SINGLE TR IA L EEG /M EG SIGNALS
By
Hamid R. Mohseni
SUBM ITTED IN PARTIAL FULFILMENT OF THE REQ UIREM ENTS FOR THE DEGREE OF
D O C T O R OF PHILOSOPHY AT
CARDIFF UNIVERSITY CARDIFF, WALES, UK
JANUARY 2010
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T a b le o f C o n te n ts iv L ist o f T a b le s vii L ist o f F ig u r e s viii A b s tr a c t x v i A c k n o w le d g e m e n ts x v iii S ta te m e n t o f O r ig in a lity x ix L ist o f A b b r e v ia tio n s x x ii N o m e n c la tu r e s x x iv 1 I n tr o d u c tio n 1
1.1 Aim and O b je c tiv e s ... 2
1.2 Thesis O u tlin e ... 4
2 I n tr o d u c tio n to E E G an d M E G 7 2.1 E le c tro e n c e p h a lo g h y ... 7
2.1.1 EEG A dvantages and L im ita tio n s ... 8
2.1.2 EEG A c q u i s i t i o n ... 8
2.1.3 R e f e r e n c in g ... 10
2.1.4 P r e p r o c e s s in g ... 10
2.2 Event-R elated P o t e n ti a ls ... 12
2.2.1 P300 ... 12
3.1.1 Single-Channel Single Trial E s tim a tio n ... 17
3.1.2 M ultichannel Single Trial E s ti m a t io n ... 22
3.2 E /M E G Dipole Source L o c a liz a tio n ... 24
3.3 Fundam entals of K F and P F ... 28
3.3.1 Problem Form ulation in S tate Space ... 29
3.3.2 K alm an F i l t e r i n g ... 30
3.3.3 Particle F i l t e r s ... 32
4 S t a te S p a ce A p p ro a c h e s for S in g le Trial E stim a tio n o f E R P s 38 4.1 I n tr o d u c tio n ... 38
4.2 Problem Form ulation in S tate Space ... 39
4.3 E xperim ents and R e s u l ts ... 41
4.3.1 Sim ulation R e s u l t s ... 42
4.3.2 Real D a ta R e s u lts ... 51
4.4 C o n c lu s io n s ... 55
5 D e fla tio n B e a m fo r m in g for E /M E G D ip o le S ou rce L o ca liza tio n 57 5.1 I n tr o d u c tio n ... 57
5.2 M e t h o d s ... 58
5.2.1 Problem F o rm u la tio n ... 58
5.2.2 B e a m f o r m in g ... 59
5.2.3 D eflation B ea m fo rm in g ... 61
5.2.4 Perform ance A n a ly sis ... 65
5.2.5 Iterativ e Localization and D e fla tio n ... 67
5.3 Num erical R e s u lts ... 68
5.3.1 Tw o Dipoles Exam ple ... 69
5.3.2 Sim ulated M EG Localization ... 75
5.4 A pplication to A uditory E R F ... 78
5.5 C o n c lu s io n s ... 80
6 D ip o le L o c a liz a tio n an d T ra ck in g o f E /M E G S ources v ia J o in t B ea m -fo rm in g - R a o -B la c k w e lliz e d P a r tic le F ilte rin g 88 6.1 In tr o d u c tio n ... 88
6.2 M e t h o d s ... 90
6.2.1 Problem Form ulation in S tate S p a c e ... 90 6.2.2 A Beamforming - R B P F in Tim e and Space Domains (B -R B PF) 93
6.4 C o n c lu s io n s ... 113
A Derivation of Solution for Equation (6.15) 117 7 V a r ia tio n a l B a y e s for S p a tio te m p o ra l Id en tifica tio n o f E R P S u b c o m p o n e n ts 119 7.1 In tr o d u c tio n ... 119
7.2 M e t h o d s ... 121
7.2.1 Problem F o r m u la tio n ... 121
7.2.2 P aram eter E stim ation by Variational Bayes ... 123
7.3 R e s u lts ... 131
7.3.1 Sim ulated D a ta Results ... 131
7.3.2 Real D a ta R e s u lts ... 135
7.4 D is c u s s io n ... 139
7.5 C o n c l u s io n s ... 142
A ML E stim atio n of a * ^ ... 143
B ML E stim ation of Q * ... 144
C D erivation of E quations (7.16) and (7 .1 7 )... 145
8 S u m m a ry , C o n c lu s io n s , an d F utu re W orks 147 8.1 Sum m ary and C o n c lu s io n s ... 147
8.2 Future W o r k ... 150
B ib lio g r a p h y 152
6.1 E rrors of two moving dipoles’ locations and m om ents using the R B PF and B -R B P F m ethods. The error of locations and moments are in m eter and /xV units, respectively...
2.1 Scalp electrode positions in a 10-20 system used for recording the avail able d a ta s e t... 9 2.2 O ddball paradigm . Left: auditory sequences in which S refers to fre
quent tones and T refers to infrequent tones. Right: thick line is the result of averaging for the infrequent tones and thin line for th e fre
quent to n es... 13
4.1 Superposition of sim ulated ER Ps and their ST estim ations SNR = - 5dB and SBR = -lOdB; (a) sim ulated ERPs, (b) noisy E R Ps (average in thick black line), (c) extracted ER Ps using the KF m ethod, and (d) extracted E R P s using th e P F method. Notice th a t the P F estim ates the E R Ps very closely w ith respect to the original ER Ps shown in (a). 45 4.2 Three exam ples for ST estim ation SNR = -5dB and SBR = -lOdB;
(a) sim ulated EE G and E R P for trial num ber 5, (b) sim ulated and estim ated E R P s using P F and K F m ethods for trial number 5, (c) and (d) show th e results for trial num ber 35, while (e) and (f) dem onstrate
the estim ates for tria l num ber 55... 46
estim ated using P F (solid line) and estim ated am plitudes using KF (dashed line) for th e positive peak, (b) estim ated values for the negative peak, (c) the error between the simulated and estim ated am plitudes for P F (black bar) and KF methods (white bar) for th e first peak, (d) m ean and stan d ard deviation of errors for each m ethod (e) the error between sim ulated and estim ated am plitudes for P F (black bar) and K F m ethods (w hite bar) for the second peak, and (f) mean and stan d a rd deviation of errors for each m ethod ... Latencies of th e first and second peaks from trial to trial and their estim ations for SN R = -5dB and SBR = -lOdB; (a) sim ulated (thick line), estim ated using P F (solid line), and estim ated latencies using K F (dashed line) for th e positive peak, (b) sim ulated (thick line), esti m ated using P F (solid line) and estimated latencies using K F (dashed line) results for th e negative peak, (c) the error between sim ulated and estim ated latencies for P F (black bar) and K F m ethods (white bar) for th e first peak, (d) th e error between simulated and estim ated latencies for P F (black bar) and KF methods (white bar) for th e second peak, (e) th e error betw een sim ulated and estim ated latencies for P F (black bar) and K F m ethods (white bar) for the second peak, and (f) mean and stan d a rd deviation of errors for each m eth o d ... Perform ance of K F and P F methods as indexed by mean square error for 100 M onte C arlo trials (a) input SNR versus o u tp ut SNR. Both m ethods, especially P F , are robust to high in p u t SNR, and (b) input SBR versus o u tp u t SBR. This decreases rapidly in comparison with
signals - there is no evident E R P signature in this image, (c) super position of estim ated E R Ps using the P F m ethod, (d) ER P image of estim ated ER Ps (P300 in red color) using the P F m ethod, (e) esti m ated P300 filtered peak am plitude versus trials using PF. There is a decreasing trend across trials, (f) superposition of estim ated E R Ps using th e K F m ethod, (g) E R P image of estim ated E R Ps with (P300 in red) using the KF m ethod, and (h) estim ated P300 filtered peak am plitude versus trials using KF. Unlike P F (see (e) above) there is no directional tren d across trials... Latency vs. am plitude in four subjects with their linear regression obtained by P F shown by black and KF shown by red. A significant negative correlation between am plitude and latency w ith average r = —0.357 (p < 0.01, two tailed) for four subjects can be seen only for th e P F m ethod (r = —0.0986, n.s.), (a) subject num ber one (PF: r = —0.472, p < 0.001) (KF: r = —0.2864, p < 0.05), (b) subject num ber two (P F : r = —0.400, p < 0.005) (KF: r — 0.0185, n.s.), (c) subject num ber th ree (PF: r = —0.227, p < 0.05) (KF: r = —0.1182, n.s.), and (d) su b ject num ber four (PF: r = —0.327, p < 0.05) (KF:
r = —0.0102, n .s.)...
