Transcript ppt - Pavel Stránský

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
mixing
ICA
Testing for periodicity (Fisher’s test, Siegel’s test)
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
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)
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
b=2
b=4
GOE
GUE
GSE
3. Little introduction to RMT
Random matrices (cont.)
• Spectral correlations (nearest neighbour spacing distribution):
Wigner distribution
1.4
GOE
GUE
1.2
GSE
Normalization
p(s)
1
Poisson
0.8
0.6
0.4
0.2
0
0
1
2
s
3
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)
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 xm 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 r(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
0.9
0.8
EEG
0.7
Wigner
p(s)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
2
1.5
s
2.5
3
3.5
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
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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)