Tutorial_Machine Learning and Signal Processing Tools for BCI

8/2/2019 Tutorial_Machine Learning and Signal Processing Tools for BCI

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Machine Learning and Signal Processing Tools for BCI

Klaus-Robert Mller, Benjamin Blankertz, Gabriel Curio et al.


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BBCI team:

Gabriel Curio

Florian LoschVolker KunzmannFrederike Holefeld

Vadim Nikulin@Charite

Andreas ZieheFlorin Popescu

Christian GrozeaSteven LemmMotoaki Kawanabe

Guido Nolte@FIRST

Yakob Badower@Pico Imaging

Marton Danozcy

Benjamin Blankertz

Michael TangermannClaudia Sannelli

Carmen VidaurreBastian VenthurSiamac FazliMartijn Schreuder

Matthias TrederStefan Haufe

Thorsten DickhausFrank MeineckePaul von Bnau

Marton DanozcyFelix BiessmannKlaus-Robert Mller@TUB

Matthias KrauledatGuido Dornhege

Roman Krepki@industry

Collaboration with: U Tbingen, Bremen, Albany, TU Graz, EPFL, Daimler, Siemens, MES, MPIs,U Tokyo, TIT, RIKEN, Bernstein Focus Neurotechnology, Bernstein Center for ComputationalNeuroscience Berlin, picoimaging, Columbia, CUNY

Funding by: EU, BMBF and DFG


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Noninvasive Brain-Computer Interface

DECODING


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Brain Pong with BBCI


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[From Birbaumer et al.] [From Pfurtscheller et al.]

Noninvasive BCI: clinical applications

Brain- Computer Interface

Signal Processing

EEGAcquisition

EEGAcquisition

ApplicationInterface

ApplicationInterface

FES DeviceGrasp-Pattern

3 channelStimulation

BBCI: Leitmotiv: let the machines learn


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EEG based noninvasive BCI


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The cerebral cocktail party problem

use ICA/NGCAprojections for artifactand noise removal

feature extraction andselection

[cf. Ziehe et al. 2000, Blanchard et al. 2006]


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BBCI paradigms

- healthy subjects (BCI untrained) perform "imaginary movements (ERD/ERS)

- instruction: imagine

- squezzing a ball,

- kicking a ball,

- feel touch

Leitmotiv: let the machines learn


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Towards imaginations: Modulation of Brain Rhythms

IMAGINATION of left arm

Single channel


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Variance I: Single-trial vs. Averaging

Single channel


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Variance II: Trial to trial variability


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Variance III: inter subject variability [l vs r]


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BCI with machine learning: training


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BBCI paradigms

- healthy subjects untrainedfor BCI

A: training 20min: right/left hand imagined movements

infer the respective brain acivities (ML & SP)

B: online feedback session

Leitmotiv: let the machines learn


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Playing with BCI: training session (20 min)


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Machine learning approach to BCI: infer prototypical pattern

Inference by CSP Algorithm


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BCI with machine learning: feedback


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Lecture Blankertz here


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BBCI Set-up

Artifact removal

[cf. Mller et al. 2001, 2007, 2008, Dornhege et al. 2003, 2007, Blankertz et al. 2004, 2005, 2006, 2007, 2008]


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What can Machine Learning tell us about physiology?

[cf. Blankertz et al. 2001, 2006]

11


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ML for knowledge discovery


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Results: Exploring the limits of untrained users


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BCI: 1st session performance for novices


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Spelling with BBCI: a communication for the disabled I


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Variance IV: covariate shift: from training to feedback

Need for adaptation !

[cf. Sugiyama & Mller 2005, Shenoy et al. 2005,Sugiyama et al. 2007]

N h i l i l l i


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Neurophysiological analysis

[cf. Krauledat et al. submitted]

W i ht d Li R i f i t hift ti


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Weighted Linear Regression for covariate shift compensation

yields unbiased estimator even underconvariate shift

, choosing

[cf. Sugiyama & Mller 2005, Sugiyama et al. 2007]

BCI Illiteracy


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y

Screening Study (N=80):

Cat I: good calibration (cb), good feedback (fb)

Cat II: good cb, no good fb

Cat III: no good cb

Percentage of ~20% of nave users:BCI accuracy does not reach level criterion,i.e., control not accurate enough to control applications

design a predictor !

SMR Predictor


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SMR-Predictor

Calculate the power spectral density (PSD) in three Laplacian channels C3, Cz,C4 under rest cond.

Model each resulting curve by g = g1 + g2, with

g1 = g1 (x,, k) =k1 +k2/ x (estimated noise)

g2 = g2(x, , , k) =k3(x ; 1 , 1) + k4 (x ; 2, 2) (2 peaks)

Proposed predictor: Average height of the larger peak

SMR Predictor (Results)


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SMR-Predictor (Results)

As much as R

2

= 30% (calibration) and R

2

= 26% (feedback) of variability in BCI accuracycan be explained by the SMR predictor in our study sample!

Pearson R = 0.55 Pearson R = 0.51

[cf. Blankertz, Kbler, Mller et al. 2009]


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Approach to Cure BCI Illiteracy

Runs1-

3

Runs4

-6

Runs

7-

8

fixed

Laplace

CSP +sel.Lap.

CSP

supervised

unsupervis

ed

Direct feedback -> Unspecific LDA classifier. Each trial, perform adaptation of the cls.

Features: log band power (alpha and beta).

Laplacian channels C3, C4 and Cz.

Compute CSP and sel. Laps. from runs 1-3. Fixed CSP filters, automated laps. selection. Each trial retrain the classifier.

Compute CSP from runs 4-6.

Perform unsupervised adaptation of pooled mean.

Update the bias of the classifier.

[cf. Vidaurre, Blankertz, Mller et al. in preparation]

Results (Grand Averages)


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Results (Grand Averages)

C


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Example: one subject of Cat. III

!Runs 1 and 2 Runs 7 and 8

[cf. Vidaurre, Blankertz, Mller et al. 2009]

Real Man Machine Interaction


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Real Man Machine Interaction

[Tangermann, Mller et al 2009]

Harvest from BCI-gaming


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Harvest from BCI-gaming

Research platform:BCI gaming +

Machine Learning +

Single Trial Analysis (MSM)

Limits of BCI-reaction time?

Mental statesJUST BEFORE

success or failure?

Dynamics:limits of temporal

control precision?

Before-after


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Future issues: sensors

Popescu et al 2007

Future Issues: Shifting distributions within experiment


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g p

Conclusion


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Conclusion

BBCI: non-invasive with high Information transfer rates for the Untrained

BBCI: Untrained, Calibration < 20min, data analysis


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BBCI team:


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Gabriel Curio

Florian Losch

Volker KunzmannFrederike Holefeld

Vadim Nikulin@Charite

Andreas ZieheFlorin Popescu

Christian GrozeaSteven LemmMotoaki Kawanabe

Guido Nolte@FIRST

Yakob Badower@Pico Imaging

Marton Danozcy

Benjamin Blankertz

Michael TangermannClaudia SannelliCarmen Vidaurre

Bastian VenthurSiamac FazliMartijn Schreuder

Matthias TrederStefan Haufe

Thorsten DickhausFrank Meinecke

Paul von BnauMarton DanozcyFelix BiessmannKlaus-Robert Mller@TUB

Matthias KrauledatGuido Dornhege

Roman Krepki@industry

Collaboration with: U Tbingen, Bremen, Albany, TU Graz, EPFL, Daimler, Siemens, MES, MPIs,U Tokyo, TIT, RIKEN, Bernstein Focus Neurotechnology, Bernstein Center for ComputationalNeuroscience Berlin, picoimaging, Columbia, CUNY

Funding by: EU, BMBF and DFG

BCI Competitions


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For BCI IV Competition see www.bbci.de


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FOR INFORMATION SEE: www.bbci.de

Machine Learning opensource software initiative:MLOSS see

www.jmlr.org


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M a c h i n e L e a r n i n g a n d S i g n a l P r o c e s s i n g T o o l s f o r

B r a i n - C o m p u t e r I n t e r f a c i n g

B e n j a m i n B l a n k e r t z

1,2

K l a u s - R o b e r t M l l e r

1

1M a c h i n e L e a r n i n g L a b o r a t o r y , B e r l i n I n s t i t u t e o f T e c h n o l o g y

2F r a u n h o f e r F I R S T ( I D A )

b l a n k e r @ c s . t u - b e r l i n . d e

0 8 - J u l - 2 0 0 9

T o p i c o f T h i s S e c t i o n
http://[email protected]/


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M e t h o d :

c l a s s i c a t i o n o f s p a t i o - t e m p o r a l f e a t u r e s ;

s h r i n k a g e o f t h e s a m p l e c o v a r i a n c e m a t r i x t o c o u n t e r b a l a n c e

t h e e s t i m a t i o n b i a s

A p p l i c a t i o n :

c l a s s i c a t i o n o f s i n g l e - t r i a l E R P s i n a n a t t e n t i o n - b a s e d s p e l l e r

S o m e N e u r o p h y s i o l o g i c a l B a c k g r o u n d


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A n i n f r e q u e n t s t i m u l u s i n a s e r i e s o f s t a n d a r d s t i m u l i e v o k e s a P 3 0 0

c o m p o n e n t a t c e n t r a l s c a l p p o s i t i o n i f a t t e n d e d :

Cz

N1

P3

Cz

T h e p r e s e n t a t i o n o f a v i s u a l s t i m u l u s e l i c i t s a V i s u a l E v o k e d

P o t e n t i a l ( V E P ) i n v i s u a l c o r t e x i f f o c u s e d :

Oz

Oz

N1

P1

E x p e r i m e n t a l D e s i g n


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C l a s s i c M a t r i x S p e l l e r A t t e n t i o n - b a s e d H e x - o - S p e l l

S e e P o s t e r W 0 7 ( T r e d e r e t a l . ) f o r a i n v e s t i g a t i o n o f o v e r t v s .

c o v e r t a t t e n t i o n a n d a c o m p a r i s o n o f t h o s e t w o s p e l l e r d e s i g n s .

E x p e r i m e n t a l D e s i g n


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C l a s s i c M a t r i x S p e l l e r A t t e n t i o n - b a s e d H e x - o - S p e l l

S e e P o s t e r W 0 7 ( T r e d e r e t a l . ) f o r a i n v e s t i g a t i o n o f o v e r t v s .

c o v e r t a t t e n t i o n a n d a c o m p a r i s o n o f t h o s e t w o s p e l l e r d e s i g n s .

S i n g l e - s u b j e c t E R P s i n H e x - o - S p e l l


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D a t a s e t f o r i l l u s t r a t i o n o f c l a s s i c a t i o n m e t h o d s :

AF3

Target

Nontarget

AF4

F7

Fz

F8

FT7

FC3 FC4

FT8

C5 Cz C6

TP7

CP3 CP4

TP8Pz

PO7 PO8

Oz5 V

+

200 ms

T o p o g r a p h i e s o f E R P C o m p o n e n t s


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T h e r e a r e s e v e r a l E R P c o m p o n e n t s t h a t c a n b e u s e d t o d e t e r m i n e

t h e a t t e n d e d s y m b o l :

100 0 100 200 300 400 500

5

0

5

Cz () PO7+PO8 ()

ms

[V]

Target

Nontarget

5

0

5

5

0

5

Target

Nontarget

vP1

115135 ms

N1

135155 ms

vN1

155195 ms

P250

205235 ms

vN2

285325 ms

P400

335395 ms

N500

495535 ms

C l a s s i c a t i o n o f T e m p o r a l F e a t u r e s


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A s a r s t s t e p : c l a s s i c a t i o n o n r a w t i m e c o u r s e s ( 1 1 5 5 3 5 m s ) i n

s i n g l e c h a n n e l s . T h e r e s u l t i s d i s p l a y e d a s s c a l p m a p :

error[%]

15

20

25

30

35

40

45

50

E x t r a c t i o n o f S p a t i a l F e a t u r e s


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22 22.2 22.4

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

dev

23.323.423.5 23.623.7

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

24.524.6 24.724.8 24.9

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

25.8 26 26.2

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

dev

27 27.2 27.4

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

E x t r a c t i o n o f S p a t i a l F e a t u r e s


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22 22.2 22.4

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

dev

23.323.4 23.523.6 23.7

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

24.5 24.6 24.7 24.8 24.9

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

25.8 26 26.2

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

dev

27 27.2 27.4

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

std

T h e r2

- M a t r i x o f D i e r e n c e s


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T h e t e m p o r a l a n d s p a t i a l s t r u c t u r e o f t h e d i e r e n c e b e t w e e n E R P s

o f d i e r e n t c o n d i t i o n s c a n b e i n v e s t i g a t e d b y t h e s i g n e d r2 - m a t r i x :

[ms]

c

anne

s

100 0 100 200 300 400 500

Fz

FCz

Cz

CPz

Pz

Oz0.15

0.1

0.05

0

0.05

0.1

0.15

115 135 ms 135 155 ms 155 195 ms 205 235 ms 285 325 ms 335 395 ms 495 535 ms

r(x) :=

N1 N2

N1 + N2

mean{xi | yi = 1} mean{xi | yi = 2}std{xi}

S p a t i a l F e a t u r e s


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#01: std #02: std #03: std #04: std #05: dev #06: std

#07: std #08: dev #09: std #10: std #11: std #12: dev

#13: std #14: std #15: std #16: std #17: dev #18: std

#19: std #20: std #21: std #22: std #23: dev #24: dev

L i n e a r C l a s s i e r a s S p a t i a l F i l t e r


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A l i n e a r c l a s s i e r t h a t w a s t r a i n e d o n s p a t i a l f e a t u r e s c a n a l s o b e

r e g a r d e d a s a s p a t i a l l t e r .