Location of sources used for different simulations; (a) axial view and (b) coronal view ... O u tp u t error vs. SNR when th e sources are S i and S2. The deflation BF (red lines) has considerably b etter performance compared to BF approach (blue lines) in th e face of different value of SNRs...
2 coordinate is decreasing. The horizontal axis is the distance between
second source and S4. Both BF and deflation BF are sensitive to the dep th of the source, however, deflation BF shows b e tte r perform ance. 72 5.4 O u tp u t error vs. a for investigation of correlation between sources.
a = 0 means two sources are uncorrelated and a = 1 m eans two sources have exactly th e same moments (i.e. completely correlated). Deflation BF (red lines) has much less sensitivity to the correlation between sources in com parison with the BF (blue lines), and its error is m onotonically increasing... 73 5.5 Investigation of distance between sources. The first source is fixed at
S3 and th e second source has the same x and 2 coordinates as those of S3, while its y coordinate is increasing. The horizontal axis is the distance betw een two sources. W hen the distance between sources is more th a n 4cm, deflation BF (red lines) has better perform ance th an BF (blue lines), however, when the sources have distance less th a n 3cm the B F outperform s deflation B F ... 75 5.6 O u tp u t error vs. SB R for investigation of effect of real noise. Real
background M EG has been added to the simulated data. Deflation BF (red lines) outperform s B F m ethod (blue lines), and indeed is robust to the non-G aussian noise... 76 5.7 Location of th ree dipoles used for MEG simulation in (a) axial view
and (b) coronal view ... 76
second and th ird dipoles, (c) deflation BF m ethod while deflating the first and th ird dipoles, and (d) deflation BF m ethod while deflating the first and second dipoles. Due to correlation between th e sources, BF m ethod shows activation between three actual sources. Three power profiles obtained by deflation BF show th a t deflation BF has been successful to identify th e three sources... 82 5.9 E stim ated m om ents for the localized sources in Fig. 5.8 by (a) BF
m ethod, and (b) deflation BF. Dotted lines are the sim ulated moments and solid lines are the estim ated moments. In contrast to the BF, the deflation B F shows tru e variation of the m om ents... 83 5.10 Power profile of th e sim ulated MEG, when one source dom inates oth
ers using (a) B F m ethod, (b) deflation BF method while deflating the second and th ird dipoles, (c) deflation BF method while deflating the first and th ird dipoles, and (d) deflation BF m ethod while deflating th e first and second dipoles. In the BF approach one source dom inates o th ers an d o ther sources are not reconstructed. T he deflation BF m ethod, however, reconstructs all the sources in three power profile images successfully... 84 5.11 E stim ated m om ents for localized sources in Fig. 5.10 by (a) BF method
and, (b) deflation BF. D otted lines are simulated m om ents and solid lines are estim ated moments. In contrast to the BF, th e deflation BF shows tru e v ariatio n of the moments... 85 5.12 The averaged au d ito ry E R Fs over 1500 trials. A O.lsec pre-stimulus
was used for baseline corrections... 85
second source in axial view, (d) deflation BF while deflating the second source in coronal view, (e) deflation BF while deflating the first source in axial view, and (f) deflation BF while deflating the first source in coronal view. T h e estim ated source locations are shown by cross m ark ers. Due to p artial correlation between the sources, the BF m ethod shows th a t th e location of dipoles are bleeding towards the center of the head. T h e deflation B F, however, shows the tru e location of dipoles on th e prim ary auditory cortex... 86 5.14 E stim ated m om ents of th e real d a ta sources obtained by BF (dotted
lines) and deflation B F (solid lines); (a) first dipole moments and (b) second dipole mom ents. Since BF m ethod localizes the sources closer to th e center of th e brain, its moments have bigger am plitudes compared to those of deflation B F ... 87
6.1 Dipole locations used in simulation studies for (a) locations in axial view, and (b) th e sam e locations in coronal view... 101 6.2 Exam ples of sim ulated data: (a) noiseless moments in x, y and z di
rections, (b) sim ulated noiseless E /M E G for 25 electrodes, and (c) sim ulated noisy E /M E G w ith SNR=-5dB. The GWN was added to the m om ents and th e signals obtained after applying the forward m atrix. 102 6.3 O u tp u t error vs. in p u t SNR for one stationary dipole... 103 6.4 O u tp u t SN R vs. in p u t SNR for two distal sources for (a) th e first
source (SI) and (b) th e second source (S2)... 105 6.5 O u tp u t error vs. in p u t SNR for two proximal sources for (a) th e first
source (S3), and (b) th e second source (S4)... 106 6.6 O u tp u t error vs. in p u t SNR for two deep sources for (a) the first source
6.8 Effect of colored noise on the output location error for (a) the first source, and (b) th e second source... 109 6.9 Effect of real noise on th e outp u t location error for (a) th e first source,
and (b) the second source... 110 6.10 O u tp u t location error vs. SNR for two sim ulated sources. In this
example it is assum ed th a t only one source is generating the the data. I l l 6.11 Original and estim ated dipole trajectories using th e R B P F and B-
R B P F m ethods in (a) axial view and (b) coronal view... 114 6.12 Original and estim ated moments using R B P F and B -R BPF: (a) mo
m ent of th e first dipole in x direction, (b) m om ent of the first dipole in
y direction, (c) m om ent of the first dipole in z direction, (d) moment of the second dipole in x direction, (e) moment of the second dipole in
y direction, and (f) m om ent of the second dipole in z direction. . . . 115
7.1 An exam ple of sim ulated EEG d ata with SNR = -5dB, (a) original num ber 5 trial for m ultichannel EEG, (b) original num ber 55 trial, (c) noisy num ber 5 trial, and (d) noisy number 55 tria l... 133 7.2 An exam ple of estim ated amplitude, latency, and w idth of two ER P
subcom ponents from sim ulated d ata with SNR = 5dB, (a) am plitude, (b) mean, and (c) variance... 134 7.3 E stim ated locations using the proposed method for Fig. 7.2 in (a) axial
and (b) sag ittal view s... 134 7.4 An example of th e estim ated amplitude, latency, and w idth of two ER P
