Bayesian common spatial patterns for multi-subject EEG classification

Abstract

Multi-subject electroencephalography (EEG) classification involves algorithm development for automatically categorizing brain waves measured from multiple subjects who undergo the same mental task. Common spatial patterns (CSP) or its probabilistic counterpart, PCSP, is a popular discriminative feature extraction method for EEG classification. Models in CSP or PCSP are trained on a subject-by-subject basis so that inter-subject information is neglected. In the case of multi-subject EEG classification, however, it is desirable to capture inter-subject relatedness in learning a model. In this paper we present a nonparametric Bayesian model for a multi-subject extension of PCSP where subject relatedness is captured by assuming that spatial patterns across subjects share a latent subspace. Spatial patterns and the shared latent subspace are jointly learned by variational inference. We use an infinite latent feature model to automatically infer the dimension of the shared latent subspace, placing Indian Buffet process (IBP) priors on our model. Numerical experiments on BCI competition III IVa and IV 2a dataset demonstrate the high performance of our method, compared to PCSP and existing Bayesian multi-task CSP models.

Introduction

Electroencephalography (EEG) is the recording of electrical potentials using multiple sensors placed on the scalp, to collect multivariate time series data involving brain activities. EEG classification enables computers to translate a subject’s intention or mental status into a control signal for a device, providing a new communication mode, known as brain–computer interface (BCI) (Cichocki et al., 2008, Dornhege et al., 2007, Ebrahimi et al., 2003). Multi-subject EEG classification considers brain signals from multiple subjects, each of whom undergoes the same mental task, where the brain waves reflect task-specific and subject-specific characteristics, as well as inter-subject variations.

Common spatial patterns (CSP) is a popular discriminative EEG feature extraction method, which has proved to be a useful subject-specific spatial filter (Blankertz et al., 2008, Koles, 1991, Müller-Gerking et al., 1999, Ramoser et al., 2000). Given two class-conditional covariance matrices, CSP seeks a discriminative subspace such that the variance for one class is maximized while the variance for the other class is minimized at the same time. The solution is determined by the simultaneous diagonalization of two class-conditional covariance matrices, which is equivalent to the generalized eigenvalue problem. Probabilistic CSP (PCSP) (Wu et al., 2009, Wu et al., 2010, Wu et al., 2011) is a probabilistic counterpart of CSP, where two linear Gaussian generative models with a shared basis matrix are jointly learned to infer spatial patterns that correspond to the column vectors of the shared basis matrix. Inherently, CSP or PCSP is a user-specific spatial filter, where the subject-relatedness is neglected. In the case where the number of training samples for a subject of interest is small, subject-specific models suffer from performance degradation. Multi-subject extensions of CSP can be found in Devlaminck, Wyns, Grosse-Wentrup, Otte, and Santens (2011), Kang, Nam, and Choi (2009), Krauledat, Tangermann, and Blankertz (2008) and Lotte and Guan (2010). In regularized CSP methods, class-conditional covariance matrices for a subject of interest are regularized by a linear combination of other subjects’ covariance matrices, in order to incorporate inter-subject relatedness (Kang et al., 2009, Lotte and Guan, 2011). Learning CSP is re-formulated as a risk minimization problem, where regularization is added to constrain spatial filters to become similar across subjects (Devlaminck et al., 2011). A multi-subject EEG classification method uses its own features other than CSP, and regularizes the learning of subject-specific parameters to become similar across subjects (Alamgir, Grosse-Wentrup, & Altun, 2010).

On the other hand, multi-task learning has been adopted in the probabilistic model for CSP, treating subjects as tasks. Bayesian multi-task learning (Heskes, 2000, Xue et al., 2007) deals with several related tasks at the same time, with the intention that the tasks will learn from each other by sharing hyperparameters (parameters of prior distributions). A Bayesian multi-task extension of CSP (BCSP) was recently developed in Kang and Choi (2011), where subject-to-subject information was transferred during the learning of model for a subject of interest by sharing hyperparameters across subjects, while treating subjects as tasks. Bayesian CSP (Kang & Choi, 2011) works better than PCSP (on subject-by-subject basis), although similarities or dissimilarities between spatial patterns are neglected because all spatial patterns are forced to share the same hyperparameters. Bayesian CSP with Dirichlet process (DP) priors (BCSP-DP) jointly learns and groups spatial patterns, so that spatial patterns in the same group, determined by the DP mixture model, share the hyperparameters of their prior distributions. Coupling similar spatial patterns in the same cluster by sharing hyperparameters facilitates information transfer among subjects with similar spatial patterns, whereas information transfer is prevented among dissimilar subjects. However, information transfer across clusters is not possible, so that the common characteristics across all the subjects are not captured.

In this paper we present a hierarchical Bayesian model where spatial patterns across subjects are assumed to share a latent subspace to capture the subject-relatedness, which is determined by measurements from a subject of interest as well as other subjects. Motivated by the idea of nonparametric Bayesian multi-task learning (Rai and Daumé, 2010, Zhang et al., 2008), we use an infinite latent feature model to automatically infer the dimension of the shared latent subspace, placing Indian Buffet process (IBP) priors (Griffiths & Ghahramani, 2006) on our model. We refer to the proposed method as BCSP-IBP and develop a variational inference algorithm.

The rest of this paper is organized as follows. In Section  2, we briefly review a few relevant existing methods, including CSP, PCSP, and Bayesian CSP models. Section  3 presents the main contribution of this paper, in which we describe Bayesian CSP with Indian Buffet process priors, where model and variational inference algorithms are illustrated. Section  4 presents numerical experiments on BCI competition III IVa and IV 2a dataset, to compare the classification performances of our proposed method to existing methods. Finally, conclusions are drawn in Section  5.