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Spatial modeling with system of stochastic partial differential equations

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To define a spatial process as the solution to a stochastic partial differential equation (SPDE) is an approach to spatial modeling that is gaining popularity. The model corresponds to a Gaussian random spatial process with Matérn covariance function. The SPDE approach allows for computational benefits and provides a framework for making valid complex models (e.g., nonstationary spatial models). Using systems of SPDEs to define spatial processes extends the class of models that can be specified as SPDEs, while the computational benefits are kept. In this study, we give an overview of the current state of spatial modeling with systems of SPDEs. Systems of SPDEs have contributed toward modeling and computational efficient inference for spatial Gaussian random field (GRF) models with oscillating covariance functions and multivariate GRF models. For multivariate GRF models special systems of SPDEs corresponding to models known from the literature are set up. Little work has been done for exploring opportunities and properties of spatial processes defined as systems of SPDEs. We also describe some of the interesting topics for further research. WIREs Comput Stat 2016, 8:112–125. doi: 10.1002/wics.1378 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
A sample of a GRF (a) from SPDE (Eq. ) with τ = 1; κ = 0.3, α = 2, its correlation function corr(h) (b), and the sparse pattern of the precision matrix Q (c).
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A sample from a bivariate GRF on the sphere (radius = 6738.1) (a, b) defined with the SPDE in Eq. with τ11 = 5, τ21 = − 1, τ22 = 20, , , and , the corresponding correlation and cross‐correlation functions (cross‐correlation functions are symmetric; corr12(h) = corr21(h)) (c), and the sparse structure of precision matrix (d).
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A sample of a bivariate GRF on ℝ2 (a, b) defined with the system of SPDEs in Eq. with τ11 = τ22 = 1, τ21 = 0.4, , , and , the corresponding correlation and cross‐correlation functions (cross‐correlation functions are symmetric; corr12(h) = corr21(h)) (c), and the sparse structure of the precision matrix (d).
[ Normal View | Magnified View ]
A sample of a Matérn GRF (a) on the sphere (radius = 6738.1) from SPDE (Eq. ) with α = 2; τ = 5, κ2 = 360, its correlation function corr(h) (b) and the sparse pattern (c) of the precision matrix Q.
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A smooth surface (a) and its local linear approximation (b) using triangulation and piecewise linear basis functions.
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Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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