subcom ponents in sim ulated d a ta with SNR = -5dB, (a) am plitude, (b) latency, and (c) w id th ... 135 7.5 E stim ated locations using th e proposed m ethod for Fig. 7.4 in (a) axial
and (b) sag ittal view s... 135
plots. P300 am p litu d e can be seen b etter in th e central sensors. . . . 137 7.7 Average re-referenced and segmented d a ta around P300 (200ms-500ms)
shown w ith red lines and fitted d ata shown w ith blue lines. The lower diagram represents the amplitude and tim e scales for all the plots. T he p aram eters of th e fitted waveforms were used for initializing the algorithm ... 138 7.8 E stim ated am p litu d e, latency, and width in real d a ta for P3a (red
m arker) and P 3 b (blue marker) in (a) axial and (b) sag ittal views. . . 139 7.9 E stim ated locations of P3a (red marker) and P3b (blue marker). . . . 139
Electroencephalography (E EG ) and magnetoencephalography (M EG), which are two of a num ber of neuroim aging techniques, are scalp recordings of the electrical activity of th e brain. EEG and M EG (E /M E G ) have excellent tem poral resolution, they are easy to acquire, and have a wide range of applications in science, medicine and engineering. These valuable signals, however, suffer from poor spatial resolution and in many cases from very low signal to noise ratios. In this study, new com putational m ethods for analyzing and improving the quality of E /M E G signals are presented. We mainly focus on single trial event-related potential (ERP) estim ation and E/M E G dipole source localization. Several m ethods basically based on particle filtering (PF) are proposed.
First, a m ethod using P F for single trial estim ation of E R P signals is considered. In this m ethod, th e wavelet coefficients of each ER P are assumed to be a Markovian process and do no t change extensively across trials. The wavelet coefficients are then estim ated recursively using P F . The results both for sim ulations and real d a ta are compared w ith those of th e well known Kalman Filtering (KF) approach. In the next m ethod we move from single trial estim ation to source localization of E /M E G signals. T he beam form ing (BF) approach for dipole source localization is generalized based on prior inform ation a b o u t th e noise. BF is in fact a spatial filter th a t minimizes the power of all th e signals a t th e o u tp u t of the filter except those th a t come from the locations of interest. In th e proposed m ethod, using two more constraints th an in the classical BF form ulation, the o u tp u t noise powers are minimized and the interference activities are stopped.
P F is also introduced for E /M E G dipole source localization. PF , which can deal with the nonlinearity of th e inverse problem, is capable of localizing and tracking E /M E G sources. T his m ethod is also combined with the B F approach and the re sults are com pared w ith those of BF and recursive applied and projected MUSIC (RAP-M USIC) techniques, which are well-established m ethods in E /M E G source lo calization. The final work is devoted to spatiotem poral separation and identification of E R P subcom ponents. T his m ethod simultaneously estim ates the single trial ER P signals and localizes th e sources. Variational Bayes implies th a t the E R P subcom ponent param eters can be estim ated separately. Therefore, E R P source locations are estim ated using P F , source am plitudes and noise covariance m atrices are estim ated using a m axim um likelihood (ML) approach, and latency and w idth of ER P subcom ponents are estim ated using the Newton-Raphson technique. T he m ethod is verified via simulations and also applied to an oddball paradigm to show its potential use in practical applications.
I would like to th a n k Dr. Sanei and Dr. Wilding, my supervisors, for their many sug gestions and co n stan t su p p o rt during this research. Also, I am thankful to Dr. Evans and Dr. M uthukum arasw am y in Cardiff University Brain Research Imaging Centre (CUBRIC) for collecting and providing the real EEG and MEG data.
Cardiff, Wales Hamid R Mohseni
O ctober 2009
C hapter 4-7 of this thesis represent my original works to my best knowledge unless it is referenced or stated . In brief, these novelties stem from the following contributions:
C h a p ter 4: S t a te S p a c e A p p ro a ch es for S in gle Trial E stim a tio n o f E R P s
Single trial estim ation of E R P is formulated in state space and its recursive solution is given using particle filter.
C h a p ter 5: D e fla tio n B ea m fo r m in g for E /M E G D ip o le S o u rce L o ca liza tio n
T he classical beam form ing is modified by incorporating m ultiple constraints to its m inim ization criterion. T h e solution can be considered as a generalization of pseudo inverse, m axim um likelihood, beamforming, and beamforming w ith loading factor, which are well established m ethods for dipole source localization in E /M E G .
C h a p ter 6: D ip o le L o c a liz a tio n and Tracking o f E /M E G S ou rces v ia J o in t B ea m fo r m in g - R a o -B la c k w e lliz e d P a rticle F ilte rin g
Dipole source localization is form ulated in state space and its solution is given using Rao-Blackwellized P article Filter. Very similar approach has been published in [12]. O ur paper, which was published in Sensor, Array and M ultichannel Signal Processing Conference (SAM), 2008 [77], however, shows th a t these two independent studies have taken place in parallel.
T he m ost im p o rtan t novelty of this chapter is the use of joint beamforming and
Rao-Blackwellized particle filtering for E/M EG source localization, which improves the performance of th e m ethod compared to the Rao-Blackwellized particle filtering method.
C h a p ter 7: V a r ia tio n a l B a y es for S p a tio tem p o ra l Id e n tifica tio n o f E R P S u b c o m p o n e n ts
A new model for tracking E R P subcomponent param eters is proposed and the solution for estim ating p aram eters are given using particle filter, m axim um likelihood, and Newton-Raphson techniques. T he proposed model is more comprehensive than the available models, since it considers not only variable am plitude, latency, and width bu t also variable locations for E R P subcomponents.
L ist o f P u b lic a tio n s
• Hamid R. M ohseni, Edw ard L. Wilding, and Saeid Sanei, ’’Variational Bayes for S patiotem poral Identification of ERP Subcom ponents,” under review, IE E E
Transaction on Biom edical Engineering, 2009.
