FORSCHUNGSZENTRUM JÜLICH GmbH Zentralinstitut für Angewandte Mathematik D Jülich, Tel. (02461)

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1 FORSCHUNGSZENTRUM JÜLICH GmbH Zentralinstitut für Angewandte Mathematik D Jülich, Tel. (2461) Interner Bericht Temporal and Spatial Prewhitening of Multi-Channel MEG Data Roland Beucker, Heidi Anne Schlitt* FZJ-ZAM-IB-9721 November 1997 (letzte Änderung: ) (*) Picker International, MR Division, 595 Miner Rd., Highland Heights, OH 44143, USA

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3 Temporal and Spatial Prewhitening of Multi-Channel MEG Data R. Beucker and H. A. Schlitt y Central Institute for Applied Mathematics Research Centre Julich GmbH D Julich, Germany y Picker International MR Division 595 Miner Rd. Highland Heights, OH 44143, USA Abstract In this paper we present a prewhitening strategy to decorrelate the noise in space and in time. We discuss in detail how the eects of the ltering can be seen in the singular value decomposition of the data and we demonstrate how the approach can be used with an objective, statistical signal subspace estimation algorithm. 1 Introduction For multiple dipole reconstruction methods the noise is assumed to be spatially and temporally white. The statistical model for Magnetoencephalography (MEG) data introduced by Mosher et al. [1] consists of time series to which spatially and temporally zero-mean white noise is added. Under ideal conditions, in the Value Decomposition (SVD) spectrum of measured MEG data a singular value with a high multiplicity represents the noise part of the spectrum. For real measured data, however, the noise is not white and the noise singular values do not have identical values. By prewhitening the data, the noise can be made to more closely meet the assumption of being spatially and temporally white. Mosher et al. [2], [1] proposed, but did not implement, prewhitening the measured MEG data before performing a reconstruction with Multiple Signal Characterization 1

4 (MUSIC). In order to use MUSIC, one has to predict the dimension of the signal subspace [2]. The dimension can be determined by visual inspection of the singular value spectrum or by applying a statistical test. We found that, in contrast to the 'Minimum Description Length' (MDL), `Akaike Information Criterion' (AIC) [3] or several other tests (e.g. sphericity test), the `Fast Subspace Decomposition' (FSD) from Xu et al. [4] produces good results [5], [6]. To improve the statistical condition for the application of the test and for the MUSIC reconstruction we present a strategy for temporal and spatial prewhitening of multi-channel MEG-data [7]. We demonstrate the eects of prewhitening on two sets of measured data. 2 Methods We use the time series model X t = m t + N t introduced by Mosher et al. [1] which consists of a deterministic trend component, m t, dependent on the time courses of the associated sources and a random noise component, N t. To temporally prewhiten the data we subtract the deterministic trend, model the noise with a vector autoregressive process (VAR), determine the model order with the 'bias corrected Akaike information criterion' (AICC), estimate the noise covariance matrix and apply the VAR model as a lter to the measured data. We determine the coecient matrices of the VAR-process by taking the multi-dimensional Yule-Walker estimator [8]. The spatial prewhitening consists of an orthogonality transformation and the normalization of the noise eigenvalues [8]. Prewhitening strategy: 1. Estimation and elimination of the deterministic trend m t Filter the measured time slices with a low pass lter Subtract the ltered data from the measured data to form an estimate of the noise ( ^N t ) 2. Model the noise by a vector autoregressive process (VAR) of order p N t = P p j=1 jn t?j + w t Determine the order, p, with the bias corrected Akaike information criterion (AICC) and the coecient matrices, j, using ^Nt Estimate covariance matrix of the white noise w t 3. Temporal prewhitening: Application of the VAR-Model as lter Filter the measured data with the help of the determined coecient matrices 1; : : : ; p 4. Spatial prewhitening: Diagonalization of 2

