Stationary Linear Discriminant Analysis - Classifying
Non-Stationary Features in Brain-Computer Interfacing
Berlin Institute of Technology, Machine Learning Group, Germany
Bernstein Center for Computational Neuroscience, Germany
Fraunhofer Institute FIRST, Intelligent Data Analysis Group, Germany
Technische Universität, Germany
In Brain-Computer Interfacing (BCI), non-stationarity may be imposed by artifacts and learning related adaptation. This can leads to a changing feature distribution and can negatively affect classification performance. In this report we propose a method called stationary Linear Discriminant Analysis (sLDA) which penalizes non-stationary directions in feature space and analyse the effects in simulations and with real BCI data .
The goal of sLDA is to find a direction in feature space which is both discriminative and stationary. To this end we optimize a trade off loss function based on the Fisher ratio used by LDA but catering for non-stationarity .
The objective function can be seen in Figure 1.
Φns is the Kullback-Leibler divergence of the average empirical Gaussian on classes and i-th epoch and α is the trade-off parameter. The empirical mean and covariance of the j-th class is denoted as μj and Σj. The optimization is conducted using gradient descent.
The simulated data consists of 6 sources: we orthogonally mix one non-stationary and five stationary sources. Besides evaluating performance in terms of the angle between the normal vectors to the decision hyperplanes and stationary directions, we also consider classification performance.
sLDA finds the correct subspace (10° - 20° accuracy), whereas LDA often chooses the wrong one. However, the overall performance highly depends on the level of non-stationarity present in the data. Furthermore if the stationary but discriminative directions are not significantly more separable than the non-stationary but discriminative direction, then improvement (in terms of classification accuracy) is not possible. On the other hand if there are a number of stationary directions which are discriminative and there is one non-stationary but moderately discriminative direction, then improvement is possible using sLDA over LDA.
We also evaluate the performance of sLDA on a BCI data set . The mean (median) error rates of LDA and sLDA are 0.295 (0.310) and 0.277 (0.265), respectively. The performance gain of sLDA is significant with p=1.96×10^(−7) according to the Wilcoxon signed-rank test.
One explanation for the improvement, not attributable to the quality of the sLDA solution, is there are BCI specific non-stationarities between the training and test data, which correlate with LDA. According to this, slight deviation from the direction chosen using LDA results in an increase in performance. The improvement of sLDA with α=1 also suggests that LDA does not choose the optimal classification directions, i.e. it may overfit.
This work was supported by the German Research Foundation (GRK 1589/1).
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Linear Discriminant Analysis,
BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011, Freiburg, Germany, 4 Oct - 6 Oct, 2011.
neurotechnology and brain-machine interface (please use "neurotechnology and brain-machine interface" as keyword)
(2011). Stationary Linear Discriminant Analysis - Classifying
Non-Stationary Features in Brain-Computer Interfacing.
Front. Comput. Neurosci.
BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011.
23 Aug 2011;
04 Oct 2011.
Mr. Duncan Blythe, Berlin Institute of Technology, Machine Learning Group, Berlin, 10587, Germany, firstname.lastname@example.org
Mr. Wojciech Samek, Berlin Institute of Technology, Machine Learning Group, Berlin, 10587, Germany, email@example.com