BEAMFORMING DETECTORS WITH SUBSPACE SIDE INFORMATION. Andrew Bolstad, Barry Van Veen, Rob Nowak

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1 BEAMFORMING DETECTORS WITH SUBSPACE SIDE INFORMATION Andrew Bolstad, Barry Van Veen, Rob Nowak University of Wisconsin - Madison 45 Engineering Drive Madison, WI akbolstad@wisc.edu, vanveen@engr.wisc.edu, nowak@engr.wisc.edu ABSTRACT We examine the problem of detecting a rank one signal immersed in colored noise of unknown covariance. The signal is assumed to lie in a known, lowdimensional (rank P subspace, and (possibly inaccurate side information regarding the direction of the signal within the subspace is available. This type of problem arises naturally in the study of brain activity via magnetoencephalography (MEG due to anatomical constraints and in wireless communications when a channel estimate is available. We compare three well known beamforming based detectors: subspace LCMV (, side information constrained LCMV (, and subspace LCMV with estimated direction postfiltering (, with a new detector, subspace LCMV with side information post-filtering (. The detector is designed to mitigate the effects of inaccuracies in the side information. We evaluate ROC curves analytically for the and detectors and via Monte Carlo simulations for the other detectors. We show that the outperforms existing detectors in many interference and noise environments, including cases for which the side information is inaccurate.. INTRODUCTION We address detection of a rank one source within a known P dimensional subspace given K measurements from an M dimensional sensor array and assume an estimate of the source orientation within the subspace is available as side information. This problem occurs in magneto/electroencephalography (M/EEG, the application in this paper, because the dipolar source model for a given location is the product of a rank P (P = 2 or 3 lead field matrix and a vector representing the dipole This work is supported in part by Lincoln Laboratory under Air Force Contract FA872-5-C-2 and the National Institutes of Health under award R Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. orientation. The dipole orientation is known to be normal to the cortical surface and thus may be estimated from anatomical information, such as an MRI. Similar problems also occur in CDMA wireless communications when a channel estimate is available as side information, and in other array applications involving imperfect knowledge of propagation characteristics. The problem we consider is similar to uncertain rank one waveform (UROW detection as described by Bose and Steinhardt [2]. In [2], the signal is assumed to lie in the mean of the data and detectors based on maximally invariant parameters are developed. The problem considered here differs in that the signal is assumed to have a significant variance component so we consider energy detectors based on four variations of linearly constrained minimum variance (LCMV beamformers. These detectors are inspired by previous work in MEG and EEG, e.g., [], [], [], as well as recent developments in multi-rank beamforming [3], [9]. LCMV beamforming requires an estimate of the data covariance matrix. Finite sample support effects have not been carefully examined in MEG and EEG applications, though methods to reduce their impact have been proposed (see e.g., [8]. The majority of analysis to date has been focused on asymptotic behavior; see, e.g., [4], [9]. The work presented here directly addresses the effect of finite sample support in covariance matrix estimation for MEG source detection using receiver operating characteristic (ROC curves. Matrix and vector quantities are denoted by boldface upper and lower case letters, respectively. Matrix inverse is denoted by superscript, while transpose is denoted by superscript T. 2. PROBLEM FORMULATION Let U be a matrix whose columns form an orthonormal basis for the known P dimensional subspace. In MEG, U can be found via a singular value decomposition of the lead field matrix H(θ, whose columns are the mea-

2 sured signal due to a dipolar source at location θ pointing in the Euclidian directions. H(θ is typically rank two in MEG. Assume the true one dimensional subspace is spanned by U m for some unit norm P vector m. Likewise, assume the estimated one dimensional subspace is spanned by U m for some unit norm P vector m. That is, m and m are the estimated and true source orientations respectively. This leads to the following hypothesis testing problem. H : y N(,R n H : y N(,R n + σ 2 su m m T U T Here U is known, while the noise covariance R n, signal power σ 2 s, and true orientation m are unknown. Notice the signal (when present is assumed to lie in the covariance matrix, not the mean (as in []. In the remainder of this paper, R refers to either R n if no signal is present or R n + σ 2 s U m m T U T if a signal is present. 