vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xi
LIST OF FIGURES
xii
LIST OF SYMBOLS AND ABBREVATIONS
xvi
LIST OF APPENDIXES
xxi
INTRODUCTION
1
1.1
Introduction
1
1.2
Background of Problems
2
1.2.1
Filtering Problems in EEG
Analysis
5
1.2.1.1 Parametric Time-varying
Spectral Estimation
6
1.2.1.2 Single-trial ERP
Estimation
6
1.3
Statement of Problems
7
1.4
Objectives of the Research
9
1.5
Scope of the Research
10
1.6
Contribution of the Research
13
viii
1.7
2
Outline of the Thesis
14
LITERATURE REVIEW
18
2.1
Introduction
18
2.2
General State-Space Models
19
2.2.1 Linear Gaussian Model
20
2.2.2 Non-Linear Non-Gaussian Model
20
2.3
Optimal Bayesian Filtering
21
2.4
Kalman Filtering
23
2.5
Sequential Monte Carlo Methods for
Filtering
25
2.5.1
Bayesian Importance Sampling
26
2.5.2
Sequential Importance Sampling
27
2.5.2.1 Re-sampling
29
2.5.2.2 Choice of Importance
Density
2.5.3
Rao-Blackwellized Particle
Filtering
2.6
2.7
30
31
Sequential Monte Carlo Methods for
Parameter Estimation
32
2.6.1
Maximum Likelihood Estimation
32
2.6.2
Bayesian Estimation
35
Analysis of EEG Signals using
Discrete-time Linear Gaussian Models
2.7.1
36
Gaussian Time-Varying
AR Models for Spectral
Estimation of EEG
2.7.2
36
Discrete-time Dynamical Models
for Estimation of Single-Trial ERPs 37
2.8
Summary
38
ix
3
SPECTRAL ESTIMATION OF
NON-STATIONARY EEG USING PARTICLE
FILTERING WITH APPLICATION TO
EVENT-RELATED DESYNCHRONIZATION 42
3.1
Introduction
42
3.2
Motivation
43
3.3
Model Formulation
45
3.4
Particle Filtering for Estimation of
3.5
3.6
4
TVAR Coefficients
47
Experimental Results
49
3.5.1
Simulation Results
49
3.5.2
ERD Tracking Results
54
3.5.2.1 Estimation of ERD
54
3.5.2.2 Dataset
55
3.5.2.3 Model Order Selection
56
3.5.2.4 Results and Discussion
58
3.5.2.5 Model Evaluation
65
Summary
67
MODELING OF SINGLE-TRIAL
EVENT-RELATED POTENTIAL DYNAMICS
BY PARTIALLY OBSERVED ORNSTEINUHLENBECK PROCESS
68
4.1
Introduction
68
4.2
Motivation
69
4.3
Partially Observed OU Modeling of ERPs
71
4.3.1
The Model
71
4.3.2
State-Space Formulation
76
4.4
4.5
Rao-Blackwellised Particle Filtering for
ERP Model
79
Experimental Results
85
x
4.5.1
Simulation Results
85
4.5.2
Estimation of ABR Dynamics
91
4.5.3
Estimation Results for Irregularly
Spaced Data
4.6
5
96
Summary
97
NON-LINEAR NON-GAUSSIAN MODELING
OF STOCHASTIC VOLATILITY
IN EEG NOISE
98
5.1
Introduction
98
5.2
Motivation
99
5.3
Non-linear Non-Gaussian SV Models of
EEG Noise
5.3.1
Non-Gaussian Random-Walk
Model
6
99
100
5.3.2 Cox-Ingersoll-Ross Process
101
5.4
Experimental Results
106
5.5
Summary
116
CONCLUSION AND FUTURE WORKS
117
6.1
Conclusion
117
6.2
Future Works
119
REFERENCES
121
Appendices A-C
132-143
xi
LIST OF TABLES
TABLES NO.
2.1
TITLE
Comparison of different studies on parametric
time-varying spectral estimation of biomedical signals.
2.2
57
Model evaluation results of Gaussian and Cauchy
models on fitting the EEG data for each subject.
5.1
41
The optimal AR and ARMA model orders using
AIC and BIC.
3.2
40
Comparison of different studies on single-trial dynamical
estimation of ERPs using optimal filtering.
3.1
PAGE
66
Model evaluation results of different SV noise models
on estimation of simulated ABRs in artifacts.
115
xii
LIST OF FIGURES
FIGURES NO.
1.1
TITLE
The 10 – 20 system of electrode placement for EEG
recording.
1.2
PAGE
2
(a) Two classes of filtering problems in EEG analysis.
(b) Contributions of the research for solving the filtering
problems in EEG analysis.
