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2013-10-03T17:47:59Z

128.9.216.52: /* Examples */ Quantified variables shouldn't be subscript. Fixed.

'''Common spatial pattern''' ('''CSP''') is a mathematical procedure used in [[signal processing]] for separating a [[Multivariate analysis|multivariate]] signal into [[Additive function|additive]] subcomponents which have maximum differences in [[variance]] between two [[Window function|window]]s.<ref>Zoltan J. Koles, Michael S. Lazaret and Steven Z. Zhou, [http://www.springerlink.com/content/l378x271583v34p7/ "Spatial patterns underlying population differences in the background EEG"], Brain topography, Vol. 2 (4) pp. 275-284, 1990</ref>

== Details ==

Let <math>\mathbf{X}_1</math> of [[dimension]] <math>(n,t_1)</math> and <math>\mathbf{X}_2</math> of dimension <math>(n,t_2)</math> be two windows of a multivariate [[Signal (electrical engineering)|signal]], where <math>n</math> is the number of signals and <math>t_1</math> and <math>t_2</math> are the respective number of samples.

The CSP algorithm determines the component <math>\mathbf{w}^T</math> such that the ratio of variance (or second-order [[Moment (mathematics)|moment]]) is maximized between the two windows:
:<math>\mathbf{w}={\arg \max}_\mathbf{w} \frac{||\mathbf{wX}_1||^2}{||\mathbf{wX}_2||^2}</math>

The solution is given by computing the two [[covariance matrix|covariance matrices]]:

:<math>\mathbf{R}_1=\frac{\mathbf{X}_1\mathbf{X}_1^T}{t_1}</math>

:<math>\mathbf{R}_2=\frac{\mathbf{X}_2\mathbf{X}_2^T}{t_2}</math>

Then, the simultaneous [[Matrix diagonalization|diagonalization]] of those two [[Matrix (mathematics)|matrices]] is realized. We find the matrix of [[eigenvector]] <math>\mathbf{P}=\begin{bmatrix} \mathbf{p}_1 & \cdots & \mathbf{p}_n \end{bmatrix}</math> and the diagonal matrix <math>\mathbf{D}</math> of eigenvalues <math>\{\lambda_1, \cdots , \lambda_n \}</math> sorted by decreasing order such that:

:<math>\mathbf{P}^{-1} \mathbf{R}_1 \mathbf{P} = \mathbf{D}</math>

and

:<math>\mathbf{P}^{-1} \mathbf{R}_2 \mathbf{P} = \mathbf{I}_n</math>

with <math>\mathbf{I}_n</math> the [[identity matrix]].

This is equivalent to diagonalize the matrix <math>\mathbf{R}_2^{-1} \mathbf{R}_1</math>:

:<math>\mathbf{R}_2^{-1} \mathbf{R}_1=\mathbf{PDP}^{-1}</math>

:<math>\mathbf{w}^T</math> will correspond the first column of <math>\mathbf{P}</math>:

:<math>\mathbf{w}=\mathbf{p}_1^T</math>

== Discussion ==

=== Relation between variance ratio and eigenvalue ===

The eigenvectors composing <math>\mathbf{P}</math> are components with variance ratio between the two windows equal to their corresponding eigenvalue:

:<math>\mathbf{\lambda}_i=\frac{||\mathbf{p}_i^T\mathbf{X}_1||^2}{||\mathbf{p}_i^T\mathbf{X}_2||^2}</math>

=== Other components ===

The [[vector subspace|vectorial subspace]] <math>E_i</math> generated by the <math>i</math> first eigenvectors <math>\begin{bmatrix} \mathbf{p}_1 & \cdots & \mathbf{p}_i \end{bmatrix}</math> will be the subspace maximizing the variance ratio of all components belonging to it:

:<math>E_i={\arg \max}_{E} \begin{pmatrix}\min_{p \in E} \frac{||\mathbf{p^TX}_1||^2}{||\mathbf{p^TX}_2||^2}\end{pmatrix}</math>

On the same way, the vectorial subpsace <math>F_j</math> generated by the <math>j</math> last eigenvectors <math>\begin{bmatrix} \mathbf{p}_{n-j+1} & \cdots & \mathbf{p}_n \end{bmatrix}</math> will be the subspace minimizing the variance ratio of all components belonging to it:

:<math>F_j={\arg \min}_{F} \begin{pmatrix}\max_{p \in F} \frac{||\mathbf{p^TX}_1||^2}{||\mathbf{p^TX}_2||^2}\end{pmatrix}</math>

=== Variance or second-order moment ===

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 optimize the ratio of second-order moment.

=== Choice of windows X<sub>1</sub> and X<sub>2</sub> ===

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.<ref name="boudet">Boudet, S., [http://www.theses.fr/2008LIL10156 "Filtrage d'artefacts par analyse multicomposantes de l'électroencephalogramme de patients épileptiques."], PhD. Thesis: Unviversité de Lille 1, 07/2008</ref> Those frequency bands can be on temporal or on frequential basis. Since the matrix <math>\mathbf{P}</math> depends only of the covariance matrices, the same results can be obtained if the processing is applied on the [[Fourier transform]] of the signals.

== Applications ==
This method can be applied to several multivariate signal but it seems that most works on it concern [[electroencephalography|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).<ref name="bci">G. Pfurtscheller, C. Gugeret and H. Ramoser [http://www.springerlink.com/content/164875w6w5211303/ "EEG-based brain-computer interface using subject-specific spatial filters"], Engineering applications of bio-inspired artificial neural networks, Lecture Notes in Computer Science, 1999, Vol. 1607/1999, pp. 248-254</ref>

It can also be used to separate artifacts from electroencephalographics signals.<ref name="boudet"/>

==See also==
*[[Blind signal separation]]

==References==
{{reflist}}

[[Category:Signal processing]]

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