T(cc)= i (W2 1), as T oo (230) 2 where OO r F= df Xf, (231) i=1 W = v2 dXj, (232) i=1 2(1)i+1 d = )' (233) 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) (234) t=22 where ^S2 7 _1, i12 (235) T 2 and is computed from (227) Tables of the critical values for the .ii1!,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 21, which lists the pvalues for .,vmptotic distributions of T(c 1) and ? corresponding to percentiles of 90, 95, and 99 percent. 2.4.2 PhillipsOuliaris 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 PhillipsOuliaris cointegration test. It is noteworthy that the null hypothesis for the PhillipsOuliaris 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 Kdimensional 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,... (236) 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 (236). 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,... (237) Then the statistic Z, in PhillipsOuliaris test is defined as follows: 1 2 2 Z = T( 1 1) (238) T2 t=2 2tI 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 (240) T T T 4Si T Yj Qt +r7"; T L s, (241) t=1 s=1 t=s+1 ,.. 1 (242) 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 (237). 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 PhillipsOuliaris test (Zn) are listed in Table 22. 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 Kdimensional 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 (224). Given a realization yt, t = 2,..., of Y(t), one seeks to determine the coefficients of the following model: Yt= Alyt1 + ... + Apyt+p+ Et, t= 1,2,..., (243) subject to the constraint rank(I) = rank(IK A ... A) = r. (244) Note that Et is assumed to be a Gaussian white noise with a nonsingular covariance matrix Es. Furthermore, the initial conditions yp+l,... ,yo are supposed to be fixed. In order to impose the cointegration constraint, the model (243) is reparameterized in the following fashion [59]: Ayt DAyt_ +... + Dp_,Aytp+l + Fytp + t, t 1,2,... (245) where Ay = yt yt1, 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 : (246) AXYtp+2 AX := [AXo,..., AX,_] , D := [D1,...,Dp1], Yp := [yp, ..., YTp] Then the loglikelihood 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 A1.:. O 7, ,.od function in (247) 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) (248) 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 (224), suppose we wish to test a hypothesis Ho against an alternative H1, where Ho : r= ro against H1 : ro < r < rl. (249) 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 (249) can be determined in the following manner: ALR(ro,r1) 2[lnLmax(ri) lnLmax(ro)] (250) 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 (250) is nonstandard. Specifically, the LR statistic is not ..i','i!iil1 l ly distributed according to x2distribution. Nevertheless, the ..imptotic 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 .Ii,iIl ,l ic distribution of ALR are given in Tables 23 and 24. Table 21. 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 22. Critical values of the ..imptotic distributions of the Z, statistic for testing the null of no cointegration (PhillipsOuliaris 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 23. 95', 20.4935 26.0943 32.0615 37.1508 41.9388 9r 28.3218 34.1686 41.1348 47.5118 52.1723 Percentage points of the .,iii! 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 24. 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 largescale integration. In fact, various neuronal groups oscillate in specific frequency bands and become phaselocked over a limited period of time. This observation has stimulated the development of robust approaches that allow one to measure the phasesynchrony 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 microelectroderecording 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 lowpass 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 50350 micrometers from the tip of the electrode [53] and slower ionic events from within 0.53 millimeters from the tip of the electrode [39]. The spike component of the signal is removed by lowpass 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 longrange 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 crosscorrelograms between spike discharges may be adequate for microelectrode studies, the quantification of phase synchrony between meso or macroelectrodes (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 nonstationary. 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 phaselocking 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 phaselocking 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, (31) 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 phaselocking between their signals. Furthermore, in noninvasive 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 (31) 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 phaselocking. Two methods for detecting phaselocking 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 realvalued function f(t) with domain T is defined as a realvalued function f(t) on T as follows: f(t) = CPV f(T)g(t r)dTr CPV g(T)h(t T)dr, (33) 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 realvalued time series X(t), the corresponding I,.,l.l.' /.::,,l is defined as the following complexvalued function: x(t) = X(t) + 2 X(t) = ax(t) exp {z. x(t)}, (34) 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 34 that the instantaneous phase Ox(t) of X(t) can be computed as: x (t) arctan () (35) X (t) A key advantage of the analytic approach is that the phase can be easily computed for an arbitrary broadband signal. On the other hand, instantaneous amplitude and phase have a clear physical meaning only if X(t) is a narrowband 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} (36) 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 ) (38) 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 () (39) 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 (39) 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 3dimensional 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 41 and 42. The surface plot in Figure 41 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 41 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 42 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 41, 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 3060 Hz, 6090 Hz, 90120 Hz, and 120150 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 alvb7 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 onedimensional 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 lefthand side of Equation (31) represents the linear combination of the respective phases Ox, (t) and x,2 (t) with integer coefficients. Also recall that condition (31), which defines phaselocking 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/; phasesynchronized, 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 (31) in the classic definition of phase synchronization, the coefficients cl and c2 in the definition (41) of generalized phasesynchrony do not need to be integer. Furthermore, it is straightforward that the new condition (41) means that a twodimensional 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 phasesynchronization, 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 Xt1= Et, t= 1,2,..., (42) where Xt represents the position of the drunk at time t, and Et is a stationary whitenoise, 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 Yt1 = t, t = 2,..., (43) where ct is a stationary whitenoise (i.e. puppy's step at time t). The wellknown 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 nonzero 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(Yt1 Xt), (44) yt yt1 = t + d(xt1 yt1), (45) at time t = 1, 2,.... Note that Et and et, as before, represent the stationary whitenoise steps of the drunk and her dog. Since Equation 44 can be easily rewritten in form of (245) as follows: Xt Et C C Xt1 A (46) t t d d y/t1 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, (44) becomes simply a system of equations (42) and (43), 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 nonzero, then by multiplying system (46) by a vector [d, c]', we have: dAxt + cAyt = dEt + et, t = t, 2,..., (47) 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 Kdimensional 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 phasesynchronization introduced above, the cointegration rank represents a fundamental measure of synchrony in the multichannel 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 spikeandwave EEG pattern at a 3 Hz frequency [73]. Figure 43 dipl'i1 a multichannel EEG recording that includes an absence seizure. The duration of the seizure is approximately 4 seconds. The figure vividly illustrates a characteristic spikeandwave 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 16channel 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 DickeyFuller 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 41, 42, and 43, 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 pvalues 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 44, 45, and 46. 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 440500 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 preseizure and postseizure 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 * preseizure 3,4,7,9,15 1*,2*,5,6*,8* ,10,11,12*, 13* ,14* ,16* seizure 13,514 4,15,16 postseizure 1,3,5,6,8,10,12,15 2*,4*,7,9,13*, 14,16* Table 41. Seizure # stationar' integrate Table 42. preseizure 3,4,7,911,13,16 1*,2*,5*,6,8*,12*,14*,15 seizure 116 none postseizure 3,7,1116 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 * preseizure 7,11,13,14 1*,2*,3,4*,5*,6*,8*,9,10*,12,15,16* seizure 116 none postseizure 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 45. Seizure # long lag short lag Table 46. Seizure # long lag short lag preseizure p = 22,r = 12 p 2,r 13 p 23, r p seizure 11, p= 20, r = 13 S2,r 16t postseizure 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 preseizure S21, r 14 2, r 16t seizure p =26, r = 12, p= 20, r 9 p =3, r = 16t postseizure 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 preseizure p 24, r 13 p 2, r 9 seizure p = 26, r = 16t, p = 20,r = 16t p 2, r = 16t postseizure 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 43. Seizure # stationary integrated Table 44. 100 0 0 Figure 41. 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 42. 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 030 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 spatiotemporal 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. REFERENCES [1] L. Baccala and K. Sameshima, "Partial directed coherence: a new concept in neural structure determination," Biol. C;,I1,,. i vol. 84, pp. 46374, 2001. [2] C. Bernasconi and P. Koenig, "On the directionality of cortical interactions studied by structural analysis of electrophysiological recordings," Biol. C;,l ,,., vol. 81, pp. 199210, 1999. [3] R. D. Bickford, "Electroencephalography," in Adelman G. ed. E,. ;; 1.>pedia of Neuroscience. Cambridge, MA: Birkhauser, pp. 371373, 1987. [4] C. D. Binnie, "Long term EEG recording and its role in clinical practice," IEE Colloquium on Data Logging of Pi,;,'..:'.....: S.: ,il pp.5/15/2, 1995. [5] P. S. Bradley, O. L. Mangasarian, and W. N. Street, "Clustering via concave minimization," in M. C. Mozer, M. I. Jordan, and T. Petsche, eds.,Advances in Neural Information Processing S;r /. vol. 9, pp. 368374, 1997. [6] P. S. Bradley and O. L. Mangasarian, 1. iI ire selection via concave minimization and support vector machines," in J. Shavlik, Machine Learning Proceedings of the Fifteenth International Conference, pp. 8290, 1998. [7] P. S. Bradley, U. M. Fayyad, and O. L. Mangasarian, \ II .!, ii ii Programming for Data Mining: Formulations and C('! 