# Classification of Event-Related Potentials with Regularized Spatiotemporal LCMV Beamforming
## 4 Discussion
### Interpreting the Weights  
The weight matrix of the STBF determines how each spatiotemporal feature of a given epoch should contribute to enhancing the SNR of the discriminating signal in the classification task. Alternatively, the activation pattern can be regarded as a forward EEG model of the activity, generating the discriminating signal and the weights as a backward model [60,64]. Regularization enables a researcher to interpret better the distribution of the weight over space and time after reshaping the weight vector \(\mathbf{w}\) to its spatiotemporal matrix equivalent, \(W\), such that \(\text{vec}(W)=\mathbf{w}\). Figure 5 compares the weights calculated in STBF-emp and STBF-shrunk with the weights from STBF-STRUCT.  
Since the linear filter's noise suppression and signal amplification functions are deeply entangled, it is not necessarily true that features with a high filter weight directly correlate to features containing discriminatory information [64]. However, it is still possible to interpret the weights in terms of which features contribute most to the classification process, be it through noise suppression, signal amplification, or--most likely--a combination of both. The weights obtained by STBF-emp seem to be randomly distributed over space and time; the regularized estimator used by STBF-shrunk and STBF-struct reveal a more interpretable weight distribution. The STBF-shrunk weights show a sparse spatial distribution, whereas the STBF-struct weights show a sparse distribution in both space and in time.  
As expected, Figure 5b and d exhibit weight around the central and parietal regions, where the P300 ERP component is present. Especially the spatial weights of STBF-shrunk in Figure 5d correspond to the spatial activation pattern in Figure 5a. This is not surprising, since shrinkage transforms the covariance matrix closer to the identity matrix and assuming identity covariance in Figure 4 yields weights identical to the activation pattern (up to a scaling factor). Additionally, Figure 5b shows that weights in the baseline interval and after 0.6 s, which should contain no response information, are close to zero for the structured  
Figure 5: Spatiotemporal beamformer weights calculated using four blocks of data (of 1215 epochs) from _Subject 01_ from 0.2 s before until 1.0 s after stimulus onset. Regularized weights show an interpretable sparse pattern, whereas the empirical weights appear noisier. (**A**) Spatiotemporal activation pattern with spatial and temporal global field power. (**B**) STBF-struct weights with spatial and temporal averages of absolute values. (**C**) STBF-shrunk weights. The shrinkage factor \(\alpha=0.05\) was determined with the closed-form LOOCV method. (**D**) STBF-emp weights.  
estimator. Meanwhile, these weights are high in the occipital region between 0.1 s and 0.2 s, containing early visual components with relatively low SNR. This high weight for the early visual components confirms the results from [PERSON] and [PERSON] [65] that state that, in addition to the P300, the early N1 and P2 ERP components are also modulated by oddball attention and contain discriminatory information between attended and non-attended stimuli.  
Using an interpretable classification model has many advantages. For instance, one can use the weight matrix to determine relevant time samples and EEG channels for per-subject feature selection to refine the model further. The number of channels is also an important cost factor in practical BCI applications. Determining which channels do not contribute to the classification accuracy helps to reduce the number of required electrodes. Spatially clustered weights indicate that some electrodes are not used by the classifier and can be discarded accordingly with no substantial accuracy reduction. As another example, information about the timing and spatial distribution of the discriminatory information in the response can be extracted from the weights and linked to neurophysiological hypotheses.