3. DETECTORS In this section, we present three existing and one new beamformer and briefly discuss their theoretical differences. A beamformer adaptively suppresses noise present in a measured signal by allowing only specific signal components to pass without distortion. In LCMV beamforming, this is achieved by minimizing output power subject to a linear constraint. The LCMV beamformer w that passes signals in a one dimensional subspace spanned by u is given by the solution of: min w E(w T Rw such that w T u = ( while the LCMV beamformer W that passes signals lying in a P dimensional subspace spanned by the columns of U is given by the solution of: min W E(tr W T RW such that W T U = I. (2 The respective output variance r = w T Rw of ( and covariance matrix R = W T RW of (2 are given by (see, e.g., [5]: r o R o = u T Ru u T Ru(u T Ru u T Ru = (u T R u (3 = U T RU U T RU(U T RU U T RU = (U T R U (4 Subspace BF Side Information Constrained LCMV (a (c P 2 2 Subspace BF Subspace BF P (b P (d v max 2 m ~ 2 Figure : Each of the four detectors consists of a beamforming spatial filter followed by an energy detector or a postfilter and energy detector. (a detector. (b detector. (c detector. (d detector. where U indicates the orthogonal complement of U. Since R is typically unknown, it must be estimated from the data. Throughout this paper, the sample covariance matrix is used; i.e., ˆR = K K k= x kx T k, where it is assumed that K > M. The four detectors considered in this paper are pictured in Figure. All four perform energy detection on a beamformer output. Subspace LCMV ( (Fig. (a passes signals that lie in the subspace U ; estimated direction postfilter ( LCMV (Fig. (b estimates the rank one subspace within U by solving an eigenproblem; side information constrained ( LCMV (Fig. (c constrains the beamformer using the prior information ( m of the rank one subspace within U ; and side-information postfilter ( LCMV (Fig. (d projects the output of the beamformer constrained to U onto the space m given by the side information. The output powers or test statistics for these four cases are given by (5 (8. r = tr((u T ˆR U (5 r = T λ max ((U ˆR U (6 r = m T U T ˆR U m (7 r = m T (U T ˆR U m (8 The method ignores the side information and the rank one nature of the signal of interest and thus is susceptible to noise that lies in the subspace U. The detector uses the rank one nature of the signal, but not the side information about the direction within subspace U, and can falsely lock onto interference that is inconsistent with the side information. The approach incorporates the side information into the LCMV constraint and is thus susceptible to signal cancellation when there is mismatch between the side information and the true signal direction within subspace U. The

3 method employs the side information after LCMV beamforming, and thus is much less sensitive to mismatch. Note that the has the most degrees of freedom available for interference suppression. These four detectors exploit the side information differently and thus the relative performance is dependent on the accuracy of the side information, the nature of the interference, and the number of data records available for sample covariance matrix construction. 4. ANALYSIS The statistics of both the and detectors can be derived in closed form using properties of Wishart matrices. The sample covariance matrix ˆR has a Wishart distribution with K degrees of freedom and covariance matrix R, denoted W M (K,R. Applying Theorem 3.2. of [7] to (7: r W (K M, ( m T U T R U m. (9 Since a one by one Wishart matrix is proportional to a chi-squared random variable, the statistic r = ( m T U T R U m χ 2 K M, where χ2 K M is a chisquared random variable with K M degrees of freedom. By the same theorem: (U T ˆR U ( W P (K M + P, (U T R U. Now using Theorem of [7] we have: of a particular phenomenon in MEG is typically small since patients grow fatigued after sitting or lying still for long periods. The ROC curves for the and statistics presented here are estimated from Monte Carlo simulations. These simulated curves are sufficient for comparisons with the other two detectors and confirm the intuition that using side information of reasonable quality improves performance. To analytically compare the and detectors, we first ignore the difference in degrees of freedom between the two. If the outperforms the without the difference in degrees of freedom, it will outperform by a wider margin once this difference is reintroduced. Letting s = U m, the comparison can now be made using only the following four constants: = m T (U T R n U m (2 A s = m T (U T (R n + σ 2 s s s T U m (3 = ( m T U T R n U m (4 B s = ( m T U T (R n + σ 2 s s s T U m (5 Then r = χ 2 when only noise is present and r = A s χ 2 when a signal is present. Likewise, r = χ 2 when only noise is present and r = B s χ 2 when a signal is present. If the true covariance matrix is known, then the output power of the is A s or, and that of the is B s or. Now the probability of false alarm for the two detectors can be written as: r = m T (U T R U mχ 2 k M+P. ( Thus the statistics of these two detectors are completely described for a given signal and noise scenario, R. Comparing the two detectors requires comparing the degrees of freedom in the chi-squared random variables as well as the constants multiplying these variables. The statistics of the and detectors are more difficult to describe. The cumulative distribution function (CDF of the detector