Brain–Computer Interfaces - Index of

336 A. Schlögl et al.
whereby T indicates the transpose operator. The variances are the diagonal elements
of the variance-covariance matrix, and the off-diagonal elements indicate the covariance
σi,j = 1 �Nt=1 �
N (xi(t) − μi) · (xj(t) − μj) � between the i-th and j-th element.
We define also the so-called extended covariance matrix (ECM) E as
Nx �
ECM(x) = Ex = [1, x(t)] T ·[1, x(t)] =
t=1
� � �
a b 1 μx
= Nx ·
c D μx T �x + μT x μ �
x
One can obtain from the ECM E the number of samples N = a, the mean
μ = b/a as well as the variance-covariance matrix Σ = D/a − (c/a) · (b/a). This
decomposition will be used later.
The adaptive version of the ECM estimator is
(16)
Ex(t) = (1 − UC) · Ex(t − 1) + UC · [1, x(t)] T · [1, x(t)] (17)
where t is the sample time, UC is the update coefficient. The decomposition of
the ECM E, mean μ, variance σ 2 and covariance matrix Σ is the same as for the
stationary case; typically is N = a = 1.
1.2.4 Adaptive Inverse Covariance Matrix Estimation
Some classifiers like LDA or QDA rely on the inverse Σ−1 of the covariance
matrix Σ; therefore, adaptive classifiers require an adaptive estimation of the inverse
covariance matrix. The inverse covariance matrix Σ can be obtained from Eq. (16)
with
Σ −1 �
= a · D − c · a −1 �−1 · b . (18)
This requires an explicit matrix inversion. The following formula shows how
the inverse convariance matrix Σ −1 can be obtained without an explicit matrix
inversion. For this purpose, the block matrix decomposition [1] and the matrix
inversion lemma (20) is used. Let us also define iECM = E −1 =
S = D − CA −1 B. According to the block matrix decomposition [1]
E −1
x =
�
A B
=
� −1
=
� A B
C D
� �
A−1 + A−1BS−1CA−1 −A−1BS−1 C D
−S−1CA−1 S−1 �
1 + μx�−1 x μx −μT x �−T
�
x
−�−1 x μTx �−1 x
� −1
with
The inverse extended covariance matrix iECM = E −1 can be obtained adaptively
by applying the matrix inversion lemma (20) to Eq. (17). The matrix inversion
lemma (also know as Woodbury matrix identity) states that the inverse A −1 of a
given matrix A = (B+UDV) can be determined by
(19)