• Hamid R. M ohseni, E dw ard L. Wilding, and Saeid Sanei, ” A Deflation Based Beamforming A pproach for EEG Multiple Dipole Source Localization,” under second review, IE E E Transaction on Biomedical Engineering, 2009.
• Hamid R. M ohseni, K inaoush Nazarpour, Edward L. W ilding, and Saeid Sanei, ’’Application of P article Filters in Single Trial Event-R elated Potential Estim a tion,” Physiol. Meas. vol. 30, pp 1101-1116, 2009.
• K. N azarpour, H .R. M ohseni, C.W. Hesse, J. A. C ham bers, and S. Sanei ”A Novel Semi-Blind Signal Extraction Approach for the Removal of Eye-Blink A rtifact from E E G s,” E U R A S IP Journal on Advances in Signal Processing, 2008.
• Hamid R. Mohseni, Edward L. Wilding, and Saeid Sanei, ” A New Beamforming- Based MEG Dipole Source Localization M ethod,” submitted to IE E E Interna tional Conf. On Acoustic, Speech and Signal Processing, IC A SS P, 2010.
• Hamid R. Mohseni, and Saeid Sanei, ” A New M ethod for Spatiotem poral Identi fication of Event-R elated Potential Subcomponents,” International Conference o f the IE E E in Medicine and Biology (EMBC), 2009 (invited paper).
• Hamid R. Mohseni, Edward L. Wilding, and Saeid Sanei, ” A Beamforming Particle F ilter for EEG Dipole Source Localization,” IE E E International C onf On Acoustic, Speech and Signal Processing, IC A SSP, 2009 (best student paper in bio-imaging and signal processing area).
• Hamid R. M ohseni, Edward L. Wilding, and Saeid Sanei ’’Sequential Monte Carlo Techniques for EEG Dipole Placing and Tracking,” Fifth IE EE Sensor A r ray and M utichanel Signal processing Signal Processing Workshop, S A M, 2008.
• Hamid R. M ohseni, Edward L. Wilding, and Saeid Sanei, ’’Single Trial Estim a tion of E vent-R elated Potentials Using Particle Filtering” IE E E International
Conf. On Acoustic, Speech and Signal Processing, IC A SSP, 2008.
• Hamid R. Mohseni, Edward L. Wilding, and Saeid Sanei ” Preprocrssing of E vent-R elated Potential Signals via Kalman Filtering and Smoothing,” 15th IE E E International Conference on Digital Signal Processing, D SP, 2007.
A R Autoregressive
B E S A Brain Electrical Source Analysis
B F Beamforming
B O L D Blood O xygenation Level Dependent B - R B P F Beamforming- RB P F
D W T Discrete W avelet Transforms E / M E G EEG and M EG
E C D Electric C u rren t Dipole E C o G Electrocorticogram E E G Electroencephalogram E O G Electro-oculogram
E R P Event-R elated Potentials
fM R I Functional M agnetic Resonance Imaging G W N Gaussian W hite Noise
H E O G Horizontal EOG
i.i.d . Independent and Identically-Distributed IC A Independent Component Analysis K F K alm an Filtering
L C M V Linearly C onstrained Minimum Variance M A P M aximum A Posteriori
M E G M agnetoencephalogram M L M aximum Likelihood
P F P article Filtering
R A P -M U S IC Recursively Applied and Projected MUSIC R B P F Rao-Blackwellized Particle Filter
S B R Signal to Background Ratio
S I R - P F Sam pling Im portance Resampling P F S N R Signal to Noise Ratio
S T - E R P Single Trial Event-Related Potential S V D Singular Value Decomposition V E O G Vertical EOG
t Discrete time index
k Trail index
(.)T Transpose operation (.)! Pseudoinverse 11.112 Frobenious norm
p(.) P robability density function
p (.|.) C onditional probability density function x t S ta te of the system at tim e t
y t M easurem ent vector a t time t (EEG and MEG)
Y*; M easurem ent m atrix a t the A;th trail (EEG and MEG)
ft and ht G eneral nonlinear functions w t S ta te noise
v t M easurem ent noise
Covariance m atrix of the state noise
Q v Covariance m atrix of the measurement noise
A f( a ; , m , P ) G aussian function with mean vector m and covariance m atrix P evaluated a t a
A f ( m , P ) G aussian distribution with mean vector m and covariance m atrix P K t K alm an gain a t tim e t
S(.) D irac d elta function
N N um ber of particles
t”) W eight of n th particle a t time t w
Pi Location of th e zth dipole
H Forward m atrix
m t Moment of th e ith dipole I Identity m atrix
W Linear spatial filter
C y Covariance m atrix of the d ata c „ Covariance m atrix of the noise r Lagrange m ultiplier
P sig n a l Power of the signal p .
1 noise Power of the noise
Rt
M atrix of all dipole locations M t M atrix of all dipole momentsme
Vector of all dipole momentsL N um ber of sensors
In tro d u ctio n
Functional brain imaging is a relatively new and multidisciplinary research field
th a t encompasses techniques devoted to a b etter understanding of the human brain
through noninvasive imaging of the electrophysiological, hemodynamic, metabolic,
and neurochem ical processes. These imaging techniques are powerful tools for study
ing neural processes in the normal and pathological brain functions. Clinical appli
cations include improved understanding and treatm ent of serious neurological and
neuropsychological disorders such as intractable epilepsy, schizophrenia, depression,
and P arkinson’s and Alzheimer’s diseases.
Images of dynam ic changes in the spatial distribution of brain metabolism and
neurochem istry can be formed using positron emission tomography (PET). These
images have sp atial resolutions as high as 2mm. Temporal resolution, however, is
limited to several m inutes because of the dynamics of the processes being studied and
photon-counting noise. For more direct studies of neural activity, local hemodynamic
changes may be investigated. As neurons become active, they induce very localized
changes in blood flow and oxygenation levels th a t can be imaged as a correlate of
neural activity. H em odynam ic changes can be detected using P E T [16], functional
magnetic resonance imaging (fMRI) [113], and transcranial optical imaging [44] m eth
ods. Of these, fMRI is currently the most widely used and can be readily performed
using a 1.5-7T clinical MRI magnet. fMRI studies are capable of producing spatial
resolutions as high as l-3mm, however, tem poral resolution is limited by the relatively
slow hem odynam ic response, compared to electrical neural activity. In addition to
lim ited tem poral resolution, interpretation of fMRI d a ta is hampered by the rather
complex relationship between the blood oxygenation level dependent (BOLD) changes
and th e underlying neural activity. Regions of BOLD changes in fMRI images do not
necessarily correspond one-to-one with regions of electrical neural activity.
Among th e available functional imaging techniques, electroencephalogram (EEG)
and m agnetoencephalogram (MEG) uniquely have temporal resolutions below 1ms.
