Stationary Linear Discriminant Analysis - Classifying
Non-Stationary Features in Brain-Computer Interfacing
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1
Berlin Institute of Technology, Machine Learning Group, Germany
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2
Bernstein Center for Computational Neuroscience, Germany
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3
Fraunhofer Institute FIRST, Intelligent Data Analysis Group, Germany
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4
Technische Universität, Germany
Introduction
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 [1].
Method
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 [2].
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.
Results
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 [1]. 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.
Acknowledgements
This work was supported by the German Research Foundation (GRK 1589/1).
References
[1] T. Dickhaus, C. Sannelli, K.-R. Müller, G. Curio, B. Blankertz. Predicting BCI performance to study BCI illiteracy.
BMC Neuroscience 2009, 10:(Suppl 1):P84, 2009.
[2] W. Samek, M. Kawanabe, C. Vidaurre. Group-wise stationary subspace analysis - a novel method for studying non-stationarities.
In Proceedings of 5th International Brain-Computer Interface Conference, 2011.
Keywords:
Brain-Computer Interfacing,
Linear Discriminant Analysis,
non-stationarity
Conference:
BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011, Freiburg, Germany, 4 Oct - 6 Oct, 2011.
Presentation Type:
Abstract
Topic:
neurotechnology and brain-machine interface (please use "neurotechnology and brain-machine interface" as keyword)
Citation:
Blythe
D,
Samek
W and
Mueller
KR
(2011). Stationary Linear Discriminant Analysis - Classifying
Non-Stationary Features in Brain-Computer Interfacing.
Front. Comput. Neurosci.
Conference Abstract:
BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011.
doi: 10.3389/conf.fncom.2011.53.00080
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Received:
23 Aug 2011;
Published Online:
04 Oct 2011.
*
Correspondence:
Mr. Duncan Blythe, Berlin Institute of Technology, Machine Learning Group, Berlin, 10587, Germany, duncan.blythe@bccn-berlin.de
Mr. Wojciech Samek, Berlin Institute of Technology, Machine Learning Group, Berlin, 10587, Germany, wojwoj@mail.tu-berlin.de