11. es," INFORMS Journal on Co,,l',/,.ul vol. 11, no. 3, pp. 217238, 1999. [8] S. L. Bressler, R. Coppola, and R. Nakamura, "Episodic multiregional cortical coherence at multiple frequencies during visual task performance," Nature, vol. 366, pp. 153156, 1993. [9] R. Brown and L. Kocarev, "A unifying definition of synchronization for dynamical systems," Chaos, vol. 10, pp. 344349, 2000. [10] T. H. Bullock and M. C. McC'lti. "Lateral coherence of the electrocorticogram: a new measure of brain synchrony," Electr. ,'.. 1,,i].r Clin N '. *'iu,;;,.: vol. 73, pp. 479498, 1989. [11] T. L. Carroll and L. M. Pecora, "Cascading synchronized chaotic systems," PI,,.. D, vol. 67, pp. 126, 1993. [12] G. C. Carter, "Coherence and time delay estimation," Proc IEEE, vol. 75, pp. 236255, 1987. [13] Y.W. C('!. t K. S. Lai, "Lag Order and Critical Values of the Augmented DickeyFuller Test," Journal of Business & Economic Statistics, vol. 13, no. 3, pp. 277280, 1995. [14] R. Cont, "Statistical Properties of Financial Time Series," in R. Cont and J. Yong editors, Mathematical Finance: Theory and Practice, Lecture Series in Applied Mathematics, vol. 1, 1999. [15] A. R. Damasio, "Synchronous activation in multiple cortical regions: a mechanism for recall," Semin Neurosci, vol. 2, pp. 287296, 1990. [16] D. A. Dickey, and W. A. Fuller, "Distribution of the Estimators for Autoregressive Time Series With a Unit Root," Journal of the American Statistical Association, vol. 74, pp. 427431, 1979. [17] M. Eichler, Graphical modelling of multivariate time series, University of Heidelberg: Preprint, 2001. [18] M. Eichler, R. Dahlhaus, and J. Sandkuhler, "Partial correlation analysis for the identification of synaptic connections," Biol. C.;Il. ,,. vol. 89, pp. 289302, 2003. [19] M. Eichler, "A graphical approach for evaluating effective connectivity in neural systems," Philos. Transact. R. Soc. B vol. 360, pp. 95367, 2005. [20] L. F 1,1 i:w, P. Colet, and R. Roy, "Coherence and phase dynamics of spatially coupled solidstate lasers," Phys. Rev. A, vol. 47, pp. 4287, 1993. [21] W. J. Freeman, "Spatial properties of an EEG event in the olfactory bulb and cortex," Electr., '1',hi .,r Clin N,jir ;.?ih.1 vol. 44, pp. 586605, 1978. [22] W. A. Fuller, Introduction to Statistical Time Series, 2nd ed. New York: John Wiley, 1996. [23] D. Gabor. "Theory of communication," in Proc. IEEE London, vol. 93, pp. 429, 1946. [24] P. Georgiev, A. Cichocki, and H. Bakardii ,i "Optimization techniques for Independent Component Analysis with Applications to EEG data," In P. M. Pardalos, J. C. Sackellares, P. R. Carney, L. D. Iasemidis, eds., Quantita tive Neuroscience: Models, Algorithms, Diagnostics, and Theraputic Applications, Kluwer Academic Publishers, pp. 5368, 2004. [25] J. Geweke, \! I, cement of linear dependence and feedback between multiple time series," J. American Statis. Association, vol. 77, pp. 304313, 1982. [26] C. W. J. Granger, "Economic Processes Involving Feedback," Information and Control, vol. 6, pp. 2848, 1963. [27] C. W. J. Granger, lii., i ii ig Causal Relations by Econometric Models and CrossSpectral Methods," Econometrics, vol. 37, pp. 424438, 1969. [28] C. W. J. Granger, "Some Properties of Time Series Data and Their Use in Econometric Model Specification," J of Econometrics, vol. 16, pp. 121130, 1981. [29] M. J. Griffiths, P. Grainger, M. V. Cox, and A. W. Preece, "Recent advances in EEG monitoring for general anaesthesia, altered states of consciousness and sports performance science," in The 3rd IEE International Seminar on Medical Applications of S.:j,,,l Processing, pp. 15, 2005. [30] J. D. Hamilton, Time Series A,..l;;'. Princeton, 1994. [31] M.J. Hoeve, R. D. Jones, G. J. Carroll, and H. Goelz, "Automated detection of epileptic seizures in the EEG," in Engineering in Medicine and B.:.. .,i'/; S '.. ./.;/ Proceedings of the :' :'.1 Annual International Conference of the IEEE, vol. 1, p1' "! ; 946, 2001. [32] R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, 2nd edition. Kluwer Academic Publishers, 2000. [33] C. Huygens. Horoloquium Oscilatorium.Paris, 1673. [34] L. D. Iasemidis, "On the dynamics of the human brain in temporal lobe epilepsy," Ph.D. Dissertation, University of Michigan, Ann Arbor, 1991. [35] L. D. Iasemidis, and J. C. Sackellares, "The evolution with time of the spatial distribution of the largest Lyapunov exponent on the human epileptic cortex," in D.W. Duke, and W.S. Pritchard, eds., Measuring chaos in the human brain, pp. 4982, World Scientific, 1991. [36] L. D. Iasemidis, D.S. Shiau, J. C. Sackellares, P. M. Pardalos, and A. Prasad, "Dynamical resetting of the human brain at epileptic seizures: application of nonlinear dynamics and global optimization techniques IEEE Transactions on Biomedical Engineering, vol. 51, no. 3, pp. 493506, 2004. [37] H. H. Jasper, "The tentwenty electrode system of the International Federation," Electr .. ., ,,]hal.,p. Clin. N, a,., ;...1 vol. 10, pp. 370375, 1958. [38] S. Johansen and K. Juselius, \!. ::i:;i ii1 Likelihood Estimation and Inference on Cointegration With Applications to the Demand for Money," Oxford Bulletin of Economics and Statistics, vol. 52, pp. 169210, 1990. [39] E. Juergens, A. Guettler, and R. Eckhorn. "Visual stimulation elicits locked and induced gamma oscillations in monkey intracortical and EEGpotentials, but not in human EEG," Experimental Brain Research, vol. 129, pp. 247259, 1999. [40] Y. Kaji, M. Akutagawa, F. Shichijo, H. N ., iiip,, Y. Kinouchi, and S. N .', iliro, "Analysis for Brain Activities during Operations Using Measured EEG," in Conf Proc IEEE Eng Med Biol Soc. vol. 6, pp. 60036006, 2005. [41] T. K i1 ii i and O. Ozdamar, \V i. !. t preprocessing for automated neural network detection of EEG spikes," Engineering in Medicine and B. :..,'i; l.if', ..:,.; IEEE, vol. 14, no. 2, pp.160166, 1995. [42] M. Kaminski, M. Ding, W. A. Truccolo, and S. L. Bressler, "Evaluating causal relations in neural systems: Granger causality, directed transfer function and statistical assessment of significance," Biol. C;I,., ,i vol. 85, pp. 14557, 2001. [43] A. Kammerdiner, P. Xanthopoulos, and P. M. Pardalos, \ii[II 1, 1 I1 limitations in application of vector autoregressive modeling and Granger causality to in 1,i; of EEG time series," In AIP Conf. Proc., vol. 953, pages 232245, 2007. [44] A. Kammerdiner, and P. M. Pardalos, "Analysis of multichannel EEG recordings based on generalized phase synchronization and cointegrated VAR," In Proc. of Conference on Computational Neuroscience :' "/'' 2008 (submitted). [45] E. Keogh, and S. Kasetty, "On the need for time series data mining benchmarks: a survey and empirical demonstration," in Proceedings of the Eighth AC'MI SIGKDD international Conference on Knowledge Discovery and Data Mining, pp. 102111, 2002. [46] M.E. Kirlangic, D. Perez, S. Kudryavtseva, G. Griessbach, G. Henning, and G. Ivanova, Fli I i! dimension as a feature for adaptive electroencephalogram segmentation in epilepsy," in Engineering in Medicine and B'.: '..,i,; S... I.; Proceed ings of the *',, Annual International Conference of the IEEE, vol. 2, pp.15731576, 2001. [47] W. A. Kittel, C. M. Epstein, and M. H. H v "EEG monitoring based on fuzzy classification," Circuits and S;,1. ', Proceedings of the 35th Midwest Symposium, vol. 1, pp.699702, 1992. [48] L. Kocarev and U. Parlitz, "General approach for chaotic synchronization with applications to communication," Phys. Rev. Lett., vol. 74, pp. 5028, 1995. [49] T. Kreuz, Measuring Synchronization in Model S, ii and Electr .. ,... .i'il' g ..J,1 . Time Series from E'.:1 /i",;/ Patients, Dissertation. NIC Series, vol. 21, 2003. [50] J. P. Lachaux, E. Rodriguez, J. Martinerie, and F. J. Varela, \!. ,iiiiig phase synchrony in brain signals," Hum. Brain Mapp., vol. 8, pp. 194, 1999. [51] J. P. Lachaux, E. Rodriguez, J. Martinerie, C. Adam, D. Hasboun and F. J. Varela, "Gammaband activity in human intracortical recordings tii'..: rI by cognitive tasks," Eur J Neurosci, vol. 12, pp. 26082622, 2000. [52] J. P. Lachaux, E. Rodriguez, M. Le Van Quyen, A. Lutz, J. Martinerie, and F. J. Varela, "Studying singletrials of phasesynchronous activity in brain," Int. J. Bifurcation Chaos Appl. Sci. Eng., vol. 10, pp. 2429, 2000. [53] A. D. Legatt, J. Arezzo, and H. G. Vaughan, "Averaged multiple unit activity as an estimate of phasic changes in local neuronal activity: effects of volumeconducted potentials," J. of Neuroscience Methods, vol. 2 no. 2,pp. 203217, 1980. [54] M. Le Van Quyen, C. Adam, J. P. Lachaux, J. Martinerie, M. Baulac, B. Renault, et al., "Temporal patterns in human epileptic activity are modulated by perceptual discriminations" Neuroreport, vol. 8, pp. 17031710, 1997. [55] M. Le Van Quyen, J. Martinerie, C. Adam, and F. J. Varela, "Nonlinear spatiotemporal interdependencies of interictal intracranial EEG recordings from patients with temporal lobe epilepsy: Localizing of epileptogenic foci," P,;/,.. D, vol. 127, pp. 25065, 1999. [56] M. Le Van Quyen, J. Foucher, J.P. Lachaux, E. Rodriguez, A. Lutz, J. Martinerie, and F. J. Varela, "Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony," Journal of Neuroscience Methods, vol. 111, pp. 8398, 2001. [57] H. Liang, M. Ding, R. Nakamura, and S. L. Bressler, "Causal influences in primate cerebral cortex during visual pattern discrimination," Neuroreport, vol. 11, pp. 287580, 2000. [58] Q. Liu, M. Sun, M. L. Scheuer, and R. J. Sclabassi, "Patient tracking for video/EEG monitoring based on change detection in DCT domain," in Bioengineering Confer ence, 2005. Proceedings of the IEEE 31st Annual Northeast, pp.114115, 2005. [59] H. Liitkepohl, Intoduction to multiple time series i,,,;l;;. Berlin, Heidelberg: SpringerV i1 ,1 1991. [60] J. G. MacKinnon, "Critical Values for Cointegration Tests," in R. F. Engle and C. W. J. Granger, eds.,LongRun Economic Relationships: Readings in Cointegra tion,New York: Oxford University Press, pp. 266276, 1991. [61] J. G. MacKinnon, "Approximate Asymptotic Distribution Functions for Unit Root and Cointegration Tests," Journal of Business & Economic Statistics, vol. 12, pp. 167176, 1994. [62] V. Menon, W. J. Freeman, B. A. Cutillo, J. E. Desmond, M. F. Ward, S.L. Bressler, et al., "Spatiotemporal correlations in human gamma band electrocorticograms," Electr ',., 'l,,/,/, Clin N, ; ',,i'i;l,'.: vol. 98, no. 2, pp. 89102, 1996. [63] M. Mezard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond. World Scientific, 1987. [64] S. Mitra and T. Acharya, Data mining: multimidea, soft .i,,;,j'li: and 1'....: .,[.,r matics. Hoboken, NJ: John Wiley & Sons, Inc., 2003. [65] F. Mormann, K. Lehnertz, P. David, and C. E. Elger, .