An inter-subject model to reduce the calibration time for motion imagination-based brain-computer interface

Abstract

A major factor blocking the practical application of brain-computer interfaces (BCI) is the long calibration time. To obtain enough training trials, participants must spend a long time in the calibration stage. In this paper, we propose a new framework to reduce the calibration time through knowledge transferred from the electroencephalogram (EEG) of other subjects. We trained the motor recognition model for the target subject using both the target’s EEG signal and the EEG signals of other subjects. To reduce the individual variation of different datasets, we proposed two data mapping methods. These two methods separately diminished the variation caused by dissimilarities in the brain activation region and the strength of the brain activation in different subjects. After these data mapping stages, we adopted an ensemble method to aggregate the EEG signals from all subjects into a final model. We compared our method with other methods that reduce the calibration time. The results showed that our method achieves a satisfactory recognition accuracy using very few training trials (32 samples). Compared with existing methods using few training trials, our method achieved much greater accuracy.

Appendix

• 1. Proof of ${W}_2^{\hbox{'}}={W}_2{T}^T$

  The process of solving ${W}_2^{\hbox{'}}$ is a parallelism of the CSP algorithm.

  • (1) Find the covariance of the two types of motion:

$$ {\mathrm{C}}_i^{\hbox{'}}={\sum}_{\mathrm{k}=1}^{\mathrm{n}}\frac{{\mathrm{D}}_{\mathrm{k}}^{\hbox{'}}\times {\mathrm{D}}_{\mathrm{k}}^{\hbox{'}\mathrm{T}}}{\mathrm{tr}\left({\mathrm{D}}_{\mathrm{k}}^{\hbox{'}}\times {\mathrm{D}}_{\mathrm{k}}^{\hbox{'}\mathrm{T}}\right)}={\sum}_{\mathrm{k}=1}^{\mathrm{n}}\frac{{\mathrm{T}\mathrm{D}}_{\mathrm{k}}^{\hbox{'}}\times {\left({\mathrm{T}\mathrm{D}}_{\mathrm{k}}^{\hbox{'}}\right)}^T}{\mathrm{tr}\left({\mathrm{D}}_k\times {\mathrm{D}}_{\mathrm{k}}^T\right)}={\mathrm{T}\mathrm{C}}_{\mathrm{i}}{\mathrm{T}}^{\mathrm{T}} $$

where n = 1, 2.

• (2) Find the whitening matrix Q^'.

$$ {\displaystyle \begin{array}{l}{\mathrm{C}}^{\hbox{'}}={{\mathrm{C}}^{\hbox{'}}}_1+{\mathrm{C}}_2^{\hbox{'}}={\mathrm{T}\mathrm{CT}}^{\mathrm{T}}=\mathrm{T}\times {\mathrm{V}}_{\mathrm{C}}\sum {\mathrm{V}}_{\mathrm{C}}^{\mathrm{T}}\times {\mathrm{T}}^{\mathrm{T}}\\ {}{\mathrm{Q}}^{\hbox{'}}={\sum}^{-\frac{1}{2}}{\mathrm{V}}_{\mathrm{C}}^{\mathrm{T}}\times {\mathrm{T}}^{\mathrm{T}}={\mathrm{Q}\mathrm{T}}^{\mathrm{T}}\end{array}} $$

• (3) Calculate the whitened matrix.

$$ {\displaystyle \begin{array}{l}{\mathrm{S}}_{\mathrm{i}}=\mathrm{Q}\times {\mathrm{C}}_{\mathrm{i}}\times {\mathrm{Q}}^{\mathrm{T}}\\ {}{\mathrm{S}}_{\mathrm{i}}^{\hbox{'}}={\mathrm{Q}}^{\hbox{'}}\times {\mathrm{C}}^{\hbox{'}}\times {\mathrm{Q}}^{\hbox{'}\mathrm{T}}={\mathrm{Q}\mathrm{T}}^{\mathrm{T}}\times {\mathrm{T}\mathrm{C}}_{\mathrm{i}}{\mathrm{T}}^{\mathrm{T}}\times {\mathrm{T}\mathrm{Q}}^{\mathrm{T}}\\ {}{\mathrm{S}}_{\mathrm{i}}={\mathrm{S}}_i^{\hbox{'}}\end{array}} $$

Then, S^' and S^' have the same eigenvector matrixU.

• (4) CalculateW^′.

$$ {W}^{\hbox{'}}={U}^T\times {Q}^{\prime }={U}^T\times Q\times T=W\times {T}^T $$

• 2. Solution of the optimization:

$$ T=\mathrm{argmin}\left({\left\Vert {W}_2-{W}_1\times {T}^T\right\Vert}_F^2+\alpha {\left\Vert T-P\right\Vert}_F^2\right), st.{T}^TT=I $$

In this part, we give the details of how to solve the optimization problem in section II.A. The problem is:

$$ T=\mathrm{argmin}\left({\left\Vert {W}_2-{W}_1\times {T}^T\right\Vert}_F^2+\alpha {\left\Vert T-P\right\Vert}_F^2\right), st.{T}^TT=I $$

To easily and effectively make the calculation, the optimization problem with regular terms was transformed to an optimization formula without the regular terms. The Lagrangian multiplier method was used.

