Simulating space-time random fields with nonseparable Gneiting-type covariance functions
TL;DR: Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite functionassociated with the temporal structure.
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Abstract: Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase The difference lies in the way to handle the temporal component The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure Both algorithms are scalable as their computational cost is proportional to the number of space-time locations that may be irregular in space and time They are illustrated and validated through synthetic examples
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Figures
Figure 2: Simulation of a Gneiting model associated with a logarithmic variogram. 
Figure 1: Simulation of a Gneiting model with linear variogram. 
Figure 6: Experimental spatial and temporal variograms (solid thin lines) for fifty realizations of the model in Fig 3 obtained with the spectral approach (top) and the substitution approach (bottom) on a 100 × 100 × 100 domain of R2 × R with spatial mesh 1 × 1 and temporal mesh 0.2. Three spatial variograms are drawn in the left column, associated with u = 0 (black), u = 0.2 (red) and u = 1.6 (blue). Three temporal variograms are drawn in the right column, associated with h = (0, 0) (black), h = (6, 6) (red) and h = (10, 10) (blue). In each case, the mean of the experimental variogram (dots) and the theoretical variograms (solid thick lines) are superimposed. 
Figure 7: Simulation of a heterogeneous Poisson point process 
Figure 3: Simulation of a Gneiting model associated with a Cauchy temporal covariance CT (u). 
Table 1: Spectral measures corresponding to selected one-dimensional variograms.
Citations
Time-dependent reliability assessment of aging structures considering stochastic resistance degradation process
TL;DR: In this article , a Gamma-based stochastic resistance degradation model is developed by incorporating the spatial degradation into a non-stationary degradation process, and a novel reliability assessment approach of aging structures is proposed considering the degradation process of resistance.
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Time-dependent reliability assessment of aging structures considering stochastic resistance degradation process
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Fully nonseparable Gneiting covariance functions for multivariate space–time data
TL;DR: In this paper , a parametric class of fully nonseparable direct and cross-covariance functions for multivariate random fields is introduced, where each component has a spatial covariance function from the Matérn family with its own smoothness and scale parameters and, unlike most of currently available models, its own correlation function in time.
Criteria and characterizations for spatially isotropic and temporally symmetric matrix-valued covariance functions
TL;DR: In this article , the authors consider spatial matrix-valued isotropic covariance functions in Euclidean spaces and provide a very short proof of a celebrated characterization result proposed by earlier literature.
Characterization theorems for pseudo cross-variograms
TL;DR: In this paper , a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram is given, and a Schoenberg-type result connecting pseudo-cross-variograms and multivariate correlation functions is provided.
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