The computational cost for inference and prediction of statistical models based on Gaussian processes with Matérn covariance functions scales cubically with the number of observations, limiting their applicability to large data sets. The cost can be reduced in certain special cases, but there are no generally applicable exact methods with linear cost. Several approximate methods have been introduced to reduce the cost, but most lack theoretical guarantees for accuracy. We consider Gaussian processes on bounded intervals with Matérn covariance functions and, for the first time, develop a generally applicable method with linear cost and a covariance error that decreases exponentially fast in the order $m$ of the proposed approximation. The method is based on an optimal rational approximation of the spectral density and results in an approximation that can be represented as a sum of $m$ independent Gaussian Markov processes, facilitating usage in general software for statistical inference. Besides theoretical justifications, we demonstrate accuracy empirically through carefully designed simulation studies, which show that the method outperforms state-of-the-art alternatives in accuracy for fixed computational cost in tasks like Gaussian process regression.

Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be unfeasible. We propose a new class of methods to overcome these challenges in the common case of sparse constraints, where one has a large number of constraints and each only involves a few elements. Our methods rely on a basis transformation into blocks of constrained versus non-constrained subspaces, and we show that the methods greatly outperform existing alternatives in terms of computational cost. By combining the proposed methods with the stochastic partial differential equation approach for Gaussian random fields, we also show how to formulate Gaussian process regression with linear constraints in a GMRF setting to reduce computational cost. This is illustrated in two applications with simulated data.

This paper presents results based on polarimetric radio channel measurements at 60 GHz in a small room and in an empty, unfurnished, medium-sized room. The measurements in the small meeting room were performed using dual-polarized virtual antenna array elements at both the transmitter and receiver sides and includes LOS and NLOS scenarios. In the unfurnished room, a directional horn antenna was scanned in the azimuth plane and the Rx antenna was an omnidirectional antenna. Based on these measurements, the paper presents experimental parameter estimates for a novel model of the polarimetric propagation paths, that includes the power-decay as well as a correlated fading for the different polarization components. This work also includes an estimation method that takes measurement limitations caused by the noise floor and the limited cross-polarization discrimination of the antennas into account using censored samples.