Let of size and of size be two windows of a multivariate signal, where is the number of signals and and are the respective number of samples. The CSP algorithm determines the component such that the ratio of variance is maximized between the two windows: The solution is given by computing the two covariance matrices: Then, the simultaneous diagonalization of those two matrices is realized. We find the matrix of eigenvectors and the diagonal matrix of eigenvalues sorted by decreasing order such that: and with the identity matrix. This is equivalent to the eigendecomposition of :
The eigenvectors composing are components with variance ratio between the two windows equal to their corresponding eigenvalue:
Other components
The vectorial subspace generated by the first eigenvectors will be the subspace maximizing the variance ratio of all components belonging to it: On the same way, the vectorial subspace generated by the last eigenvectors will be the subspace minimizing the variance ratio of all components belonging to it:
CSP can be applied after a mean subtraction on signals in order to realize a variance ratio optimization. Otherwise CSP optimizes the ratio of second-order moment.
Choice of windows X1 and X2
The standard use consists on choosing the windows to correspond to two periods of time with different activation of sources.
It is also possible to choose the two windows to correspond to two different frequency bands in order to find components with specific frequency pattern. Those frequency bands can be on temporal or on frequential basis. Since the matrix depends only of the covariance matrices, the same results can be obtained if the processing is applied on the Fourier transform of the signals.
Y. Wang has proposed a particular choice for the first window in order to extract components which have a specific period. was the mean of the different periods for the examined signals.
This method can be applied to several multivariate signals but it seems that most works on it concern electroencephalographic signals. Particularly, the method is mostly used on brain–computer interface in order to retrieve the component signals which best transduce the cerebral activity for a specific task. It can also be used to separateartifacts from electroencephalographics signals. The common spatial pattern needs to be adapted for the analysis of the event-related potentials.
