Brain–Computer Interfaces - Index of

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components is larger than the number of observed mixtures and the components can
be dependent to each other. Interested readers can refer to [17, 18] for more details
on SCA.
2.1.6 Common Spatial Patterns (CSP)
In contrast to PCA, which maximizes the variance of the first component in the
transformed space, the common spatial patterns (CSP) maximizes the variance-ratio
of two conditions or classes. In other words, CSP finds a transformation that maximizes
the variance of the samples of one condition and simultaneously minimizes
the variance of the samples of the other condition. This property makes CSP one of
the most effective spatial filters for BCI signal processing provided the user’s intent
is encoded in the variance or power of the associated brain signal. BCIs based on
motor imagery are typical examples of such systems [19–21]. The term conditions
or classes refer to different mental tasks; for instance, left hand and right hand motor
imagery. The CSP algorithm requires not only the training samples but also the
information to which condition the samples belong to compute the linear transformation
matrix. In contrast, PCA and ICA do not require this additional information.
Therefore, PCA and ICA are unsupervised methods, whereas CSP is a supervised
method, which requires the condition or class label information for each individual
training sample.
For a more detailed explanation of the CSP algorithm, we assume that W ∈ Rn×n is a CSP transformation matrix. Then the transformed signals are WX, where X is
the data matrix of which each row represents an electrode channel. The first CSP
component, i.e. the first row of WX, contains most of the variance of class 1 (and
least of class 2), while the last component i.e. the last row of WX contains most of
the variance of class 2 (and least of class 1), where classes 1 and 2 represent two
different mental tasks.
The columns of W−1 are the common spatial patterns [22]. As explained earlier,
the values of the columns represent the contribution of the CSP components to the
channels, and thus can be used to visualize the topographic distribution of the CSP
components. Figure 4 shows two common spatial patterns of an EEG data analysis
example on left and right motor-imagery tasks, which correspond to the first and the
last columns of W−1 respectively. The topographic distributions of these components
correspond to the expected contralateral activity of the sensorimotor rhythms
induced by the motor imagery tasks. That is, left hand motor imagery induces sensorimotor
activity patterns (ERD/ERS) over the right sensorimotor areas, while right
hand motor imagery results in activity patterns over the left sensorimotor areas.
CSP and PCA are both based on the diagonalization of covariance matrices.
However, PCA diagonalizes one covariance matrix, whereas CSP simultaneously
diagonalizes two covariance matrices R1 and R2, which correspond to the two
different classes. Solving the eigenvalue problem is sufficient for PCA. For CSP,
the generalized eigenvalue problem with R −1
1 R1 has to be solved [12] togive
the transformation matrix W that simultaneously diagonalizes the two covariance
matrices: