Common spatial pattern

The eigenvectors composing ${\displaystyle \mathbf {P} }$ are components with variance ratio between the two windows equal to their corresponding eigenvalue:

${\displaystyle \mathbf {\lambda } _{i}={\frac {\left\|\mathbf {p} _{i}^{\text{T}}\mathbf {X} _{1}\right\|^{2}}{\left\|\mathbf {p} _{i}^{\text{T}}\mathbf {X} _{2}\right\|^{2}}}}$

The vectorial subspace ${\displaystyle E_{i}}$ generated by the ${\displaystyle i}$ first eigenvectors ${\displaystyle {\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{i}\end{bmatrix}}}$ will be the subspace maximizing the variance ratio of all components belonging to it:

${\displaystyle E_{i}={\arg \max }_{E}{\begin{pmatrix}\min _{p\in E}{\frac {\left\|\mathbf {p^{\text{T}}X} _{1}\right\|^{2}}{\left\|\mathbf {p^{\text{T}}X} _{2}\right\|^{2}}}\end{pmatrix}}}$

On the same way, the vectorial subspace ${\displaystyle F_{j}}$ generated by the ${\displaystyle j}$ last eigenvectors ${\displaystyle {\begin{bmatrix}\mathbf {p} _{n-j+1}&\cdots &\mathbf {p} _{n}\end{bmatrix}}}$ will be the subspace minimizing the variance ratio of all components belonging to it:

${\displaystyle F_{j}={\arg \min }_{F}{\begin{pmatrix}\max _{p\in F}{\frac {\left\|\mathbf {p^{\text{T}}X} _{1}\right\|^{2}}{\left\|\mathbf {p^{\text{T}}X} _{2}\right\|^{2}}}\end{pmatrix}}}$

CSP can be applied after a mean subtraction (a.k.a. "mean centering") on signals in order to realize a variance ratio optimization. Otherwise CSP optimizes the ratio of second-order moment.

• The standard use consists on choosing the windows to correspond to two periods of time with different activation of sources (e.g. during rest and during a specific task).
• 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.[2] Those frequency bands can be on temporal or on frequential basis. Since the matrix ${\displaystyle \mathbf {P} }$ 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 [3] has proposed a particular choice for the first window ${\displaystyle \mathbf {X} _{1}}$ in order to extract components which have a specific period. ${\displaystyle \mathbf {X} _{1}}$ was the mean of the different periods for the examined signals.
• If there is only one window, ${\displaystyle \mathbf {R} _{2}}$ can be considered as the identity matrix and then CSP corresponds to Principal component analysis.