5 Calculate the eigendecomposition of = V D V T Apply D? 1 2 V T to the temporally prewhitened measured data It is common to estimate the noise covariance matrix from data measured before stimulus onset or to measure the environment noise for several hundreds of milliseconds without a subject [9]. Because the noise conditions change with time, we decided to subtract the deterministic trend of the time interval which contains the evoked response. If one assumes the noise is not strongly correlated over time, one can directly calculate the noise covariance matrix and multiply the data with D? 1 2 V T (spatial prewhitening) [9]. This has the eect that the singular values which one assumes to belong to the noise are normalized (gure 2). The spatial prewhitening also eliminates the spatial dependencies of the sensors between each other (orthogonality transformation). The temporal prewhitening eliminates dependencies in the noise in time and gives the researcher an indication which singular values represented dependencies in the noise after subtracting the global trend. 3 Results We rst analyzed data from an auditory experiment to show the eect of spatial prewhitening on the SVD. These data were measured with the Neuromag 122 channel whole-head array [1], but we restricted our analysis to one of the two sets of orthogonal gradient directions (61 channels) to reduce the spatial dependence of the noise. The measured excitation was evoked by a 5 ms 1 khz sinusoidal sound (left ear). The measurement was started 1 ms before stimulus onset. The sampling frequency was 297 Hz and a band-pass lter (.3-9 Hz) was applied to the data. For our further analysis we averaged 95 epochs. One epoch lasted 6 ms (18 time slices, gure 1). The SVD of the auditory data (gure 2) shows that the small singular values continue to decrease instead of leveling out as they would if the noise were white. Before applying the FSD to these data we decorrelate the noise in space. As an estimate for the noise we simply took the residuals after subtracting the deterministic trend from the measured data and calculated the noise covariance matrix. After spatial prewhitening, however, the small singular values have approximately the same size. The signal subspace dimension estimated with the FSD is 4. Before the spatial prewhitening the estimated signal subspace dimension was 5. Because of the fact that our chosen interval contains a long pre- and poststimulus intervals, background activity is also represented by singular values which can be seen as signals. If the interval size is reduced to include only the evoked response, the estimated signal subspace dimension becomes smaller. Temporally prewhitening of the data was 3

6 8 6 A m 4 p l 2 i t u d -2 e -4 f T / cm ms f T 4 3 A 2 m p 1 l i t -1 u d-2 e ms Figure 1: On the left side data from an auditory experiment is presented. 6ms with a sampling rate from 297Hz are presented. On the right 768ms of averaged data from a somato-sensory experiment are presented not possible because the sampling rate was too low to estimate parameters for the autoregressive model. 7 1 of Values of Values Noise Part estimated by the FSD Figure 2: On the left side the SVD of 61 channels of the auditory data is shown. The smaller singular values fall o strongly. On the right the eect of spatial prewhitening can be seen. Because of the normalization the singular values have a dierent scaling. The smaller singular values of the whole spectrum have approximately the same size. The second data set was from a somato-sensory experiment measured with a BTi 37 channel array. The measured excitation was evoked by stimulating the thumb with pus of air. The epochs were 2 s long, the measurement was started 1 ms before stimulus onset, and the response occurred at about 13 ms post-stimulus. The sampling frequency was 52 Hz and 141 time slices were recorded per epoch. During the measurement, a high pass lter with a cut-o frequency of 1 Hz was applied. We averaged 2 epochs and chose a subinterval from 1 ms to 768 ms (4 time slices) for our further analysis. This subinterval is shown in gure 1. 4

7 7 6 5 of 4 Values Figure 3: In this gure the complete SVD spectrum of the non-ltered somatosensory data is shown of Values 4 Temporally Prewhitened of Values Temporally Prewhitened Spatially Prewhitened Figure 4: In this gure on the left side the SVD spectrum of the temporally prewhitened time series from the somato-sensory experiment is shown. On the right side the spectrum of the temporally and spatially prewhitened time series is shown. In gure 3 the SVD of the somato-sensory data is shown. One cannot clearly see which singular values belong to the signal part and which to the noise part. In gure 4 we plotted the singular values from the temporally prewhitened data and those ones from the temporally and spatially prewhitened data. After applying the rst step of our algorithm (temporal prewhitening, model order p = 2) the distance from the sixth to the seventh singular value of the temporally prewhitened data decreases. This indicates that the noise residuals resulting after subtracting the global trend from the data were not independent. The decorrelation in time separates the spectrum which makes it much easier to decide which singular values belong to which part. After spatial prewhitening the normalization eect can clearly be seen. The estimated signal subspace dimension of the chosen interval of the somatosensory data was 15. After temporal and spatial prewhitening the estimated signal subspace dimension of the chosen interval was 14. 5

