With the neuroelectromagnetic inverse problem, the optimal choice of the number of sources is a difficult problem, especially in the presence of correlated noise. In this paper we present a number of information criteria that help to solve this problem. They are based on the probability density function of the measurements or their eigenvalues. Make use of the Akaike or MDL (minimum description length) correction term and all employ some sort of noise information. By extensive simulations we investigated the conditions under which these criteria yield reliable estimations. We were able to quantify two major factors of influence: (1) the precision of the noise information and (2) the signal-to-noise ratio (SNR). Here defined as the ratio of the smallest signal eigenvalues and the average of the noise eigenvalues. Furthermore, we found that the Akaike correction term tends to overestimate, due to its greater sensibility to the precision of the noise informatio

Berends, E.M.

Jagers, Bert

Knosche, T.R.

Peters, M.J.

English

With the neuroelectromagnetic inverse problem, the optimal choice of the number of sources is a difficult problem, especially in the presence of correlated noise. In this paper we present a number of information criteria that help to solve this problem. They are based on the probability density function of the measurements or their eigenvalues. Make use of the Akaike or MDL (minimum description length) correction term and all employ some sort of noise information. By extensive simulations we investigated the conditions under which these criteria yield reliable estimations. We were able to quantify two major factors of influence: (1) the precision of the noise information and (2) the signal-to-noise ratio (SNR). Here defined as the ratio of the smallest signal eigenvalues and the average of the noise eigenvalues. Furthermore, we found that the Akaike correction term tends to overestimate, due to its greater sensibility to the precision of the noise information.<br/

Jagers, H.R.A.

University of Twente Research Information

Information criteria determine the number of active sources

NARCIS 

Berends, E.

