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

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

3. Measurement and Structure of EEG Signal1
Measurement and Structure of EEG Signal

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

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

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

7. Statistical Approach to EEG Data2.
Statistical Approach to EEG Data

8. 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, t1, t2

9. 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)
mixing
ICA

10. 3. Small introduction to random matrix theory (RMT)

11. 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
Hermitian self-dual matrices, symplectic transformations
Hermitian matrices, unitary transformations
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)

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

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

14. 3. Little introduction to RMT
Random matrices (cont.)
Other distributions (taking into account correlations for longer distances)
S2 statistics (number variance)
D3 statistics (spectral rigidity)

15. 4. Results, outlook

16. Correlation analysis of EEG Data
4. Results, outlook
Correlation analysis of EEG Data
Dividing EEG signal from M channels x1, ..., xM into cells of constant time length T
Computing correlation matrix Cm for the mth cell with normalizing mean and variance:
Finding eigenvalues xm of all correlation matrices Cm

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

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

19. Outlook 4. Results, 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

20. Literature 4. Results, outlook
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)