Tensor Decomposition for EEG Signal Retrieval

Tensor Decomposition for EEG Signals Retrieval Zehong Cao , Member I EEE, Mukesh Prasad , Member IEEE, M. Tanvee r , Senior Member IEEE , C hin - Teng Lin , Fellow IEEE Abstract - Prior studies have proposed methods to recov er multi - channel e lectroencephalograph y (EEG) signal ensembles from their partially sample d entries. These methods depend on spatial scenarios, yet few approaches aiming to a tem poral reconstruction with lower loss . The goal of this study i s to retrieve the temporal EEG signals independently which was ov erlooked in data pre - processing . We conside red EEG signals are impinging on tensor - based approach, named n onlinear Canonical P olyadic D ecomposition (CPD) . In this study, w e collected EEG signals during a resting - state task. Then, we defined that t he source signals are original EEG signal s and t he generated tensor is perturbed by Gaussian noise with a signal - to - noise ratio of 0 dB. The sources are se p arated using a basic n on - negative CPD an d the relative errors on the estimates of the factor m atrices . Compar ing the similarities between t he source signa ls and their recovered versions, the results showed significantly high correlation over 95 %. Our findings reveal the possibility of recovera ble temporal sign al s in EEG applications. Keywords - EEG , Tensor, Nonli near, CPD, Recov er y . I. INTRODUCTIO N Nowadays, considerable inte rest has been dedicated to t he development of several wearable electroencephalography (EEG) system s with dry sensors that collect and rec ord different vital signs for an extended period. The long - term recording EEG data depends on lo w- power communication and transmission protocols. However, the per formance of wearable EEG s ystems is bottlenecked mainly by the limited lifespan of batteri es . Therefore, explorin g data compression techniques can reduce the number of the data transmitted from the EEG systems to the clouds. Compre ssive sensing (CS) , a novel data sampling paradigm that merges the acquisition and the compression pr ocesses , provides the best trade - off between reconstruction quality and low - power consumption compared to conventional compression approaches [1]. The CS suggests reconstructing a signal from its partial observations if it enjoys a sparse representation in some tra nsform domain and the observation operator satisfi es some incoherence conditions. Recently, recove ring a spectrall y temporal and sp at ial signal becomes of great i nterest in signal processing community [2 ]- [3]. The spectrally sp at ial signal can be sparse i n the dis crete Fourier transform domain if the frequencies are aligned well with the di screte freq uencies. In this case, signals can be recovered from few measurements by enforcing the sparsity in the discrete Fourier domain [4]. However , frequenc y information in practical applicatio ns generally take fe wer values compared to the temporal domain, and leads to the loss of sparsity and hence worsens the performance of compressed sensing. To address this problem, total variation or atomic norm [ 5 ] minimization methods were proposed to deal with signal recovery with continuous sinusoids or exponential signal s [6 ] , but these methods did not touch to temp oral EEG signals. Therefore , t he signal reconstr uction from its temporal sampled paradigm is recognized as a cha llenge of EEG signal processing. II. MULTI - VIEW EEG SIGNALS Given a time serie s recorded p hysiological data, all data samples were carried by a vector . T he power spectrum analysis of t he time series has often been applied for investigatin g p hysiological ( e.g., EEG ) oscillations by computational intelligence models [7- 14 ] and associated healthcare applications [15 - 20 ] . Recently, multiple electrodes are often used to collect EEG data in the experim ent. Indeed, in EEG experiments, there are high - order modes than t he two modes of time and space. For in stance, analysi s of EEG signals may compare responses recorded in different subject groups or event - related potentials (ERPs) as trial s, which indicates the brain data collected by EEG techniques can be na turally fit into a multi - way arr ay including multiple modes. The m ulti - way array is a tensor, a new way to represen t EEG signals. Tensor decompositi on inherently expl oits the interactions among multiple modes of the tensor. I n an EEG experiment, potential ly, there could be even seven modes Zehong Cao are wi th Disc