Event Abstract

Fom classical to Bayesian estimators in the interpretation of MEG and EEG measurements

  • 1 Aalto University , Athena project team, Finland
  • 2 Aalto University , Department of Biomedical Engineering and Computational Science, Finland

MEG and EEG measurements can be modeled by the equation y = A x + e. Here y is the data vector, A is the observation matrix, x is the unknown parameter vector, and e is the noise vector. Due to noise or possible illposedness of A, the equation cannot be solved for x exactly, and the solution x(y) becomes an estimator to the true x. Now a practical question is how to choose a good solving method, i.e. a good estimator x(y). In this paper we show how a search for better estimators leads from the classical to Bayesian estimators. The usual choice is a least-squares estimator, which, however, due to the possible illposedness of A, is often unstable, and regularization is needed. The classical regularized estimators are the truncated-singular-value, Tikhonov and Wiener estimators. The amount of regularization can be determined, e.g., by Morozov's principle. Usually, however, one has prior knowledge on the unknown x, which can be used to find better estimators than Morozov's principle yields. We show that in a fairly elementary way one can, for these estimator types, find linear minimum mean square error (MMSE) estimators which make full use of both the data y and the prior knowledge on x.This result raises the question whether one can still find better estimators if linearity is not required. A complete answer is given by the Bayesian approach: the Bayesian posterior mean estimator always is the best possible MMSE estimator. If the prior knowledge and the noise are Gaussian, this estimator is linear and includes the MMSE Tikhonov and Wiener ones as special cases. We also show how the hierarchical Bayesian approach can be used to still improve the Tikhonov estimator. The various estimators will be illustrated by numerical examples with simulated measurement data.

Conference: Biomag 2010 - 17th International Conference on Biomagnetism , Dubrovnik, Croatia, 28 Mar - 1 Apr, 2010.

Presentation Type: Poster Presentation

Topic: MEG Modeling

Citation: Sarvas J and Ilmoniemi R (2010). Fom classical to Bayesian estimators in the interpretation of MEG and EEG measurements. Front. Neurosci. Conference Abstract: Biomag 2010 - 17th International Conference on Biomagnetism . doi: 10.3389/conf.fnins.2010.06.00063

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Received: 21 Mar 2010; Published Online: 21 Mar 2010.

* Correspondence: Jukka Sarvas, Aalto University, Athena project team, Espoo, Finland, jukka.sarvas@tkk.fi