L e t w b e t h e L D A w e i g h t v e c t o r a n d X R# c h a n s # t i m e p o i n t s b ec o n t i n u o u s E E G s i g n a l s . T h e n

Xf := wX R1#t i m e p o i n t s

i s t h e r e s u l t o f s p a t i a l l t e r i n g : e a c h c h a n n e l o f

Xi s w e i g h t e d w i t h

t h e c o r r e s p o n d i n g c o m p o n e n t o f w

a n d s u m m e d u p .

T h e w e i g h t v e c t o r o f t h e

c l a s s i e r c a n b e d i s p l a y a s

s c a l p m a p :

0.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

C l a s s i c a t i o n R e s u l t s f o r S p a t i a l F e a t u r e s


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5

0

5

vP1 N1 vN1 P250 vN2 P400 N500

115135 ms 135155 ms 155195 ms 205235 ms 285325 ms 335395 ms 495535 ms

10

15

20

25

30

35

error[%]

E x t r a c t i o n o f S p a t i o - T e m p o r a l F e a t u r e s


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22 22.1 22.2 22.3 22.4 22.5 22.6

FC3

FCz

FC4

C3

Cz

C4

CP3

CPz

CP4

Pz

[s]

dev

S p a t i o - T e m p o r a l F e a t u r e s


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S p a t i o - t e m p o r a l f e a t u r e s a r e t y p i c a l l y h i g h - d i m e n s i o n a l ( h e r e

5 9 E E G c h a n n e l s

7 t i m e i n t e r v a l s = 4 1 3 d i m e n s i o n a l f e a t u r e s ) :

mean of time interval [ms]

space

[channe

ls]

125 145 175 220 305 365 515

Fz

FCz

Cz

CPz

Pz

Oz

[V]

15

10

5

0

5

10

15

C l a s s i c a t i o n R e s u l t f o r S p a t i o - T e m p o r a l F e a t u r e s


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5

0

5

vP1 N1 vN1 P250 vN2 P400 N500

115135 ms 135155 ms 155195 ms 205235 ms 285325 ms 335395 ms 495535 ms

10

15

20

25

30

35

error[%]

Combined

A l t h o u g h i n f o r m a t i o n w a s a d d e d , c l a s s i c a t i o n o n t h e c o n c a t e n a t e d

f e a t u r e b e c o m e s w o r s e : o v e r t t i n g .

B i a s i n E s t i m a t i n g C o v a r i a n c e s


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L e t x1, . . . ,xn Rd b e n v e c t o r s d r a w n f r o m a d - d i m e n s i o n a l G a u s s i a n d i s t r i b u t i o n

N(,)

.

F o r c l a s s i c a t i o n a n d h a v e t o b e e s t i m a t e d f r o m t h e d a t a :

= 1n

nk=1 xk

= 1n1

nk=1(xk )(xk )

B u t , i f t h e n u m b e r o f s a m p l e s n i s n o t l a r g e r e l a t i v e t o t h e d i m e n s i o n d, t h e e s t i m a t i o n i s e r r o r - p r o n e . T h e r e i s a s y s t e m a t i c a l b i a s :

L a r g e E i g e n v a l u e s o f a r e t o o l a r g e

S m a l l E i g e n v a l u e s o f a r e t o o s m a l l

T h i s a e c t s , e . g . , c l a s s i c a t i o n w i t h L D A :

N o r m a l v e c t o r o f L D A : w = 1(1 2) .

B i a s i n E s t i m a t i n g C o v a r i a n c e s ( 2 )


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2 1 0 1 2

1.5

1

0.5

0

0.5

1

1.5 true covempirical cov

0 50 100 150 200

0

50

100

150

200

250

index of Eigenvector

Eigen

value

true

N=500

N=200

N=100

N= 50

A R e m e d y f o r C l a s s i c a t i o n


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A s i m p l e w a y t h a t c a n p a r t l y x t h e b i a s i s s h r i n k a g e : t h e

e m p i r i c a l c o v a r i a n c e m a t r i x i s m o d i e d t o b e m o r e s p h e r i c a l .

I n L D A t h e e m p i r i c a l c o v a r i a n c e m a t r i x i s r e p l a c e d b y

() = (1 )+ If o r a [0, 1] a n d d e n e d a s a v e r a g e E i g e n v a l u e trace(Si)/d.

A R e m e d y f o r C l a s s i c a t i o n


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A s i m p l e w a y t h a t c a n p a r t l y x t h e b i a s i s s h r i n k a g e : t h e

e m p i r i c a l c o v a r i a n c e m a t r i x i s m o d i e d t o b e m o r e s p h e r i c a l .

I n L D A t h e e m p i r i c a l c o v a r i a n c e m a t r i x i s r e p l a c e d b y

() = (1 )+ If o r a [0, 1] a n d d e n e d a s a v e r a g e E i g e n v a l u e trace(Si)/d.S i n c e i s p o s i t i v e s e m i - d e n i t e w e c a n h a v e a n E i g e n v a l u e

d e c o m p o s i t i o n

= VDVw i t h o r t h o n o r m a l

Va n d d i a g o n a l

D.

F r o m

= (1 )VDV + I = V ((1 )D+ I)V

w e s e e t h a t

() a n d h a v e t h e s a m e E i g e n v e c t o r s ( c o l u m n s o f V )

e x t r e m e E i g e n v a l u e s ( l a r g e / s m a l l ) a r e s h r u n k / e x t e n d e d

t o w a r d s t h e a v e r a g e .

= 0y i e l d s L D A w i t h o u t s h r i n k a g e ,

= 1a s s u m e s s p h e r i c a l

c o v a r i a n c e m a t r i c e s .

M o d e l s e l e c t i o n


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L D A w i t h s h r i n k a g e o f t h e e m p i r i c a l c o v a r i a n c e m a t r i x h a s o n e f r e e

p a r a m e t e r ( ) , a l s o c a l l e d h y p e r p a r a m e t e r , t h a t n e e d s t o b e s e l e c t e d . T h e r e i s n o g e n e r a l w a y t o d o i t .

