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'''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]] |
Revision as of 19:47, 3 October 2013
Common spatial pattern (CSP) is a mathematical procedure used in signal processing for separating a multivariate signal into additive subcomponents which have maximum differences in variance between two windows.[1]
Details
Let of dimension and of dimension 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 (or second-order moment) 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 eigenvector and the diagonal matrix of eigenvalues sorted by decreasing order such that:
and
with the identity matrix.
This is equivalent to diagonalize the matrix :
Discussion
Relation between variance ratio and eigenvalue
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 subpsace generated by the last eigenvectors will be the subspace minimizing the variance ratio of all components belonging to it:
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 X1 and X2
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 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 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).[3]
It can also be used to separate artifacts from electroencephalographics signals.[2]
See also
References
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- ↑ Zoltan J. Koles, Michael S. Lazaret and Steven Z. Zhou, "Spatial patterns underlying population differences in the background EEG", Brain topography, Vol. 2 (4) pp. 275-284, 1990
- ↑ 2.0 2.1 Boudet, S., "Filtrage d'artefacts par analyse multicomposantes de l'électroencephalogramme de patients épileptiques.", PhD. Thesis: Unviversité de Lille 1, 07/2008
- ↑ G. Pfurtscheller, C. Gugeret and H. Ramoser "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