T his tem poral precision allows us to explore dynamics of neural networks or cell as
semblies th a t occur a t typical time scales in the order of tens of milliseconds [83]. EEG
and MEG (E /M E G ) are two complementary techniques th a t measure, respectively,
th e scalp electric potentials and the magnetic induction outside the head produced
by electrical activity in neural cell assemblies. They directly measure the electrical
brain activity and offer the potential for superior tem poral resolution when compared
to P E T or fMRI. Sampling of electromagnetic brain signals at millisecond intervals is
readily achieved and is limited only by the multichannel analog-to-digital conversion
ra te of th e measurements.
1 .1
A im a n d O b je c tiv e s
There are drawbacks, however, th a t lim it the application of E/M E G . One of the
E /M E G resolution is limited by both the number of spatial measurements and their
location over the scalp. The only way to localize the putative electric sources in the
brain is through the solution of the inverse problem, a problem th a t can only be
solved by introducing a priori assumptions on the generation of E /M E G signals. The
more appropriate these assumptions are the more trustable are the source estima
tions. D uring the last two decades different such assum ptions have been formulated
and im plem ented in inverse solution algorithms. They range from single equivalent
current dipole estim ations to the calculation of three-dimensional (3D) current den
sity distributions. Each approach uses different m athem atical, biophysical, statistical,
anatom ical or functional constraints.
T he m ain shortcom ing of the current source localization methods is th a t the signal
should be statio n ary during the time of processing. Here, using particle filtering (PF)
new m ethods th a t can be applied to the non-stationary signals are proposed. The
conventional beam form ing (BF) approach for source localization is also extended
when the location and covariance m atrix of the noise is known. Furthermore, a new
model for analyzing event-related potentials (ERP) is given and its param eters are
estim ated separately using different methods. In fact, this m ethod is a single trial
estim ation th a t can also localize the source activities.
A nother m ain drawback of E/M E G is its low signal to noise ratio. Conventional
m ethods to improve E R P signal quality involve averaging time-locked segments of the
EEG signals over many trials. These m ethods assume th a t the statistical parameters
of the E R P waves of a given kind remain the same over time and the background EEG
is a random process th a t is attenuated by averaging over trials. There is evidence,
many approaches for estimation of single trials, novel m ethods to investigate the
variability of E R Ps across trials are proposed.
Several techniques are proposed in this study and in the following, a brief overview
of them is presented.
1 .2
T h e s is O u tlin e
T he layout of the thesis is as follows. In C hapter 2, an introduction to EEG and its
acquisition and specifications is presented followed by a brief introduction to ERP
and its application focussing on the P300 component. This chapter ends with an
overview of M EG and its comparison with the EEG signals.
Bayesian filtering and mathematical frameworks for Kalm an filtering (KF) and
P F , which are the m ain bases of different proposed approaches in this study, are
presented in C h ap ter 3.
In C h ap ter 4, an approach for estimation of single trial event-related potentials
(ST -E R Ps) using P F is presented. The method is based on recursive Bayesian mean
square estim ation of E R P wavelet coefficients, which are estim ated sequentially by
their previous estim ates as prior information. To enable a performance evaluation
for th is approach in th e Gaussian and non-Gaussian distributed noise conditions,
G aussian w hite noise (GWN) and real electroencephalogram (EEG) signals is added
to th e sim ulated E R P s and the results are compared to th a t of KF approaches.
In C h ap ter 5, a deflation scheme based on BF for multiple dipole source local
ization of surface E /M E G d a ta is considered. Two more constraints are added to
the conventional B F form ulation and a closed-form solution is given. The first con
be considered as a generalization of pseudo-inverse, maximum likelihood, and loading
factor m ethods which have been applied effectively for E /M E G source localization.
By adding another constraint to the BF formulation, the identified dipoles are de
flated and, simultaneously, the location of the next dipole is identified. This method
is called deflation BF and is capable of detecting highly correlated sources, as well as
sources w ith small power th a t are dominated by other sources. An iterative deflation
and localization method is also proposed to improve the accuracy of the method.
In addition, in C hapter 6, a method based on Rao-Blackwellized particle filtering
(R B P F ) and BF for E /M E G dipole source localization and tracking is presented.
T he localization problem is formulated in state space and P F is employed to pro
vide a recursive nonlinear and non-Gaussian Bayesian solution. The use of Rao-
Blackwellization in combination with P F improves the performance and reduces the
com putational costs. In this approach, the nonlinear part of the model (location) is
estim ated by P F and th e linear p art of the model (the moments) is marginalized out
and estim ated using KF. Further performance improvement can be obtained via joint
beam form ing-R B PF (B-RBPF). In this approach, it is assumed th a t the d a ta is sta
tionary w ithin a window around the current time sample. R B PF and B-RBPF were
applied to different kinds of simulated d a ta and the results were compared with those
obtained by using RAP-M USIC and BF algorithms, which are two well-established
m ethods for dipole source localization.
Finally, in C h ap ter 7 a novel m ethod for the detection and tracking of ERP sub
com ponents from trial to trial is proposed. The ERP subcomponent sources are
(am plitude, latency, and width) are estimated and tracked from trial to trial. Vari
ational Bayes implies th a t the param eters can be estim ated separately using the
likelihood function of each parameter. Estimations of dipole locations, which have
nonlinear relation to the measurement, are given by PF, estim ations of amplitude and
noise covariance m atrix of the measurement are optimally given by maximum likeli
hood (ML) approach, and estimations of latency and width of the Gaussian functions
are given by Newton-Raphson technique. New recursive m ethods are introduced for
both ML and Newton-Raphson approaches to prevent the divergence of the filtering
in the presence of very low signal to noise ratio (SNR). The main advantage of the
In tro d u ctio n to E E G and M E G
In this chapter, an introduction to EEG and its acquisition is given first and then
a brief review of ER Ps focusing on the P300 component is presented. In the final
section, MEG is introduced and compared with EEG.
2 .1
E le c t r o e n c e p h a lo g h y
EEG is the recording of electrical activity from over the scalp produced by firing
th e neurons w ithin the brain. These electrical activities can provide records of brain
activity a t any reasonable scale of temporal resolution. In comparison with the other
neuroim aging techniques, EEG can be acquired easily and inexpensively. Because of
these advantages, it has been widely employed in hum an monitoring and research by
workers from different fields. For instance, EEG has been used for clinical diagnosis
of epilepsy. It has also attracted engineers to provide a method for interfacing brain
and machine, which can improve the life style of disabled people. In addition to the
medical applications, EEG has been recently employed in game industries and many
commercial p ro d u cts based on EEG have been produced.
EEG has several advantages as a tool for investigation of the brain activities. For
instance, EEG is non-invasive, convenient to acquire and inexpensive. EEG also
has a high tem poral resolution compared to other techniques such as fMRI and is
capable of detecting changes in electrical activity in the brain in a millisecond time
scale. Furtherm ore, EEG measures the brain ’s electrical activity directly, while other
m ethods record the changes in blood flow (e.g., SPECT, fMRI) or metabolic activity
(e.g. P E T ), which are indirect markers of brain electrical activity.