N u m e r o u s s t r a t e g i e s w i t h d i e r e n t p r o p e r t i e s e x i s t , e . g .

e m p i r i c a l B a y e s s h r i n k a g e e s t i m a t o r

M D L : M i n i m u m D e s c r i p t i o n L e n g t h

M o d e l - s e l e c t i o n b a s e d o n c r o s s - v a l i d a t i o n .

. . .

A n e a s y ( a n d a l s o t i m e - c o n s u m i n g ) w a y i s m o d e l - s e l e c t i o n b a s e d o n

c r o s s - v a l i d a t i o n .

R e g u l a r i z e d L D A a t W o r k


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C r o s s - v a l i d a t i o n r e s u l t s f o r d i e r e n t s i z e s o f t r a i n i n g d a t a ( 2 5 0 ,

5 0 0 , 2 0 0 0 ) f o r d i e r e n t v a l u e s o f t h e r e g u l a r i z a t i o n p a r a m e t e r

( x - a x i s ) . F e a t u r e s v e c t o r s h a v e 2 5 0 d i m e n s i o n s .

0 0.2 0.4 0.6 0.8 15

10

15

20

25

regularization parameter

crossvalidationerror[%]

250

0 0.2 0.4 0.6 0.8 15

10

15

20

25



500

0 0.2 0.4 0.6 0.8 15

10

15

20

25



2000

I n v e s t i g a t i n g t h e I m p a c t o f S h r i n k a g e


67/106

L D A : w = 1(1

2) ; s h r i n k a g e : () = (1

)+ I

= 1

w 1 2



68/106

L D A : w = 1(1

2) ; s h r i n k a g e : () = (1

)+ I

= 0

w 1(1 2) = 1

w 1 2



69/106

L D A : w = 1(1

2) ; s h r i n k a g e : () = (1

)+ I

= 0

w 1(1 2)

a c c o u n t i n g f o r

s p a t i a l s t r u c t u r e o f

t h e n o i s e

= 1

w 1 2

E R P a n d N o i s e

S i m p l e a s s u m p t i o n f o r E R P s : s i n g l e t r i a l xk(t) i s c o m p o s e d o f a n


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( )E R P s(t) a n d G a u s s i a n ` n o i s e ' nk(t) :

xk(t) = s(t) + nk(t) f o r a l l t r i a l s k = 1, . . . , K

10 5 0 5 108

6

4

2

0

2

4

6

8

10

potential at FCz [V]

potentialatCPz

[V

]

mean: phaselocked ERP

cov: nonphaselocked noise

S p a t i a l S t r u c t u r e o f t h e N o i s e


71/106

T h e t w o s t r o n g e s t p r i n c i p a l c o m p o n e n t s o f t h e n o i s e ( c o v a r i a n c e

m a t r i x ) i n t h i s d a t a s e t :

T r i a l - t o - t r i a l v a r i a t i o n o f P 3

V i s u a l a l p h a

U n d e r s t a n d i n g S p a t i a l F i l t e r s

( a )

ith littl di t b

( b )


72/106

10 5 0 5 108

6

4

2

0

2

4

6

8

10


potentialatCPz

[V]

with little disturbance

0.2

0.4 CPz

FCz

Oz

U n d e r s t a n d i n g S p a t i a l F i l t e r s

( a )

i h li l di b

( b )

ith di t b f O


73/106

10 5 0 5 108

6

4

2

0

2

4

6

8

10


potentialatCPz

[V]

with little disturbance

10 5 0 5 108

6

4

2

0

2

4

6

8

10


potentialatCPz

[V]

with disturbance from Oz

T w o c h a n n e l c l a s s i c a t i o n o f ( a ) : 1 5 % e r r o r , ( b ) : 3 7 % e r r o r

W h e n d i s t u r b i n g c h a n n e l O z i s a d d e d t o t h e d a t a ( 3 D ) : 1 6 % e r r o r .

H e r e , c h a n n e l O z i s r e q u i r e d f o r g o o d c l a s s i c a t i o n a l t h o u g h i t s e l f i s

n o t d i s c r i m i n a t i v e .

I m p a c t o f S h r i n k a g e o n t h e S p a t i a l F i l t e r s


74/106

W i t h i n c r e a s i n g s h r i n k a g e , t h e s p a t i a l l t e r s ( c l a s s i e r ) l o o k

s m o o t h e r , b u t c l a s s i c a t i o n m a y d e g r a d e w i t h t o o m u c h s h r i n k a g e .

= 0 = 0.001 = 0.005 = 0.05 = 0.5 = 1

15

20

25

30

35

40

er

ror[%]

M a p s o f s p a t i a l l t e r s f o r d i e r e n t v a l u e s o f .

A N o v e l A n a l y t i c a l M e t h o d


75/106

R e c e n t l y , a m e t h o d t o a n a l y t i c a l l y c a l c u l a t e t h e o p t i m a l s h r i n k a g e

p a r a m e t e r w a s p u b l i s h e d ( [ 1 ] ) .

T h a n k s t o N i c o l e K r m e r f o r p o i n t i n g t h e B B C I g r o u p t o t h i s

m e t h o d .

O p t i m a l S e l e c t i o n o f S h r i n k a g e P a r a m e t e r

L e t x1, . . . ,xn Rd b e n f e a t u r e v e c t o r s a n d l e t = 1nn

k=1 xk


76/106

b e t h e e m p i r i c a l m e a n .

A i m : g e t a b e t t e r e s t i m a t e o f t h e t r u e c o v a r i a n c e m a t r i x

( e s p e c i a l l y i n c a s e n < d ) t h a n t h e s a m p l e c o v a r i a n c e m a t r i x = 1

n1

nk=1(xk )(xk ) b y s e l e c t i n g a i n

() := (1 )+ I.

O p t i m a l S e l e c t i o n o f S h r i n k a g e P a r a m e t e r

L e t x1, . . . ,xn Rd b e n f e a t u r e v e c t o r s a n d l e t = 1nn

k=1 xk


77/106

b e t h e e m p i r i c a l m e a n .

A i m : g e t a b e t t e r e s t i m a t e o f t h e t r u e c o v a r i a n c e m a t r i x

( e s p e c i a l l y i n c a s e n < d ) t h a n t h e s a m p l e c o v a r i a n c e m a t r i x = 1

n1

nk=1(xk )(xk ) b y s e l e c t i n g a i n

() := (1 )+ I.W e d e n o t e b y (xk)i r e s p . ()i t h e i - t h e l e m e n t o f t h e v e c t o r xkr e s p .