\!, i phase coherence as a measure for phase synchronization and its application to the EEG of epileptic patients," Ph;I;... D, vol. 144, pp. 358369, 2000. [66] M. P. Murray "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction," The American Statistician, vol. 48, no. 1 pp. 3739, 1994. [67] E. Niedermeyer, F. H. Lopes da Silva, Electr ,** ',' i.',., l''i,,'i,." Basic principles, clinical applications and related fields, 3rd edition. Philadelphia: Lippincott, Williams & Wilkins, 1993. [68] P. L. Nunez, Neocortical D;,i/,i,. and Human EEG Rl.iln,, New York: Oxford Univ. Press, 1995. [69] N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, "Geometry from a Time Series," Phys. Rev. Lett., vol. 45, no. 9, pp.712716, 1980. [70] P. Panter. Modulation, noise, and spectral ,.iil,.: New York: McGrawHill, 1965. [71] H. S. Park, Y.H. Lee, S. Doo S, et al., "Detection of epileptiform activity using wavelet and neural network," Engineering in Medicine and B.:,. .,i/;/, S1 ,,. .I;, vol.3, pp. 1194, 1997. [72] U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, "Experimental observation of phase synchronization," Phys. Rev. E, vol. 54, pp. 2115, 1996. [73] J. B. Patten, N, ,. ,',,.: ,l Differential Diagnosis, 2nd ed., SpringerV, i1 .: 1996. [74] D. W. Peterman, M. Ye, and P. E. Wigen, "High frequency synchronization of chaos," Phys. Rev. Lett., vol. 74, pp. 1740, 1995. [75] P. C. B. Phillips, "Time series regression with unit root," Econometrica, vol. 55, pp. 277301, 1987. [76] P. C. B. Phillips and S. Ouliaris, "Asymptotic Properties of Residual Based Tests for Cointegration," Econometrica, vol. 58, pp. 165193, 1990. [77] A. S. Pikovsky, "Phase synchronization of chaotic oscillations by a periodic external field," Sov J Commun Technol Electron, vol. 30, pp. 85, 1985. [78] A. S. Pikovsky, M. G. Rosenblum, and J. Kurths. Synchronization. A universal concept in nonlinear sciences. Cambridge, UK: Cambridge Univ. Press, 2001. [79] O. A. Prokopyev, V. Boginski, W. C'! ,valitwongse, P. M. Pardalos, J. C. Sackellares, and P. R. Carney, \, Iworkbased Techniques in EEG Data Analysis and Epileptic Brain Modeling," in P. M. Pardalos, V. Boginski and A. Vazacopoulos, eds., Data Mining in Biomedicine, Springer, pp. 559573, 2007. [80] K. Pyragas, "Continuous control of chaos by selfcontrolling feedback," Phys. Lett. A, vol. 170, pp. 421, 1992. [81] R. Quian Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, "Performance of different synchronization measures in real data: A case study on electroencephalographic signals," Phys. Rev. E, vol. 65, pp. 041903, 2002. [82] E. Rodriguez, N. George, J. P. Lachaux, J. Martinerie, and F. J. Varela, "Perceptions shadow: Longdistance synchronization in the human brain," Na ture, vol. 397, pp. 340343, 1999. [83] P. R. Roelfsema, A. K. Engel, P. Konig, and W. Singer, "Visuomotor integration is associated with zero timelag synchronization among cortical areas," Nature, vol. 385, pp. 157161, 1997. [84] E. Rosa Jr., W. B. Pardo, C. M. Ticos, J. A. Walkenstein, and M. Monti, "Phase synchronization of chaos in a plasma discharge tube," Int. J. Bifurc. Chaos, vol. 10, pp. 2551, 2000. [85] M. Rosenblum, A. Pikovsky, J. Kurths, "Phase synchronization of chaotic oscillators," Phys Rev Lett, vol. 76, pp. 18041807, 1996. [86] M. G. Rosenblum, A. S. Pikovsky, C. Schafer, P. Tass, and J. Kurths, "Phase synchronization: From theory to data analysis," in: S. Gielen, F. Moss F, eds. Handbook of B.: '1.:'..rl PhI,'. ; vol. 4. Elsevier Science, Neuroinformatics, 1999. [87] R. Roy and K. S. Thornburg, "Experimental synchronization on chaotic lasers," Phys. Rev. Lett., vol. 72, pp. 2009, 1994. [88] N. F. Rulkov, L. S. TiIvir,. and H. D. I. Abarbanel, "Tracking unstable orbits in chaos using dissipative feedback control," Phys. Rev. E, vol. 50, pp. 314, 1994. [89] J. C. Sackellares, L. D. Iasemidis, R. L. Gilmore, and S. Roper, "Epilepsy when chaos fails," in K. Lehnertz, J. Arnold, P. Grassberger, and C. E. Elger, eds., Chaos in the brain?, World Scientific, 2002. [90] J. C. Sackellares, L. D. Iasemidis, D.S. Shiau, P. M. Pardalos, and P. R. Carney, "Spatiotemporal transitions in temporal lobe epilepsy," in P. M. Pardalos, J. C. Sackellares, P. R. Carney, L. D. Iasemidis, eds., Quantitative Neuroscience: Models, Algorithms, Diagnostics, and Theraputic Applications, Kluwer Academic Publishers, pp. 223238, 2004. [91] S. E. Said and D. A. Dickey, "Testing for Unit Roots in AutoregressiveMoving Average Models of Unknown Order," Biometrika, vol. 71, pp. 599607, 1984. [92] K. Sameshima and L. Baccala, "Using partial directed coherence to describe neuronal ensemble interactions," J. Neurosci. Methods, vol. 94, pp. 93103, 1999. [93] C. Schafer, M. G. Rosenblum, J. Kurths, and H. H. Abel, "Heartbeat synchronized with ventilation," Nature, vol. 392, pp. 239240, 1998. [94] F. Lopez de Silva, "EEG analisys:Theory and practice; Computerassisted EEG diagnosis: Pattern recognition techniques," in E. Niedermeyer, F. H. Lopes da Silva, eds. Electr '.. '1'1A.. ',i''l,/;, Basic principles, clinical applications and related fields, pp. 871919, 1987. [95] C. A. Sims, "Money, Income and Causality," American Economic Review, vol. 62, pp. 540552, 1972. [96] W. Singer and C. M. Gray, "Visual feature integration and the temporal correlation hypothesis," Annu Rev Neurosci, vol. 18, pp. 555586, 1995. [97] C. TallonBaudry, O. Bertrand, C. Delpuech, and J. Pernier, "Oscillatory gammaband (3070 Hz) activity induced by a visual search task in human," J Neurosci, vol. 17, pp. 722734, 1997. [98] D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Experimental evidence of frequency entrainment between coupled chaotic oscillations," Phys. Rev. E, vol. 57, no. 3, pp. 3649, 1998. [99] P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, et al., "Detection of n:m phase locking from noisy data: application to magnetoencephalography," Phys Rev Lett, vol. 81, pp. 32913294, 1998. [100] M. Teplan, "Fundamentals of EEG measurement," Measurement science review, vol. 2, no. 2, pp. 11223, 2002. [101] G. Tononi and G.M. Edelman, "Consciousness and complexity," Science, vol. 282, pp. 18461851, 1998. [102] R. D. Traub, M. A. Whittington, E. Buhl, J. G. Jefferys, and H. J. Faulkner, "On the mechanism of the gamma to beta frequency shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation," J Neurosci, vol. 19, pp. 10881105, 1999. [103] B. van der Pol and J. van der Mark, "The heartbeat considered as a relaxation oscillation, and an electrical model of the heart," Phil. Mag., vol. 6, pp. 763, 1928. [104] F. J. Varela, "Resonant cell assemblies: A new approach to cognitive functions and neuronal synchrony," Biol Res, vol. 28, pp. 8195, 1995. [105] F. J. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, "The brain web: phase synchroniztion and largescale integration," Nature Rev Neurosci, vol. 2, pp. 22939, 2001. [106] N. Wiener, "The Theory of Prediction," in E. F. Beckenbach, ed. Modern Mathe matics for the Engineer, New York: McGrawHill, 1956. [107] A. Wolf, J. B Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponent from a time series," P/,,K.:. a D, vol. 16, pp. 285317, 1985. [108] K. Yang, and C. Shahabi, "On the Stationarity of Multivariate Time Series for CorrelationBased Data Analysis," in Proceedings of Fifth IEEE International Conference on Data Mining (ICDM'05), pp. 805808, 2005. 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. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IwouldliketoshowmyheartfeltappreciationtomyadvisorDr.PanosM.Pardalosforhissupportandmentoring.Workingwithhimhashelpedmegrownotonlyprofessionally,butalsoasaperson.Iamalsoverygratefultoothermemberswhoservedonmysupervisorycommittee,J.ColeSmith,WilliamW.Hager,VladimirL.Boginski,andH.EdwinRomeijn,fortheirvaluablecommentsonmyresearchforthisdissertation.Lastbutnotleast,IthankmyhusbandJason,myparentsOlgaandAleksandr,mybrotherMikhail,andalltherestofmygreatfamilyfortheirunconditionalloveandsupport. 4 PAGE 5 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 2.4.1AugmentedDickeyFullerTestforTestingtheNullHypothesisofthePresenceofaUnitRoot ...................... 40 2.4.2PhillipsOuliarisCointegrationTest .................. 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 PAGE 6 ....................................... 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 PAGE 7 Table page 21ADFtest:CriticalvaluesoftheT(bc1)andbstatistics ............. 49 22PhillipsOuliarisdemeaned:CriticalvaluesoftheZstatistic .......... 49 23Johansentest:CriticalvaluesoftheLR(r;K)statistic .............. 49 24Johansentest:CriticalvaluesoftheLR(r;r+1)statistic ............ 50 41Seizure1:ResultsoftheADFunitroottests ................... 70 42Seizure2:ResultsoftheADFunitroottests ................... 70 43Seizure3:ResultsoftheADFunitroottests ................... 70 44Seizure1:ResultsoftheJohansencointegrationranktests ............ 70 45Seizure2:ResultsoftheJohansencointegrationranktests ............ 70 46Seizure3:ResultsoftheJohansencointegrationranktests ............ 70 7 PAGE 8 Figure page 11TheInternational1020systemforplacementofEEGelectrodes ......... 27 41Rawdata:NumbersofunstablerootsfordierentTandp 71 42Filtereddata:NumbersofunstablerootsfordierentTandp 71 43EEGsegmentwithabsenceseizure ......................... 72 8 PAGE 9 Mostofthenotationisunambiguouslydenedinthetextwhereitisintroduced.Toprovidesomegeneralguidelines,weincludethefollowinglistofcommonlyusedsymbols. ARIMA(p;d;q)autoregressiveintegratedmovingaverageprocessAR(p)autoregressiveprocessoforderpCPVCauchyprincipalvalue 9 PAGE 11 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 PAGE 12 Dataminingtogetherwithdatapreprocessingconstitutethecentralpartofamoregeneralprocessofknowledgediscoveryindatabases(KDD).TheKKDprocesscanbedescribedasasequenceofactions,whichselectstherawdataindatawarehousesandtransformstheselecteddatainordertodiscovervalid,understandable,novelandpotentiallyusefulknowledgefromthedata.Datapreprocessingthatisappliedtorawdatatoimprovethequalityofthedataofteninuencestheselectionandfacilitatestheapplicationofdataminingtechniques.Properpreprocessingofrawdataleadstoadecreaseinthetimeneededtominethedata,andbooststheoverallminingeciency. Thetechniquesusedfordatapreprocessingcanberoughlysubdividedintodatacleaning,dataintegrationanddatareduction[ 64 ].Datacleaningtechniqueshandletheproblemofincomplete,inconsistentanderroneousdata,removethenoiseinherentlypresentintherawdata,minimizeredundancyinthedata,etc.Dataintegrationisconcernedwithcombiningheterogeneousdatacollectedfromdierentsourcestoforma 12 PAGE 13 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 ]demonstratedthataproblemofminimizingthenumberofmisclassiedpointsintwoclassclassicationcanbeviewedasalinearprogramwithequilibriumconstraints(LPEC).LPECisalinearprogram(LP)withasinglecomplementarityconstraint.Suchconstraintimposesaconditionoforthogonalitybetweentwolinearfunctions.LPECformulationarisesintheinstanceswhentheconstraintsoftheproblemincludeanotherLPproblem. Inaddition,theproblemoffeatureselectionintwoclassclassicationbyndingaseparatingplanethatutilizesminimumnumberoffeaturescanbegivenamathematicalprogrammingformulationasaparametricproblem.Furthermore,theclassicationviasupportvectormachines(SVMsthatndtheseparatingplanemaximizingthe 13 PAGE 14 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,thestudyofseismicactivityrelatedtoearthquakesproducestwodimensionaltimeseries,whereeachmeasurementconsistsofthetimeandthemagnitudeofaregisteredseismicevent.Manybiomedicalsignals,suchaselectrocardiogram(ECG),electroencephalogram(EEG)andelectrooculogram(EOG)representtimeseriesthatcanbeinterpretedviaapplicationofregression,segmentation,neuralnetworks,andotherdataminingmethodologies.Soundsignalsareanotherexampleoftimeseriesthatareeectivelyanalyzedusingdierentdataminingtechniques. 14 PAGE 15 14 ],althoughthenormaltimescaleisaverynaturalchoiceofparameterforthetimeseriesdescribingphysicalprocesses,theregulartimeloosesitsnaturalmeaningwhendealingwithmanynancialtimeseries.Becausemostofnancialtimeseriesareirregularlyspacedinphysicaltime,theconceptof\businesstime"or\intrinsictime"isintroducedtorepresentanewtimeparameterwithrespecttowhichtimeseriesareregularlyscaled.Thisprocedureoftimedeformationallowstherelabeledtimeseriestobeviewedasstationaryonanewtimescale.Financialtimeseriesalsooftenexhibitclearseasonaltrends,whichobviouslycannotbefoundwhenexaminingtimeseriesproducedbyspeech. StatisticaltestingofseveralmultivariatetimeseriesdeterminedthatthetimeseriesfromAUSLANandBCIdatasetscanbeconsideredstationary,whereasBCIMPIandEEGcontainednonstationarytimeseries[ 108 ].Asaresultofinherentdierencesintimeseriesdatafromdiversesources,somedataminingmethodsthataresuccessfullyappliedtotimeseriesinoneresearchareamaynotnecessarilybeapplicabletoanalysisoftimeseriesthatstemfromanotherappliedeld. Timeseriesobtainedfromelectroencephalogram(EEG)recordingshaveseveralinterestingpropertiesthatdistinguishthemfromothertimeseries.AlthoughsomestudiesapplyonedimensionalmodelingbyconsideringonechannelinEEGrecordingatatime,ingeneral,EEGdatashouldbetreatedasmultivariatetimeseries.ThemultivariateapproachbecomesespeciallyimportantinviewofitsabilitytoinvestigatespatiotemporaldependenciesintheEEGdataincontrasttobeinglimitedtoonlytemporalrelationsinaonedimensionalcase.ThereisadisagreementamongresearchersstudyingEEGdataonwhethertheseriesshouldbemodeledbyanonlinearstochasticprocessortheycanbebetterdescribedbyadeterministicchaoticdynamicalsystem. 