Slice Oriented Tensor Decomposition of EEG Data for Feature Extraction in Space, Frequency and Time Domains

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1 Slice Oriented Tensor Decomposition of EEG Data for Feature Extraction in Space, and Domains Qibin Zhao, Cesar F. Caiafa, Andrzej Cichocki, and Liqing Zhang 2 Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, Saitama, Japan 2 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 Introduction Tensors (also known as n-way arrays) are used in a variety of applications ranging from neuroscience and psychometrics to chemometrics [ 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. qbzhao@brain.riken.jp. On leave from Engineering Faculty, University of Buenos Aires, ARGENTINA.

2 2 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, ], tensor subspace analysis [ 3], treat original data as second- or high-order tensors. For supervised feature classification [4], the tensor factorization can lead to structured dimensionality reduction by learning multiple interrelated subspaces. Discriminant analysis using tensor representation [5] 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- 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- 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. SOD model In [6], 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 R I J K by decomposing it as a sum of elemental (simple) tensors: P Q R Ŷ = H p + L q + F r = p= q= r= P Q R H p u p + L q 2 v q + F r 3 w r, () p= where matrices H p, L q and F r are called matrix components, vectors u p, v q and w r are called vector components, H, L, F R I J K and n is the n-mode outer product (n =, 2 or 3) defined as follows: q= r= [H] ijk = [H u] ijk = h jk u i, (2) [L] ijk = [L 2 v] ijk = l ik v j, (3) [F] ijk = [F 3 w] ijk = f ij w k. (4)

3 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. -(a) the equation () is illustrated while in Fig. -(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 [6]. (a) SOD as sum of simple tensors I J K P L q p= u p H p Q q= v q R r= w r F r (b) Compact representation of SOD P horizontal components ( J K) (, P<<I) H p I J K Tensor R frontal components ( I J) (, R<<K) F r Q lateral components ( I K) (, Q<<J) L q Fig.. Slice Oriented Decomposition (SOD) model 2.2 Feature extraction To apply SOD for extracting slice features along horizontal, lateral and frontal directions, a tensor data X can be projected on slice matrix components of H p, L q and F r along each tensor mode as: û p = X () vect(h p ), ˆv q = X (2) vect(l q ),ŵ r = X (3) vect(f r ). (5) Then the correlation coefficients between û p, ˆv q,ŵ r and u p,v q,w r calculated by R U Û, R V ˆV and R WŴ can be combined as: f = [diag(r U Û ); diag(r V ˆV ); diag(rwŵ)], (6) where R U Û denotes the correlation matrix between matrix U with each column of u p, p =...P and Û with each column of û p, p =...P. Thus the vector

4 4 Qibin Zhao, Cesar F. Caiafa, Andrzej Cichocki, and Liqing Zhang f denotes the similarity between original Y and X at the linear combination patterns for all slice matrix components. Therefore, the vector f can also be interpreted as a group of projecting coefficients on all slice-based tensor basis. For the classification in the multi-class case, we first obtain class-specific slice components group by applying SOD on the c-th class averaged tensor data Y c, c =...C. Then we obtain a slice components group for each class. Thus, the features of new tensor data calculated by Eq.(6) can be used to train a linear classifier. (a) Left hand (b) Right hand Fig. 2. Averaged 3-way tensors of space-frequency-time representation for EEG signals during MI tasks. The size of tensor data is (i.e., channels frequency time). (a) for left hand class and (b) for right hand class. 3 Experiments and Results In our application, EEG signals with only 5 electrodes (i.e., C3, Cp3, Cz, Cp4, C4) over the motor cortex were recorded from the scalp at a sampling rate of 256Hz for 2 classes MI-based BCI experiments. In the experimental sessions used for the present study, labeled trials of EEG signals were recorded in the following way: the subjects were sitting in a comfortable chair with arms lying relaxed on the armrests. Each trial consists of 2s for relaxation and 4s for movement imagination (i.e., left hand or right hand) tasks following visual cue stimulus. The EEG data are transformed from the time-domain to the time-frequency domain using a complex Morlet continuous wavelet transform (CWT) with center frequency ω c = and bandwidth parameter ω b = 2. The frequency range from 6Hz to 3Hz at.5hz step are focused in our application. Thus, we obtain EEG tensor representation X R N d N f N t which is a 3-way time-varying EEG wavelet coefficients array, where N d, N f, N t are the number of channels, frequency bins, and time points respectively. In our application, we only consider the time-frequency power features of EEG trials, hence a square operation is performed on X in advance. In order to find the invariable feature structure through all trials, we first preprocessed EEG tensors by averaging the same class