• (1) First, the target function is transformed.

${\left\Vert {W}_2-{W}_1\times {T}^T\right\Vert}_F^2= tr\left({W_1}^T{W}_1+{W_2}^T{W}_2\right)-2\times tr\left(T{W_1}^T{W}_2\right){\left\Vert T-P\right\Vert}_F^2= tr\left({\left(T-P\right)}^T\left(T-P\right)\right)= tr\left({T}^TT-{T}^TP-{P}^TT+{P}^TP\right)$The constraint condition is T^TT = I. Then,

${\left\Vert T-P\right\Vert}_F^2= tr\left(I-{T}^TP-{P}^TT+{P}^TP\right)$We used trace to describe the orthogonality constraints. The Lagrange function is presented as follows.

L =  − tr(TA) + αtr(−T^TP − P^TT + I + P^TP) + λtr((Q^TQ − I)^T(Q^TQ − I)) where $A={W}_1^T{W}_2$.

• (2) The minimization condition is

$$ \Big\{{\displaystyle \begin{array}{c}\frac{\partial L}{\partial T}=0\\ {}\frac{\partial L}{\partial \lambda }=0\end{array}} $$

For the case,

$$ {\displaystyle \begin{array}{l}\frac{\partial L}{\partial T}=-\frac{\partial }{\partial T}\left( tr(TA)\right)+\alpha \frac{\partial }{\partial T}\left( tr\left(-{T}^TP-{P}^TT\right)\right)\\ {}\kern2em +\lambda \frac{\partial }{\partial T}\left( tr\left({\left({T}^TT-I\right)}^T\left({T}^TT-I\right)\right)\right)\end{array}} $$

Then,

$$ {\displaystyle \begin{array}{l}\frac{\partial L}{\partial T}=-{A}^T-2\alpha P+\lambda \frac{\partial }{\partial T}\left( tr\left({\left({T}^TT-I\right)}^T\left({T}^TT-I\right)\right)\right)\\ {}\kern1.5em ={B}^T+\lambda \frac{\partial }{\partial T}\left( tr\left({\left({T}^TT-I\right)}^T\left({T}^TT-I\right)\right)\right)\end{array}} $$

where, B = (A^T + 2αP)^T = A + 2αP^T

• (3) For optimization problems without the regular terms α and P, $\mathrm{T}=\mathrm{argmin}\left({\left\Vert {\mathrm{W}}_2-{\mathrm{W}}_1\times {\mathrm{T}}^{\mathrm{T}}\right\Vert}_{\mathrm{F}}^2\right),\mathrm{st}.{\mathrm{T}}^{\mathrm{T}}\mathrm{T}=\mathrm{I}$, we have a similar transformation with the previous derivation.

$$ {L}_1=- tr(TA)+\lambda tr\left({\left({T}^TT-I\right)}^T\left({T}^TT-I\right)\right) $$

The minimization condition is

$$ \Big\{{\displaystyle \begin{array}{c}\frac{\partial {L}_1}{\partial T}=0\\ {}\frac{\partial {L}_1}{\partial \lambda }=0\end{array}},\frac{\partial {L}_1}{\partial T}={A}^T+\lambda \frac{\partial }{\partial T}\left( tr\left({\left({T}^TT-I\right)}^T\left({T}^TT-I\right)\right)\right) $$

• (4) By comparing the formulas in (2) and (3), the optimization problem of the regular term can be transformed into the optimization problem without the regular term.

$$ \mathrm{T}=\mathrm{argmax}\left(\mathrm{tr}\left(\mathrm{TB}\right)\right),\mathrm{st}.{\mathrm{T}}^{\mathrm{T}}\mathrm{T}=\mathrm{I} $$

where B = A + 2αP^T.

Then, the new problem is solved as follows.

• (1) The matrix B is decomposed by the SVD method.

B = UΣV^T, tr(TB) = tr(TUΣV^T) = tr(V^TTUΣ).

Let Z = V^TTU Z = V^TTU,

then Z is an orthogonal matrix with the condition

|z_ij| ≤ 1, ∀ i, j.

• (2) Then,

$$ {\displaystyle \begin{array}{l} tr\left({V}^T TU\varSigma \right)= tr\left( Z\varSigma \right)={z}_{11}{\sigma}_1+{z}_{22}{\sigma}_2+\Big\lfloor +{z}_{nn}{\sigma}_n\\ {}\kern8.25em \le {\sigma}_1+{\sigma}_2+\Big\lfloor +{\sigma}_n\end{array}} $$

when Z = I, the equation is satisfied.

Hence, V^TTU = I, T = VU^T.

• (3) The final spatial filter is T = VU^TT = VU^T, where V, U is the result of the SVD decomposition on B.