Linear Discriminant Analysis (LDA) is characterized by means (µ1,µ2) for each class and the feature covariance matrix Σ, which is considered to be equal in both classes. Mean and covariance are usually estimated during a calibration session and transformed into feature weights ω and bias b, resulting in a discriminative function D(x). Given a feature vector x, class membership can be estimated. In an assistive BCI setting, the classification output can be used for communication or to control a device. In a restorative BCI setting, the classification output is used to control the feedback. This allows operant conditioning of the desired pattern of brain modulation.
Unsupervised or unregularized calibration is problematic for restorative BCIS. Essentially, the subject has to learn the modulation, or the modulation might be impaired by lesions. This means the desired modulation is not fully expressed during the calibration sessions. Additionally, the subject will try out different mental strategies. This can result in a noisy covariance and mean estimation. When the subject learns the desired modulation, the mean vector µ2 will change location. Unsupervised weight and bias estimation will be affected by this noisy or misaligned mean and covariance estimates. In our lab, we therefore decided to developed a different classification approach. It is only based on the mean and covariance of the rest condition, and we believe it might be suited better for restorative BCIs.
Because the bias is only defined by the mean of the rest condition, an additional threshold θ is beneficial for controlling sensitivity and specificity of the feedback. Following item response theory, we interpret this parameter as the difficulty of the restorative BCI . Consider also that according to cognitive load theory, the difficulty, the zone of proximal development and the instructional efficiency are intrinsically linked . While the bias b can therefore be adapted in an unsupervised fashion to maximize classification accuracy, the difficulty θ should be controlled by the therapist to maximize instructional efficiency .
Put shortly: Basing feature weights only on the rest condition allows to tackle covariance and mean estimation, when the desired brain modulation is not yet sufficiently expressed. Furthermore, the introduction of a difficulty threshold θ in addition to bias b allows the therapist to match difficulty and ability to increase the instructional efficiency of a restorative BCI.
 Bauer, R., & Gharabaghi, A. (2015a). Estimating cognitive load during self-regulation of brain activity and neurofeedback with therapeutic brain-computer interfaces. Frontiers in Behavioral Neuroscience, 9(21). http://doi.org/10.3389/fnbeh.2015.00021
 Bauer, R., & Gharabaghi, A. (2015b). Reinforcement learning for adaptive threshold control of restorative brain-computer interfaces: a Bayesian simulation. Front. Neurosci. http://doi.org/10.3389/fnins.2015.00036