In this paper, we propose spatial filters for a linear regression model, which are based on the minimum-variance pseudo-unbiased reduced-rank estimation (MV-PURE) framework. As a sample application, we consider the problem of reconstruction of brain activity from electroencephalographic (EEG) or magnetoencephalographic (MEG) measurements. The proposed filters come in two versions depending on whether or not the EEG/MEG forward model explicitly considers interfering activity in the way of brain activity originating in regions different to those of main interest, but measured as correlated with signals of interest by the EEG/MEG sensor array. In both cases, the proposed filters are equipped with a rank-selection criterion minimizing the mean-square error (MSE) of the filter output. Therefore, we consider them as novel nontrivial generalizations of well-known linearly constrained minimum variance and nulling filters. In order to facilitate reproducibility of our research, we provide (jointly with this paper) comprehensive simulation framework that allows for estimation of error of signal reconstruction for a number of spatial filters applied to MEG or EEG signals. Based on this framework, chief properties of proposed filters are verified in a set of detailed simulations.
