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there is only one covariance matrix and the average covariance matrix is itself. How-
ever, in BCI, many trials of each class are required to properly train the CSP. Thus, it
is possible that the values in a covariance matrix of a trial differ significantly from the
values in the covariance matrix of another trial in the same class and therefore differ
from the values in the averaged covariance matrix. In other words, the averaging
process removes some information about stability of each spatial filter from CSP
projection matrix computation.
Recall that the main idea of CSP is to find a projection that maximizes variance of
one class and minimizes variance of the other class. In 2 dimension cases, the projec-
tion along the first spatial filter maximizes the variance of the first class and mini-
mizes the variance of the second class while the projection along the second spatial
filter maximizes the variance of the second class and minimizes the variance of the
first class. In general, spatial filters in CSP projection matrix gradually shift the vari-
ances of projected data of two classes. Hence, the first and last spatial filters are the
most useful when applied to averaged signals. However, this may not be true for in
each individual trial since the signals can significantly differ from the average. This is
why more spatial filters are needed. Furthermore, we found that for these 2 spatial
filters, one or both of them may not localize the brain area around the related motor
imaginary task, while the desired spatial filters are something else.
To address such problem, the coefficient of determination (
r
2
) was considered. It
takes into consideration the stability of data between trials. We have used this pa-
rameter in both of our selection approaches to determine whether all training trials
can be discriminated well when projected onto the prominent spatial filter dimen-
sions.
3 Proposed Methods
We propose 2 selection approaches to select prominent spatial filter set: the first ap-
proach is automatic selection; the second approach is semi-automatic selection. The
main idea is to extend the selection of spatial filters into ones with less significant
eigenvalue but have high stability. In section 3.3, we provide a working example of
selecting the value of
m
by both approaches.
3.1 Automatic Selection Approach
The detailed steps can be expressed as follows:
1.
Adjusts the sample length of each trial. We assume that all trials have the same
number of samples. If this is not the case, we trim out some less-important periods
such as the beginning of imagery in order to make their lengths equal. Divides the
trials into training and testing sets.
2.
Trains the CSP using training sets and obtains the full projection matrix
W
(size
N
N
). Projects all training data with this projection matrix. Each row of the pro-
jected data, called
component
, is the linear combination of channels.
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