moabb.pipelines.classification.SSVEP_TRCA#

class moabb.pipelines.classification.SSVEP_TRCA(n_fbands=5, is_ensemble=True, method='original', estimator='scm')[source]#

Task-Related Component Analysis (TRCA) method for SSVEP detection [1].

TRCA is a data-driven spatial filtering approach that enhances SSVEP detection by maximizing the reproducibility of task-related EEG components across multiple trials. Unlike CCA which uses predefined sinusoidal references, TRCA learns optimal spatial filters directly from the training data.

Mathematical Formulation

Given \(N_t\) training trials \(\mathbf{X}^{(h)} \in \mathbb{R}^{N_c \times N_s}\) for a stimulus frequency, TRCA finds the optimal spatial filter \(\mathbf{w}\) that maximizes the inter-trial covariance while constraining the variance:

\[\hat{\mathbf{w}} = \arg\max_{\mathbf{w}} \frac{\mathbf{w}^T \mathbf{S} \mathbf{w}}{\mathbf{w}^T \mathbf{Q} \mathbf{w}}\]

Inter-trial Covariance Matrix S

The matrix \(\mathbf{S}\) captures the covariance between different trials:

\[S_{j_1, j_2} = \sum_{h_1=1}^{N_t} \sum_{h_2=1, h_2 \neq h_1}^{N_t} \text{Cov}(x_{j_1}^{(h_1)}(t), x_{j_2}^{(h_2)}(t))\]

where \(x_j^{(h)}(t)\) is the signal from channel \(j\) in trial \(h\).

Variance Constraint Matrix Q

The matrix \(\mathbf{Q}\) represents the pooled variance across all trials:

\[\mathbf{Q} = \sum_{h=1}^{N_t} \mathbf{X}^{(h)} (\mathbf{X}^{(h)})^T\]

Generalized Eigenvalue Problem

The optimization is solved as a generalized eigenvalue problem:

\[\mathbf{S} \mathbf{w} = \lambda \mathbf{Q} \mathbf{w}\]

The eigenvector corresponding to the largest eigenvalue gives the optimal spatial filter \(\hat{\mathbf{w}}\).

Template Construction

For each stimulus frequency \(f_n\), the template is the average of spatially filtered training trials:

\[\bar{\mathbf{X}}_n = \frac{1}{N_t} \sum_{h=1}^{N_t} \mathbf{X}_n^{(h)}\]

Ensemble TRCA

The ensemble method combines spatial filters from all stimulus frequencies into a filter bank \(\mathbf{W} = [\mathbf{w}_1, \mathbf{w}_2, ..., \mathbf{w}_{N_f}]\) for improved robustness.

Filter Bank Approach

To capture harmonic components, EEG signals are decomposed into \(N_m\) sub-bands using a filter bank. The correlation coefficient for sub-band \(m\) is:

\[r_n^{(m)} = \rho\left((\mathbf{X}^{(m)})^T \mathbf{W}^{(m)}, (\bar{\mathbf{X}}_n^{(m)})^T \mathbf{W}^{(m)}\right)\]

Classification Rule

The final feature combines sub-band correlations with weights \(a^{(m)} = m^{-1.25} + 0.25\):

\[\rho_n = \sum_{m=1}^{N_m} a^{(m)} \cdot (r_n^{(m)})^2\]

The predicted class is: \(\hat{\tau} = \arg\max_n \rho_n\)

Parameters:

n_fbands (int) – Number of sub-bands \(N_m\) for the filter bank decomposition. Each sub-band captures different harmonic components of the SSVEP. Defaults to 5.
is_ensemble (bool) – If True, use ensemble TRCA which combines spatial filters from all stimulus frequencies for improved robustness. If False, use only the class-specific spatial filter. Defaults to True.
method (str) –
Method for computing the inter-trial covariance matrix \(\mathbf{S}\). Defaults to 'original'.
- 'original': Euclidean mean as in the original paper [1].
- 'riemann': Geodesic (Riemannian) mean, more robust to outliers but sensitive to ill-conditioned matrices.
- 'logeuclid': Log-Euclidean mean, a computationally stable alternative to the Riemannian mean.
estimator (str) –
Covariance estimator for regularization. Defaults to 'scm'. Options include:
- 'scm': Sample covariance matrix (no regularization, as in [1]).
- 'lwf': Ledoit-Wolf shrinkage estimator.
- 'oas': Oracle Approximating Shrinkage estimator.
- 'schaefer': Schäfer-Strimmer shrinkage estimator.

fb_coefs#

Sub-band weights \(a^{(m)} = m^{-1.25} + 0.25\) for filter bank fusion, of length n_fbands.

Type:: list of float

classes_#

Encoded class labels extracted at fit time, of shape (n_classes,).

Type:: numpy.ndarray

n_classes#

Number of unique stimulus frequencies/classes.

Type:: int

templates_#

Average templates \(\bar{\mathbf{X}}_n^{(m)}\) for each class and sub-band, of shape (n_classes, n_fbands, n_channels, n_samples).

Type:: numpy.ndarray

weights_#

Spatial filter weights \(\mathbf{w}_n^{(m)}\) for each sub-band and class, of shape (n_fbands, n_classes, n_channels).

Type:: numpy.ndarray

freqs_#

List of stimulus frequencies from training data.

Type:: list of str

peaks_#

Numeric frequency values for filter bank design, of shape (n_classes,).

Type:: numpy.ndarray

sfreq_#

Sampling frequency of the training data.

Type:: float

References

[1] (1,2,3,4)

M. Nakanishi, Y. Wang, X. Chen, Y.-T. Wang, X. Gao, and T.-P. Jung, “Enhancing detection of SSVEPs for a high-speed brain speller using task-related component analysis”, IEEE Trans. Biomed. Eng, 65(1):104-112, 2018. https://doi.org/10.1109/TBME.2017.2694818