can be written as an infinite sum of zonal polynomials, whereas the CDF of the statistic can be expressed exactly in terms of a hypergeometric function of a matrix argument [7] or as alternating series in some special cases [6]. In general these are difficult to evaluate numerically. More tractable expressions exist for the asymptotic distributions of these statistics (as K M, but it can be difficult to obtain many samples in certain applications. In particular, the number of samples P FA = P(r > γ H = P(χ 2 > γ P FA = P(r > γ H = P(χ 2 > γ Equating false alarm rates for the two detectors implies γ = γ. Next consider the probabilities of detection: P D = P(r > γ H = P(χ 2 > γ A s P D = P(r > γ H = P(χ 2 > γ B s When the false alarm rates are equal, i.e. γ = γ, the will be a better detector if PD PD. Making this substitution in the last two equations, the detector is more powerful when:

4 A s B s (6 where the greater than or equal sign is a result of reintroducing the P degree of freedom difference between the two detectors. Note this is a sufficient condition, but is not necessary. This condition may be applied to various noise scenarios with additional definitions. First, let M represent the P by P orthogonal complement of m. In the P = 2 case (e.g. MEG, this is simply a two vector. Define V = U M so that [U m,v ] forms an orthonormal basis for the subspace spanned by U. In the white noise case (R n = σn 2 I, it is straightforward to show that the detector outperforms the detector for any SNR and any m such that m T m >. (If m T m =, both reduce to coin flipping. This is not surprising when considering that a beamformer cannot adaptively cancel white noise. In this case, the beamformer wastes P more degrees of freedom trying to cancel noise, whereas the detector uses these degrees of freedom to stabilize the χ 2 random variable. It is straightforward to show that in the white noise case: A s = + η(m T m 2 (7 B s η(m T m 2 = + + ηm T MM T (8 m where η = σ2 s σ is the signal to noise ratio. The second n 2 factor in (8 represents adaptive signal cancellation due to mismatch between the assumed and actual orientations. Since M is the orthogonal complement of m, the mismatch term m T MM T m is small for good estimates ( m m. As the mismatch approaches zero, the performance of the two detectors converge, but the detector remains superior due to the additional degrees of freedom. Next we analyze a more complicated noise scenario. In the dominant interferer model, the noise is white except for an additional rank one term. This case may occur when there is one strong source of interference. The noise covariance matrix in this case is of the form R n = σn 2I + σ2 i fft, where f represents the sensor measurements due to the interferer. Derivation of the constants A s,, B s, and is more complicated than the white noise case, but can still be found via (3 and (4 with appropriate substitutions. For the MEG case (P = 2, it can be shown that the sufficient condition (6 holds when V T f is small. That is, when the portion of the interference that lies in the subspace U but orthogonal to m is small. Without specifying a formal probability density for the interference, it is reasonable to assume that the component of an interferer in any one direction is likely to be small. For a known subspace with dimension P > 2, this suggests the will continue to perform better than the when the interferer has most of its power outside the subspace U (or parallel to m. Note that as P increases, the amount of interference in the subspace will increase, and the performance of the will eventually suffer. This loss is balanced somewhat by the increasing difference in degrees of freedom between the two detectors if the number of samples is low. One more case is briefly studied before considering simulation results. Suppose the true one dimensional subspace containing the signal is known exactly; i.e., m = m. In this no mismatch case, it seems the detector would be the best choice since it suffers no signal cancellation and can adaptively cancel the most noise. Surprisingly, the detector can outperform the detector in this case too for certain noise scenarios. This is a result of the extra degree(s of freedom in the chisquared random variable associated with the. 5. RESULTS In this section, receiver operating characteristic (ROC curves are presented comparing the four detectors described in Section 3. The curves for the and detectors are based on chi-squared distributions, while the and curves are based on 5, point Monte Carlo simulations of the CDFs. The signal model is taken from the MEG lead field based on the MRI of a healthy adult with the 74-channel Magnes II Biomagnetometer sensor configuration. The known subspace U is spanned by the lead field matrix corresponding to a location on the somatosensory cortex. The true orientation, m, is taken to be the normal vector at that location. To simulate error in the orientation estimate, m is taken as the true orientation vector rotated by an angle θ. The ROCs assume K = samples and M = 74. First, the four detectors are compared in the case of white noise. The results are shown in Figures 2(a and 2(b. Here the noise power is equal to the signal power at the sensors. As mentioned in Section 4, the detector is more powerful than the in this case. This is evident in the ROC plots of Figure 2, especially when the mismatch increases. The simulations confirm that both the and outperform the and filters when the mismatch is relatively small (θ o. At θ = 2 o, however, the detector performance drops below the and remains slightly above the detector, while the remains more powerful than both

