Slice Oriented Tensor Decomposition of EEG Data for Feature Extraction in Space, Frequency and Time Domains Qibin Zhao1⋆ , Cesar F. Caiafa1⋆⋆ , Andrzej Cichocki1 , and Liqing Zhang2 1
2
Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, Saitama, Japan http://www.bsp.brain.riken.jp/ Department of Computer Science and Engineering, Shanghai Jiao Tong University Shanghai, China
Abstract. In this paper we apply a novel tensor decomposition model of SOD (slice oriented decomposition) to extract slice features from the multichannel time-frequency representation of EEG signals measured for MI (motor imagery) tasks in application to BCI (brain computer interface). The advantages of the SOD based feature extraction approach lie in its capability to obtain slice matrix components across the space, time and frequency domains and the discriminative features across different classes without any prior knowledge of the discriminative frequency bands. Furthermore, the combination of horizontal, lateral and frontal slice features makes our method more robust for the outlier problem. The experiment results demonstrate the effectiveness and robustness of our method. Key words: Tensor decomposition, EEG, BCI
1
Introduction
Tensors (also known as n-way arrays) are used in a variety of applications ranging from neuroscience and psychometrics to chemometrics [1–3]. From a viewpoint of data analysis, tensor decomposition is very attractive because it takes into account spatial and temporal correlations between variables more accurately than 2D matrix factorizations, and it usually provides sparse common factors or hidden components with physiological meaning and interpretation. In most applications, especially in neuroscience (EEG, fMRI), the standard PARAFAC and Tucker models were used[4–6]. Feature extraction for high dimension data and high noise data plays an important role in machine learning and pattern recognition. In the real world, the extracted feature of an object often has some specialized structures and such ⋆ ⋆⋆
Corresponding author. Email:
[email protected]. On leave from Engineering Faculty, University of Buenos Aires, ARGENTINA.
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Qibin Zhao, Cesar F. Caiafa, Andrzej Cichocki, and Liqing Zhang
structures are in the form of 2nd or even higher-order tensor. Recently, multilinear algebra, the algebra of high-order tensors, was applied for analyzing the multifactor structure image ensembles, EEG signals [7] and etc. These methods, such as tensor PCA [8], tensor LDA [9, 10], tensor subspace analysis [11–13], treat original data as second- or high-order tensors. For supervised feature classification [14], the tensor factorization can lead to structured dimensionality reduction by learning multiple interrelated subspaces. Discriminant analysis using tensor representation [15] can avoid the curse of dimensionality dilemma and overcome the small sample size problem. In the most existing tensor decomposition models, high-dimension tensors are decomposed to many rank-1 vector components on each mode. PARAFAC model can be explained as a special case of Tucker model in which the core tensor is reduced to a super-diagonal tensor. Unlike most existing models such as PARAFAC, Tucker and HOSVD, our SOD model is to represent a 3D tensor by outer product of slice matrices and corresponding vectors on each tensor mode rather than rank-1 components. Therefore, the structure of tensor data associated to its horizontal, lateral and frontal slices can be captured. Based on the SOD model, we developed a feature extraction framework for single-trial EEG classification. This paper is organized as follows: in section 2, SOD model and its main properties are introduced briefly, then the feature extraction framework based on SOD are proposed; in section 3, data analysis results on EEG data are presented and discussed; in section 4, the main conclusions and future perspectives of improvement are presented.
2
Method
2.1
SOD model
In [16], the Slice Oriented Decomposition (SOD) model was recently proposed as a decomposition method of 3-way tensors that captures the structure of data slices providing also a compact representation. SOD takes into account the interactions among the three modes of a tensor Y ∈ RI×J×K by decomposing it as a sum of elemental (simple) tensors: b = Y
P X p=1
Hp +
Q X q=1
Lq +
R X r=1
Fr =
P X p=1
Hp ◦1 up +
Q X
Lq ◦2 vq +
q=1
R X
Fr ◦3 wr , (1)
r=1
where matrices Hp , Lq and Fr are called matrix components, vectors up , vq and wr are called vector components, H, L, F ∈ RI×J×K and ◦n is the n-mode outer product (n = 1, 2 or 3) defined as follows: [H]ijk = [H ◦1 u]ijk = hjk ui , [L]ijk = [L ◦2 v]ijk = lik vj ,
(2) (3)
[F]ijk = [F ◦3 w]ijk = fij wk .
(4)
Title Suppressed Due to Excessive Length
3
The effect of the n-mode outer product is to create simple tensors where slices are scaled versions of a basic matrix. In Fig. 1-(a) the equation (1) is illustrated while in Fig. 1-(b) the SOD compact representation is shown. When vector and matrix components are constrained to be nonnegative we arrive to the Non-negative SOD (NN-SOD) for which an Alternate Least Squared (ALS) Newton based algorithm is available [16]. (a) SOD as sum of simple tensors P I´J´K
@å p=1
Hp up
Q
+å L
q
q=1
vq
R
+å r=1
wr Fr
(b) Compact representation of SOD P horizontal components (J´K) (Hp, P