16-17
2.1
Graphical representation of a HMM.
20
2.2
(a) Single-trial ABRs. (b) Image and epoch
plots for single trials. (b) Average over 500 trials.
3.1
Tracking of TVAR coefficients by Gaussian and
Cauchy model.
3.2
51
Estimated filtered density p ( at ,1 | y1:t ) by
Gaussian model.
3.4
50
MSE performance (in log scale) versus time for
Gaussian and Cauchy model.
3.3
38
52
Estimated filtered mean and the 5th, 25th, 75th,
95th percentiles by Gaussian model.
52
xiii
3.5
Kernel smoothed filtered density p (at ,1 | y1:t ) by
Cauchy model.
3.6
Estimated filtered mean and the 5th, 25th, 75th
, 95th percentiles by Cauchy model.
3.7
53
53
The AIC and BIC values as a function of AR order p
for AR(p) on 1 s EEG segments before and after
the event at position C4 for subject 3.
3.8
57
Comparison of time-varying spectral representations
of ERD for average EEG of subject 1 at position C4
for left-hand motor imagery.
3.9
60
(a) Average EEG measured at position C4 for left-hand
motor imagery, with its spectrum, band power and ERD
estimates by Cauchy TVARMA model. The horizontal
line marks band power in reference period. (b) Estimated
residuals vˆt , posterior mean of dispersion of Cauchy AR
state noise qct , and variance of Gaussian MA state
noise σ gt2 .
3.10
62
ERD estimates by Cauchy TVARMA model and
Gaussian TVAR model for average EEGs of subject 1
at position C4 and C3 for left-hand motor imagery.
3.11
63
ERD estimations by Cauchy TVARMA model
, Cauchy TVAR model and Gaussian TVAR model for
subject 1, subject 2, and subject 3 on left-hand
motor imagery EEG at position C4.
4.1
Estimation of ABRs through subcomponent parameters
64
xiv
for a simulated single-trial ABR data.
4.2
Estimated time-varying log-variance φvn and variance
σ v2n for random-walk SV model on simulated data.
4.3
95
Estimated time-varying log-variance φvn and variance
σ v2n by random-walk SV model for subject 2.
4.7
94
Estimated time-varying log-variance φvn and variance
σ v2n by random-walk SV model for subject 1.
4.6
93
Estimation of ABR dynamics for single-trial recordings
from subject 2 with normal hearing.
4.5
90
Estimation of ABR dynamics for single-trial recordings
from subject 1 with normal hearing.
4.4
89
95
Estimation of ABR dynamics for recordings from
subject 1 in Figure 4.5 with simulated missing
observations.
5.1
Gamma stationary pdf of CIR process with parameters
θ1 = 0.7 , θ 2 = 1 and θ3 = 0.55 .
5.2
96
103
Noncentral chi-squared conditional densities of CIR
process p(σ v2n | σ v2n −1 ) for σ v2n −1 = 0.1 , σ v2n −1 = 0.5 , and
σ v2n −1 = 1 .
5.3
(a) Image plot for trials of background noises. (b)
Sample variances for each trial, σˆ v2n , sample ACF and
103
xv
partial ACF of σˆ v2n .
5.4
104
Marginal densities of σˆ v2n (histogram, estimated (by
maximum likelihood methods) Gamma and Gaussian
densities).
5.5
Effect of varying θ1 on marginal and conditional densities
of CIR SV process.
5.6
107
Effect of varying θ3 on marginal and conditional densities
of CIR SV process.
5.7
105
108
Estimation of ABRs through subcomponent parameters
for a simulated single-trial ABR data with artifacts using
Gaussian random-walk SV noise model.
5.8
110
Estimation of ABRs through subcomponent parameters
for a simulated single-trial ABR data with artifacts using
alpha-stable random-walk SV noise model.
5.9
111
Estimation of ABRs through subcomponent parameters
for a simulated single-trial ABR data with artifacts using
CIR SV noise model.
5.10
112
(Top) Sample variances over each trial for artifact
segment. (Bottom) Filtered mean of the time-varying
variance of observation noise, E[σ v2n | y1:n ] by the
Gaussian random-walk, alpha-stable random-walk
and CIR SV model.
5.11
113
E[σ v2n | y1:n ] by CIR SV model with θ3 = 0.4 , θ3 = 0.55
and θ3 = 0.7 .
114
xvi
LIST OF SYMBOLS AND ABBREVATIONS
ABR
-
Auditory brainstem response
AIC
-
Akaike information criterion
APF
-
Auxiliary particle filter
AR
-
Autoregressive
ARMA
-
Autoregressive moving-average
BCI
-
Brain computer interface
BIC
-
Bayesian information criterion
BS
-
Bowman-Shenton
CIR
-
Cox-Ingersoll-Ross
dB
-
decibel
EEG
-
Electroencephalography
EKF
-
Extended Kalman filter
EM
-
Expectation-maximization
ERD
-
Event-related desynchronization
ERP
-
Event-related potential
ERS
-
Event-related synchronization
ESS
-
Effective sample size
HMM
-
Hidden Markov model
i.i.d.
