Brain–Computer Interfaces - Index of

Adaptive Methods in BCI Research - An Introductory Tutorial 343
Hyperplane:
b+x*w T =0
μ j
μ i −μ j
w
Fig. 3 Concept of classification with LDA. The two classes are reprensented by two ellipsoids
(the covariance matrices) and the respective class mean values. The hyperplane is the boundary of
decision, with D(x) = b + x · W T = 0. A new observation x is classified as follows: if D(x) is
greater than 0, the observation x is classified as class i and otherwise as class j. The normal vector
to the hyperplane, w, is in general not in the direction of the difference between the two class means
heightheight
Fig. 4 Paradigm of cue-based BCI experiment. Each trial lasted 8.25 s. A cue was presented at
t = 3s, feedback was provided from t=4.25 to 8.25 s
The methods to adapt LDA can be divided in two different groups. First, using
the estimation of the covariance matrices of the data, for which the speed of adaption
is fixed and determined by the update coefficient. The second group is based
on Kalman Filtering and has the advantage of having a variable adaption speed
depending on the properties of the data.
Fixed Rate Adaptive LDA Using (19), it can be shown that the distance function
(Eq. 39) is
μ i
D(xk) = [bk, wk] · [1, xk] T
= bk + wk · x T k
(42)
(43)
=−�μ k · � −1
k · μ T k + �μ k · � −1
k · x T k (44)
= [0, μ {i},k − μ {j},k] · E −1
k · [1, xk] (45)
with �μ k = μ {i},k − μ {j},k, b =−�μ(t) · �(t) −1 · μ(t) T and w = �μ(t) · � −1 .