15 PAGE 16 100 ]. Generallyspeaking,EEGrepresentsadigitaloragraphicrecordoftheelectricalactivityinthebrain,andcanbemeasuredbyeithernoninvasiveorinvasivemethods.EEG(obtainedduringanoninvasiveprocedure)isdenedasarecordofelectricalactivityofanalternatingtypemeasuredfromthescalpsurfaceafterbeingpickedupbymetalelectrodesandconductivemedia[ 67 ].TherearetwotypesofEEGproducedbyinvasiveprocedures,theelectrocorticogram,whichmeasuresthebrain'selectricalactivitydirectlyfromthecorticalsurface,andtheelectrogram,whichisanEEGobtainedusingdeepprobes. EEGestimatesandrecordstherelativechangeinelectricpotentialsproducedbyalargenumberofelectricdipolesduringaperiodofneuralexcitations.Theactivationofneurons(braincells)generateslocalcurrentowsinthebrain.EEGrecordsmostlytheelectricalcurrentsthatowduringsynapticexcitationsofthedendritesofnumerouspyramidalneuronsinthecerebralcortex.EEGrecordedfromthescalpsurfacecanonlydetecttheelectricalactivityproducedbymassivepopulationsofactiveneurons.Ontheotherhand,EEGrecordedusingdeepprobeelectrodesimplantedintothebraincanpickupasignalfromasmallgroupofneurons,whichcanbefurtherlteredouttoobtainthe 16 PAGE 17 68 ].Thisgivesanaveragedensityofabout104neuralcellspercubicmillimeterofthebrain.Neuralcellsareinterconnectedthroughsynapticconnectionsinthebrainintoneuralnets.Thebrainofanaverageadultcontainsapproximately500trillionsynapses.Thetotalnumberofneuronsdecreaseswithage.Asaresultthetotalnumberofsynapticconnectionsdeclineswithaging,eventhoughthenumberofsynapsesperoneneuronincreaseswithage. ToensuretheconsistencyinreferencinglocationsofelectrodesinEEGexperiments,theInternational1020systemforEEGelectrodeplacementwasdeveloped[ 37 ].The1020EEGsystemisusedtodescribetherespectivelocationsofscalpelectrodesduringEEGrecordinginrelationtotheunderlyingareaofcerebralcortex. Accordingtothe1020system,anatomicallandmarksofaskull,suchnasion,inionandpreauricularpoints,areidentiedforconsecutiveplacementoftheelectrodesatxeddistancesfromthesepointsinstepsofeither10or20percent.Thisapproachisdevisedtotakeintoaccountpossiblevariationsofheadsize.Inaddition,themethodiseasilyapplicableinpracticaluse.Asaresult,the1020EEGsystembecameverywidelyusedforpositioningelectrodes. Inthe1020system,thepointsaredenotedwithoneortwoletters,andcanbealsofollowedbyanumber(asshownonFigure 11 ).Thelettersroughlyrepresentthelobelocation(withexceptionoflettersCandZ),whereasthenumbersserveforidentifyingthecorrespondinghemisphere.Morespecically,thepointslocatedonthelefthemisphereofthebrainarerepresentedbyoddnumbers(1,3,5,and7),andthesitesontherighthemispherearemarkedwithevennumbers(2,4,6,and8).Thesiteslocatedonthefrontal,temporal,parietalandoccipitallobesaredenotedbythecorrespondinginitials 17 PAGE 18 EEGsignalresemblesacollectionofsinusoidsofvariousamplitudeandfrequency.PowerspectrumisextractedfromtherawEEGdatausingFouriertransformtoobtaintheinformationaboutthecontributionofsinusoidalwavesofdierentfrequency.ThepowerspectrumofEEGiscontinuous,rangingfrom0Hzuptoahalfofthesamplingfrequency.Dependingonthestateofthebrain,certainfrequenciesappeartobemoreprevalent.Therearefourmajorfrequencybands,alpha,beta,deltaandtheta,whichpresenceinEEGduringvariousstatesofconsciousnesshasbeenextensivelystudied.Thesebandsrepresentsinewavesofrelativelylowfrequency,withdeltarangingfrom0.5to4Hz,theta4{8Hz,alpha8{13Hz,andbetaover13Hz.AlphawavesdiscoveredbyAdrianandMatthewsin1934arethebestknownandthemoststudiedamongthefourfrequencybands[ 100 ].Theyareinducedbyclosingeyesandbyrelaxation,andterminatedwitheyesopeningorduetothinking,calculating,andotheranalyticalactivities.Inparticular,inmostpeople,eyeclosingproducesrapidchangesinbrainactivitymanifestingthemselvesinEEGasanadjustmentofthedominantfrequencybandfrombetatoalpha.EEGiscapableofdiscriminatingbetweendierentstates,suchasresting,alertness,stressstate,varioussleepstages,hypnosis,etc.Presenceofbetabandisdominantduringthestateofalertnesswitheyesopen.Drowsinessortherestingareusuallycharacterizedbytheriseinalphaactivity.Duringthesleep,presenceoflowerfrequencywavesbecomesmoreapparent.AhigherproportionofdeltabandfrequenciesisobservedduringstagesIIIandIVofthenonrapideyemovementsleep(NREM).EEGrecordedfromdistinctregionsinthebrainexhibitsdierentspectrumofwavefrequencies.Inaddition,thebrainpatternsareuniqueforeveryindividual. PracticalapplicationsofEEGincludeepilepsyresearchandlocalizationofthefocusofepilepticseizures,testingofepilepsydrugeects;determiningareasofdamagedue 18 PAGE 19 3 ]. 94 ].Inordertoproperlyreectthespatiotemporalpropertiesofbraindynamics,theanalysisofEEGdatamustinvolveasimultaneousinvestigationofthedependenciesacrosschannelswithrespecttotime. DierentfeatureshavebeenproposedforanalysisofEEGtimeseries,includingFouriertransform,wavelets,crosscorrelation,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 PAGE 20 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,thecrosscorrelation,coherenceandGrangercausalitymeasuretheinterdependencybetweendierentchannels.Thecrosscorrelationfunctionquantiesthelinearcorrelationbetweentwoprocesses.Giventwonormalizedsignalsx(t)andy(t)withzeromeansandunitvariances,thecrosscorrelationbetweenthesesignalsisestimatedas: 20 PAGE 21 Thecoherencefunctionisafrequencydomainanalogofthecrosscorrelationmeasure.CoherenceisobtainedfromcrosscorrelationbyapplyingFouriertransformto( 1{3 ).Theestimateofthecoherencespectrumoftwosignalsiscalledperiodogram.Theperiodogramiscalculatedbysubdividingthesignalsintoanumberofepochsofthesamelength,andthenapplyingthefollowingformula: where Whilethecrosscorrelationandcoherencearefeatures,whichreectthelineardependencybetweentwochannelsinthedata,theconceptofGrangercausalityiscapableofnotonlyestablishingthelineardependency,butalsospecifyingthedirectionofsuchdependency.Inotherwords,byapplyingGrangercausality,itbecomespossibletoidentifycausalrelationshipamongthechannelsofEEG.Grangercausalityisbasedonthemultivariateautoregressivemodelingoftimeseries.Ithasalsoreceivedanalternativereformulationinthefrequencydomainviaspectraldecompositionforstochasticprocesses. 21 PAGE 22 35 89 ].Inchaoticsystems,trajectoriesoriginatingfromverycloseinitialconditionsdivergeexponentially.Thesystemdynamicsarecharacterizedbytherateofthedivergenceofthetrajectories,whichismeasuredbyLyapunovexponentsanddynamicalphase. ShorttermlargestLyapunovexponent(denotedSTLmax),whichisanestimateofthemaximumLyapunovexponentfornonstationarydata,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 PAGE 23 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 PAGE 24 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 PAGE 25 43 ]. ComprehensivestatisticaltestingisnecessaryinordertomakeconclusionsonwhatmultivariatemodelsarethemostappropriateforextractingthedirectionaldependenciesbetweenchannelsinafrequencydomainfrommultichannelEEGdata. 15 101 104 105 ].Actually,itwasfoundthatoscillationofvariousneuronalgroupsingivenfrequencybandsleadstotemporaryphaselockingbetweensuchgroupsofneurons.ThisobservationhasstimulatedthedevelopmentofrobustapproachesthatallowonetomeasurethephasesynchronyinagivenfrequencybandfromexperimentallyrecordedbiomedicalsignalssuchasEEG. Inparticular,theimportanceofsynchronizationofneuronaldischargeshasbeenshownbyavarietyofanimalstudiesusingmicroelectroderecordingsofbrainactivity[ 83 96 ],andevenatcoarserlevelsofresolutionbyotherstudiesinanimalsandhumans[ 21 ]. ThephasesynchronizationinthebrainextractedfromEEGdatausingHilbertorwavelettransformshasrecentlybeenshowntobeanespeciallypromisingtoolinanalysisofEEGdatarecordedfrompatientswithvarioustypesofepilepsy[ 86 ]. Inourrecentstudy[ 44 ],weintroduceanovelconceptofgeneralizedphasesynchronization,whichisbasedonvectorautoregressivemodeling.Thisnewnotionofphasesynchronizationisconstructedasanextensionoftheclassicaldenitionofphasesynchronizationbetweentwosystems.Infact,thephasesynchronizationisusuallydenedastheconditionthatsomeintegercombinationoftheinstantaneousphasesoftwosignalsisconstant.Oftenthisconditionisrelaxedbyallowingforaboundedlinearcombinationoftwophases,inorder 25 PAGE 26 Toconstructamoregeneralmultivariateconceptofphasesynchronization,weextendedtheclassicaldenitionbyconsideringalinearcombinationofphasesforanitenumberofsignalsthatrepresentsastationaryprocess.Alltheindividualsignalstogetherformacommonsystemdescribedbysomemultivariateprocess.Wenotethatavectorprocess,suchthatalinearcombinationofitsindividualcomponentsisastationaryprocess,canbemodeledasacointegratedvectorautoregressivetimeseries. Furthermore,itiseasytosee(asshowninSection 4.2.1 )thatthecointegratedrankoftheregressiondetermineshowrestrictedthebehaviorofsuchsystemis.Thismeansthattherankrofcointegratedautoregressivemodel,estimatedfromthemultipletimeseriesoftheinstantaneousphases,measureshowlargethevectorsubspace,whichgeneratesthechangesinthephasevalues,is. ThisnewmeasureofcointegrationwasappliedtoabsenceepilepsyEEGdatain[ 44 ].Thedatasetscollectedfromthepatientswithothertypesofepilepsyarecurrentlybeinginvestigated. 26 PAGE 27 Bproleview TheInternational1020systemforplacementofEEGelectrodes 27 PAGE 28 26 27 ]formeasuringlineardependenceandfeedbackineconomictimeseries.Later,thisideawasfurtherextendedbyJohnGeweke[ 25 ],whoproposedanequivalentmeasurebasedonthespectralrepresentationoftimeseries.BothGranger'sandGeweke'sapproachesemploythevectorautoregressivemodelingtoderiveestimatesofunderlyingcausalrelationsinthedata.However,thelatterapproachisfoundparticularlyusefulforanalysisofEEGtimeseries,sinceitinvestigatesthecausalrelationinthefrequencydomaininsteadofthetimedomainasintheformerapproach.Inparticular,thespectralmeasureofGrangercausalityproposedbyJohnGewekewasemployedonintracorticallocaleldpotentialsrecordedfrom8electrodesduringgo/nogotrialsofcat'svisualresponses[ 2 ].Anotherstudy[ 57 ]utilizedasimilarmethodofdirectedtransferfunction(whichisequivalenttothespectralmeasureofGrangercausality)toexaminecausalinuencesintheprimatevisualcortexduringthetaskofvisualpatternrecognition.ThedirecttransferfunctionapproachtoGrangercausalitywasalsoappliedtoanalyzingbrainconnectivitypatternsonhumanEEGdatarecordedduringstage2sleep[ 42 ]. MichaelEichlerproposedagraphicalapproachformodelingGrangercausalrelationshipsinmultivariatetimeseries[ 17 ]andlaterappliedthismethodtostudyingconnectivityinneuralsystems[ 18 19 ].LuizBaccalaandKoichiSameshimaintroducedaconceptofthepartialdirectedcoherenceforinferenceofGrangercausalityinthefrequencydomainbasedonthelinearvectorautoregressivemodeling,andappliedittoinvestigatingthefunctionalinteractionsamongdierentbrainstructures[ 1 92 ]. 28 PAGE 29 30 ]. LetX(t)andY(t),t2Zdenotetwotimeseries(ordiscretetimestochasticprocesses)withthecorrespondingrealizationsxtandyt,t2Z.SupposethatX;tandY;tdenotealltheinformationabouttherealizationsofprocessesXandY,respectively,uptotimet.Then,therelationshipofGrangercausalitybetweensuchseriescanbeformallydenedasfollows: AtimeseriesX(t)issaidtoGrangercauseY(t)ifthereexistsp=1;2;:::suchthatthemeansquarederror(MSE)ofthepstepforecastofY(t)basedontheinformationX;tandY;tissmallerthantheMSEofthepstepforecastofY(t)basedonY;talone,i.e. whereY(pj)istheMSEofthepstepforecastofY(t)basedoninformation. Usingtheabovedenition,wenowpresentthetestforGrangercausalitybasedonthebivariateautoregressivemodel.Supposethatforsomeintegerlagparameterp>0,therealizationsoftimeseriesY(t)aregivenbythemodel where"tisastandardwhitenoise(orinnovationprocess,i.e."thaszeromeanandzeroautocorrelation). 