5 Title Suppressed Due to Excessive Length 5 H H2 H H2.5 u C3 Cp3 Cz Cp4 C4.5 u2 C3 Cp3 Cz Cp4 C4.5 u C3 Cp3 Cz Cp4 C4.5 u2 C3 Cp3 Cz Cp4 C4 (a) Horizontal Slice (b) Horizontal Slice L L2 L L2 Channel Channel v v (Hz) (Hz) (c) Lateral Slice v v (Hz) (Hz) (d) Lateral Slice F F2 F F2 Channel Channel.5 w (s) w (s)..5 w (s) w (s) (e) Frontal Slice (f) Frontal Slice Fig.3. Results of NN-SOD with P = Q = R = 2 applied to class-specific EEG tensors. (a)(c)(e) are slice components for left class, (b)(d)(f) are slice components for right class. (a)(b) are horizontal slice matrix H p and combination vector u p; (c)(d) are Lateral slice matrix L q and combination vector v q; (e)(f) are frontal slice matrix F r and combination vector w r. The tensor size is , i.e., 5 channels, 49 frequency bins and 24 sample points.

6 6 Qibin Zhao, Cesar F. Caiafa, Andrzej Cichocki, and Liqing Zhang as Y c = M Σ i class c X i, M is the trial number of c-th class. Fig.2 shows the 3D averaged tensors for each class. The SOD model with non-negative constraints was performed for slice decomposition on each of class-specific tensors (i.e., the space, frequency and time domain). Fig.3 presents the decomposition results as the horizontal, lateral and frontal slices with 2 components on each mode, i.e., P = Q = R = 2. In the horizontal slice components (Fig.3(a)) for left class, the time-frequency matrix H mainly focuses around Hz (µ-rhythm) throughout the whole 4s duration of one trial. Then the vector u which represents the space distribution of the corresponding slice demonstrates that the slice H is decreased from channel C3 to C4. This is the ERS phenomena. Meanwhile the slice H 2 and corresponding vector u 2 demonstrate the ERD phenomena of decreasing power of µ-rhythm on right motor area of brain. Similar with left class, Fig.3(b) shows the time-frequency slice components and the distribution on channels domain, H denotes interrupt of µ-rhythm on left and right hemisphere of brain, H 2 denotes low β-rhythm in the frequency of 6-2Hz on right hemisphere of brain. Therefore, the significance of ERD/ERS for left hand and right hand are not same for specific subject. Similar to the horizontal slices, Fig.3(c),3(d) present the space-time lateral slice components and distribution vectors in the frequency domain. Fig.3(e),3(f) present the space-frequency frontal slice components and distribution vectors in the time domain. In Fig.3(e), F denotes the continuous β-rhythm focused on the left hemisphere of brain..8 Slice components from left class Slice components from right class r 2 value Components Fig.4. r 2 -value of slice components. The first 6 components are obtained by left class tensor, and the last 6 components are obtained by right class tensor. The order of each 6 components are H,H 2,L,L 2,F,F 2. In order to find the most discriminative slice components for two mental tasks, the r 2 -value are calculated for each slice components. Fig.4 shows r 2 - value along slice components, the first 6 components for left class and the last

