Interactions Between Pre-Processing and Classification Methods for Event-Related-Potential Classification

Abstract

Detecting event related potentials (ERPs) from single trials is critical to the operation of many stimulus-driven brain computer interface (BCI) systems. The low strength of the ERP signal compared to the noise (due to artifacts and BCI irrelevant brain processes) makes this a challenging signal detection problem. Previous work has tended to focus on how best to detect a single ERP type (such as the visual oddball response). However, the underlying ERP detection problem is essentially the same regardless of stimulus modality (e.g. visual or tactile), ERP component (e.g. P300 oddball response, or the error-potential), measurement system or electrode layout. To investigate whether a single ERP detection method might work for a wider range of ERP BCIs we compare detection performance over a large corpus of more than 50 ERP BCI datasets whilst systematically varying the electrode montage, spectral filter, spatial filter and classifier training methods. We identify an interesting interaction between spatial whitening and regularised classification which made detection performance independent of the choice of spectral filter low-pass frequency. Our results show that pipeline consisting of spectral filtering, spatial whitening, and regularised classification gives near maximal performance in all cases. Importantly, this pipeline is simple to implement and completely automatic with no expert feature selection or parameter tuning required. Thus, we recommend this combination as a “best-practice” method for ERP detection problems.

Appendix: Feature Rotation Invariance of Quadratically Regularised Linear Classifiers

A quadratically regularised linear classifier finds its solution by minimising an objective function with the form,⁷

$$ J(w) = \lambda w^{{\top}} w + \sum\limits_{i=1..N}\mathcal{L}(x_i^{{\top}} w,y_i) $$

(4)

where, $x_i\in \mathbb{R}^d$ is the ith training example with d features. w ∈ ℝ^d are linear classifier weights. $\mathcal{L}$ is the classification loss function, which penalises differences between the classifier predictions ($x_i^{{\top}} w$) and the examples true class y _i . Depending on the choice of loss function $\mathcal{L}$ one obtains different classifiers, e.g. for logistic regression $\mathcal{L}=(1+\exp(-y_i x_i^{{\top}} w))^{-1}$, or a least squares classifier $\mathcal{L}=(y_i-x_i^{{\top}} w)^2$ (which can be used to implement LDA section “LDA”). $w^{{\top}} w$ is the quadratic regularisation penalty, which penalises “complex” solutions and λ the relative strength of this penalty.

Taking derivatives with respect to w and setting equal to zero one finds the optimal solution, w ^*, is given by;

$$ 2 \lambda w^* + \sum\limits_{i=1..N} \mathcal{L}'(x_i^{{\top}} w^*,y_i)x_i = 0, \label{eqn:raw} $$

(5)

where $\mathcal{L}'$ is the derivative of loss function $\mathcal{L}$.

If one rotates the features with an arbitrary rotation matrix R such that $\hat{x} = Rx$ the solution, $\hat{w}^*$, to this rotated problem is given by;

$$2 \lambda \hat{w}^* + \sum\limits_{i=1..N} \mathcal{L}'(\hat{x}_i^{{\top}}\hat{w}^*,y_i)\hat{x}_i = 0,$$

(6)

$$2 \lambda \hat{w}^* + \sum\limits_{i=1..N} \mathcal{L}'(x_i^{{\top}} R^{{\top}}\hat{w}^*,y_i) Rx_i = 0,$$

(7)

$$2 \lambda R^{{\top}}\hat{w}^* + \sum\limits_{i=1..N} \mathcal{L}'(x_i^{{\top}} R^{{\top}}\hat{w}^*,y_i)x_i = 0, $$

(8)

where we have used the property that the inverse of a rotation is its transpose, i.e. $R^{{\top}} R=I$. Making the substitution $R^{{\top}}\hat{w}^*=w^*$ one sees that Eqs. 5 and 8 are identical with the same solution, demonstrating that the only effect of rotation of the features is to rotate the optimal solution in the opposite direction.