EEG has several limitations. The most im portant one is its poor spatial resolution
which is lim ited by the number and location of the electrodes. Another im portant
lim itation is th a t some particular sets of neurons make more contribution to EEG
signals (those which are located in the superficial layers of cortex and generate radial
currents tow ard the skull) than the others (those which are located in deep structures
such as the hippocam pus and produce currents tangential to the skull).
2 .1 .2
E E G A c q u isitio n
In conventional scalp EEG, the recording is obtained by placing silver/silver chloride
electrodes on th e scalp using a conductive gel or paste, usually after preparing the
scalp area by light abrasion to reduce the impedance. Electrode locations and names
are specified by th e international 10-20 system in most clinical and research applica
tions. An exam ple of th e electrode locations and names in a two-dimensional map
has been shown in Fig. 2.1.
E x tra electrodes are sometimes used for the measurement of EOG, ECG, and EMG
were placed above and below the eye for vertical EOG (VEOG) and two others placed
left and right of the eye for horizontal EOG (HEOG). VEOG and HEOG usually are
used as a cue to remove eye blink and movement related artifacts from the signals.
Figure 2.1: Scalp electrode positions in a 10-20 system used for recording the available d a ta set.
As an example in epilepsy surgery, it may be necessary to insert electrodes near the
surface of th e brain. This is referred to as electrocorticography (ECoG), intracranial
EEG (I-EEG) or subdural EEG (SD-EEG). The ECoG signal is processed in the
same m anner as digital scalp EEG, with a couple of caveats. ECoG is typically
recorded a t higher sam pling rates than scalp EEG because of the requirements of
Nyquist theorem - the subdural signal is composed of a higher predominance of higher
frequency com ponents. Also, many of the artifacts which affect scalp EEG do not
im pact ECoG, and therefore the linear filtering in preprocessing stage is often not
2 .1 .3
R e fe r e n c in g
Electric potentials are defined only with respect to a reference, i.e., an arbitrarily
chosen zero level. The choice of the reference may differ depending on the purpose of
the recording and has to be selected in advance. In referential montage, each channel
represents the difference between a certain electrode and a predefined reference elec
trode. T here is no standard position a t which this reference is always placed. Midline
positions are often used since they do not amplify the signal in one hemisphere versus
th e other. In this work, the Fz (midline frontal) site as the reference is used in on
line recording. This referencing was changed (off-line) to another popular reference
called linked mastoids - a m athematical average of electrodes attached to both left
and right m astoids. W ith digital EEG, all signals are typically digitized and stored
usually w .r.t a particular referential montage and the signals using other montages
can be m athem atically constructed.
A nother ty p e of montage is average reference montage, which is especially useful
for localization of the sources. In this montage, the outputs of all of the channels are
averaged, and th e averaged signal is used as a common reference for each channel.
2 .1 .4
P r e p r o c e s s in g
A linear bandpass filter is usually used to remove the noise. Typical settings for the
highpass and lowpass cut-off frequencies are 0.5-1 Hz and 35-70 Hz, respectively. The
highpass p a rt typically filters out the slow artifacts, such as electrogalvanic signals
and m ovem ent-related artifact, whereas the lowpass part filters out the high-frequency
artifacts, such as electromyogram signals. An additional notch filter is sometimes
the available recording, the frequency bandwidth of the linear bandpass filter was
0.03-40Hz and the sampling rate was set to 250Hz. The d a ta are digitized using a 16
bit analog to digital convertor with 250Hz sampling rate which satisfies the Nyquist
theorem for an EEG with a bandwidth within 0.03-40Hz.
D uring the recording, EEG signals undergo slow shifts over tim e such th a t the
zero level might differ considerably across channels. These signal shifts can be due
to brain activity, but can also be caused by sweating (in the case of EEG), muscle
tension, or other noise sources. It would be therefore desirable to have a time range
where one can reasonably assume th a t the brain is not producing any stimulus related
activity, and th a t any shift from the zero line is likely due to noise. In most studies,
this baseline interval is defined as several tens or hundreds of milliseconds preceding
th e stim ulus - in the available d ata set, a 0.15sec pre-stimulus interval was used for
baseline correction. For each recorded channel, the mean signal over this interval
is com puted, and subtracted from the signal at all time points. This procedure is
usually referred to as baseline correction.
Eye blink is one of the m ajor artifacts in a laboratory environment. In many
applications of ERP, the recorded trials are visually inspected and those containing
eye blink are removed from analysis. Another useful and semi autom atic method,
which has been used in th e available d ata set, is independent component analysis
(ICA) [112]. In this m ethod, after applying ICA, the component(s) including eye
blink are set to zero and they are back projected to obtain a set of eye blink-free
2 .2
E v e n t - R e la t e d P o t e n t ia ls
E R P is any measured brain response th a t is directly the result of a thought or percep
tion. M ore formally, it is any stereotyped electrophysiological response to an internal
or external stimulus. ERP experiments usually involve a subject being provided a
stim ulus to which s/h e reacts either overtly or covertly. There are often at least two
conditions th a t vary in some manner of interest to the researcher. As this stimulus-
response is going on, an EEG is being recorded from the subject. The ERP is obtained
by segm enting and averaging the EEG signal for each of the trials within a certain
condition; averages from one stimulus-response condition can then be compared with
averages from th e responses from other stimulus.
T here is considerable interest in the EEG techniques concerned with ER P record
ing and analysis since they deal with brain functional and mental abnormalities and
can be used as indicators of cognitive processes and dysfunctions which are not ac
cessible in behavioral testings.
2 .2 .1
P 3 0 0
In this section P300 which is a well-known ER P component is explained. This com
ponent is used to validate the proposed methods in this thesis. P300 is a positive
wave th a t occurs approxim ately 300ms after a rare or task-relevant stimulus, which
can be auditory, visual, or somatosensory. P300 is the most widely used ERP because
of th e relatively large am plitude (5 — 20/xV) and easy acquisition.
T h e paradigm used to elicit a P300 is the presentation of unexpected and infre
quent stim uli random ly interspersed between frequent stimuli to an attentive sub
stim uli, i.e. to count the infrequent target, or to press a button whenever an infre
quent stim ulus occurs. An example of auditory oddball paradigm in which frequent
and infrequent tones are presented to participants has been shown in Fig.2.2.
ODDBALL P300
/
E a s y S tim u lu s D iscrim in atio nrS
s
Figure 2.2: O ddball paradigm. Left: auditory sequences in which S refers to frequent tones and T refers to infrequent tones. Right: thick line is the result of averaging for the infrequent tones and thin line for the frequent tones.