7 Title Suppressed Due to Excessive Length 7 6 components for right class. It can be clearly seen that H of left class and H 2 of right class have most discriminative ability, which illustrates that most discriminative information between left and right class lies in the space distribution of time-frequency slices. Therefore, BCI researches always obtained high performance only by spatial filters, e.g., CSP method. However, the CSP algorithm is also known for its tendency to overfit, i.e., to learn the non-discriminative brain rhythm which has an overlapping frequency range with most discriminative brain rhythm. Especially in the small training samples case, CSP is suffered for the outlier problem because of high dependence upon the distribution properties of training data. As compared with CSP, our method are more stable in case of small training samples and more robust to deal with the nonstationary of EEG signals. To prove that, we has trained a SOD model and SVM classifier on first experiment run and tested our method on several subsequent runs. Fig.5 presents the classification performance of runs for subject A and 8 runs for subject B. The results demonstrate that the relative stable performance can be obtained by our method when compared with CSP. Therefore, the generalization ability of our method seems to be more suitable for online BCI system. 8 8 Accuracy (%) CSP SOD Accuracy (%) CSP SOD Runs (a) Subject A Runs (b) Subject B Fig.5. The classification performance of subsequent runs based on the model trained on the first run. One experiment run contains only 2 trials for each class and the duration of mental tasks is 4s for each trial. (a) for subject A and (b) for subject B. 4 Conclusions In this study, we have presented a novel tensor feature extraction framework for EEG classification based on SOD algorithm. Through applying the non-negative SOD, the slice features on each tensor mode can be easily obtained. Data analysis on EEG signals from BCI experiments demonstrates the effectiveness of our method. Compared with traditional tensor learning methods, our method is able to extract slice matrices from tensor data on multi-mode simultaneously, hence the space-frequency, space-time, and time-frequency structure features

8 8 Qibin Zhao, Cesar F. Caiafa, Andrzej Cichocki, and Liqing Zhang can be captured from 3D tensor data. Classification performance on several experiment runs also confirmed the robustness of our method. To further improve the discriminative ability, the class information will be additionally considered in the cost function and the semi-supervised feature extraction method will be studied in next step. References. Smilde, A., Bro, R., Geladi, P.: Multi-way Analysis with Applications in the Chemical Sciences. Wiley (24) 2. Heiler, M., Schnorr, C.: Controlling Sparseness in Non-negative Tensor Factorization. Computer Vision ECCV 26: 9th European Conference on Computer Vision, Graz, Austria, May 7-3, 26: Proceedings (26) 3. Cichocki, A., Zdunek, R., Choi, S., Plemmons, R., Amari, S.: Non-Negative Tensor Factorization using Alpha and Beta Divergences. Acoustics, Speech and Signal Processing, 27. ICASSP 27. IEEE International Conference on 3 (27) 4. Mørup, M., Hansen, L., Herrmann, C., Parnas, J., Arnfred, S.: Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG. Neuroimage 29(3) (26) Miwakeichi, F., Martínez-Montes, E., Valdés-Sosa, P., Nishiyama, N., Mizuhara, H., Yamaguchi, Y.: Decomposing EEG data into space time frequency components using Parallel Factor Analysis. Neuroimage 22(3) (24) Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. Wiley (November 29) 7. Mřrup, M., Hansen, L., Arnfred, S.: ERPWAVELAB A toolbox for multi-channel analysis of time frequency transformed event related potentials. Journal of Neuroscience Methods 6(2) (27) Yang, J., Zhang, D., Frangi, A., J.Y., Y.: Two-dimensional pca: A new approach to appearance-based face representation and recognition. IEEE Trans. Pattern Anal. Mach. Intell. 26() (24) Tao, D., Li, X., Wu, X., Maybank, S.: General Tensor Discriminant Analysis and Gabor Features for Gait Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 29() (27) Ye, J., Janardan, R., Li, Q.: Two-dimensional linear discriminant analysis. NIPS (24). Wang, X., Tang, X.: A unified framework for subspace face recognition. IEEE Trans. Pattern Analysis and Machine Intelligence 26(9) (24) Fu, Y., Huang, T.: Image classification using correlation tensor analysis. IEEE Transaction on Image Processing 7(2) (28) He, X., Cai, D., Niyogi, P.: Tensor subspace analysis. NIPS6 (26) 4. Tao, D., Li, X., Hu, W., Maybank, S., Wu, X.: Supervised tensor learning. Proceedings of the Fifth IEEE International Conference on Data Mining (ICDM 5) (27) 5. Yan, S., Xu, D., Q., Y., Zhang, L., X., T., Zhang, H.: Discriminant analysis with tensor representation. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 5) (25) 6. Caiafa, C.F., Cichocki, A.: Slice Oriented Decomposition (SOD): A New Tensor Decomposition for Representation of 3-way Data. submitted February 29.