8 4 Discussion In this paper we presented a method for temporal and spatial prewhitening. After subtracting the global trend we showed how to model the noise by a vector autoregressive process and how the spectrum is changed after applying the VAR-process as a lter to the measured data. We showed that in case of lower temporal sample rates the spatial prewhitening can be used to equalize the small singular values of the spectrum. Our prewhitening strategy improves the statistical conditions so that statistical methods for estimating the dimension of the signal subspace (e.g. FSD) and multiple dipole reconstruction methods (e.g. MUSIC) can produce better results. The spatial prewhitening normalizes the variances and makes the small singular values more equal. The noise covariance matrix, which must be known for the spatial prewhitening, can be determined with the help of the VAR-model. If the data has been sampled at a high enough rate in time, then temporal prewhitening can be used to improve the noise statistics. If there are not enough time samples, the coecient matrices cannot be estimated in a stable fashion. In this case the noise covariance matrix can be estimated either by collecting a prestimulus interval [9] or by subtracting the global trend of the time interval which contains the evoked response or by using a recursive least squares algorithm [11]. We recommend the prewhitening before applying multiple dipole methods because, in general, the measurement noise is correlated. Furthermore, one should use a sampling rate from 512 Hz so that one can use approximately 4 time slices for 37 channels for the calculation of the coecient matrices of the VAR-model. 5 Acknowledgments The authors are indebted to W. Meyer (Central Institute for Applied Mathematics, Research Centre Julich) and J. C. Mosher (Biophysics Group, Los Alamos National Laboratory) for giving many valuable comments. We would like to thank R. Hari (Low Temperature Laboratory, Helsinki University of Technology) and K. Wassmuth (Biomagnetic Technologies, Aachen) for generously sharing MEG data with us, and F. Hofeld (Central Institute for Applied Mathematics, Research Centre Julich) for his support. References [1] J. C. Mosher and R. M. Leahy. EEG and MEG source localization using recursively applied (RAP) MUSIC. Proceedings of the Thirtieth Annual Asilomar Conference on Signals, Systems, and Computers, Pacic Grove, California, USA, November

9 [2] J. C. Mosher, P. S. Lewis, and R. M. Leahy. Multiple dipole modeling and localization from spatio-temporal MEG data. IEEE Trans. Biomed. Eng., 39:541{ 557, [3] M. Wax and T. Kailath. Detection of signals by information theoretic criteria. IEEE Trans. Acoust. Speech Signal Process., 33:387{392, [4] G. Xu and T. Kailath. Fast estimation of principal eigenspace using Lanczos algorithm. SIAM J. Matrix Anal. Appl., 15:974{994, [5] R. Beucker and H. A. Schlitt. Objective signal subspace determination for meg. Technical Report FZJ-ZAM-IB-9715, Research Centre Julich, Julich, Germany, [6] R. Beucker and H. A. Schlitt. Comparison of statistical tests for determining the number of MEG sources. In Abstract Book, Eighth World Congress of the International Society for Brain Electromagnetic Topography, Zurich, Switzerland, March [7] R. Beucker and H. A. Schlitt. Spatial and temporal prewhitening of multichannel EEG/MEG data. In Abstract Book, Sixth German EEG/EP Mapping Meeting, Gieen, Germany, September [8] P. J. Brockwell and R. A. Davis. Time Series: Theory and Methods. Springer, second edition, [9] K. Sekihara, D. Poeppel, A. Marantz, H. Koizumi, and Y. Miyashita. Noise covariance incorporated MEG-MUSIC algorithm: A method for multiple-dipole estimation tolerant of the inuence of background brain activity. IEEE Trans. Biomed. Eng., 44:839{847, [1] M. S. Hamalainen. Functional localization based on measurement with a wholehead magnetometer system. Brain Topogr., 7:283{289, [11] M. Campi. Performance of RLS identication algorithms with forgetting factor: A -mixing approach. J. Math. Systems Estim. Control, 7:29{53,