Local linear estimators for the bioelectromagnetic inverse problem

Linear estimators have been used widely in the bioelectromagnetic inverse problem, but their properties and relationships have not been fully characterized. Here, we show that the most widely used linear estimators may be characterized by a choice of norms on signal space and on source space. These norms depend, in part, on assumptions about the signal space and source space covariances. We demonstrate that two estimator classes (standardized and weight vector normalized) yield unbiased estimators of source location for simple source models (including only the noise-free case) but biased estimators of source magnitude. In the presence of instrumental (white) noise, we show that the nonadaptive standardized estimator is a biased estimator of source location, while the adaptive weight vector normalized estimator remains unbiased. A third class (distortionless) is an unbiased estimator of source magnitude but a biased estimator of source location.