Quantitative EEG-Based Brain-Computer Interface - Quantitative EEG Analysis Methods and Clinical Applications - page 209

where U 0 is the eigenvector matrix, and is a diagonal matrix with corresponding
eigenvalues as its diagonal elements. The variance in the space spanned by U 0 com-
ponents can be equalized by the following whitening matrix P :

− 12 T

P = Σ U (8.10)

0

It can be shown that, if R L and R R are transformed into S L and S R by whitening
matrix P :

T T

S RPS RP
L

= = (8.11)

L R R

then S L and S R will share common eigenvalues. This means, given the SVD of S L and
S R ,

T T

SUUSU U
L

= Σ = Σ (8.12)

L L L R R R R

the following equation holds true:

UUU = = ΣΣ + = I (8.13)

L R L R

Thus,

L and R may look like the following diagonal matrix:

⎡ ⎤

m

m m C

L R

⎢
⎢
⎢

⎥
⎥
⎥

Σ = diag 11 σσ 00

L 1 m

C

⎣ ⎦

(8.14)

⎡ ⎤

m m m C

L R

⎢
⎢
⎢

⎥
⎥
⎥

Σ = diag 00 δδ

1

11

R m

C

⎣ ⎦

Because the sum of corresponding eigenvalues in L and R is always 1, the big-
gest eigenvalue of S L corresponds to the smallest eigenvalue of S R. The eigenvectors
in L corresponding to the first m eigenvalues in L are used to form a new transform
matrix U l , which makes up the spatial filter with whitening matrix P , for extracting
the so-called source activity of left-hand imagery. The spatial filters for the left and
the right cases are constructed as follows:

T T

FUPFUP
L

= = (8.15)

r

l R

Then the source activities Y L and Y R are derived by applying the preceding spa-
tial filter on bandpass-filtered EEG data matrix X , that is,

YFXYFX
L

=⋅ =⋅ (8.16)

L R R

Because of the way in which the spatial filter is derived, the filtered source activ-
ities Y L and Y R are expected to be better features for discriminating these two imag-
ery tasks, compared with the original EEGs. Usually, the following inequation holds