. F u r t h e r m o r e w e d e n o t e b y

sij t h e e l e m e n t i n t h e i- t h r o w

a n d j - t h c o l u m n o f . W e d e n e

zij(k) = ((xk)i ()i) ((xk)j ()j)T h e n t h e o p t i m a l s h r i n k a g e p a r a m e t e r f o r w h i c h () = argminS||S||2F c a n b e a n a l y t i c a l l y c a l c u l a t e d ( [ 2 ] ) a s

=n

(n 1)2

di,j=1 vark(zij(k))

i=j s2ij +

i(sii )2

R e s u l t o f C l a s s i c a t i o n w i t h S h r i n k a g e


78/106

5

0

5

vP1 N1 vN1 P250 vN2 P400 N500

115135 ms 135155 ms 155195 ms 205235 ms 285325 ms 335395 ms 495535 ms

10

15

20

25

30

35

error[%]

Combined

LDA

with

shrinkage 5%

U s i n g s h r i n k a g e t h e c l a s s i c a t i o n e r r o r c o u l d b e d r a s t i c a l l y r e d u c e d

t o 4 % .

R e s u l t s f o r t h e C l a s s i c M a t r i x S p e l l e r

100bae


79/106

1 2 3 4 5 6 7 8 9 100

20

40

60

80

acc

uracy[%]

intensification [#]

sab

sac

sad

sae

saf

sag

sah

saisaj

sak

daj

sal

mean

A c c u r a c y i n l e t t e r s e l e c t i o n , c h a n c e l e v e l : 3 . 3 3 % .

R e s u l t s f o r t h e C l a s s i c M a t r i x S p e l l e r

100bae


80/106

1 2 3 4 5 6 7 8 9 100

20

40

60

80

acc

uracy[%]

intensification [#]

sab

sac

sad

sae

saf

sag

sah

saisaj

sak

daj

sal

median

S e e P o s t e r W 0 7 ( T r e d e r e t a l ) , a n d f o r o t h e r a p p l i c a t i o n s o f t h i s

t e c h n i q u e A 0 2 ( T h u r l i n g s e t a l ) , W 1 0 ( H h n e e t a l ) .

S u m m a r y o f S p a t i o - T e m p o r a l C l a s s i c a t i o n

L i n e a r c l a s s i c a t i o n w i t h s h r i n k a g e i s a p o w e r f u l m e t h o d .


81/106

C o m p l e t e s h r i n k a g e ( = 1) m e a n s n e g l e c t i n g t h e s t r u c t u r e o f

t h e n o i s e . I n t h i s c a s e t h e c l a s s i e r i s t h e d i e r e n c e o f t h e

E R P s .

T h e a p p r o p r i a t e n e s s o f a l i n e a r s e p a r a t i o n d e p e n d s o n t h e w a y

f e a t u r e s a r e e x t r a c t e d a n d t r a n s f o r m e d .

I n c o n t r a s t t o n o n - l i n e a r c l a s s i e r s , t h e w e i g h t s o f a l i n e a r

c l a s s i e r a r e i n f o r m a t i v e .

T h e w e i g h t s o f t h e t r a i n e d c l a s s i e r c a n b e v i s u a l i z e d a s a s e q u e n c e

o f s c a l p t o p o g r a p h i e s :

[250 ms] [150 ms] [50 ms] [50 ms] [150 ms] [250 ms]

0.02

0

0.02

T o p i c o f T h i s S e c t i o n


82/106

M e t h o d :

C l a s s i c a t i o n o f s p e c t r a l f e a t u r e s , n a m e l y m o d u l a t i o n s o f t h e

a m p l i t u d e i n s p e c i c f r e q u e n c y b a n d s .

I n p a r t i c u l a r , C o m m o n S p a t i a l P a t t e r n ( C S P ) a n a l y s i s t o

c l a s s i f y d i e r e n t c o n d i t i o n s t h a t a r e c h a r a c t e r i z e d b y a

m o d u l a t i o n o f t h e a m p l i t u d e o f b r a i n r h y t h m s ( [ 3 , 4 ] ) .

A p p l i c a t i o n :

C l a s s i c a t i o n o f m o t o r i m a g e r y c o n d i t i o n s i n a B C I p a r a d i g m .

N e u r o p h y s i o l o g y : S e n s o r i m o t o r R h y t h m s


83/106

right hemisphereleft hemisphere

right

hand hand

left

20 30 4010 50SpectralPo

wer[dB]

Frequency [Hz]

~ 1/f

Spectrum over left hand area

hand at rest

C3 C4Cz

N e u r o p h y s i o l o g y : S e n s o r i m o t o r R h y t h m s


84/106

right hemisphereleft hemisphere

right

hand hand

left

20 30 4010 50

Spectrum over left hand area

imagined hand movement

SpectralPo

wer[dB]

Frequency [Hz]

~ 1/f

C3 C4Cz

I m a g i n i n g a m o v e m e n t o f a l i m b c a u s e s a l o c a l b l o c k i n g o f t h e

c o r r e s p o n d i n g s e n s o r i m o t o r r h y t h m ( S M R ) , s e e [ 5 , 6 , 7 ] .

A v e r a g e T o p o g r a p h y o f I d l e S M R

F o r e a c h L a p l a c e l t e r e d c h a n n e l i n a r e l a x r e c o r d i n g , t h e s t r e n g t h


85/106

o f t h e l o c a l r h y t h m w a s e s t i m a t e d . T h e g r a n d a v e r a g e o v e r 8 0

p a r t i c i p a n t s i s d i s p l a y e d a s t o p o g r a p h i c m a p p i n g :

[dB]

3.5

4

4.5

5

5.5

6

6.5

C o n c l u s i o n

L o c a t i o n s C 3 a n d C 4 a r e g o o d c a n d i d a t e s t o o b s e r v e S M R

m o d u l a t i o n s . T h e s e c o v e r t h e s e n s o r i m o t o r a r e a s o f t h e r i g h t a n d

t h e l e f t h a n d .

S p a t i a l S m e a r i n g

R a w E E G s c a l p p o t e n t i a l s a r e k n o w n t o b e a s s o c i a t e d w i t h a


86/106

l a r g e s p a t i a l s c a l e o w i n g t o v o l u m n e c o n d u c t i o n .

I n a s i m u l a t i o n o f N u n e z e t a l [ 8 ] o n l y h a l f t h e c o n t r i b u t i o n t o

o n e s c a l p e l e c t r o d e c o m e s f r o m s o u r c e s w i t h i n a 3 c m r a d i u s .

