Brain–Computer Interfaces - Index of

342 A. Schlögl et al.
d{i}(x) = ((x − μ {i}) T · � −1
{i} · (x − μ {i})) 1/2
where μ {i} and �{i} are the mean and the covariance, respectively, of the class samples
from class i.IfD(x) is greater than 0, the observation is classified as class i and
otherwise as class j. One can think of a minimum distance classifier, for which the
resulting class is obtained by the smallest Mahalanobis distance argmin i(d{i}(x)).
As seen in Eq. (35), the inverse covariance matrix (16) is required. Writing the
mathematical operations in Eq. (35) in matrix form yields:
with
F{i} =
d{i}(x) = ([1; x] · F{i} · [1; x] T ) 1/2
�
−μT �
{i} · Σ
I
−1
{i} · � �
−μ �I i
� =
�
μT {i} � −1
{i} μ {i} −μT {i} �−T
{i}
−� −1
{i} μ {i}
� −1
{i}
�
(35)
(36)
(37)
Comparing Eq. (19) with (37), we can see that the difference between F{i} and
E −1
{i} is just a 1 in the first element of the matrix, all other elements are equal.
Accordingly, the time-varying Mahalanobis distance of a sample x(t) toclassi is
where E −1
{i}
d{i}(xk) =
� �
[1, xk] · E −1
{i},k −
� ��
1 01×M
· [1, xk]
0M×1 0M×M
T
�1/2 can be obtained by Eq. (25) for each class i.
1.4.2 Adaptive LDA Estimator
Linear discriminant analysis (LDA) has linear decision boundaries (see Fig. 3). This
is the case when the covariance matrices of all classes are equal; that is, Σ{i} = Σ
for all classes i. Then, all observations are distributed in hyperellipsoids of equal
shape and orientation, and the observations of each class are centered around their
corresponding mean μ {i}. The following equation is used in the classification of a
two-class problem:
D(x) = w · (x − μ x) T = [b, w] · [1, x] T
w = �μ · � −1 = (μ {i} − μ {j}) · � −1
b =−μ x · w T =− 1
2 · (μ {i} + μ {j}) · w T
where D(x) is the difference in the distance of the feature vector x to the separating
hyperplane described by its normal vector w and the bias b. IfD(x) is greater than
0, the observation x is classified as class i and otherwise as class j.
(38)
(39)
(40)
(41)