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Analysis of Human EEG Data Pavel Stránský Supervisor: Prof. RNDr. Petr Šeba, DrSc.

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Content 1.Measurement and structure of EEG signal 2.EEG as a multivariate time series, statistical approach to EEG data processing 3.Small introduction to random matrices theory 4.My present results and outlook

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1 Measurement and Structure of EEG Signal

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1. Measurement and Structure of EEG Signal Cerebral Electric Activity EEG = Electro-encephalography, Electro-encephalogram

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1. Measurement and Structure of EEG Signal Location of the Electrodes (10-20 system, 21 electrodes)

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1. Measurement and Structure of EEG Signal An Example of EEG Measurement Alpha waves Beta, theta, delta waves Other graphoelements Artefacts

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2. Statistical Approach to EEG Data

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2. Statistical Approach to EEG Data Modelling and processing time series Vector Autoregression VAR(p) Stacionarity (Covariance – stacionarity): for all t and any j White noise: for all t, t 1, t 2

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2. Statistical Approach to EEG Data Modelling and processing time series (cont.) Other ways of treating with time series: Principal component analysis Independent component analysis Testing for periodicity (Fisher’s test, Siegel’s test) mixingICA

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3. Small introduction to random matrix theory (RMT)

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3. Small introduction to RMT Random matrices Study of excitation spectra of compound nuclei The same behaviour like eigenvalues of random matrices 3 principal ensembles: GOE, GUE, GSE Def: Gaussian othogonal ensemble is defined in the space of real symmetric matrices by two requirements: 1. Invariance (O is orthogonal matrix) 2. Elements are statistically independent which means that, where (probablity density function) Hermitian matrices, unitary transformations Hermitian self-dual matrices, symplectic transformations

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3. Small introduction to RMT Random matrices (cont.) Universality classes: GUEHamiltonians without time reversal symmetry GOEHamiltonians with time reversal symmetry and WITHOUT spin-1/2 interactions GSEHamiltonians with time reversal symmetry and WITH spin-1/2 interactions Universal law for joint probability density function: For energies (eigenvalues of H) = 1GOE = 2GUE = 4GSE

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3. Little introduction to RMT Random matrices (cont.) Spectral correlations (nearest neighbour spacing distribution): Wigner distribution Normalization

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3. Little introduction to RMT Random matrices (cont.) Other distributions (taking into account correlations for longer distances) statistics (number variance) 3 statistics (spectral rigidity)

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4. Results, outlook

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Correlation analysis of EEG Data Dividing EEG signal from M channels x 1,..., x M into cells of constant time length T Computing correlation matrix C m for the m th cell with normalizing mean and variance: Finding eigenvalues m of all correlation matrices C m

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4. Results, outlook Correlation analysis (cont.) Unfolding the spectra: (after unfolding all eigenvalues are "equally important", the resulting eigenvalue density (x) is constant) Finding nearest neighbour distribution p(s) for the unfolded spectra:

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4. Results, outlook Correlation analysis (cont.) Comparing computed spacing distribution with theoretical Wigner curve

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4. Results, outlook Outlook Use more subtle method from RMT and time series analysis to analyze the correlations and also autocorrelations (correlations in time) Find significant and reproducible variables for standard EEG measured on healthy subjects Deviations are expected if there was some neural disease

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4. Results, outlook Literature P. Šeba, Random Matrix Analysis of Human EEG Data, Phys. Rev. Lett. 91, (2003) T. Guhr, A. Müller-Groeling, H. A. Weidenmüller, Random Matrix Theories in Quantum Physics: Common Concepts, Phys. Rep. 299, 189 (1998) M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic Press (1967) H. J. Stöckmann, Quantum Chaos: An Introduction, Cambridge University Press (1999) A. F. Siegel, Testing for Periodicity in a Time Series, JASA 75, 345 (1980) J. D. Hamilton, Time Series Analysis, Princeton University Press (1994) A. Jung, Statistical Analysis of Biomedical Data, Dissertation, Universität Regensburg (2003) J. Faber, Elektroencefalografie a psychofyziologie, ISV nakladatelství Praha (2001)

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