0

0.1667

0.3333

0.5

0.6667

0.8333

1

T h e N e e d f o r S p a t i a l F i l t e r i n g

raw: CP4 bipolar: FC4CP4 CAR: CP4


87/106

5 10 15 20 25 30

15

20

25

30

[Hz]

[dB]

5 10 15 20 25 30

10

15

20

25

[Hz]

[dB]

5 10 15 20 25 30

10

15

20

25

[Hz]

[dB]

5 10 15 20 25 30

0

5

10

15

[Hz]

[dB]

Laplace: CP4

5 10 15 20 25 30

0

5

10

[Hz]

[dB]

CSP

leftright

0

0.2

0.4

0.6r2

A n a l y s i s o f M o t o r I m a g e r y C o n d i t i o n s : S p e c t r a

VPk 08 08 07/i VPk l ft / i ht N 66/63 [5 40] H [ 15 25] dB


88/106

left

right

10 dB+

10 Hz

P6 larP4 lar

10 20 30 40

Pz larP3 larP5 lar

CP4 larCP2 larCP1 larCP3 lar

C4 larC2 larCz larC1 larC3 lar

F4 larFz larF3 lar

VPkg_08_08_07/imag_arrowVPkg: left / right, N= 66/63, [5 40] Hz [15 25] dB

0.2

0

0.2

sgn r2

F i r s t s t e p : d e t e r m i n e a s u i t a b l e f r e q u e n c y b a n d t h a t s h o w s g o o d

d i s c r i m i n a t i o n b e t w e e n t h e c o n d i t i o n s .

E R D C u r v e s o f M o t o r I m a g e r y

F4 larFz larF3 lar

VPkg_08_08_07/imag_arrowVPkg: left / right, N= 66/63, [500 6000] ms [2 1] V

sgn r2


89/106

left

right1 V

+

2000 ms

P6 larP4 larPz larP3 larP5 lar

CP4 larCP2 larCP1 larCP3 lar

C4 larC2 larCz larC1 larC3 lar

0.20.10

0.10.2

sgn r

S e c o n d s t e p : d e t e r m i n e a s u i t a b l e t i m e i n t e r v a l d u r i n g w h i c h

d i s c r i m i n a t i o n i s m o s t p r o m i n e n t .

R e m a r k : S i m u l t a n e o u s s e l e c t i o n o f f r e q u e n c y b a n d a n d i n t e r v a l i s

m o r e a p p r o p r i a t e .

C o m m o n S p a t i a l P a t t e r n ( C S P ) A n a l y s i s

G o a l : F i n d s p a t i a l l t e r s t h a t o p t i m a l l y c a p t u r e m o d u l a t i o n s o f


90/106

b r a i n r h y t h m s

O b s e r v a t i o n : p o w e r o f a b r a i n r h y t h m v a r i a n c e o f b a n d - p a s s l t e r e d s i g n a l .

min variancefor

min variancefor

unknownsources

discriminativesignals

V1

left

right

EEGno classspecificinfluenceon variance

csp:L1,2,...

csp:R1,2,...

V

projection filter

forward

model model

backward

observedsignals

C S P A n a l y s i s

T h e g o a l o f C S P :

right left right


91/106

2425 2430 2435 [s]

csp:R

csp:L

right left right

C S P a n a l y s i s y i e l d s s p a t i a l l t e r s t h a t c a n b e v i s u a l i z e d :

CSP

filter

right

CSP

filter

left

C S P M o r e P r a c t i c a l

E E G - s i g n a l s d u r i n g m o t o r i m a g e r y , b a n d - p a s s l t e r e d ( h e r e 9 1 3 H z ) :

left rightright


92/106

C4

C3

728 732 734 736 738730

left rightright

[s]

L

:= XL

XL

R

:= XR

XR

X L

( T

c h a n s ) ( T

c h a n s )

X R

r i g h t d a t a l e f t d a t a

VL

V = D & V(L

+R

)V = I

1 ) c h o o s e e i g e n v e c t o r vi f r o m V h a v i n g a l a r g e e i g e n v a l u e di w . r . t . L .

CSP

728 730 732 734 736 738 [s]

right rightleft

var(XL

vi) = dil a r g e

var(XR

vi) = 1di s m a l l

I n M a t l a b : [ V , D ] = e i g ( S i g m a 1 , S i g m a 1 + S i g m a 2 ) .

C S P M o r e P r a c t i c a l

E E G - s i g n a l s d u r i n g m o t o r i m a g e r y , b a n d - p a s s l t e r e d ( h e r e 9 1 3 H z ) :

left rightright


93/106

C4

C3

728 732 734 736 738730

left rightright

[s]

L

:= XL

XL

R

:= XR

XR

XL

( T

c h a n s ) ( T

c h a n s )

XR

r i g h t d a t a l e f t d a t a

VL

V = D & V(L

+R

)V = I

2 ) c h o o s e e i g e n v e c t o r vi f r o m V h a v i n g a s m a l l e i g e n v a l u e di w . r . t . L .

CSP

728 730 732 734 736 738 [s]

left right right

var(XL

vi) = dis m a l l

var(XR

vi) = 1di l a r g e

I n M a t l a b : [ V , D ] = e i g ( S i g m a 1 , S i g m a 1 + S i g m a 2 ) .

T r a i n i n g C S P - b a s e d C l a s s i c a t i o n

T o o b t a i n f e a t u r e s f r o m t h e C S P l t e r e d E E G , i n e a c h c h a n n e l a n d

t r i a l , t h e v a r i a n c e a c r o s s t i m e i s c a l c u l a t e d a n d t h e l o g a r i t h m i s


94/106

a p p l i e d . T h i s i s a s c a t t e r p l o t o f t h e r e s u l t i n g C S P f e a t u r e s :

3 2 1 0 1

3

2

1

0

1

log(var(CSPL))

log(var(CSP

R))

leftright

H e r e , o n l y t w o d i m e n s i o n s a r e s h o w n . N o t e , t h a t a p p l y i n g t h e

l o g a r i t h m t o t h e b a n d p o w e r f e a t u r e s m a k e s t h e d i s t r i b u t i o n m o r e

G a u s s i a n a n d t h e r e f o r e e n h a n c e s l i n e a r s e p a r a b i l i t y .

T r a i n i n g C S P - b a s e d C l a s s i c a t i o n


95/106

D e t e r m i n e m o s t d i s c r i m i n a t i v e f r e q u e n c y b a n d ,

b a n d - p a s s l t e r E E G i n t h a t b a n d ,

e x t r a c t s i n g l e t r i a l s u s i n g a n a p p r o p r i a t e t i m e i n t e r v a l ,

c a l c u l a t e a n d s e l e c t C S P l t e r s ,

a n d a p p l y t h e m t o E E G s i n g l e t r i a l s ,

c a l c u l a t e t h e l o g v a r i a n c e w i t h i n t r i a l s .

T h i s r e s u l t s i n a l o w d i m e n s i o n a l f e a t u r e v e c t o r f o r e a c h t r i a l

( d i m e n s i o n a l i t y e q u a l s n u m b e r o f s e l e c t e d C S P l t e r s ) .

