Common spatial pattern

Common Spatial Pattern (CSP) is a mathematical procedure for separating a multivariate signal into additive subcomponents which have most differences in variance (or order 2 moment) between two windows.

Details:

Let's suppose two windows of a multivariate signals $\mathbf {X} _{1}$ of dimension $(n,t_{1})$ and $\mathbf {X} _{2}$ of dimension $(n,t_{2})$. $n$ is the number of signals and $t_{1}$, $t_{2}$ are the respective number samples.

The CSP consists in looking the component $\mathbf {w} ^{T}$ such the ratio of variance (or order 2 moment) is maximum between the two windows: $\mathbf {w} ={\arg \max }_{\mathbf {w} }{\frac {||\mathbf {wX} _{1}||^{2}}{||\mathbf {wX} _{2}||^{2}}}$

The solution is given by computing the two covariance matrices:

$\mathbf {R} _{1}={\frac {\mathbf {X} _{1}\mathbf {X} _{1}^{T}}{t_{1}}}$ $\mathbf {R} _{2}={\frac {\mathbf {X} _{2}\mathbf {X} _{2}^{T}}{t_{2}}}$

Then, the simultaneous diagonalization of those two matrices is realized. We find the matrix of eigen vector $\mathbf {P} ={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n}\end{bmatrix}}$ and the diagonal matrix $\mathbf {D}$ of eigen values $\{\lambda _{1},\cdots ,\lambda _{n}\}$ sorted by decreasing order such that:

$\mathbf {P} ^{-1}\mathbf {R} _{1}\mathbf {P} =\mathbf {D}$

and

$\mathbf {P} ^{-1}\mathbf {R} _{2}\mathbf {P} =\mathbf {I} _{n}$

with $\mathbf {I} _{n}$ the identity matrix.

This is equivalent to diagonalize the matrix $\mathbf {R} _{2}^{-1}\mathbf {R} _{1}$:

$\mathbf {R} _{2}^{-1}\mathbf {R} _{1}=\mathbf {PDP} ^{-1}$

$\mathbf {w} ^{T}$ will correspond the first column of $\mathbf {P}$:

$\mathbf {w} =\mathbf {p} _{1}^{T}$

Discussion

Relation between variance ratio and eigen value

The eigen vectors composing $\mathbf {P}$ are components with variance ratio between the two windows equal to their corresponding eigen value:

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

Other components

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

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

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

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

Relation between variance ratio and eigen value

You can apply the CSP after a mean subtraction (a.k.a. "mean centering") on signals in order to realize a variance ratio optimization. Otherwize the CSP opitmize the ratio of order-2 moment

Choice of windows $\mathbf {X} _{1}$ and $\mathbf {X} _{2}$

The standard practice consists on choosing the windows to correspond to two period of time with different activation sources (e.g. during rest and during a specific tasks).

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. Since the matrix $\mathbf {P}$ depends only of the covariance matrices, the same results can be obtained if the processing is applied on the Fourrier transform of the signals.

Applications:

This method can be applied to several multivariate signal 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 signal which best transduce the cerebral activity for a specific task (e.g. hand movement).

It can also be used to separate artifacts form elegtroencephalographics signals.