i f  / I  18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Amsterdam 1996 3.5.2: EEG/MEG - Inverse Problem and Mapping INFORMATION CRITERIA DETERMINE THE NUMBER OF ACTIVE SOURCES T.R. Knosche, E.M. Berends, H.R.A. Jagers, and M.J' Peters LOW Temperature Divisiozi, (Jniversify of Twezite, Posthm 21 7, 7500 AE Emchede, The Netherlarids enmil: t. r. kiosche@tn. utMJente. z d  Abstract With the neuroelectromagnetic inverse problem, the optimal choice of the number of sources is a difficult problem, especially in the presence of correlated noise. In this paper we present a number of information criteria that help to solve this problem. They are based on the probability density function of the measurements or their eigenvalues. make use of the Akaikc or MDL (minimum description Icngth) correction term and all employ sonic sort of noisc information. By esteiisivc simulations we invcstigatcd the conditions under which these criteria yield rcliable estimations. We were able to quantify two major factors of influence: (1) the precision of tlie noise information and (2) the signal-to-noise ratio (SNR). here defined as the ratio of the smallcst signal eigenvalues and the average of the noise eigenvalues. Furthermore. we found that the Akaikc correction term tends to overestimate. duc to its greater sensibility to the precision of the noise information. Introduction The reconstruction of intracranial activity froni EEG and MEG measurements, also referred to as the invcrsc problem. is generally non-unique. In order to obtain a solution anlway. it is necessary to make a modcl of the sources. Obviously. the number of free parameters of this modcl must not esceed the numbcr of indcpcndcnt mcasurcmcnts. I n  order to achieve this. certain assuniptions about tlie espected activity have to be made. Many inverse algorithms are bascd on n priori knowledge on the numbcr of indcpendent sources. For the well-known spatiotemporal dipolc fit mcthod [ I ]  this nicaiis i t  must be known. how many current dipoles are simultaneously active. If there was not any noise. the number of non-zcro cigcnvalucs of the spatial covariance niatris of the measured data would equal the number of active sources (see example in  figure 1.a). If we added Gaussian uncorrelatcd noise and were able to obtain the exact data covariancc matris. the spectrum would look like the one i n  figure 1.b. In this casc. wc are still able to determine the number of source components that stick out of the uniform noisc floor. If. however, we have to be content with the saniplc covariance matris aiid tlie noise is more realistic. meaning e.g. spatially and temporally correlated, a picture like in figure 1.c occurs. I f  thc noise is small. iihc largest eigenvalues that differ significantly from the rest. still represent the sources. However. i n  many cases tlie eigenvalues decrease smoothly, making thc choicc of tlic number of components very subject ive (a) N o  noisc. (I)) Ilnifbnn. Gaussian. (c) Correlated noise, uncorrslatod iioiso: sample covariance c'xact covariance matrix. inatris. Figure 1 : Esamplc of eigenvalue spectra of noise mcasurcmcnt covaria~icc matriccs. First 10  cigcnvalucs shoivn. units arbitrary. There are 2 signal components. I t  is obvious thal a source niodel consisting of a larger number of coinponelids can esplain more of the mcasurcnicnts. Thcrcforc. if the rcsidual variance was thc only criterion for the choice of the number of sources in  the prcscnce of noisc. i t  would always equal the rank of the data matrix. Many of tlie sources would then merely explain noisc. In this paper. M'C prcsent information criteria that may help to niakc an niorc objective choice. Some of thcm have can be found i n  lilcraturc[ 2-41. others were and are being published elscwhcrc IS). These crilcria have been esplored by means of simulations with respect to their reliability. Methods A sct of mcasurcnicnls may bc described as superposition of a linear combination of the source signals and additive noise. -I' rcprcscnts thc data n i a t r n  with one row for each of the tti clianncls and onc column for cacli of thc 17 timc saniplcs. .I' stands for the source strengths and has k rows and f columns N contains additivc nOt!iC (Gaussian. itncorrclatcd i n  timc) and .-I symbolises tlic transfer matrix between source 0-7803-381 1- 1/97/$10.00 QIEEE 819 Authorized licensed use limited to: UNIVERSITY OF TWENTE.. Downloaded on February 25,2022 at 14:24:03 UTC from IEEE Xplore.  Restrictions apply. 18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Amsterdam 1996 3.5.2: EEG/MEG - Inverse Problem and Mapping Criteria hased on density hnction of' riieasurzmeiits all eigenvalues noise cigsnvalues the probability average noisz sigenvalue strengths and measured values. taking into account the source and sensor positions. thc volume conduction of the head, etc. No assumptions are made about the spatial shape, position, and temporal behaviour of the sourccs. The criteria explored in this paper generally consist of two terms: (1) a log likelihood function that reflects how wcll the observations g can be explained by a certain statistical model. and (2) a correction term that somehow rcpresents Ihc complcsity of the model. - CW( K) = -2 log f e z  (x) + C( v, ,n) (2) Spatial tioisz tnodsl white m i s s  COlT&Ib2:d noise level noisc lcvol iioisz 11 tlblowll hiiowii c1 c2 c3 C4a CSa CGb C7a C8a C9 n/C9 I) C10 c11 The first term is monotone decreasing with the number of sources K.  The maximum likelihood function 0, is the parameters vector of the statistical model that best explains the data with K sources. The correction term depends on the number of parameters for the particular stalistical modcl vK and the number of observations 17. I (  i s  an increasing function. The minimum of the sum of both should be at the correct number of sources (sec figure 2). Figure 2: The combination of log likelihood function and correction term yields a critcrion with a minimum at the correct number of sources (here 3). The critcria diffcr in ( 1 )  the stochastic variables the probability density function is based upon. (2) the statistical model of the noise. and (3) the type of correction Ierm. Table 1: Overview of the information criteria. Thcy can be used with Akaike (CsA) or MDL (CsR) correction terin. Table 1 gives an overview. Each critcrion can bc irscd with the Akaike correction term 161. denoted CsA. Al~erniitively. the minimum description length (MDL) 171 correction term can be used, denotcd C a .  