Common spatial pattern

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.

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. \mathbf{w}={\arg \max}_\mathbf{w} \frac{ \left\| \mathbf{wX}_1 \right\| ^2 } { \left\| \mathbf{wX}_2 \right\| ^2 } The solution is given by computing the two covariance matrices: \mathbf{R}_1=\frac{\mathbf{X}_1\mathbf{X}_1^\text{T}}{t_1} \mathbf{R}_2=\frac{\mathbf{X}_2\mathbf{X}_2^\text{T}}{t_2} Then, the simultaneous diagonalization of those two matrices (also called generalized eigenvalue decomposition) is realized. We find the matrix of eigenvectors \mathbf{P}=\begin{bmatrix} \mathbf{p}_1 & \cdots & \mathbf{p}_n \end{bmatrix} and the diagonal matrix \mathbf{D} of eigenvalues \{\lambda_1, \cdots , \lambda_n \} sorted by decreasing order such that: \mathbf{P}^{\mathrm{T}} \mathbf{R}_1 \mathbf{P} = \mathbf{D} and \mathbf{P}^{\mathrm{T}} \mathbf{R}_2 \mathbf{P} = \mathbf{I}_n with \mathbf{I}_n the identity matrix. This is equivalent to the eigendecomposition of \mathbf{R}_2^{-1} \mathbf{R}_1 : \mathbf{R}_2^{-1} \mathbf{R}_1=\mathbf{PDP}^{-1} \mathbf{w}^\text{T} will correspond to the first column of \mathbf{P} : \mathbf{w}=\mathbf{p}_1^\text{T} \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 } 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 F_j generated by the j last eigenvectors \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: 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} ==See also== Blind signal separation ==References==

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