29 PAGE 30 againstthealternative Notethatifthenullhypothesisisaccepted,thenatimeseriesY(t)isbelievedtobeNOTGrangercausedbyX(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 ByconductingtheFtestofthenullhypothesis,onecanndtheteststatistic RSS1=(T2p1)Fp;T2p1:(2{7) IftheteststatisticS1exceedsthespeciedcriticalvalue,thenthenullhypothesisthatX(t)doesnotGrangercausesY(t)isrejected.Otherwise,H0isaccepted. AnasymptoticallyequivalenttestofGrangercausalityisgivenbythefollowingstatistic 30 PAGE 31 ThebivariateapproachtotestingGrangercausalitycanbenaturallyextendedtothemultivariatecasebypartitioningthevectorautoregressiveprocessZ(t)intotwocomponentsX(t)andY(t),sothatZ(t)=(X(t);Y(t)),andthentestingthesuitablezeroconstraintsonthecoecientsofvectorautoregression.ForthederivationoftheWaldstatisticandtheFstatisticfortestingGrangercausalityinthemultivariatecase,seethebookonmultipletimeseriesbyLutkepohl[ 59 ]. 2.3.1MethodsforVARParameterEstimation Letpdenoteapositiveinteger,andletytdenotetheKvariatetimeseries(i.e.realizationsofKdimensionalprocessY(t)).Avectorautoregressivemodeloforderp,denotedVAR(p),isformallydenedasfollows: whereyt=(y1t;:::;yKt)0isa(K1)randomvector,=(1;:::;K)0isaxed(K1)vectorrepresentinganonzeromeanEY(t),theAi,i=1;:::;parexed(KK)dimensionalcoecientmatrices,and"t=("1t;:::;"Kt)0isaKdimensionalwhitenoiseprocess(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 PAGE 32 59 ].UndertheassumptionsofstabilityandGaussiandistribution,theseapproachesleadtoestimatorswiththesameasymptoticproperties.However,theasymptoticresultsshouldbeusedcautiouslyininferencefromsmallsamples.Asaresult,dierentapproachesmaysometimesleadtodierentresultswhenestimatingthemodelparametersusingsmallsamples. Letusnowbrieypresentthemultivariateleastsquaresestimation,whichisahigherdimensionalextensionofthewellknownmethodofordinaryleastsquares.Formoredetaileddiscussion,referto[ 59 ]. Supposethattheavailabledatainclude(T+p)successiverealizationsofestimatedmultipletimeseriesrepresentedbyKdimensionalvectorsyp+1;:::;y0;y1;:::;yT PAGE 33 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 PAGE 34 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 PAGE 35 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),theHannanQuinncriterionisconsistent(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 PAGE 36 det(IKA1z:::Apzp)6=0forcomplexz:jzj1:(2{20) Inotherwords,theVAR(p)process( 2{9 )satisesthestabilityconditionwhenitsreversecharacteristicpolynomial(givenbydet(IKA1z:::Apzp))hasnorootsonandinsidethecomplexunitcircle. Thestabilityconditionguaranteesthatthereexistsamovingaverage(MA)representationfortheVAR(p)process.Alsostabilityensuresthattheprocessisawelldenedstochasticprocesswiththedistributionsofitsunivariatecomponentsandjointdistributionoftheprocessytuniquelydeterminedbytheinnovationprocess"t.ForastableVAR(p)process,boththeprocessmeanandtheautocovariancearetimeinvariant(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,theexplosivetimeseriesarenotaswellstudied,becauseitisbelievedthatanexponentialincreaseinthevarianceoftheeconomictimeseriesisnotwellfounded.AsonecanseethestabilityassumptionplaysanimportantroleinVAR(p). AwidesensestationarityforstochasticprocessesisimposedontheVARtimeseriesasfollows.AstochasticprocessY(t)isconsideredstationaryif 36 PAGE 37 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.anunstabletimeseriesisnotnecessarynonstationary. 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 PAGE 38 Thisrendersthefollowingdenitionofanintegratedprocess. Aonedimensionalprocesswithdrootsontheunitcircleissaidtobeintegratedoforderd(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).TheonedimensionalprocessY(t)issaidtobeARIMA(p,d,q),ifZ(t):=(1L)dY(t)isastationaryautoregressivemovingaverageARMA(p,q)process,i.e. where"tj'sareindependentnormallydistributedrandomvariableswithmean0andvariance2,andListhedierencingoperatorintroducedabove. Itisnoteworthytopointoutthattakingdierencesmaydistorttherelationshipamongthevariables(i.e.onedimensionalcomponents)insomeVAR(p)models.Inparticular,thisisthecaseforsystemswithcointegratedvariables.ItturnsoutthatttingVAR(p)modelafterdierencingtheoriginalcointegratedprocessproducesinadequateresults.Next,wediscusssuchprocesses. 38 PAGE 39 SupposethatsampledvaluesyitofKdierentvariablesofinterestYi(t)arecombinedintotheKdimensionalvectorsyt=(y1t;:::;yKt)0.Inaddition,supposethatthevariablesareinalongrunequilibriumrelation wherec=(c1;:::;cK)0isaKdimensionalrealvector.Duringanygiventimeinterval,therelation( 2{22 )maynotnecessarilybesatisedpreciselybythesampleyt,insteadwemayhave: where"tisastochasticprocessthatdenotesthedeviationfromtheequilibriumrelationattimet.IfourassumptionaboutthelongrunequilibriumamongindividualvariablesYi(t),i=1;:::;KisvalidthenitisreasonabletoexpectthatthevariablesYi(t)movetogether,i.e.thestochasticprocess"tisstable.Ontheotherhand,thisdoesnotcontradictthepossibilitythatthevariablesdeviatesubstantiallyasagroup.Therefore,itispossiblethatalthougheachindividualcomponentYi(t)isintegrated,thereisalinearcombinationofYi(t),i=1;:::;K,whichisstationary.Integratedprocesseswithsuchpropertyarecalledcointegrated. Withoutlossofgenerality,weassumethatallindividualonedimensionalcomponentsYi(t)(i=1;:::;K)areeitherI(1)orI(0)processes.ThenthecombinedKdimensionalVAR(p)process 39 PAGE 40 =IKA1:::Ap(2{25) hasrankr. SincesomeonedimensionalcomponentsofthecointegratedVAR(p)processareintegratedprocesses,onemaybeinterestedintestingthepresenceofaunitrootintheunivariateseries.Inthefollowingsection,wepresentacommonlyusedunitroottest,whichwasderivedbyDickeyandFuller[ 16 ]. 16 ],anditcanbeshownthatthisdistributionisthesamefork>1andfork=1.FullertabulatedtheapproximatecriticalvaluesfortheADFtestwithk1andpk1forspecicsamplesizes. FinitesamplecriticalvaluesfortheADFtestforanysamplesizewereobtainedbymeansofresponsesurfaceanalysisbyMacKinnon[ 60 ],whoalsoshowedthatanapproximateasymptoticdistributionfunctionforthetestcanbederivedviaresponsesurfaceestimationofquantiles[ 61 ]. AlthoughtheasymptoticdistributionoftheADFteststatisticdoesnotdependonthelagorder,itisnotedbyCheungetal.[ 13 ]thatempiricalapplicationsmustdealwithnitesamples,inwhichcasethedistributionoftheADFteststatisticcanbesensitiveto 40 PAGE 41 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 PAGE 42 =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 PAGE 43 andbciscomputedfrom( 2{27 ) TablesofthecriticalvaluesfortheasymptoticdistributionsoftheADFteststatisticT(bc1)andthestatisticbcanbefoundinFuller[ 22 ].WesummarizesomeoftheinformationinTable 21 ,whichliststhepvaluesforasymptoticdistributionsofT(bc1)andbcorrespondingtopercentilesof90,95,and99percent. 75 ].Inparticular,inhisstudyPhillipsrstconsideredtwostatisticsZandZtfortestingthenullofnocointegrationintimeseries. Becausemanyunitroottests,constructedbefore1987,werefoundedontheassumptionthattheerrorsintheregressionareindependentwithcommonvariance(whichisrarelymetinpractice),Phillipswantedtorelaxtheratherstrictconditionthatthetimeseriesaredrivenbyindependentidenticallydistributedinnovations.Inotherwords,hewantedtodevelopthetestingproceduresbasedontheleastsquaresregressionestimationandtheassociatedregressiontstatistic,whichwouldallowforrathergeneralweaklydependentandheterogeneouslydistributedsequenceoferrorterms. ThepropertiesofasymptoticdistributionsofresidualbasedtestsforthepresenceofcointegrationinmultipletimeserieswerethoroughlyinvestigatedbyPhillipsandOuliaris[ 76 ].Thecharacteristicfeatureofthesetestsisthattheyutilizetheresidualscomputedfromregressionsamongtheunivariatecomponentsofmultivariateseries.TheresidualbasedproceduresdevelopedbyPhillipsandOuliarisaredesignedtotestthenullofnocointegrationbymeansoftestingthenullhypothesisoftheunitrootpresenceintheresidualsagainstthealternativeofarootthatliesinsidethecomplexunitcircle.ThehypothesisH0oftheabsenceofcointegrationisrejected,ifthenullofaunitrootinthe 43 PAGE 44 Asnotedin[ 76 ],theresidualbasedunitroottestsareasymptoticallysimilar,andcanberepresentedviathestandardBrownianmotion.Moreover,theADFandZttestsareprovedtobeasymptoticallyequivalent.However,thesetwotestsarenotaspowerfulasthetestbasedonstatisticZ,becauseitwasshownbyPhillipsandOuliaris[ 76 ]thattherateofdivergenceundercointegrationassumptionisslowerfortheADFandZtthanothertests,suchastheZstatistictest.Thelatertest(i.e.thecointegrationtestbasedonZ)isalsowidelyknownasthePhillipsOuliariscointegrationtest. ItisnoteworthythatthenullhypothesisforthePhillipsOuliaristestisthatofnocointegration(insteadofcointegration).Thisformulationischosenbecauseofsomemajorpitfallsfoundinproceduresthataredesignedtotestthenullofcointegrationinmultipletimeseries.Thesedefects(discussedinmoredetailin[ 76 ])aresignicantenoughtobeastrongargumentagainsttheindiscriminateuseofthetestformulationsbasedonthenullofcointegration,andtosupportthecontinuinguseofresidualbasedunitroottests. ConsidertheKdimensionalvectorautoregressiveprocessY(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 PAGE 45 ThenthestatisticZinPhillipsOuliaristestisdenedasfollows: 2s2Tls2% whereastheZtstatisticisgivenbythefollowingformula: 2(b1) 2s2Tls2% 2;(2{39) where l+1: Notethats2%andsTlareconsistentestimatorsforthevariance2%of%tandthepartialsumvariance2=limT!1E1 2{37 ). ThecriticalvaluesforZandZtstatisticscanbefoundin[ 76 ](TablesIandII).PhillipsandOuliaristabulatedthevaluesforcointegratingregressionswithatmost5explanatoryvariables.SomeestimatesofthecriticalvaluesforthePhillipsOuliaristest(Z)arelistedinTable 22 45 PAGE 46 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]: PAGE 47 lnl=KT 2trace(YDX+HCYp)01"(YDX+HCYp): TheproofofthefollowingtheoremonthemaximumlikelihoodestimatorsofacointegratedVARprocesscanbefoundin[ 59 ](Proposition11.1). 59 ]) Dene Denote1:::KtobetheeigenvaluesofGS10S100S01G0, and Thentheloglikelihoodfunctionin( 2{47 )ismaximizedfor PAGE 48 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,theLRstatisticisnotasymptoticallydistributedaccordingto2distribution.Nevertheless,theasymptoticdistributionofthecointegrationrankteststatisticLRdependsonlyontwofactors: 48 PAGE 49 Asaresult,theselectedpercentagepointsoftheasymptoticdistributionoftheteststatisticLRweretabulatedbyJohansenandJuseliusin[ 38 ].ThepercentagepointsofasymptoticdistributionofLRaregiveninTables 23 and 24 Table21. CriticalvaluesoftheasymptoticdistributionsoftheT(bc1)andbforperformingunitrootcheckbytheADFtest(reproducedfrom[ 22 ]) Statistic90%95%99% Table22. CriticalvaluesoftheasymptoticdistributionsoftheZstatisticfortestingthenullofnocointegration(PhillipsOuliarisdemeaned,reproducedfrom[ 76 ]).Parametern(n=K1)representsthenumberofexplanatoryvariables 117.039020.493528.3218222.194826.094334.1686327.584632.061541.1348432.738237.150847.5118537.007441.938852.1723 Table23. PercentagepointsoftheasymptoticdistributionsoftheLR(r;K)fortestingthecointegrationrank(reproducedfrom[ 38 ]) 16.698.0811.58215.5817.8421.96328.4431.2637.29 49 PAGE 50 PercentagepointsoftheasymptoticdistributionsoftheLR(r;r+1)fortestingthecointegrationrank(reproducedfrom[ 38 ]) 16.698.0811.58212.7814.6018.78318.9621.2826.15 50 PAGE 51 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 PAGE 52 15 101 104 105 ].Neuralsynchronyplaysavitalroleinsuchlargescaleintegration.Infact,variousneuronalgroupsoscillateinspecicfrequencybandsandbecomephaselockedoveralimitedperiodoftime.Thisobservationhasstimulatedthedevelopmentofrobustapproachesthatallowonetomeasurethephasesynchronyinagivenfrequencybandfromexperimentallyrecordedbiomedicalsignals,suchasEEG. Inparticular,theimportanceofsynchronizationofneuronaldischargeshasbeenshownbyavarietyofanimalstudiesusingmicroelectroderecordingsofbrainactivity[ 83 96 ].Thendingsinthemicroelectroderecordingstudiesarealsosupportedatcoarserlevelsofresolutionbyotherstudiesinanimalsandhumans[ 21 ]. Anelectrophysiologicalsignalisrecordedviaalowimpedanceextracellularmicroelectrodebyplacingthemicroelectrodesucientlyfarfromindividuallocalneuronsinordertopreventanyparticularcellfromdominatingthesignal.Next,toobtainthelocaleldpotential(LFP),thesignalislowpassltered,withacutoatapproximately300Hz.Duetothelowimpedanceandpositioningofthemicroelectrode,therecordedsignalispredominantlyinducedbytheactivityofalargenumberofneurons.Theunlteredsignalreectsthesumofactionpotentialsfromcellswithinapproximately50{350micrometersfromthetipoftheelectrode[ 53 ]andslowerioniceventsfromwithin0.5{3millimetersfromthetipoftheelectrode[ 39 ].Thespikecomponentofthesignalisremovedbylowpasslter,whereasthelowerfrequencysignal,theLFP,ispreservedinthesignal.Itisassumedthatthelocaleldpotentialcharacterizesthesynchronizedinputintotheobservedarea,inthecontrasttothespikedata,whichrepresentstheoutputfromthearea. Localeldpotentials(LFPs)ofvariousdegreesofspatialresolutioncanberecordedbyscalpEEGorMEG.Infact,studieshaveshownthatthepresenceofgammaandbeta 52 PAGE 53 97 ]andinsubduralelectrocorticograms[ 50 54 ].Inaddition,somerecentndingssuggestthatlongrangesynchronizationanalogoustotheonefoundinmicroelectrodestudiesinanimalscanalsobedetectedbetweensurfacerecordings[ 82 ]. Ithasbeenshownthatsynchronizationisasignicantattributeofthesignalrecordedfromthepatientsaectedbyseveralneurologicaldisorders.Inparticular,researchershavefoundthatepilepsy[ 65 ]andParkinsonsdisease[ 99 ]manifestasapathologicalformofthesynchronizationprocess. Itisnotedin[ 56 ]thatalthoughthecrosscorrelogramsbetweenspikedischargesmaybeadequateformicroelectrodestudies,thequanticationofphasesynchronybetweenmesoormacroelectrodes(i.e.EEG/MEG,intracranialrecordings)callsforentirelydierentmethods.Therefore,theyemphasizeanimportanceofclearlydistinguishingbetweensynchronyasanappropriateestimateofphaserelation,andtheclassicalmeasuresofcoherenceorspectralcovariancethathavebeenextensivelyusedinneuroscience[ 8 10 62 ].LeVanQuyenetal.discusstwoimportantlimitationsofcoherence[ 56 ]. Therstlimitationarisesbecausethestandardapproachesformeasuringcoherence[ 12 ]basedonFourieranalysisareknowntobehighlydependentonthestationarityofthemeasuredsignal,whereasthesignalsrecordedfromthebrain,suchasEEG,appeartobeclearlynonstationary.Applyingthetimefrequencyestimationmethod,whichisnotfoundedontheassumptionofstationarity,couldimprovethislimitationtowardsestimatingastable,instantaneouscoherenceaswellassynchronybetweentwoconcurrentbrainsignals. Thesecondlimitationstemsfromthefactthatclassicalcoherenceisameasureofspectralcovariance.Hence,itisnotabletoseparatetheeectsofamplitudeandphaseintherelationsbetweentwosignals.Becauseweareconcernedwithexaminingthespecichypothesisthatphaselockingsynchronyisthepertinentbiologicalmechanismoftransientintegrationinthebrain,coherenceservesonlyasanindirectmeasure.Inorder 53 PAGE 54 86 ].Inotherwords,eventhoughourdiscussionofthephasesynchronyisfocusedonEEGdata,itsapplicationscanalsobeextendedtotheeldsotherthanneuroscience. Classicalconceptofthesynchronizationoftwooscillatorsisdescribedasanactiveadjustmentoftheirrhythmicitythatmanifestsinphaselockingbetweenthesynchronizedoscillators.Specically,giventwosignalsX1(t)andX2(t),andtheircorrespondinginstantaneousphases1(t)and2(t),thebasicdenitionofthephaselockingstatesthat whereintegersnandmspecifythephaselockingratio. Wheninvestigatingphasesynchronyinneurophysiologicalsignals,onemustassumethattheconstantphaselockingratioisvalidwithinalimitedtimeintervalT,whichusuallymeansafewhundredsofmilliseconds.Whenexaminingneuralsignals,onemustkeepinmindthatdiscoveringthepresenceofthephaselockingbetweenEEGrecordingsfromtwodistantpartsofthebrainisnotstraightforward.Thedetectionofphasesynchronyinneuralsignalsisproblematicbecauseofseveralfactorsparticularlywhenworkingnotonthelevelofasingleneuron,butratherwithlargeneuronalpopulations,whoseactivityispickedupbymacroscopicormesoelectrodes.Asnotedin[ 56 ],asaconsequenceofvolumeconductioneectsinbraintissues,theactivityofasingleneuronalpopulationcanberecordedbytwodistantelectrodes,whichresultsinspuriousphaselockingbetweentheirsignals.Furthermore,innoninvasiveEEG,thetruesynchroniesarehiddeninasignicantbackgroundnoise,andso,inthesynchronous 54 PAGE 55 3{1 )mustbeadjustedtoaccountforthenoiseasfollows: whereCdenotesapositiveconstant. Thefollowingtwokeystepsareinstrumentalininvestigatingthephasesynchrony: 1. estimateinstantaneousphaseofeachsignal; 2. provideastatisticalcriteriatoquantifythedegreeofphaselocking. Twomethodsfordetectingphaselockingappliedtoneuronalsignalshaverecentlybeenconsideredintheliterature.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 PAGE 56 ^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. GivenanarbitrarycontinuousrealvaluedtimeseriesX(t),thecorrespondinganalyticsignalisdenedasthefollowingcomplexvaluedfunction: wheretdenotestime,{isaunitonthecomplexaxis,^X(t)denotestheHilberttransformofthetimeseriesX(t),aX(t)isthecorrespondinginstantaneousamplitude,andX(t)representstheinstantaneousphaseofthesignalviaHilbertconvolution. ItfollowsfromEquation 3{4 thattheinstantaneousphaseX(t)ofX(t)canbecomputedas: Akeyadvantageoftheanalyticapproachisthatthephasecanbeeasilycomputedforanarbitrarybroadbandsignal.Ontheotherhand,instantaneousamplitudeandphasehaveaclearphysicalmeaningonlyifX(t)isanarrowbandsignal.Therefore,ltrationisrequiredinordertoseparatethefrequencybandofinterestfromthebackgroundbrainactivity. 56 PAGE 57 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 PAGE 58 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 PAGE 59 49 56 ].However,mostofthecurrentlyusedmeasuresofphasesynchronizationarebasedonbivariateindexes. InChapter 4 ,weproposeanovelmultivariateapproachtodetectingphasesynchronizationinthephasesextractedfrommultipletimeseries,suchasmultichannelEEG. 59 PAGE 60 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 PAGE 61 41 and 42 ThesurfaceplotinFigure 41 supportsthefact[ 59 ]thatthesamplesizeparametershouldsignicantlyexceedthelaglengthp,i.e.Tp.Ontheotherhand,itcanbeseenfromFigure 41 thatevenforTp,thestabilityassumptionsmaystillbeviolated.Indeed,forT=132considerablylargerthanp=2,theestimatedVARhastwoRCProotsonorinsidetheunitcircle(n=2). Figure 42 showsthatfortheltereddata,the(T;p)region,wherethestabilityconditionoftheestimatedVAR(p)modelisviolated,coversalmostthewholedomain.Whereasthe(T;p)regionthatcorrespondstostableVAR(p)modelsismuchsmallerthanthestableregioninFigure 41 ,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 PAGE 62 Variousmodicationsofvectorautoregression(whichrelaxthestabilityconditionofnorootsinsideunitcircle)aredevelopedforanalysisofeconomictimeseries.AlthoughBernasconiandKonig[ 2 ]examinedthestationarityofdierentEEGdata,andconcludedthatatimeintervalofapproximatelyonesecondistheintervalduringwhichtheEEGtimeseriescanbeconsideredstationary,theunderlyingassumptionsoftheVARmodelingsuchasstationarityandstabilityarerarelystatisticallytestedinapplicationstoEEGdata.TheresultsofourexperimentsindicatethatthestabilityassumptionsofvectorautoregressivemodelareoftenviolatedinapplicationtotheEEGdata.Thisobservationsuggeststhatsuitableextensionsofmultivariateautoregressiontounstableprocessesmaytthedatabetter,andasaresult,suchextensionsofVARmaybemoreappropriatefortheEEGtimeseriesanalysisthanthestandardlinearvectorautoregressivemodeling. AdditionalstatisticaltestingisrequiredinordertomakeconclusionsonwhatmultivariatemodelsarethemostsuitableforextractingthedirectionaldependenciesbetweenchannelsinafrequencydomainfrommultichannelEEGdata. GiventhesignalrepresentedformallyasamultipletimeseriesX(t),onecanextracttheinstantaneousphasesXi(t)fromeachonedimensionalcomponentXi(t)ofthesignalasshowninChapter3(eitherviaaconvolutionwiththeMorletwaveletorbyapplyingtheHilberttransform).ThephaseextractionprocedureproducesanewmultipletimeseriesX(t)ofthecorrespondentphases. Next,wederivenewmeasuresofphasesynchronyofthesignalbasedontheconceptsintroducedinChapter 3 .LetusobservethatthelefthandsideofEquation( 3{1 ) 62 PAGE 63 3{1 ),whichdenesphaselockingbetweentwosignalsX1(t)andX2(t),needstobemodiedinpracticetoaccountforthenoiseinthesignal.Takingintoaccountpresenceofthestochasticnoiseinthephaseseries,letusintroduceamodiedconceptofthephasesynchronybetweentwosignalsbyrelaxingtheintegralityconditiononthecoecientsinthelinearcombinationasfollows. TwosignalsX1(t)andX2(t)areconsideredtobegenerallyphasesynchronized,ifthecorrespondentinstantaneousphasesX1(t)andX2(t)satisfytheconditionbelow: whereztN(C;0)isastochasticvariablethatrepresentsthedeviationfromtheconstantlevelCasaresultofthenoise.Noticethatinthecontrasttocondition( 3{1 )intheclassicdenitionofphasesynchronization,thecoecientsc1andc2inthedenition( 4{1 )ofgeneralizedphasesynchronydonotneedtobeinteger. Furthermore,itisstraightforwardthatthenewcondition( 4{1 )meansthatatwodimensionalprocessX(t)=(X1(t);X2(t))0iscointegrated.Basedonthisobservation,wecanextendourmodiedconceptofphasesynchronizationbetweentwosignalstothemultivariatecaseinthefollowingmanner. ThemultichannelsignalX(t)=(X1(t);:::;XK(t))isconsideredtobephasesynchronizedofrankr,iftheprocessX(t)composedofthecorrespondentinstantaneousphasesXi(t),i=1;:::;Kiscointegratedofrankr. Inthesubsequentsubsections,werstdiscusstheroleofthecointegrationrankintheframeworkofmultivariatephasesynchronization,andthenapplythisapproachtomultichannelEEGdatacollectedfromthepatientswithabsenceepilepsy. 63 PAGE 64 66 ],MichaelMurrayusedanexampleofdrunkardandherdogtoillustratetheconceptofthecointegration.Toexplainourreasoningbehindtherankofcointegrationasameasureofsynchrony,webrieysummarizeandthenfurtherextendhisanalogy. Thenonstationaryprocesses(suchasarandomwalk)areoftenintroducedbyteachersofstatisticsbycomparingthem(it)withthedrunkard'swalk.Thedrunkardwondersaimlessly,sothatthedirectionofeachstepisrandomandcompletelyindependentofherprevioussteps.Inotherwords,themeanderingofthedrunkardisdescribedbyarandomwalk: wherextrepresentsthepositionofthedrunkattimet,and"tisastationarywhitenoise,whichmodelsthedrunk'sstepattimet). AsMurraynoticed[ 66 ],anunleashedpuppyisanothercreature,whosebehaviorremindsarandomwalk.Indeed,eachnewscentthatpuppy'snosecomesupondictatesadirectionforthepup'snextstepsostronglythatthelastscentalongwithitsdirectionisforgottenassoonasthenewscentappears.Havingshownthatthepuppiesfollowtherandomwalkyt,t=1;2;:::,letusrepresentthepuppy'swalkas: wheretisastationarywhitenoise(i.e.puppy'sstepattimet). Thewellknownfeatureofarandomwalkisthatthebestpredictorofthefuturevalueisthemostrecentlyobservedone.Inotherwords,thelongerithasbeensincewehadseenthedrunk,orthedog,thefurtherawayfromtheinitialplace,onaverage,they 64 PAGE 65 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(withnonzeroweights)thatisI(0),i.e.stationary. Mathematically,thecointegratingrelationshipbetweenaladyandherpuppycanbewrittenas: attimet=1;2;:::.Notethat"tandt,asbefore,representthestationarywhitenoisestepsofthedrunkandherdog. SinceEquation 4{4 canbeeasilyrewritteninformof( 2{45 )asfollows: 264xtyt375=264"tt375264ccdd375264xt1yt1375; 65 PAGE 66 =264ccdd375; Notethatrank()=0,ifandonlyifc=d=0.Insuchcase,( 4{4 )becomessimplyasystemofequations( 4{2 )and( 4{3 ),whichmodelstwoindependentrandomwalksdrivenbyindependentwhitenoiseprocesses"and.Ontheotherhand,whenatleastoneofthecoecientscanddisnonzero,thenbymultiplyingsystem( 4{6 )byavector[d;c]0,wehave: whichmeansthatthemodelisdrivenbyasinglecommonstochastictrendd"t+ct. AlthoughtheexampledescribedbyMurrayisclearlyabivariatecointegratedVAR(1),itcanbeextendedtoanillustrationofthemultivariatecointegratedprocess.Consider,forexample,aheardofsheepguardedbytwodogs,wherethesheepwonderaimlesslyintheeld,whilethedogsrunaroundandbringthesheepthathavestrayedtoofarbackintotheock.Say,forexample,afasterdogguardssheepfromtheeast,south,andwest,whereasaslowerdogfromthenorth,thenthecointegratedprocessappeartohavethecointegrationrankof2.Clearly,twodogsareabletokeepaockofsheepclosertogether,thanasingledogcan.Inotherwords,thehighercointegrationrankthemorerestrictiveitis. Infact,letusconsideraKdimensionalcointegratedvectorautoregressiveprocess,andletrdenotethecointegrationrankoftheprocess.Similarlytothebivariateexampleabove,wecanseethatwhentherankiszero(r=0),theunivariatecomponentsoftheprocessareindependent,andthemodelisdrivenbyKindependentwhitenoiseprocesses(i.e.thereisnocointegration).Inthecaseofr=1,wecandecomposethemultivariate 66 PAGE 67 Therefore,thesmalleristhecointegrationrankr,thelargeristhenumberKroftheunderlyingindependentstochastictrends,andso(thelarger)isthevectorspaceinwhichourcointegratedmodelcantravel.Andtheotherwayaround,increasingthecointegrationrankofthemodelshrinkstheunderlyingdomainoftheprocess,i.e.makesitboundedtoasmallerhyperplane.Forr=K,theVAR(p)isastableprocess,whichclearlyhasthemostconstraineddomain.Forr=0,theVARprocessisnotcointegratedandunrestricted. Thus,intheframeworkofgeneralizedphasesynchronizationintroducedabove,thecointegrationrankrepresentsafundamentalmeasureofsynchronyinthemultichannelsignal,suchasEEG.Inparticular,wesaythatthesignaliscompletelyasynchronous,ifthecointegrationrankriszero.Ontheotherhand,whenthemultivariateprocessisstable(i.e.therankcoincideswiththedimensionoftheprocess,r=K),thesignalissaidtobeperfectlysynchronous. 