cmL = v, logn (3) Thc lowcrcasc lctters n and h in the codcs for thc criteria 4, 5, and 6 stand for statistical assumptions on the sources (n - stochastic, b - deterministic). For the criteria 7, 8, and 9 the letter a indicates that the equality of the eigenvalues is tested against the alternative that they are not all equal to the noise level. The lcttcr h means that the equality of the eigenvalues is tested against the alternative that any deviation from the noise level is positive. Although some of the criteria assume that the noise is spatially uncorrelated, they can also be used for correlated noise if the data covariance matrix is corrected in advance using the known noise covariance matrix [5]. Results The criteria have becn 1csted on simulations of electrical brain activity. Current dipoles with fised positions and time- varying monicnts have been placed iii  a volume conductor modcl consisting of three concentric spheres. The electric potential was computed at clcctrode positions distributed over the upper hemispherc and Gaussian noise was added. I n  order to test the rcliability of thc criteria we varied noise Icvcl. accuracy of the noise information. number of observations (n). number of channels (nr), noise type (corrclated and uncorrclatcd). and dipole configuration. The signal-to-noise ratio ( S N R )  was defined as ratio of smallest signal eigenvalue to average noise eigenvalue. This value depends not only on the noise level but also on dipole configuration. 111. and I?. I t  turned out that the performance of the criteria was independently limited by two factors: the SNR and the accuracy of thc noisc information. For both. we found maximum values that must not be esceeded. Correlated and uncorrelated noise yielded the same results, indicating the correction procedure worked effectively. Figurc 3 sho\vs the minimum SNR for which the criteria cstimatcd the correct value in 90 '%, of all cases (23 trials). Figure 3. Minimum SNR for at least 90 YO success (2 dipoles w i t h  1 coinponcnts. 27 electrodcs. 550 time steps). 820 Authorized licensed use limited to: UNIVERSITY OF TWENTE.. Downloaded on February 25,2022 at 14:24:03 UTC from IEEE Xplore.  Restrictions apply. 18th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Amsterdam 1996 3.5.2: EEGIMEG - Inverse Problem and Mapping It can be seen that criteria using the MDL correction generally need a higher SNR to respond correctly. In order to test the influence of n and tti separately, we did not look at the SNR (since it depends on M and 1) ) ) .  but at the absolute variance of the noise. It was found that the maximum variance for which a criterion still works reliably, increases with both n and in.  In figure 4 we plolted the niaximum noise level that could be tolerated by the Akaike criteria against the number of time samples. n o  ” - a  Figure 4. Maximum noise variance ( i n  V’) for which thc Akaike criteria responded correctly (90 ‘% of cases) against the number of tiine samples (4 components. 27 clectrodcs) The other crucial factor for the perforinancc of the criteria is the accuracy of the provided iioisc information. Since the noise covariances must be estimated from a limited number of samples. certain errors are inevitablc. If the noise is Gaussian and the noise covariances is coinpiitcd from certain signal-free parts of each trial (pre-stimulus intervals). then the relative error of the noise variancc is dcfincd as follolvs. with n being the number of time samples in  thc pre-stimulus interval. q the number of trials. uva, the standard deviation of the noise kariance, and o2 the noise mriance itself We found that the maximum error that I S  tolcratcd (for 90 ?A of cases) by the criteria 1. 1. 7. and 10 (thosc critcria using noise iilformation only for N prior, correction. sec tablc 1) is independent of both SNR and noise variancc The values are presented in table 2 For thc other critcria. no gencral rule could be established so far Table 2: Maximum error of the noise variance for which the criteria 1. 4, 7. and 10 estimated the right number of sources. From table 2 we can see that the MDL criteria are more stable against i naccurate noise information. Discussion and Conc1,usions We investigated the reli,ability of 12 different information criteria with respect to the nuinbcrs of time samples ( I? )  and channels ( t u )  in the data. the SNR (defined as ratio of the smallcst signal and the average noise eigeiivaluc). and the accuracy of the providled noise information. Wc may summarise the following findings: 1. The masirnum noise lcvcl that could be tolerated increased with growing 111 and 17. 2. MDL based criteria generally nced a higher SNR than Akaike based ones. 3. For all criteria that use noise information only for a pi-iot-i corrcction of the data ( I .  4. 7. IO) the maximum tolerable crror i n  the noisc \>ariance nas  independent of thc iioisc level and the SNR. 4. MDL criteria are niorc stable against inaccurate noise informaiion. I n  practice. the stability against inaccurate noise information seems the most crucial fxlor. From equation 4 and table 2 one can concludc that e.g. critcrion c lR  necds t?q = 11 12 for a reliable estimate (e.g. 1,OO trials and 12 samples), while a typical Akaikc criterion such as c lA  needs i7q = ‘40001 (e.g. 500 trials and 80 samples). The latter casc may prove problcinatic, especially if‘ only a short pre-stimulus interval is available or i f  the noise information must be estimated from tlic average (just I trial thcn). We also pcrfornxd analyses of real measurements. The results. allhough not prt:sented here due to limited space. suggcsl that thc MDL vcrsions of thc criteria 2. 3. bb, and 9b produce the most consisiient and convincing results. These are criteria that use noisc information explicitly. suggesting that thc n priori compensation kvorks less well here (possibly due to non-Gaussian distribution of the noise). References [ 11 hl. Sclirrg and 1’. Berg. Use cfprior knowledge in  bruin elecrroningnefic l3r.aiii ‘I’opograpliy. 4( 199 1 ) T. li:iilaili. Detection ofsignols h ) ,  in/brinurion criteria. IEEE .. I rails. hcoiist.SpcccIi Signill I’l-occsaing. 33( 1985) 131 G.J.H. lliien ;iiid :\. v m  Ooslcrocn, The pei:/orniance q/in/brninrron- theoretic ciirerici in rlerecttng !he niinihai. of .sr,oncr/.s I17 ~ir i~l i i leud ECG ‘ .Ne t l i .  Inlhnn. Mcd . 31(1992) 141 K.M. M’ong Q.T. Aiaiig. J.I”.I<ci11y. mid P.C. Yip. O n  inforiiioiron theoretic criteria fbr detei-niinins fhe  n i o i i b w  of .rignuls in high resoli4rion arruyproce.s.sing. IKEE Trans. .Acoiisl. Specch Signal Processing, 38( 1990) (51 1I.R.A. .lagers, T. Kiidsclie. h1.J. I’elers. f<.stiriia/ion qfnitriiher of s: In/orniurtoii crireriu ccin help IO solve inverse problem. ‘rails. Signal I’rocessitig, 19YZ [6] 11. Akaikr. Extenston of the j”x~tnii~ii i  likelihoodprinciple. in B.N. Petrov Ak:idL:mini Kiadd. Ihdnpest. 19’73 [7] J. Riss:tnen. Afodelling h,jr shortest dom pi-tncip/e.Autoiiialica. 14( 1978) and F. Ca&i (+&) 2nd Intemorionnl Spyx.s, ,~ui on I?fon>icllion Theory. 82 1 Authorized licensed use limited to: UNIVERSITY OF TWENTE.. Downloaded on February 25,2022 at 14:24:03 UTC from IEEE Xplore.  Restrictions apply. 

Information criteria determine the number of active sources

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