Analysis of Human EEG Data

Pavel Stránský

Supervisor: Prof. RNDr. Petr Šeba, DrSc.

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

1

Measurement and Structure of EEG Signal

1. Measurement and Structure of EEG Signal

Cerebral Electric Activity

EEG = Electro-encephalography, Electro-encephalogram

1. Measurement and Structure of EEG Signal

Location of the Electrodes(10-20 system, 21 electrodes)

1. Measurement and Structure of EEG Signal

An Example of EEG

Measurement

•Alpha waves

•Beta, theta, delta waves

•Other graphoelements

•Artefacts

2.

Statistical Approach to EEG Data

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

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

3. Small introduction to random matrix theory

(RMT)

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

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 interactionsGSE Hamiltonians with time reversal symmetry and WITH spin-1/2 interactions

• Universal law for joint probability density function:

For energies (eigenvalues of H)= 1 GOE

= 2 GUE

= 4 GSE

3. Little introduction to RMT

Random matrices (cont.)

• Spectral correlations (nearest neighbour spacing distribution):Wigner distribution

Normalization

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3

s

p(s)

GOE

GUE

GSE

Poisson

3. Little introduction to RMT

Random matrices (cont.)

• Other distributions (taking into account correlations for longer distances)statistics (number variance)

3 statistics (spectral rigidity)

4.Results, outlook

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 m of all correlation matrices Cm

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:

4. Results, outlook

Correlation analysis (cont.)

• Comparing computed spacing distribution with theoretical

Wigner curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3 3.5

s

p(s)

EEG

Wigner

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

4. Results, outlook

Literature

• P. Šeba, Random Matrix Analysis of Human EEG Data, Phys. Rev. Lett. 91, 198104 (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)