67 PAGE 68 73 ].Thesebriefseizurescanhappenseveraltimesduringaday,butinsomepatients,thefrequencyofabsenceseizurescanbeashighashundredoftimesaday,whichinterfereswiththedailyactivitiesofachildsuchasschool.Insomecasesofchildhoodabsenceepilepsy,theseizuresstopwhenachildreachespuberty.AbsenceseizuresexhibitacharacteristicspikeandwaveEEGpatternata3Hzfrequency[ 73 ]. Figure 43 displaysamultichannelEEGrecordingthatincludesanabsenceseizure.Thedurationoftheseizureisapproximately4seconds.Thegurevividlyillustratesacharacteristicspikeandwaveactivityduringtheseizure. First,themultipletimeseriesoftheinstantaneousphaseswereextractedfromtherawEEGdatausingtheHilberttransformapproachasdescribedinSection 3.2 .Inparticular,wetookadvantageofthefunctionshilbertandanglereadilyavailableintheMATLABR2006aenvironment. TheVARmodelingandtestingwereimplementedusingtheR2.6.1statisticalsoftware.Inouranalysisoftheinstantaneousphases,weincorporatedar,adf.test,po.test,cajolstandotherfunctionsfoundinpackagestseriesandurca. Next,weillustrateourapproachontheexampleoftheEEGdatalethatincludesthreeseizureintervals.Thelecontainsa16channelrecordingofscalpEEGsampledatthe200Hzfrequencyaswellastwoauxiliarychannels,whichwerediscarded.TheinstantaneousphasevalueswereestimatedfromtheEEGtimeseriesbymeansofHilberttransform,andtheresultingphaseseriesweretestedusingtheADFtestintroducedinSection 2.4.1 .Specically,weappliedtheAugmentedDickeyFullerproceduretotestthe 68 PAGE 69 Theresultsofourexperimentsforseizures1,2,and3arepresentedinTables 41 42 ,and 43 ,respectively.Thechannels,forwhichtheADFunitroottesthasdetectedapresenceofaunitrootatthesignicancelevel=0:01,arelistedasintegrated.Whereasthechannels,forwhichthenullhypothesisofaunitroothasbeenrejectedbytheADFatthe1percentlevel,aredenotedbystationary.ThechannelsforwhichthepvaluesoftheADFtestexceed2.5%aremarkedwith*.Noticethatallthreeseizuresegmentsareconsideredstable,whentheADFisappliedata0.025signicancelevel. Next,wetvectorautoregressiontothemultipletimeseriesofphaseestimates,foreachofthreedierentsegments(before,during,andafteraseizure)inordertodetermineappropriatelaglengthparameterp.Tondappropriatelagsp,theAkaikeInformationCriteria(AIC)wasused.Thisledustochooseseverallaglengthforeachsegmentandeachseizure.Finally,Johansencointegrationrankprocedurewasappliedtodeterminethevaluesofcointegrationrankrforeachcase.TheresultsaresummarizedinTables 44 45 ,and 46 Noticethatduringtheseizurethesystembecomesstable,especiallywhenmodeledusingashortestimateofthelagparameter.Sincethedurationsoftheseizure1andseizure2arerathershort,andonlyinclude440500samplepoints,themodelsestimatedunderalonglagparametermaynotadequatelyrepresenttheunderlyingprocessesinseizure1and2.Ontheotherhand,seizure3isestimatedbasedonalmost1200samplevalues,andthereforethelonglagmodelofalongerseizure3maybemorerealistic,thanthelonglagmodelsforshorterseizures1and2.Overall,themodelsbasedonashortlagpforallthreeseizuresprovideanevidenceofabsolutesynchronizationamongthechannels.Whereas,thethepreseizureandpostseizuremodelsaremorelikelytobelessrestricted,andseemtoexhibitacointegrationrankbetween9and16. 69 PAGE 70 ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure1.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#preseizureseizurepostseizure stationary3,4,7,9,1513,5141,3,5,6,8,10,12,15integrated1*,2*,5,6*,8*,10,11,12*,13*,14*,16*4,15,162*,4*,7,9,13*,14,16* Table42. ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure2.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#preseizureseizurepostseizure stationary3,4,7,911,13,161163,7,1116integrated1*,2*,5*,6,8*,12*,14*,15none1*,2,4*,5,6*,8,9*,10 Table43. ResultsoftheADFunitroottestsforeachchannelduringthreesegments(2secondsimmediatelybeforeseizure,duringseizure,and2secondsafterseizure)forseizure3.Notethatthesignicanceat2.5%levelisdenotedby* Seizure#preseizureseizurepostseizure stationary7,11,13,141162,4,5,11,13,15integrated1*,2*,3,4*,5*,6*,8*,9,10*,12,15,16*none1,3*,6*,7,8,9*,10*,12,14*,16 Table44. 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 Table45. 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 Table46. 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 PAGE 71 Numbersnofrootsofthereversecharacteristicpolynomial(RCP)forVAR(p),whichlieonandinsidethecomplexunitcircle,computedfordierentsamplesizesTandfordierentmodelorderspusingtherawdata Numbersnofrootsofthereversecharacteristicpolynomial(RCP)forVAR(p),whichlieonandinsidethecomplexunitcircle,computedfordierentsamplesizesTandfordierentmodelorderspusingthe0{30Hzbandltereddata 71 PAGE 72 SegmentofmultichannelEEGrecordingthatcontainsafoursecondlongabsenceseizure PAGE 73 InvestigationofspatiotemporalpropertiesoftheEEGdatabydataminingandoptimizationapproachespostsvariouschallenges.NumerousfeaturesandmethodshavebeenproposedforstudyingthemultivariateseriesthatisEEG.TheanalysisofEEGtimeseriesisoftenapproachedfromtwodierentpointsofview,theonethattreatsEEGdataasproducedbyadeterministicchaoticdynamicalsystem,andtheothermoretraditionalapproachoflinearautoregressivemodeling. Inthiswork,weinvestigatedseveralstatisticalapproachesthatarerecentlyintroducedfordataminingbraindynamics.Inparticular,weexaminedtheapplicationofvectorautoregressivemodelingandlinearGrangercausalitytorawandlteredEEGdata. Motivatedbyrecentsuccessinapplicationofphasesynchronizationtoanalysisofdynamicprocessesinepilepticbrain,wedevelopedaconceptofgeneralizedsynchronizationbasedonthenovelideaofextendingtheclassicalsynchronizationconditionofaboundedlinearcombinationoftwophases.Thissimplebivariateconditioninthemultivariatecaseisanalogoustoacointegratingrelationshipinthemultipletimeseries.Thus,wecananalyzethesynchronyamongdierentpartsofthecommoninterrelatedsystem(suchasahumanbrain),bymodelingthephasesextractedfromanitenumberofsignalsinthesystemsbymeansofcointegratedvectorautoregression.Moreover,weshowedthatthecointegrationrankinthecointegratedVARmodelofthephasetimeseriescanbeviewedasameasureofsynchronyamongthephasesofdierentcomponentsoftheEEGsignal. Notonlythisnewmeasureofmultivariatephasesynchronycanbetestedonvariousbiomedicaldata,suchasmultichannelEEGrecordedfromanepilepticbrain,butalsothenewmultiplephasesynchronizationcanbeemployedindierentareasofappliedandtheoreticresearch(includingphysics,communication,electronics,laserdynamics,and 73 PAGE 74 74 PAGE 75 [1] L.BaccalaandK.Sameshima,\Partialdirectedcoherence:anewconceptinneuralstructuredetermination,"Biol.Cybern.,vol.84,pp.46374,2001. [2] C.BernasconiandP.Koenig,\Onthedirectionalityofcorticalinteractionsstudiedbystructuralanalysisofelectrophysiologicalrecordings,"Biol.Cybern.vol.81,pp.199{210,1999. [3] R.D.Bickford,\Electroencephalography,"inAdelmanG.ed.EncyclopediaofNeuroscience.Cambridge,MA:Birkhauser,pp.371{373,1987. [4] C.D.Binnie,\LongtermEEGrecordinganditsroleinclinicalpractice,"IEEColloquiumonDataLoggingofPhysiologicalSignals,pp.5/15/2,1995. [5] P.S.Bradley,O.L.Mangasarian,andW.N.Street,\Clusteringviaconcaveminimization,"inM.C.Mozer,M.I.Jordan,andT.Petsche,eds.,AdvancesinNeuralInformationProcessingSystems,vol.9,pp.368{374,1997. [6] P.S.BradleyandO.L.Mangasarian,\Featureselectionviaconcaveminimizationandsupportvectormachines,"inJ.Shavlik,MachineLearningProceedingsoftheFifteenthInternationalConference,pp.82{90,1998. [7] P.S.Bradley,U.M.Fayyad,andO.L.Mangasarian,\MathematicalProgrammingforDataMining:FormulationsandChallenges,"INFORMSJournalonComputing,vol.11,no.3,pp.217{238,1999. [8] S.L.Bressler,R.Coppola,andR.Nakamura,\Episodicmultiregionalcorticalcoherenceatmultiplefrequenciesduringvisualtaskperformance,"Nature,vol.366,pp.153156,1993. [9] R.BrownandL.Kocarev,\Aunifyingdenitionofsynchronizationfordynamicalsystems,"Chaos,vol.10,pp.344349,2000. [10] T.H.BullockandM.C.McClune,\Lateralcoherenceoftheelectrocorticogram:anewmeasureofbrainsynchrony,"ElectroencephalogrClinNeurophysiol,vol.73,pp.479498,1989. [11] T.L.CarrollandL.M.Pecora,\Cascadingsynchronizedchaoticsystems,"PhysicaD,vol.67,pp.126,1993. [12] G.C.Carter,\Coherenceandtimedelayestimation,"ProcIEEE,vol.75,pp.236255,1987. [13] Y.W.Cheung,K.S.Lai,\LagOrderandCriticalValuesoftheAugmentedDickeyFullerTest,"JournalofBusiness&EconomicStatistics,vol.13,no.3,pp.277{280,1995. 75 PAGE 76 R.Cont,\StatisticalPropertiesofFinancialTimeSeries,"inR.ContandJ.Yongeditors,MathematicalFinance:TheoryandPractice,LectureSeriesinAppliedMathematics,vol.1,1999. [15] A.R.Damasio,\Synchronousactivationinmultiplecorticalregions:amechanismforrecall,"SeminNeurosci,vol.2,pp.287296,1990. [16] D.A.Dickey,andW.A.Fuller,\DistributionoftheEstimatorsforAutoregressiveTimeSeriesWithaUnitRoot,"JournaloftheAmericanStatisticalAssociation,vol.74,pp.427{431,1979. [17] M.Eichler,Graphicalmodellingofmultivariatetimeseries,UniversityofHeidelberg:Preprint,2001. [18] M.Eichler,R.Dahlhaus,andJ.Sandkuhler,\Partialcorrelationanalysisfortheidenticationofsynapticconnections,"Biol.Cybern.,vol.89,pp.289302,2003. [19] M.Eichler,\Agraphicalapproachforevaluatingeectiveconnectivityinneuralsystems,"Philos.Transact.R.Soc.Bvol.360,pp.95367,2005. [20] L.Fabiny,P.Colet,andR.Roy,\Coherenceandphasedynamicsofspatiallycoupledsolidstatelasers,"Phys.Rev.A,vol.47,pp.4287,1993. [21] W.J.Freeman,\SpatialpropertiesofanEEGeventintheolfactorybulbandcortex,"ElectroencephalogrClinNeurophysiol,vol.44,pp.586605,1978. [22] W.A.Fuller,IntroductiontoStatisticalTimeSeries,2nded.NewYork:JohnWiley,1996. [23] D.Gabor.\Theoryofcommunication,"inProc.IEEELondon,vol.93,pp.429,1946. [24] P.Georgiev,A.Cichocki,andH.Bakardjian,\OptimizationtechniquesforIndependentComponentAnalysiswithApplicationstoEEGdata,"InP.M.Pardalos,J.C.Sackellares,P.R.Carney,L.D.Iasemidis,eds.,QuantitativeNeuroscience:Models,Algorithms,Diagnostics,andTheraputicApplications,KluwerAcademicPublishers,pp.53{68,2004. [25] J.Geweke,\Measurementoflineardependenceandfeedbackbetweenmultipletimeseries,"J.AmericanStatis.Association,vol.77,pp.304{313,1982. [26] C.W.J.Granger,\EconomicProcessesInvolvingFeedback,"InformationandControl,vol.6,pp.28{48,1963. [27] C.W.J.Granger,\InvestigatingCausalRelationsbyEconometricModelsandCrossSpectralMethods,"Econometrics,vol.37,pp.424{438,1969. [28] C.W.J.Granger,\SomePropertiesofTimeSeriesDataandTheirUseinEconometricModelSpecication,"JofEconometrics,vol.16,pp.121{130,1981. 