ipline of ICT, Sc hool of Technology, Environments and Desi gn, College of Sciences and Engine ering, University of Tasmania, TAS, Australia , and S chool of Software, Faculty of Engineering and Information Technology, University of Technology Sydney , NSW, Australia. (corresponding author to zehong.cao@utas.edu.au). Mukesh Prasad and Chin - Teng Lin are with Centre for Artificial Intelligence, Faculty of Engineering and Inform ation Technology, University of Technol ogy Sydney, NSW, Australi a. M. Tanveer is with Disciplin e of Mathematics, Indian Institute of Technology Indore, India. including time, frequency , space, trial, condition, sub ject, and group. In the past ten years, there have been many reports about tensor decomposition for processing and analyzing EEG signals [21 - 22] . However, there is n o study particularly for tensor decomposition of EEG sig nals retrieval yet. III. MULTIDIMENS IONAL HARMONIC RETRIEVAL The fundamental model s for tensor decomposition are Canonical P olyadic Decomposition (CP D) [2 3] , and we expanded this framework to Nonlinear Canonical P olyadic Decomposition (NCPD) to fi t EEG signals [2 4]. A. Definition Given a th ird - order tensor, a two - component canonical polyadic decomposition (CPD) is shown below: X = a1 ◦ b1 ◦ c1 + a2 ◦ b2 ◦ c2 + E ≈ a1 ◦ b1 ◦ c1 + a2 ◦ b2 ◦ c2 = X1 + X2 . (1) After the two - component CPD is applied on the tensor, two temporal, two spectral, and two spatial components ar e extracted . T he first temporal component a1 , the first spectral component b1 , and the first spatial component c1 are associated w ith one another, and their outer product produces rank - one tensor X1 . The se cond components in the time , frequency, and space modes are associated with one another, and their outer product generates rank - one tensor X2 . The sum of rank - one tensors X1 and X2 appr oximates original tensor X . Therefore, CPD is the sum of some rank - one tensors plus the error tensor E . Generally, for a given Nth - order tensor X ∈ R I 1 ×I 2 … ×I N , the CPD is defined as X = ∑ ( $ %& ' u (1) ◦u (2) ◦ ···◦u (N) ) +E = Xˆ +E ≈ X ˆ . (2) where X =u (1) ◦ u (2) ◦ ··· ◦ u (N) , r=1, 2, ···, R; Xˆ approximates tensor X , E ∈ R I 1 × I 2 × ··· × I N ; and ||u (n) || = 1, for n=1, 2, ···, N − 1 . U ( n ) = u (n) , u (n) , · · ·, u (n) ∈ R I n × R denotes a component matrix for mode n, and n=1, 2, ···, N.  In the tensor - matrix produc t form, Eq . (2) tr ansforms i nto X =I × 1 U ( 1 ) × 2 U ( 2 ) × 3 ··· × N U ( N ) +E= X ˆ +E . (3) where I is an identity tensor, which is a diago nal tensor with a diagonal entry of one. Here, we used Tensorlab [25 ] for signal processing and tensor compositions. The batch algorithms , nonlinear least squares (NLS) algorithm, called cpd_nls , compute the CPD of the tensor form ed by the slices in the window. B. Data One ma n with age 25 participated in the resting - state experiment with reco rding EEG signals at O1, Oz, and O2 channels , who were asked to read and sign an i nformed consent form before part icipating in the EEG experiment . This study was appr oved by t he Institut ional Review Board of the Veterans General Hospital, Taipei, Taiwan. Three sources impinge on EEG signal s with azimut h angles of 10°, 30° and 70°, respectively, and with elevation ang les of 20°, 30° and 40°, respectively. We observe 200 - time samples, such that a tensor T ∈ℂ 10×10×15 is obtained with t ijk the observed signal sampled at time instance k . Each source contributes a rank - 1 term to the tensor. The vectors in the first and second mode are Vandermonde and the third mode co ntains the r espective source signal s mult iplied by attenuation factors. Hence, the factor matrices in the first and second mode denoted as A and E , are Vandermonde matrices and t he factor matrix in the third mode is the matrix containing the attenuated sources, denot ed by S is the EEG raw data (source signal). Addit ionally, we defi ned t he generated tensor is perturbed by Gaussian noise wit h a signal - to - noise ratio of 0 dB. C. Signal separation and direction - of - arrival estimation The sour ces are separated by means of a basic CPD, wi thout using the Vandermonde structure. The relative errors on t he estimates of the factor matrices can be calculated with e rrors b etween factor mat rices in a CPD (ERRCPD), called cpderr in Tensorlab . The ERRCPD comp utes the relative diff erence in Frobenius norm between the factor matrix U n and the estimated factor matrix Ues t n as : ERRCPD n = Norm(U n - Uest n × P × D n )/ N orm(Un) (4) W here the matrices P and D n are a permutation and scaling matrix such that the estimated factor matrix Ues t n is optimally permuted and scaled to fit U n . The optimal ly permuted and scal ed version i s returned as fourth output argument. If size (Uest n ,2) > size(U n ,2), then P selects size (U n ,2) rank - one terms of Uest that best match those in U . If size (Uest n ,2) < size(U n ,2), then P pads the rank - one terms of Uest with rank - zero terms. Furthermore, it is important to note t hat the diagonal matrices D n are not constrained to multiply to the identity matrix. In other words, ERRCPD n returns the relative error bet ween U n and Ues t n independently from the relative error between U m and Ues t m where m ~= n. IV. RESULTS A. The source of EEG signals The source of E EG signals is shown in Fig. 1 , which includes the O 1, O z and O2 chan nels corresponding to sour ce 1, 2 and 3. Figure 1 The three sou rces of EEG signals . B. Visualisation of the tensor Here, as shown in Fig. 2, we visualized the third - order tensor T by drawing its mode 1, 2, and 3 slices using sliders to define their respective indices . The index i, j, and k indicate the representation sc ales for the third - order tensor. Figure 2 Visuali ze a third - order t ensor with slices. C. Observed si gnals with and without noise We generated tensor per turbed by Gaussian noise with a signal - to - noise ratio of 0 dB. As shown in Fig. 3, we gave three observed signals with and without noise, for source EEG signals. Fig. 3 Three observed s ignals with and without noise. D. Signal separation The relative errors on the estimates of the factor matrices can be calculated with ERRCPD , which are 0.0504, 0.0487 and 0.1634 , respectively. The ERRCPD also returns estimates of the permutation matrix and scaling matrices, which can be used to f ix the ind eterminacies. Th e source signals and their recovered versions are compared in Fig. 4 . Figure 4 The original and recovered sour ce signals. 0 50 100 150 200 t -20 0 20 Source 1 0 50 100 150 200 t -20 0 20 Source 2 0 50 100 150 200 t -20 0 20 Source 3 10 1 9 7 3 200 199 5 i 166 5 Slices of the observed tensor j 3 133 k 100 1 7 67 9 34 10 1 i j k 0 50 100 150 200 t -20 0 20 observed partial tensor with and without noise Exact signal Noisy signal 0 50 100 150 200 t -20 0 20 observed partial tensor with and without noise 0 50 100 150 200 t -20 0 20 observed partial tensor with and without noise 0 50 100 150 200 n -20 0 20 Source 1 Exact Estimated 0 50 100 150 200 n -20 0 20 Source 2 0 50 100 150 200 n -20 0 20 Source 3 Additionall y, we have conducted the correlation between origin al and recovered source signals, and the outcome showed the over 95% correlation with the significance level ( p < 0.05). E. Direction - of - arrival estimation and missing values due to broken sensors The direction - of - arrival angles can be determined using the shift - invariance property of the individual Vanderm onde vectors. This gives relative errors for the azimuth angles of 0.03 0 3, 0. 0069 and 0.0058, and fo r the elevation angles of 0.0061 , 0.0 114 and 0.0098 . Since Tensorlab is enable to process full, sparse and incomplete tensors, t he missing entries can be indi cated by empty values. We consider the equivalent of a deactivated sensor, a sensor that breaks halfway the experiment, and a sensor that starts to work halfway the experiment. The incomplete t ensor is visualized in Fig 5. Figure 5 Visuali zation of t he data tens or in the case of broken sensors . V. CONCLUSION T his is the first study to retrieve the temporal EEG s ignals independently. In this study, we collected EEG signals during a resting - state task and investigated EEG signals impingin g o n tensor - based approach, named nonlinear CPD . The source signals are separated using a basic CPD and the re lative err ors on the estimates of the factor matrices of tensors . Compar ing the similariti es between the source signals and their recovered versions, the results showed s ignif icantly high correlation over 95 %. Our f indings reveal the possibili ty of reco verable temporal signals in EE G applications. ACKNOWLEDGEMENT This work was sup ported in pa rt by the Australian Research Council (ARC) under disco very grant DP180100670 and DP180100656. The resear ch was als o sp onsored in part by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF - 10 -2- 0022 and W911NF - 10 -D- 0002/TO 0023. 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