T h e am plitude of P300 is inversely related to the stimulus occurrence probability
and directly related to the task difficulty. The latency of P300 correlates to some
extent w ith categorization or evaluation of the stimulus and consequently is related
to th e task difficulty. Concerning subject param eters, the latency of P300 shows
a positive correlation with age and negative correlation with level of attention and
vigilance. For more detailed information on P300 the reader may refer to a number
of com prehensive reviews [18, 28, 50].
T he topography of P300 recorded by surface electrodes shows a maximum over
the m idline of centro-parietal regions (Pz site). The generator sites of the P300 are
not known w ith certainty, bu t the available d a ta leads to the conclusion th a t several
cortical and sub-cortical structures contribute to this positive wave [87].
In clinical studies a prolongation of the latency has been reported in dementia [36],
Park in so n ’s disease [105], H untington’s disease [43] as well as in patients with chronic
reported in schizophrenics [93], in depressed patients, and in chronic alcoholics [17].
Note th a t m any of these findings are restricted to the group of patients and they can
not be used to detect individual dysfunctions.
P 3 0 0 su b c o m p o n e n t
T h e origin and number of responsible sources of P300 are unknown, however, it
is assumed P300 has two subcomponents: P3a and P3b. There is good evidence to
believe th a t P300 subcomponents arise from interactions between frontal and parietal
neural sources - P 3a is in frontal and P3b is in parietal loci. There is also localization
work which justifies this assumption th at the P300 d a ta can be modeled accurately
by sources placed in anterior and in posterior-parietal cortex [42],
2 .3
M a g n e to e n c e p h a lo g r a m a n d it s C o m p a r is o n w ith
E E G
MEG is an imaging technique used to measure the magnetic fields produced by elec
trical activity in th e brain via extremely sensitive devices such as super conducting
quantum interference devices (SQUIDs). These measurements are commonly used
in both research and clinical settings. There are many uses for the MEG, including
assisting surgeons in localizing a pathology, assisting researchers in determining the
function of various p arts of th e brain, neuro-feedback, and many others.
Although EEG and MEG are generated by the same neurophysiologic processes,
there are im p o rtan t differences concerning the neurogenesis of MEG and EEG [17].
the skull and scalp, which results in a better spatial resolution of the MEG. As
electric and magnetic fields are oriented perpendicular to each other, the directions of
highest sensitivity, usually the directions of the field maxima, are orthogonal to each
other. W hereas scalp EEG is more sensitive to radial than tangential components of
a cu rren t source in a spherical volume conductor, MEG detects only its tangential
com ponents. This shows th at MEG selectively measures the activity in the sulci,
whereas scalp EEG measures activity both in the sulci and a t the top of the cortical
gyri b u t appears to be dominated by radial sources.
Scalp EEG is sensitive to extracellular volume currents produced by postsynaptic
potentials, MEG prim arily detects intracellular currents associated with these synap
tic potentials because the field components generated by volume currents tend to
cancel out in a spherical volume conductor [7]. The decay of magnetic fields as a
function of distance is more pronounced than for electric fields. MEG is therefore
more sensitive to superficial cortical activity, which should be useful for the study of
neocortical epilepsy. Finally, MEG is reference-free which is in contrast to scalp EEG
where an active reference can lead to serious difficulties in the interpretation of the
d ata.
Despite th e above differences between EEG and MEG data, the mathematical
form ulation of bo th d a ta provided in this thesis is the same, and the proposed method
can similarly be applied to both data. Throughout this thesis, therefore, E/M EG
abbreviation which stan ds for both EEG and MEG d ata is used. Depending on the
application, however, the proposed methods are applied either to the real EEG or
S in g le Trial E stim a tio n and D ip o le
S ou rce L o ca liza tio n for E /M E G : a
L itera tu re S u rvey
This chapter is devoted to the literature survey of two major approaches for E/M EG
signal processing. The first approach is single trial estimation and the next is dipole
source localization. In addition, in Section 3.3 mathematical theory of KF and PF,
as th e foundations of proposed approaches in the following chapters, are presented.
T hroughout this thesis, plain italics indicate scalars, lowercase boldface italics indicate
vectors, uppercase boldface indicates m atrices and tiled uppercase boldface indicates
higher order m atrices.
3.1
S in g le T ria l E s tim a tio n o f E R P s
Conventional m ethods for analyzing ERPs typically involve time-locked averaging
over many trials. The assum ption underlying this approach is th a t the background
EEG as a random process is atten u ated by averaging, if the results of averaging is to
be an accurate reflection of the activity elicited on individual trials, the positive- and
negative-going E R P m odulations (components) for all the trials m ust have the same
onset latencies, durations, and amplitudes. Recent studies have indicated th a t there is
single-trial variability in E R P s due to environmental and cognitive factors th a t might
include fatigue, h ab itu ation , and changes in levels of attention [51]. These factors are
of course not m utually exclusive, b u t the key point here is th a t the ST-ERP enables
capturing the changes in th e signals of interest due to neurophysiological changes,
which are lost when using conventionally averaged measures. Therefore, researchers
in the signal processing com m unity have proposed a number of m athem atical meth
ods to e x tra ct th e E R P inform ation as much as possible. The proposed approaches
for detection of E R P param eters (basically amplitude and latency) can generally be
categorized into single- and multi-channel based methods. In the following sections
an overview of m ajor approaches in each category is presented.
3 .1 .1
S in g le -C h a n n e l S in g le Trial E stim a tio n
There are num erous studies reporting approaches in the past three decades for ERP
detection and tracking using only one channel. Several of these approaches are cate
T he sim plest model for the ER P can be
y* = sfc + v fc (3.1)
where and Sfc are the measured EEG and the ER P at the A;th trial, respectively,
and Vfc is th e additive noise. If it is assumed th a t Sfc remains constant from trial
to trial and the noise Vfc is Gaussian zero mean, the result of ensemble averaging of
yfc over a num ber of trials is an optim um solution for Sfc and also can be considered
as th e ML estim ation. However, in the laboratory environment, Vfc is generally the
background EEG which is a non-stationary and non-Gaussian noise.
Pioneering studies such as [70] employed a time-invariant linear minimum mean
squared error filter to estim ate Sfc in equation (3.1) based on the auto- and cross
covariance values of th e background EEG (as noise) and ERPs (as signals of interest).
Tim e-varying [122] and W iener [13] filtering are the other approaches th a t have been
widely utilized. T h e fundam ental problem in these methods, however, is obtaining
an estim ato r for th e cross-covariance m atrix between the d a ta Sfc and the noise Vfc.
A nother intensively studied technique for estim ation of Sfc based on equation (3.1)
is adaptive filtering. M ost attention has been paid to the use of least mean square
(LMS) algorithm in th e filtering problem (e.g. [14] and [106]).
T h e pitfall of such approaches, however, lies in considering the ERP signal as
a statio n ary process: E R Ps are superpositions of transient responses with changing
tem poral and spectral components.
T h e next model for ER P is
Here only the latency rk is varying and shape of ERPs are constant. Equation (3.2)
is the m ost utilized model in analyzing ERPs after the model in equation (3.1).
