The SPDE model with transparent barriers

Elias T Krainski

October-2024

The transparent barrier model

This model considers an SPDE over a domain $\Omega$ which is partitioned into $k$ subdomains $\Omega_d$, $d\in\{1,\ldots,k\}$, where $\cup_{d=1}^k\Omega_d=\Omega$. A common marginal variance is assumed but the range can be particular to each $\Omega_d$, $r_d$.

From Bakka et al. (2019), the precision matrix is $$\mathbf{Q} = \frac{1}{\sigma^2}\mathbf{R}\mathbf{\tilde{C}}^{-1}\mathbf{R} \textrm{ for } \mathbf{R}_r = \mathbf{C} + \frac{1}{8}\sum_{d=1}^kr_d^2\mathbf{G}_d , \;\;\; \mathbf{\tilde{C}}_r = \frac{\pi}{2}\sum_{d=1}^kr_d^2\mathbf{\tilde{C}}_d$$ where $\sigma^2$ is the marginal variance. The Finite Element Method - FEM matrices: $\mathbf{C}$, defined as $$\mathbf{C}_{i,j} = \langle \psi_i, \psi_j \rangle = \int_\Omega \psi_i(\mathbf{s}) \psi_j(\mathbf{s}) \partial \mathbf{s},$$ computed over the whole domain, while $\mathbf{G}_d$ and $\mathbf{\tilde{C}}_d$ are defined as a pair of matrices for each subdomain $$(\mathbf{G}_d)_{i,j} = \langle 1_{\Omega_d} \nabla \psi_i, \nabla \psi_j \rangle = \int_{\Omega_d} \nabla \psi_i(\mathbf{s}) \nabla \psi_j(\mathbf{s}) \partial \mathbf{s}\; \textrm{ and }\; (\mathbf{\tilde{C}}_d)_{i,i} = \langle 1_{\Omega_d} \psi_i, 1 \rangle = \int_{\Omega_d} \psi_i(\mathbf{s}) \partial \mathbf{s} .$$

In the case when $r = r_1 = r_2 = \ldots = r_k$ we have $\mathbf{R}_r = \mathbf{C}+\frac{r^2}{8}\mathbf{G}$ and $\mathbf{\tilde{C}}_r = \frac{\pi r^2}{2}\mathbf{\tilde{C}}$ giving $$\mathbf{Q} = \frac{2}{\pi\sigma^2}( \frac{1}{r^2}\mathbf{C}\mathbf{\tilde{C}}^{-1}\mathbf{C} + \frac{1}{8}\mathbf{C}\mathbf{\tilde{C}}^{-1}\mathbf{G} + \frac{1}{8}\mathbf{G}\mathbf{\tilde{C}}^{-1}\mathbf{C} + \frac{r^2}{64}\mathbf{G}\mathbf{\tilde{C}}^{-1}\mathbf{G} )$$ which coincides with the stationary case in Lindgren and Rue (2015), when using $\tilde{\mathbf{C}}$ in place of $\mathbf{C}$.

Implementation

In practice we define $r_d$ as $r_d = p_d r$, for known $p_1,\ldots,p_k$ constants. This gives $$\mathbf{\tilde{C}}_r = \frac{\pi r^2}{2}\sum_{d=1}^kp_d^2\mathbf{\tilde{C}}_d = \frac{\pi r^2}{2} \mathbf{\tilde{C}}_{p_1,\ldots,p_k} \textrm{ and } \frac{1}{8}\sum_{d=1}^kr_d^2\mathbf{G}_d = \frac{r^2}{8}\sum_{d=1}^kp_d^2\mathbf{\tilde{G}}_d = \frac{r^2}{8}\mathbf{\tilde{G}}_{p_1,\ldots,p_k}$$ where $\mathbf{\tilde{C}}_{p_1,\ldots,p_k}$ and $\mathbf{\tilde{G}}_{p_1,\ldots,p_k}$ are pre-computed.

References

Bakka, H., J. Vanhatalo, J. Illian, D. Simpson, and H. Rue. 2019. “Non-Stationary Gaussian Models with Physical Barriers.” Spatial Statistics 29 (March): 268–88. https://doi.org/https://doi.org/10.1016/j.spasta.2019.01.002.

Lindgren, Finn, and Havard Rue. 2015. “Bayesian Spatial Modelling with R-INLA.” Journal of Statistical Software 63 (19): 1–25.