-
Independent identically distributed
IS
-
Importance sampling
xvii
KF
-
Kalman filter
LB
-
Ljung-Box
LMSE
-
log mean square error
MC
-
Monte Carlo
ML
-
Maximum likelihood
MLE
-
Maximum likelihood estimation
MMSE
-
Minimum mean-squared error
MSE
-
Mean square error
OU
-
Ornstein-Uhlenbeck
PF
-
Particle filter
RBPF
-
Rao-Blackwellized particle filter
RLS
-
Recursive least square
SDE
-
Stochastic differential equation
SIR
-
Sequential importance sampling-resampling
SIS
-
Sequential importance sampling
SMC
-
Sequential Monte Carlo
SNR
-
Signal-to-noise ratio
SSM
-
State-space model
STFT
-
Short-time Fourier transform
SV
-
Stochastic volatility
TVAR
-
Time-varying autoregressive
TVARMA
-
Time-varying autoregressive moving-average
{xt }
-
Hidden states
{y t }
-
Observations
x0
-
Initial state
xviii
x0:t = {x0 ,… , xt }
-
Sequence of states until time t
y 0:t = {y 0 ,… , y t }
-
Sequence of observations until time t
vt
-
Observation noise
wt
-
State noise
θ
-
Model parameters
θˆ
-
Estimated model parameters
σ v2
-
Variance of observation noise
σ w2
-
Variance of state noise
σ g2
-
Variance of driving noise for moving-average
(MA) coefficient
qc2
-
Dispersion parameter of Cauchy distribution
q( x) and π ( y | x)
-
Importance density
fθ ( x ' | x)
-
State transition density
gθ ( y | x)
-
Observation density
p(x0:t | y 0:t )
-
Posterior density
p (xt | y 0:t )
-
Filtering density
p(xt | y 0:t −1 )
-
One-step ahead prediction density
pθ (y 0:t )
-
Marginal likelihood of y 0:t given θ
p(y t | y 0:t −1 )
-
Predictive likelihood
-
Log-likelihood of y 0:T given θ
xˆ t|t −1
-
Mean of p(xt | y 0:t −1 )
Pt|t −1
-
Covariance of p(xt | y 0:t −1 )
T
(θ )
xix
xˆ t|t
-
Mean of p (xt | y 0:t )
Pt|t
-
Covariance of p (xt | y 0:t )
θt
-
Mean of p(θt | y1:t )
Vt
-
Covariance of p(θt | y1:t )
xt(i )
-
Particle or sample i at time t
N
-
Number of samples
N eff
-
Effective sample size
Nˆ eff
-
Estimated effective sample size
W (i )
-
Normalized weights
w( i )
-
Unnormalized weights
p
-
Order of AR model
q
-
Order of MA model
δ
-
Discounting factor
b ( X t , θ)
-
drift
σ ( X t , θ)
-
diffusion coefficient
Wt
-
Standard Brownian motion
β
-
Rate of reversion
U [ a, b]
-
Uniform distribution over interval[a, b]
N (μ ,σ 2 )
-
Univariate Gaussian distribution with mean
μ and variance σ 2
N (μ, Σ)
-
Univariate Gaussian distribution with mean
vector μ and covariance matrix Σ
xx
C (μ c , Q c )
-
Multivariate Cauchy distribution with location
μ c and dispersion matrix Q c
Sα S (α , γ )
-
Symmetric alpha-stable distribution with
characteristic exponent α and scale
parameter γ
R
-
Number of realizations
k
-
Number of estimated parameters
T
-
Observation length
Δt
-
Time step
δ (.)
-
Delta function
∇
-
Gradient
xxi
LIST OF APPENDICES
APPENDIX
A.1
TITLE
PAGE
Stationary AR model fitting to assumed locally
stationary short-time segment of motor-imagery
EEG for different subjects.
A.2
132
Stationary ARMA model fitting to assumed locally
stationary short-time segment of motor-imagery
EEG for different subjects.
B.1.1
Effect of varying the variance of the Gaussian state noise
on the TVAR(4) coefficient tracking for Section 3.5.1.
B.1.2
C
135
Average MSEs and fitted log-likelihoods for the Cauchy
model using different number of particles N.
B.2
133
136
Evaluation of OU noise variances for dynamical ABR
estimation in Section 4.5.2 for subject 1.
137
Publications
143