Unified Framework

LS framework

Various CA-based SSVEP spatial filtering methods can be formulated as one kind of RRR problems:

\[\arg\min_{\mathbf{W},\mathbf{M}} \left\| \mathbf{E}\mathbf{W}\mathbf{M}^T-\mathbf{T} \right\|^2_F,\]
\[\text{subject to } \mathbf{M}^T\mathbf{M}=\mathbf{I},\]

where \(\mathbf{W}\) is the spatial filter. In principle, the matrix \(\mathbf{E}\) presents the combined EEG features of inter-classes, i.e.,

\[\mathbf{E}=\mathbf{L}_\mathbf{E}\mathbf{P}_\mathbf{E}\mathbf{Z},\]

where \(\mathbf{L}_\mathbf{E}\) denotes the combination matrix of inter-class EEG features, \(\mathbf{P}_\mathbf{E}\) denotes the orthogonal matrix applied for generating inter-class EEG features, and \(\mathbf{Z}\) denotes the EEG data. In addition, the matrix \(\mathbf{T}\) is

\[\mathbf{T}=\mathbf{E}\,\mathcal{S}\!\left(\mathbf{K}\right).\]

Similarly as \(\mathbf{E}\), \(\mathbf{K}\) presents the combined EEG features of intra-classes, i.e.,

\[\mathbf{K}=\mathbf{L}_\mathbf{K}\mathbf{P}_\mathbf{K}\mathbf{Z},\]

where \(\mathbf{L}_\mathbf{K}\) denotes the combination matrix of intra-class EEG features, and \(\mathbf{P}_\mathbf{K}\) denotes the orthogonal matrix applied for generating intra-class EEG features.

Parameters in LS framework

In the LS framework,

  1. The matrices \(\mathbf{L}_\mathbf{E}\) and \(\mathbf{P}_\mathbf{E}\) are related to the inter-class features.

  2. The matrices \(\mathbf{L}_\mathbf{K}\) and \(\mathbf{P}_\mathbf{K}\) are related to the intra-class features.

  3. \(\mathbf{L}_\mathbf{E}\) and \(\mathbf{L}_\mathbf{K}\) present how to combine EEG features of inter- and intra-classes, respectively.

  4. \(\mathbf{P}_\mathbf{E}\) and \(\mathbf{P}_\mathbf{K}\) are the orthogonal projection matrices that extract EEG features of inter- and intra-classes, respectively.

  5. The combined intra-class features \(\mathbf{K}\) can be regarded as the normalization item of the combined inter-class features \(\mathbf{E}\) in the output of the LS regression \(\mathbf{T}\).

The parameters in the propsoed LS framework of the CA-based methods are summarized in the following table:

../_images/lsframework_table.png

Spatial filtering computation based on LS framework

  1. Initilize \(\mathbf{M}\):

    \[\mathbf{M}=\mathbf{V}^{\left( \mathbf{T} \right)}\text{ where }\mathbf{V}^{\left( \mathbf{T} \right)}\text{ is obtained from the SVD of }\mathbf{T}.\]

  2. Update \(\mathbf{W}\):

    \[\mathbf{W}=\left(\mathbf{E}^T\mathbf{E}\right)^{-1}\mathbf{E}^T\mathbf{T}\mathbf{M}.\]

  3. Update \(\mathbf{M}\):

    \[\mathbf{M}=\mathbf{U}^{\left(\mathbf{P}\right)}\left(\mathbf{V}^{\left(\mathbf{P}\right)}\right)^T.\text{ where }\mathbf{U}^{\left(\mathbf{P}\right)}\text{ and }\mathbf{V}^{\left(\mathbf{P}\right)}\text{ are obtained from the SVD of }\mathbf{P}\]

  4. Repeat the steps 2 and 3 until \(\mathbf{W}\) and \(\mathbf{M}\) are converged.