76 PAGE 77 M.J.Griths,P.Grainger,M.V.Cox,andA.W.Preece,\RecentadvancesinEEGmonitoringforgeneralanaesthesia,alteredstatesofconsciousnessandsportsperformancescience,"inThe3rdIEEInternationalSeminaronMedicalApplicationsofSignalProcessing,pp.1{5,2005. [30] J.D.Hamilton,TimeSeriesAnalysis.Princeton,1994. [31] M.J.Hoeve,R.D.Jones,G.J.Carroll,andH.Goelz,\AutomateddetectionofepilepticseizuresintheEEG,"inEngineeringinMedicineandBiologySociety,Proceedingsofthe23rdAnnualInternationalConferenceoftheIEEE,vol.1,pp.943{946,2001. [32] R.Horst,P.M.Pardalos,andN.V.Thoai,IntroductiontoGlobalOptimization,2ndedition.KluwerAcademicPublishers,2000. [33] C.Huygens.HoroloquiumOscilatorium.Paris,1673. [34] L.D.Iasemidis,\Onthedynamicsofthehumanbrainintemporallobeepilepsy,"Ph.D.Dissertation,UniversityofMichigan,AnnArbor,1991. [35] L.D.Iasemidis,andJ.C.Sackellares,\TheevolutionwithtimeofthespatialdistributionofthelargestLyapunovexponentonthehumanepilepticcortex,"inD.W.Duke,andW.S.Pritchard,eds.,Measuringchaosinthehumanbrain,pp.49{82,WorldScientic,1991. [36] L.D.Iasemidis,D.S.Shiau,J.C.Sackellares,P.M.Pardalos,andA.Prasad,\Dynamicalresettingofthehumanbrainatepilepticseizures:applicationofnonlineardynamicsandglobaloptimizationtecniques,"IEEETransactionsonBiomedicalEngineering,vol.51,no.3,pp.493{506,2004. [37] H.H.Jasper,\ThetentwentyelectrodesystemoftheInternationalFederation,"Electroencephalogr.Clin.Neurophysiol.,vol.10,pp.370{375,1958. [38] S.JohansenandK.Juselius,\MaximumLikelihoodEstimationandInferenceonCointegrationWithApplicationstotheDemandforMoney,"OxfordBulletinofEconomicsandStatistics,vol.52,pp.169{210,1990. [39] E.Juergens,A.Guettler,andR.Eckhorn.\VisualstimulationelicitslockedandinducedgammaoscillationsinmonkeyintracorticalandEEGpotentials,butnotinhumanEEG,"ExperimentalBrainResearch,vol.129,pp.247259,1999. [40] Y.Kaji,M.Akutagawa,F.Shichijo,H.Nagashino,Y.Kinouchi,andS.Nagahiro,\AnalysisforBrainActivitiesduringOperationsUsingMeasuredEEG,"inConfProcIEEEEngMedBiolSoc.vol.6,pp.6003{6006,2005. [41] T.KalayciandO.Ozdamar,\WaveletpreprocessingforautomatedneuralnetworkdetectionofEEGspikes,"EngineeringinMedicineandBiologyMagazine,IEEE,vol.14,no.2,pp.160{166,1995. 77 PAGE 78 M.Kaminski,M.Ding,W.A.Truccolo,andS.L.Bressler,\Evaluatingcausalrelationsinneuralsystems:Grangercausality,directedtransferfunctionandstatisticalassessmentofsignicance,"Biol.Cybern.,vol.85,pp.145{57,2001. [43] [44] [45] E.Keogh,andS.Kasetty,\Ontheneedfortimeseriesdataminingbenchmarks:asurveyandempiricaldemonstration,"inProceedingsoftheEighthACMSIGKDDinternationalConferenceonKnowledgeDiscoveryandDataMining,pp.102{111,2002. [46] M.E.Kirlangic,D.Perez,S.Kudryavtseva,G.Griessbach,G.Henning,andG.Ivanova,\Fractaldimensionasafeatureforadaptiveelectroencephalogramsegmentationinepilepsy,"inEngineeringinMedicineandBiologySociety.Proceedingsofthe23rdAnnualInternationalConferenceoftheIEEE,vol.2,pp.1573{1576,2001. [47] W.A.Kittel,C.M.Epstein,andM.H.Hayes,\EEGmonitoringbasedonfuzzyclassication,"CircuitsandSystems,Proceedingsofthe35thMidwestSymposium,vol.1,pp.699{702,1992. [48] L.KocarevandU.Parlitz,\Generalapproachforchaoticsynchronizationwithapplicationstocommunication,"Phys.Rev.Lett.,vol.74,pp.5028,1995. [49] T.Kreuz,MeasuringSynchronizationinModelSystemsandElectroencephalographicTimeSeriesfromEpilepsyPatients,Dissertation.NICSeries,vol.21,2003. [50] J.P.Lachaux,E.Rodriguez,J.Martinerie,andF.J.Varela,\Measuringphasesynchronyinbrainsignals,"Hum.BrainMapp.,vol.8,pp.194,1999. [51] J.P.Lachaux,E.Rodriguez,J.Martinerie,C.Adam,D.HasbounandF.J.Varela,\Gammabandactivityinhumanintracorticalrecordingstriggeredbycognitivetasks,"EurJNeurosci,vol.12,pp.26082622,2000. [52] J.P.Lachaux,E.Rodriguez,M.LeVanQuyen,A.Lutz,J.Martinerie,andF.J.Varela,\Studyingsingletrialsofphasesynchronousactivityinbrain,"Int.J.BifurcationChaosAppl.Sci.Eng.,vol.10,pp.2429,2000. [53] A.D.Legatt,J.Arezzo,andH.G.Vaughan,\Averagedmultipleunitactivityasanestimateofphasicchangesinlocalneuronalactivity:eectsofvolumeconductedpotentials,"J.ofNeuroscienceMethods,vol.2no.2,pp.203{217,1980. 78 PAGE 79 M.LeVanQuyen,C.Adam,J.P.Lachaux,J.Martinerie,M.Baulac,B.Renault,etal.,\Temporalpatternsinhumanepilepticactivityaremodulatedbyperceptualdiscriminations,"Neuroreport,vol.8,pp.17031710,1997. [55] M.LeVanQuyen,J.Martinerie,C.Adam,andF.J.Varela,\NonlinearspatiotemporalinterdependenciesofinterictalintracranialEEGrecordingsfrompatientswithtemporallobeepilepsy:Localizingofepileptogenicfoci,"PhysicaD,vol.127,pp.25065,1999. [56] M.LeVanQuyen,J.Foucher,J.P.Lachaux,E.Rodriguez,A.Lutz,J.Martinerie,andF.J.Varela,\ComparisonofHilberttransformandwaveletmethodsfortheanalysisofneuronalsynchrony,"JournalofNeuroscienceMethods,vol.111,pp.8398,2001. [57] H.Liang,M.Ding,R.Nakamura,andS.L.Bressler,\Causalinuencesinprimatecerebralcortexduringvisualpatterndiscrimination,"Neuroreport,vol.11,pp.2875{80,2000. [58] Q.Liu,M.Sun,M.L.Scheuer,andR.J.Sclabassi,\Patienttrackingforvideo/EEGmonitoringbasedonchangedetectioninDCTdomain,"inBioengineeringConference,2005.ProceedingsoftheIEEE31stAnnualNortheast,pp.114{115,2005. [59] H.Lutkepohl,Intoductiontomultipletimeseriesanalysis.Berlin,Heidelberg:SpringerVerlag,1991. [60] J.G.MacKinnon,\CriticalValuesforCointegrationTests,"inR.F.EngleandC.W.J.Granger,eds.,LongRunEconomicRelationships:ReadingsinCointegration,NewYork:OxfordUniversityPress,pp.266{276,1991. [61] J.G.MacKinnon,\ApproximateAsymptoticDistributionFunctionsforUnitRootandCointegrationTests,"JournalofBusiness&EconomicStatistics,vol.12,pp.167{176,1994. [62] V.Menon,W.J.Freeman,B.A.Cutillo,J.E.Desmond,M.F.Ward,S.L.Bressler,etal.,\Spatiotemporalcorrelationsinhumangammabandelectrocorticograms,"ElectroencephalogrClinNeurophysiol,vol.98,no.2,pp.89102,1996. [63] M.Mezard,G.Parisi,andM.A.Virasoro,Spinglasstheoryandbeyond.WorldScientic,1987. [64] S.MitraandT.Acharya,Datamining:multimidea,softcomputing,andbioinformatics.Hoboken,NJ:JohnWiley&Sons,Inc.,2003. [65] F.Mormann,K.Lehnertz,P.David,andC.E.Elger,\MeanphasecoherenceasameasureforphasesynchronizationanditsapplicationtotheEEGofepilepticpatients,"PhysicaD,vol.144,pp.358369,2000. [66] M.P.Murray\ADrunkandHerDog:AnIllustrationofCointegrationandErrorCorrection,"TheAmericanStatistician,vol.48,no.1,pp.37{39,1994. 79 PAGE 80 E.Niedermeyer,F.H.LopesdaSilva,Electroencephalography:Basicprinciples,clinicalapplicationsandrelatedelds,3rdedition.Philadelphia:Lippincott,Williams&Wilkins,1993. [68] P.L.Nunez,NeocorticalDynamicsandHumanEEGRhythms.NewYork:OxfordUniv.Press,1995. [69] N.H.Packard,J.P.Crutcheld,J.D.Farmer,andR.S.Shaw,\GeometryfromaTimeSeries,"Phys.Rev.Lett.,vol.45,no.9,pp.712{716,1980. [70] P.Panter.Modulation,noise,andspectralanalysis.NewYork:McGrawHill,1965. [71] H.S.Park,Y.H.Lee,S.DooS,etal.,\Detectionofepileptiformactivityusingwaveletandneuralnetwork,"EngineeringinMedicineandBiologySociety,vol.3,pp.1194,1997. [72] U.Parlitz,L.Junge,W.Lauterborn,andL.Kocarev,\Experimentalobservationofphasesynchronization,"Phys.Rev.E,vol.54,pp.2115,1996. [73] J.B.Patten,NeurologicalDierentialDiagnosis,2nded.,SpringerVerlag,1996. [74] D.W.Peterman,M.Ye,andP.E.Wigen,\Highfrequencysynchronizationofchaos,"Phys.Rev.Lett.,vol.74,pp.1740,1995. [75] P.C.B.Phillips,\Timeseriesregressionwithaunitroot,"Econometrica,vol.55,pp.277{301,1987. [76] P.C.B.PhillipsandS.Ouliaris,\AsymptoticPropertiesofResidualBasedTestsforCointegration,"Econometrica,vol.58,pp.165193,1990. [77] A.S.Pikovsky,\Phasesynchronizationofchaoticoscillationsbyaperiodicexternaleld,"SovJCommunTechnolElectron,vol.30,pp.85,1985. [78] A.S.Pikovsky,M.G.Rosenblum,andJ.Kurths.Synchronization.Auniversalconceptinnonlinearsciences.Cambridge,UK:CambridgeUniv.Press,2001. [79] O.A.Prokopyev,V.Boginski,W.Chaovalitwongse,P.M.Pardalos,J.C.Sackellares,andP.R.Carney,\NetworkbasedTechniquesinEEGDataAnalysisandEpilepticBrainModeling,"inP.M.Pardalos,V.BoginskiandA.Vazacopoulos,eds.,DataMininginBiomedicine,Springer,pp.559{573,2007. [80] K.Pyragas,\Continuouscontrolofchaosbyselfcontrollingfeedback,"Phys.Lett.A,vol.170,pp.421,1992. [81] R.QuianQuiroga,A.Kraskov,T.Kreuz,andP.Grassberger,\Performanceofdierentsynchronizationmeasuresinrealdata:Acasestudyonelectroencephalographicsignals,"Phys.Rev.E,vol.65,pp.041903,2002. 80 PAGE 81 E.Rodriguez,N.George,J.P.Lachaux,J.Martinerie,andF.J.Varela,\Perceptionsshadow:Longdistancesynchronizationinthehumanbrain,"Nature,vol.397,pp.340343,1999. [83] P.R.Roelfsema,A.K.Engel,P.Konig,andW.Singer,\Visuomotorintegrationisassociatedwithzerotimelagsynchronizationamongcorticalareas,"Nature,vol.385,pp.157161,1997. [84] E.RosaJr.,W.B.Pardo,C.M.Ticos,J.A.Walkenstein,andM.Monti,\Phasesynchronizationofchaosinaplasmadischargetube,"Int.J.Bifurc.Chaos,vol.10,pp.2551,2000. [85] M.Rosenblum,A.Pikovsky,J.Kurths,\Phasesynchronizationofchaoticoscillators,"PhysRevLett,vol.76,pp.18041807,1996. [86] M.G.Rosenblum,A.S.Pikovsky,C.Schafer,P.Tass,andJ.Kurths,\Phasesynchronization:Fromtheorytodataanalysis,"in:S.Gielen,F.MossF,eds.HandbookofBiologicalPhysics,vol.4.ElsevierScience,Neuroinformatics,1999. [87] R.RoyandK.S.Thornburg,\Experimentalsynchronizationonchaoticlasers,"Phys.Rev.Lett.,vol.72,pp.2009,1994. [88] N.F.Rulkov,L.S.Tsimring,andH.D.I.Abarbanel,\Trackingunstableorbitsinchaosusingdissipativefeedbackcontrol,"Phys.Rev.E,vol.50,pp.314,1994. [89] J.C.Sackellares,L.D.Iasemidis,R.L.Gilmore,andS..Roper,\Epilepsywhenchaosfails,"inK.Lehnertz,J.Arnold,P.Grassberger,andC.E.Elger,eds.,Chaosinthebrain?,WorldScientic,2002. [90] J.C.Sackellares,L.D.Iasemidis,D.S.Shiau,P.M.Pardalos,andP.R.Carney,\Spatiotemporaltransitionsintemporallobeepilepsy,"inP.M.Pardalos,J.C.Sackellares,P.R.Carney,L.D.Iasemidis,eds.,QuantitativeNeuroscience:Models,Algorithms,Diagnostics,andTheraputicApplications,KluwerAcademicPublishers,pp.223{238,2004. [91] S.E.SaidandD.A.Dickey,\TestingforUnitRootsinAutoregressiveMovingAverageModelsofUnknownOrder,"Biometrika,vol.71,pp.599607,1984. [92] K.SameshimaandL.Baccala,\Usingpartialdirectedcoherencetodescribeneuronalensembleinteractions,"J.Neurosci.Methods,vol.94,pp.93103,1999. [93] C.Schafer,M.G.Rosenblum,J.Kurths,andH.H.Abel,\Heartbeatsynchronizedwithventilation,"Nature,vol.392,pp.239240,1998. [94] F.LopezdeSilva,\EEGanalisys:Theoryandpractice;ComputerassistedEEGdiagnosis:Patternrecognitiontechniques,"inE.Niedermeyer,F.H.LopesdaSilva,eds.Electroencephalography:Basicprinciples,clinicalapplicationsandrelatedelds,pp.871{919,1987. 81 PAGE 82 C.A.Sims,\Money,IncomeandCausality,"AmericanEconomicReview,vol.62,pp.540{552,1972. [96] W.SingerandC.M.Gray,\Visualfeatureintegrationandthetemporalcorrelationhypothesis,"AnnuRevNeurosci,vol.18,pp.555586,1995. [97] C.TallonBaudry,O.Bertrand,C.Delpuech,andJ.Pernier,\Oscillatorygammaband(3070Hz)activityinducedbyavisualsearchtaskinhuman,"JNeurosci,vol.17,pp.722734,1997. [98] D.Y.Tang,R.Dykstra,M.W.Hamilton,andN.R.Heckenberg,\Experimentalevidenceoffrequencyentrainmentbetweencoupledchaoticoscillations,"Phys.Rev.E,vol.57,no.3,pp.3649,1998. [99] P.Tass,M.G.Rosenblum,J.Weule,J.Kurths,A.Pikovsky,J.Volkmann,etal.,\Detectionofn:mphaselockingfromnoisydata:applicationtomagnetoencephalography,"PhysRevLett,vol.81,pp.32913294,1998. [100] M.Teplan,\FundamentalsofEEGmeasurement,"Measurementsciencereview,vol.2,no.2,pp.11223,2002. [101] G.TononiandG.M.Edelman,\Consciousnessandcomplexity,"Science,vol.282,pp.18461851,1998. [102] R.D.Traub,M.A.Whittington,E.Buhl,J.G.Jeerys,andH.J.Faulkner,\Onthemechanismofthegammatobetafrequencyshiftinneuronaloscillationsinducedinrathippocampalslicesbytetanicstimulation,"JNeurosci,vol.19,pp.10881105,1999. [103] B.vanderPolandJ.vanderMark,\Theheartbeatconsideredasarelaxationoscillation,andanelectricalmodeloftheheart,"Phil.Mag.,vol.6,pp.763,1928. [104] F.J.Varela,\Resonantcellassemblies:Anewapproachtocognitivefunctionsandneuronalsynchrony,"BiolRes,vol.28,pp.8195,1995. [105] F.J.Varela,J.P.Lachaux,E.Rodriguez,andJ.Martinerie,\Thebrainweb:phasesynchroniztionandlargescaleintegration,"NatureRevNeurosci,vol.2,pp.22939,2001. [106] N.Wiener,\TheTheoryofPrediction,"inE.F.Beckenbach,ed.ModernMathematicsfortheEngineer,NewYork:McGrawHill,1956. [107] A.Wolf,J.BSwift,H.L.Swinney,andJ.A.Vastano,\DeterminingLyapunovexponentfromatimeseries,"PhysicaD,vol.16,pp.285{317,1985. [108] K.Yang,andC.Shahabi,\OntheStationarityofMultivariateTimeSeriesforCorrelationBasedDataAnalysis,"inProceedingsofFifthIEEEInternationalConferenceonDataMining(ICDM'05),pp.805{808,2005. 82 PAGE 83 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 