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The SPDE model with transparent barriers

Elias T Krainski

October-2024

The transparent barrier model

This model considers an SPDE over a domain Ω which is partitioned into k subdomains Ωd, d{1,,k}, where kd=1Ωd=Ω. A common marginal variance is assumed but the range can be particular to each Ωd, rd.

From Bakka et al. (2019), the precision matrix is Q=1σ2R˜C1R for Rr=C+18kd=1r2dGd,˜Cr=π2kd=1r2d˜Cd where σ2 is the marginal variance. The Finite Element Method - FEM matrices: C, defined as Ci,j=ψi,ψj=Ωψi(s)ψj(s)s, computed over the whole domain, while Gd and ˜Cd are defined as a pair of matrices for each subdomain (Gd)i,j=1Ωdψi,ψj=Ωdψi(s)ψj(s)s and (˜Cd)i,i=1Ωdψi,1=Ωdψi(s)s.

In the case when r=r1=r2==rk we have Rr=C+r28G and ˜Cr=πr22˜C giving Q=2πσ2(1r2C˜C1C+18C˜C1G+18G˜C1C+r264G˜C1G) which coincides with the stationary case in Lindgren and Rue (2015), when using ˜C in place of C.

Implementation

In practice we define rd as rd=pdr, for known p1,,pk constants. This gives ˜Cr=πr22kd=1p2d˜Cd=πr22˜Cp1,,pk and 18kd=1r2dGd=r28kd=1p2d˜Gd=r28˜Gp1,,pk where ˜Cp1,,pk and ˜Gp1,,pk 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.