T r a i n a l i n e a r c l a s s i e r l i k e L D A o n t h e f e a t u r e s .

( S i n c e t h e s e f e a t u r e s a r e l o w - d i m e n s i o n a l , s h r i n k a g e i s t y p i c a l l y

n o t n e c e s s a r y . )

S u m m a r y : T r a i n i n g C S P - b a s e d C l a s s i c a t i o n

12

C4 lap

0.4

C4 lap


96/106

( 1 )

5 10 15 20 25 30 35 40 45

2

4

6

8

10

12

[Hz]

[dB]

left

right

( 2 )

0 1000 2000 3000 4000 5000

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

[ms]

[V]

left

right

( 4 )

3 2 1 0 1

3

2

1

0

1

log(var(CSPL))

log(var(CSPR

)

)

leftright

( 3 )

csp1 [0.68] csp2 [0.64] csp3 [0.60] csp4 [0.71] csp5 [0.68] csp6 [0.61]

28.2 17.0 39.3 15.5

A p p l y i n g C S P - b a s e d C l a s s i c a t i o n


97/106

P r o j e c t E E G w i t h s p a t i a l C S P l t e r s a n d a p p l y b a n d - p a s s l t e r ,

c a l c u l a t e t h e v a r i a n c e i n s h o r t w i n d o w s ( e . g . l a s t 5 0 0 m s ) ,

t a k e t h e l o g a r i t h m ,

a n d a p p l y t h e c l a s s i e r w e i g h t i n g .

R e m a r k : O n e n i c e f e a t u r e o f C S P i s t h a t t h e l e n g t h o f t h e

c l a s s i c a t i o n w i n d o w c a n b e c h a n g e d a t r u n t i m e ( i . e . d u r i n g

f e e d b a c k ) .

F o r m o r e t h e o r e t i c a l c o n s i d e r a t i o n s a s w e l l a s p r a c t i c a l h i n t s s e e [ 3 ] .

S e c t i o n : C a v e a t s i n V a l i d a t i o n

W h e n m a c h i n e l e a r n i n g t e c h n i q u e s a r e u s e d f o r c l a s s i c a t i o n o f


98/106

E E G s i n g l e - t r i a l s , t h e e x p e c t e d p e r f o r m a n c e o f a m e t h o d h a s t o b e

e v a l u a t e d c a r e f u l l y , a n d t h e r e a r e s e v e r a l p o s s i b l e p i t f a l l s .

T h e e s t i m a t i o n o f g e n e r a l i z a t i o n p e r f o r m a n c e r e q u i r e s a t r a i n i n g

a n d a t e s t s e t . T h e e s t i m a t i o n i s o n l y p r o p e r

i f t h e t e s t s e t w a s n o t u s e d i n a n y w a y t o d e t e r m i n e

p a r a m e t e r s o f t h e m e t h o d , a n d

i f t h e s a m p l e s i n t h e t e s t s e t a r e i n d e p e n d e n t f r o m t h e s a m p l e s

i n t h e t r a i n i n g s e t .

A l t h o u g h t h e s e p r i n c i p l e s a r e q u i t e o b v i o u s , i t h a p p e n s t h a t t h e y

a r e v i o l a t e d .

U n f o r t u n a t e l y , e v e n s o m e p u b l i s h e d j o u r n a l a r t i c l e s l a c k a p r o p e r

v a l i d a t i o n o f t h e p r o p o s e d m e t h o d s .

H a l l o f P i t f a l l s i n S i n g l e - T r i a l E E G A n a l y s i s


99/106

p r e p r o c e s s i n g m e t h o d s t h a t u s e s t a t i s t i c s o f t h e w h o l e d a t a s e t

l i k e I C A , o r n o r m a l i z a t i o n o f f e a t u r e s

( p a r t i c u l a r l y s e v e r e f o r m e t h o d s t h a t u s e l a b e l i n f o r m a t i o n )

f e a t u r e s a r e s e l e c t e d o n t h e w h o l e d a t a s e t , i n c l u d i n g t r i a l s

t h a t a r e l a t e r i n t h e t e s t s e t

s e l e c t p a r a m e t e r s b y c r o s s v a l i d a t i o n o n t h e w h o l e d a t a s e t

a n d r e p o r t t h e p e r f o r m a n c e f o r t h e s e l e c t e d v a l u e s

a r t i f a c t s / o u t l i e r s a r e r e j e c t e d f r o m t h e w h o l e d a t a s e t

( r e s u l t i n g i n a s i m p l i e d t e s t s e t )

u n s u c i e n t v a l i d a t i o n f o r p a r a d i g m s w i t h b l o c k d e s i g n

I n t h i s p r e s e n t a t i o n w e h i g h l i g h t t h e l a s t i s s u e .

B l o c k D e s i g n


100/106

A s s u m e t h e t a s k i s t o d i s c r i m i n a t e b e t w e e n m e n t a l s t a t e s i n

d i e r e n t c o n d i t i o n s .

W e s a y t h a t a n e x p e r i m e n t h a s a b l o c k d e s i g n , i f t h e p e r i o d s f o r

w h i c h t h e r e i s n o a l t e r n a t i o n b e t w e e n c o n d i t i o n s a r e l o n g e r t h a n

t h e i n t e n d e d c h a n g e o f s t a t e s i n o n l i n e o p e r a t i o n .

cond #1 cond #1cond #2 cond #2cond #1 cond #2 cond #1

block #1 block #2 block #3 block #4 block #5 block #6 block #7

A p r o b l e m a r i s e s , i f t h e p e r f o r m a n c e i s e s t i m a t e d f o r s u c h a d a t a

s e t b y c r o s s v a l i d a t i o n .

S l o w l y C h a n g i n g V a r i a b l e s

Training set


101/106

Test set

I n E E G t h e r e a r e m a n y s l o w l y c h a n g i n g v a r i a b l e s o f b a c k g r o u n d

a c t i v i t y , t h e r e f o r e t h e s i n g l e - t r i a l s a r e n o t i n d e p e n d e n t . F o r a n

o r d i n a r y c r o s s v a l i d a t i o n i n a b l o c k d e s i g n d a t a s e t , t h e r e q u i r e m e n t

o f i n d e p e n d e n c e b e t w e e n t r a i n i n g a n d t e s t s e t i s v i o l a t e d .