Peak picking [9] is a straightforw ard method for finding the latency variability
(jitter). This m ethod simply looks for the largest positive peak in the ER P component
tim e interval in the lowpass filtered trials. The latency of this peak is then considered
to be the single-trial latency.
A well-known approach for ST -ER P estimation based on equation (3.2) has been
proposed by Woody [118]. His approach was to cross correlate a predetermined tem
plate against a sequence of samples of the response in order to estim ate the signal
latency. The individual responses were then corrected for their average latency varia
tions and an average response computed. The Woody technique results in an average
signal which is obtained by removing the estimated latency variation before aver
aging. A lthough this m ethod of analysis represents a significant step forward over
conventional averaging, some inform ation inherent in the signal, such as independent
shifts in latency and am plitude in the components of the individual responses are still
buried in the noise.
Similarly, in [86] th e ML estim ator of the model in equation (3.2) in the frequency
dom ain was form ulated yielding estim ators for latency variability.
In [48] th e perform ances of the above three methods ([9, 118, 86]) as well as an
extension of th e la tte r m ethod were studied. Performance of all methods critically
depended on th e signal-to-noise ratio, however, there was some advantage for the
more sophisticated m ethods (particularly [86] and its extension), when signal-to-noise
In the next model E R Ps have invariant shapes, while their latencies and ampli
tudes can vary across trials. These assumptions results
where ak is th e am plitude of E R P component at the kth. trial. The ML estimation of
this equation has been given in [47]. A restriction of this paper is th a t it is based on
the assum ption of uncorrelated background noise, whereas it is known for instance
th a t the alpha rhythm is bo th tem porally and spatially correlated over channels (e.g.
[80] and [81]). Moreover, in [107], Truccolo et al. dem onstrate th a t neglecting trial-
to-trial shape variations results in an estim ate of the background noise of which the
variance is non-stationary over th e time interval of interest.
In the next model th e E R P a t each trial is assumed to be superimposed of several
components:
where q is the num ber of components. Each component is assumed to have a trial-
invariant waveform, b u t a trial-dependent amplitude scaling factors and latency shifts.
A m aximum a posteriori (M AP) solution of such a multi-component ERP model via
an iterative algorithm is presented in [108]. The method is called differentially variable
com ponent analysis (dVCA). The dVCA method is implemented in the time domain
and due to th e m atched-filtering operation in the method, the white noise assumption
may affect th e perform ance of the estimator.
T he frequency-dom ain solution of the dVCA method has been carried out in [119].
T he ongoing activity is herein assumed to be an autoregressive (AR) random process.
y k = aks(t -I- T*) + v fc (3.3)
(3.4)
This assum ption reduces the number of unknowns significantly and in contrast to the
dVCA solution, the resulting ML estim ator provides a certain solution.
T here are several other studies which consider a linear observation model for ERP
as:
y k = U 0 k + v fc (3.5)
T his model connects the vector of sampled measurements y k with the param eters 9 k
and th e m easurem ent errors v. H is a m atrix th a t does not contain param eters to
be estim ated and should be selected a priori. W ith a linear observation model the
choice of the observation m atrix H has a significant role in ST-ERP estimation. For
instance, to consider accurately any nonlinear p art of the model such as latency may
considerably increase the size of H .
A Bayesian approach for realizing a robust estim ator for param eters of equation
(3.5) has been carried out in [53]. In this approach, using subspace regularization, the
second-order statistical information extracted from a set of measured potentials was
used to represent a priori knowledge in the estim ation procedure. It was shown th at,
using this approach, the latencies of some E R P peaks can be estimated accurately.
T his approach does not, however, yield reliable estimates of peak amplitudes, mainly
due to unpredictable fluctuations of the background EEG.
K F has also been employed for estim ation of the ERPs based on equation (3.5)
in [34]. One of th e main advantages of the proposed method is considering the non-
statio n arity of E R P s from trial to trial. The method uses, however, a trivial model
assum ing th a t H is an identity m atrix. Furthermore, the method utilizes the entire
trial as th e in p u t to the filter and this high dimensional input would impair the
and th e results com pared with those of the conventional averaging method in [76].
O ther techniques employed for ST-ERP estimation include param etric methods [59,
60], clustering analysis [38], m atching pursuit [121], matched filtering [19, 94], and
wavelet transform s [8, 68, 90].
It is noteworthy th a t all these techniques focus solely on characterizing the tem
poral trial to trial signal variability and ignore spatial information whereby inefficient
and less accurate estim ates are achieved.
3 .1 .2
M u ltic h a n n e l S in g le Trial E stim a tio n
Some of the approaches explained in the previous section for single channel have been
extended to m ultichannel m easurements. For instance, a multichannel extension of
ML estim ator [47] for th e model in equation (3.3) was applied in [22], where the
ongoing brain activity was modeled with a stationary autoregressive (AR) tim e series.
The estim ator accounts for spatially and temporally correlated background noise th at
is superim posed on th e brain responses.
Furtherm ore, a m ultichannel extension of the dVCA method for the multiple com
ponent model (equation (3.4)) has recently appeared in [57]. Multichannel subspace
regularization [53] given for equation (3.5) has also been carried out in [91].
O ther m ultichannel estim ation of ER P includes principal component analysis
(PCA ) m ethods [3, 15]. It is evident th a t the performances of these methods are
somehow lim ited by some signal source properties such as stationarity and these
m ethods are also unable to retain their performance when the SNR is low.
ICA based m ethods [52, 61, 94] have also been widely exploited in ERP extrac
which can be well violated in presence of intrinsic correlated noise sources.
A nother im p o rtan t approach for single trial estim ation is considering the spa-
tiotem poral inform ation of ERP data. Several methods have been devoted to this
idea by modeling the E R P sources using the ECD model:
<7
Y * = H (pw )M fcii(ptjj) + N* (3.6)
1 = 1
where p k i and are th e location and moments of the zth ECD a t the A;th trial,
respectively. N*, is th e spatiotem poral additive noise at the kth trial. Moment is
a m atrix th a t shows th e strength and orientation of the ECD in three dimensional
space for each trial.
In [23] and [97] spatiotem poral measurements are incorporated in a spherical head
model using th e ECD model in equation (3.6). The dipoles are assumed to have fixed
locations and orientations, whereas their strengths are allowed to change in time
according to either a param etric [97] or nonparam etric [23] model. In [98] only the
dipole position is fixed, and the orientation and strength are allowed to vary in time
according to a p aram etric model. A method is also presented in [27] for spatially
correlated noise betw een the sensors with unknown spatial covariances. Similar to
this approach, a m ethod was presented in [80], in which the noise covariance and the
dipole param eters are estim ated separately.
T he m ain draw back of all of these spatiotem poral m ethods is th a t the dipole loca
tions are assum ed to rem ain the same during the course of recording. In Chapter 7,
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