Abstract — In electromagnetic source analysis it is necessary to determine how many sources are required to describe the EEG or MEG adequately. Model selection procedures (MSP’s, or goodness of fit procedures) give an estimate of the required number of sources. Existing and new MSP’s are evaluated in different source and noise settings: two sources which are close or distant, and noise which is uncorrelated or correlated. The commonly used MSP residual variance is seen to be ineffective, that is it often selects too many sources. Alternatives like the adjusted Hotelling’s test, Bayes information criterion, and the Wald test on source amplitudes are seen to be effective. The adjusted Hotelling’s test is recommended if a conservative approach is taken, and MSP’s such as Bayes information criterion or the Wald test on source amplitudes are recommended if a more liberal approach is desirable. The MSP’s are applied to empirical data (visual evoked fields). I.

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In [1] we proposed to analyze cross-spectrum matrices obtained from electro- or magneto-encephalographic (EEG/MEG) signals, to obtain estimates of the EEG/MEG sources and their coherence. In this paper we extend this method in two ways. First, by modelling such interactions as linear filters, and second, by taking the mean of the signals across different trials into account. To obtain estimates we propose a stochastic maximum likelihood (SML) method, and obtain the concentrated likelihood that includes the trial means.

To test whether the model fits the data well, a goodness-of-fit (GOF) test can be used. The chi-square GOF test is often used to test the null hypothesis that a function describes the mean of the data well. The null hypothesis with this test is rejected too often, however, because the nominal significance level (usually 0.05) is exceeded. Alternatively, the level of Hotelling’s test is accurate if a fixed hypothesis for the mean is available. In many situations, however, only an estimate of the mean is available, and so the level of Hotelling’s test may also be incorrect. An approximate version of Hotelling’s test is suggested as a GOF test. It is shown that this requires only an adjustment of the degrees of freedom of Hotelling’s original test. GOF tests assume that the model is either correct or incorrect whereas in model specification it is often assumed that the model is an approximation. Consequently, for approximate models a GOF test will mostly indicate that the model does not fit. It is therefore suggested that a measure of approximation to the true model could be used to get an indication of how bad the approximate model is. It is also shown that correct confidence intervals can be obtained from when using an approximate model. The results are applied to data from the daily news memory test. 1

Abstract—By using signal processing techniques, an estimate of activity in the brain from the electro- or magneto-encephalogram (EEG or MEG) can be obtained. For a proper analysis, a test is required to indicate whether the model for brain activity fits. A problem in using such tests is that often, not all assumptions are satisfied, like the assumption of the number of shells in an EEG. In such a case, a test on the number of sources (model order) might still be of interest. A detailed analysis is presented of the Wald test for these cases. One of the advantages of the Wald test is that it can be used when not all assumptions are satisfied. Two different, previously suggested, Wald tests in electromagnetic source analysis (EMSA) are examined: a test on source amplitudes and a test on the closeness of source pairs. The Wald test is analytically studied in terms of alternative hypotheses that are close to the null hypothesis (local alternatives). It is shown that the Wald test is asymptotically unbiased, that it has the correct level and power, which makes it appropriate to use in EMSA. An accurate estimate of the Cramér–Rao bound (CRB) is required for the use of the Wald test when not all assumptions are satisfied. The sandwich CRB is used for this purpose. It is defined for nonseparable least squares with constraints required for the Wald test on amplitudes. Simulations with EEG show that when the sensor positions are incorrect, or the number of shells is incorrect, or the conductivity parameter is incorrect, then the CRB and Wald test are still good, with a moderate number of trials. Additionally, the CRB and Wald test appear robust against an incorrect assumption on the noise covariance. A combination of incorrect sensor positions and noise covariance affects the possibility of detecting a source with small amplitude. Index Terms—Approximate model, constrained optimization, Fisher information with constraints, model checking, parameter covariance, separable least squares, source localization. I.

Most of the current methods to assess effective connectivity from functional magnetic resonance imaging (fMRI) rely on the assumption that all relevant brain regions are entered into the analysis. If this assumption is untenable, which we believe is most often the case, then spurious connections between brain regions can appear. In this paper we propose to use an ancestral graph to model connectivity, which provides a way to avoid spurious connections. The ancestral graph is determined from trial-by-trial variation and not from the time series. A random effects model is defined for ancestral graphs which allows for individual differences. The framework of local misspecification in the random effects model is used, which allows for modeling errors in connections and brain regions. The framework of local misspecification additionally provides a test on parameters in the graph which is robust against model misspecification. The test can be used to find differences in connection strength between, for example, conditions. Monte Carlo simulations show that the ancestral graph is appropriate to use even with modeling errors. To assess the accuracy further, the proposed method was applied to real fMRI data to determine how brain regions interact during speech monitoring.

discussions on this topic. The research of HMH is funded by an NWO-VIDI grant.

présentée à la Faculté des Sciences de l'Université de Genève pour obtenir le grade de Docteur ès Sciences, mention Informatique par