A V a l i d a t i o n T e s t

T o d e m o n s t r a t e i m p a c t o f b l o c k d e s i g n i n c r o s s v a l i d a t i o n , w e

p e r f o r m c r o s s v a l i d a t i o n i n t h e f o l l o w i n g s e t t i n g . T a k i n g a n


102/106

a r b i t r a r y E E G d a t a s e t , w e a s s i g n f a k e l a b e l s ( r e g a r d l e s s o f w h a t

h a p p e n e d d u r i n g t h e r e c o r d i n g ) l i k e t h i s :

n B l o c k s P e r C l a s s = 1 :

class #2class #1


class #1 class #2class #1class #2


class #1 class #2 class #1 class #2class #2 class #1

a n d s o o n .

R e s u l t s o f t h e V a l i d a t i o n T e s t

F r o m e a c h b l o c k s i n g l e - t r i a l s a r e e x t r a c t e d o f l e n g t h 1 s . T h i s

p r o c e d u r e w a s p e r f o r m e d f o r 8 0 E E G d a t a s e t s . B l u e b o x p l o t s s h o w


103/106

t h e r e s u l t s o f c r o s s - v a l i d a t i o n :

1 2 3 4 5 10

0

10

20

30

40

50

60

70

testerror

[%]

number of blocks per class

R e s u l t s o f t h e V a l i d a t i o n T e s t

F r o m e a c h b l o c k s i n g l e - t r i a l s a r e e x t r a c t e d o f l e n g t h 1 s . T h i s

p r o c e d u r e w a s p e r f o r m e d f o r 8 0 E E G d a t a s e t s . B l u e b o x p l o t s s h o w


104/106

t h e r e s u l t s o f c r o s s - v a l i d a t i o n :

1 2 3 4 5 10

0

10

20

30

40

50

60

70

testerror

[%]

number of blocks per class

F o r c o m p a r i s o n , r e s u l t s f o r l e a v e - o n e - b l o c k - o u t v a l i d a t i o n a r e

s h o w n i n g r e e n .

F u r t h e r C o m m e n t s a n d S u m m a r y


105/106

T h e s e v e r e n e s s o f t h e u n d e r e s t i m a t i o n o f t h e t r u e e r r o r

d e p e n d s o n t h e c o m p l e x i t y o f t h e f e a t u r e s a n d t h e c l a s s i e r .

C r o s s v a l i d a t i o n i n b l o c k d e s i g n d a t a m i g h t a l s o g i v e t h e

c o r r e c t r e s u l t b u t a l t e r n a t i v e e v a l u a t i o n i s r e q u i r e d .

T h e s i t u a t i o n g e t s w o r s e i f t r i a l s a r e e x t r a c t e d f r o m

o v e r l a p p i n g s e g m e n t s .

T h e m o s t r e a l i s t i c v a l i d a t i o n i s t o t r a i n t h e m e t h o d s o n t h e

r s t N 1 r u n s a n d t o e v a l u a t e o n t h e l a s t r u n . L e a v e - o n e - b l o c k - o u t a n d l e a v e - o n e - r u n - o u t h a v e l a r g e r

s t a n d a r d e r r o r s t h a n c r o s s v a l i d a t i o n .

R e f e r e n c e s

O . L e d o i t a n d M . W o l f , A w e l l - c o n d i t i o n e d e s t i m a t o r f o r l a r g e - d i m e n s i o n a l c o v a r i a n c e m a t r i c e s ,

J M u l t i v a r A n a l , 8 8 ( 2 ) : 3 6 5 4 1 1 , 2 0 0 4 .

J . S c h f e r a n d K . S t r i m m e r , A S h r i n k a g e A p p r o a c h t o L a r g e - S c a l e C o v a r i a n c e M a t r i x E s t i m a t i o n

a n d I m p l i c a t i o n s f o r F u n c t i o n a l G e n o m i c s , S t a t i s t i c a l A p p l i c a t i o n s i n G e n e t i c s a n d M o l e c u l a r


106/106

B i o l o g y , 4 ( 1 ) , 2 0 0 5 .

B . B l a n k e r t z , R . T o m i o k a , S . L e m m , M . K a w a n a b e , a n d K . - R . M l l e r , O p t i m i z i n g S p a t i a l F i l t e r s

f o r R o b u s t E E G S i n g l e - T r i a l A n a l y s i s , I E E E S i g n a l P r o c M a g a z i n e , 2 5 ( 1 ) : 4 1 5 6 , 2 0 0 8 , U R L

h t t p : / / d x . d o i . o r g / 1 0 . 1 1 0 9 / M S P . 2 0 0 8 . 4 4 0 8 4 4 1 .

H . R a m o s e r , J . M l l e r - G e r k i n g , a n d G . P f u r t s c h e l l e r , O p t i m a l s p a t i a l l t e r i n g o f s i n g l e t r i a l E E G

d u r i n g i m a g i n e d h a n d m o v e m e n t , I E E E T r a n s R e h a b E n g , 8 ( 4 ) : 4 4 1 4 4 6 , 2 0 0 0 .

C . N e u p e r a n d W . K l i m e s c h , e d s . , E v e n t - r e l a t e d D y n a m i c s o f B r a i n O s c i l l a t i o n s , E l s e v i e r , 2 0 0 6 .

G . P f u r t s c h e l l e r , C . B r u n n e r , A . S c h l g l , a n d F . L . d a S i l v a , M u r h y t h m ( d e ) s y n c h r o n i z a t i o n a n d

E E G s i n g l e - t r i a l c l a s s i c a t i o n o f d i e r e n t m o t o r i m a g e r y t a s k s , N e u r o I m a g e , 3 1 ( 1 ) : 1 5 3 1 5 9 ,

2 0 0 6 .

G . P f u r t s c h e l l e r a n d F . L o p e s d a S i l v a , E v e n t - r e l a t e d E E G / M E G s y n c h r o n i z a t i o n a n d

d e s y n c h r o n i z a t i o n : b a s i c p r i n c i p l e s , C l i n N e u r o p h y s i o l , 1 1 0 ( 1 1 ) : 1 8 4 2 1 8 5 7 , 1 9 9 9 .

P . L . N u n e z , R . S r i n i v a s a n , A . F . W e s t d o r p , R . S . W i j e s i n g h e , D . M . T u c k e r , R . B . S i l b e r s t e i n ,

a n d P . J . C a d u s c h , E E G c o h e r e n c y I : s t a t i s t i c s , r e f e r e n c e e l e c t r o d e , v o l u m e c o n d u c t i o n ,

L a p l a c i a n s , c o r t i c a l i m a g i n g , a n d i n t e r p r e t a t i o n a t m u l t i p l e s c a l e s , E l e c t r o e n c e p h a l o g r C l i n

N e u r o p h y s i o l , 1 0 3 ( 5 ) : 4 9 9 5 1 5 , 1 9 9 7 .
http://dx.doi.org/10.1109/MSP.2008.4408441

Tutorial_Machine Learning and Signal Processing Tools for BCI

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