Spatial statistics often involves Cholesky decomposition of covariance matrices. To ensure scalability to high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional-independence assumptions imply sparsity in the Cholesky factors of both the precision and the covariance matrix. This remarkable property is crucial for applications to high-dimensional spatiotemporal filtering. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide extensions to nonlinear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, including a filtering analysis of satellite data, our methods strongly outperformed alternative approaches.

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Purpose The aim of this paper is presenting a new application of material obtained from the acrylonitrile butadiene styrene (ABS) recycling process from electronic equipment housings. Elements of computer monitors were used to prepare re-granulate, which in turn was used to manufacture a filament for fused filament fabrication (FFF) additive manufacturing technology. Design/methodology/approach The geometry of test samples (i.e. dumbbell and bar) was obtained in accordance with the PN-EN standards. Samples made with the FFF technology were used to determine selected mechanical properties and to compare the results obtained with the properties of ABS re-granulate mould pieces made with the injection moulding technology. The GATE device manufactured by 3Novatica was used to make the prototypes with the FFF technology. Processing parameters were tested with the use of an Aflow extrusion plastometer manufactured by Zwick/Roell and other original testing facilities. Tests of mechanical properties were performed with a Z030 universal testing machine, a HIT 50P pendulum impact tester and a Z3106 hardness tester manufactured by Zwick/Roell. Findings The paper presents results of tests performed on a filament obtained from the ABS re-granulate and indicates characteristic processing properties of that material. The properties of the new secondary material were compared with the available original ABS materials that are commonly used in the additive technology of manufacturing geometrical objects. The study also presents selected results of tests of functional properties of ABS products made in the FFF technology. Originality/value The test results allowed authors to assess the possibility of a secondary application of used elements of electronic equipment housings in the FFF technology and to compare the strength properties of products obtained with similar products made with the standard injection moulding technology.

We propose an approximation to the forward filter backward sampler (FFBS) algorithm for large-scale spatio-temporal smoothing. FFBS is commonly used in Bayesian statistics when working with linear Gaussian state-space models, but it requires inverting covariance matrices which have the size of the latent state vector. The computational burden associated with this operation effectively prohibits its applications in high-dimensional settings. We propose a scalable spatio-temporal FFBS approach based on the hierarchical Vecchia approximation of Gaussian processes, which has been previously successfully used in spatial statistics. On simulated and real data, our approach outperformed a low-rank FFBS approximation.