5 (a (b Figure 2: ROC curves comparing the,,, and detectors in the case of white noise (SNR= db with (a o mismatch and (b 2 o mismatch (a (b Figure 3: ROC curves comparing the,,, and detectors in the dominant interferer case (SINR= 3.3 db with (a o mismatch and (b 2 o mismatch. the and. This illustrates the detector s robustness to mismatch. Notice the detector outperforms the detector in the white noise case. This follows from the fact that the assumes the signal lies in a one dimensional subspace. The will only declare a signal present if the energy in the subspace is large in one direction, whereas the will declare a detection if the total power is large. Note the performance of the and do not change as the mismatch increases because they do not use the estimate m. Next, the four detectors are compared when the dominant interferer model is used for the noise. In the plots shown here, f is the lead field corresponding to another dipolar source on the cortex. The signal, interference, and white noise power are all equal. As seen in Figure 3, the and are robust to this additional interference component due to the use of m, whereas the and both suffer. Interestingly, the performance of the slips below that of the when this interferer is present. This is because the interferer can create a large maximum eigenvalue in the signal subspace when no signal is present, but the total energy in the subspace remains too low for the to declare a detection. It may be of interest to know how much mismatch can be tolerated before the estimate m should be ignored. The precise answer depends on the noise coloration and subspace of interest. For the dominant interferer model used here, our simulations indicate that at θ = 3 o, the four detectors give roughly the same performance, while the is most powerful for θ > 3 o. As explained in Section 4, having perfect knowledge of m does not necessarily make the more powerful than the. Figure 4 illustrates the performance of the

6 (a (b Figure 4: ROC curves comparing the,,, and detectors in the dominant interferer case with no mismatch and two different interferers. four detectors using the dominant interferer model with two different interferers. The and detectors use m = m in both cases. In Figure 4(a, the detector is the most powerful, while the is most powerful in Figure 4(b. In either case, knowledge of the true source orientation gives the and detectors a strong advantage over the and detectors. 6. CONCLUSION The detector is shown to be the most powerful detector for typical beamformer-based MEG source detection scenarios. Specifically, it was shown to be theoretically superior to the detector in white noise, as well as in the dominant interferer model when the component of the interferer lying in the signal subspace but orthogonal to the assumed moment is small. Monte Carlo simulations provided verification that the and detectors outperform the and detectors when the estimate m is reasonably close to m. 7. REFERENCES [] S. Baillet, J.C. Mosher, and R.M. Leahy. Electromagnetic brain mapping. IEEE Signal Processing Magazine, 8(6:4 3, 2. [2] S. Bose and A. Steinhardt. Adaptive array detection of uncertain rank one waveforms. IEEE Trans. Signal Proc., 44(:28 289, 996. [3] H. Cox, A. Pezeshki, L.L. Scharf, O. Besson, and H. Lai. Multi-rank adaptive beamforming with linear and quadratic constraints. In Conference Record of the Thirty- Ninth Asilomar Conference on Signals, Systems and Computers, 25., October 25. [4] D. Gutiérrez, A. Nehorai, and A. Dogandžić. Performance analysis of reduced-rank beamformers for estimating dipole source signals using eeg/meg. IEEE Trans. Biomed. Eng., 53(5:84 844, 26. [5] Simon Haykin. Adaptive Filter Theory, 4th Edition. Prentice Hall, Upper Saddle Ridge, NJ, 22. [6] I. M. Johnstone. On the distribution of the largest eigenvalue in principle component analysis. Annals of Statistics, 29(2: , 2. [7] R. Muirhead. Aspects of multivariate statistical theory. John Wiley & Sons, Inc., New York, 982. [8] A. Rodríguez-Rivera, B. Baryshnikov, B. Van Veen, and R. Wakai. MEG and EEG source localization in beamspace. IEEE Trans Biomed. Eng., 53(3:43 44, 26. [9] L.L. Scharf, A. Pezeshki, and M. Lundberg. Multirank adaptive beamforming. In IEEE/SP 3th Workshop on Statistical Signal Processing, 25., July 25. [] K. Sekihara, S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita. Reconstructing spatio-temporal activities of neural sources using an MEG vector beamformer technique. IEEE Trans. Biomed. Eng., 48(7:76 77, 2. [] B. Van Veen, W. Van Drongelen, M. Yuchtman, and A. Suzuki. Localization of brain electrical activity via linearly constrained minimum variance spatial filtering. IEEE Trans. Biomed. Eng., 44(9:867 88, 997.