Simpson 2016 - Examples BCI Sampling Effort and Ocean

Summary

Three INLA/R-INLA case studies: (1) 4,294 Protium tenuifolium trees on Barro Colorado Island with a soil-phosphorus covariate (SPDE vs lattice, both about 25 s); (2) simulated LGCP with an unsampled hole handled by $\lambda (s)=S(s)e^{Z(s)}$ and a coarsened mesh (37 s to 24 s); (3) 913-point LGCP on the world’s oceans with Neumann boundaries.

Overview

All use $\alpha =2$, independent $N(0,100)$ priors on $\log \kappa ,\log \tau$ unless stated.

Main Content

7.2 Rainforest (BCI) vs lattice

50 ha plot, 319 species; 4,294 Protium tenuifolium. Latent field $Z(s)=\mu +\beta P(s)+x(s)$, $P$ = soil phosphorus on an interpolated grid, $x$ an approximately intrinsic SPDE field with $\kappa =0.0014$ (range far larger than the domain), vague prior on $\log \tau$ (variance 1000). Lattice comparison: intrinsic 2nd-order CAR, $\tau \sim \text{Ga}(1,10^{-5})$. Both took about 25 s; posterior mean spatial effects (Fig. 3) are centred at the same location; differences attributed to different priors and precisions. Posterior of the phosphorus effect: Fig. 2(b). (Same BCI plot as PyMC E86.)

7.3 Variable sampling effort

Sampling effort model (Chakraborty et al. 2011)

$\lambda (s)=S(s)\exp \{Z(s)\}$, $S$ known effort; here $S=0$ in an unsampled rectangle and 1 elsewhere, so the hole drops out of the integral in eq. 1.

Simulation: LGCP on $[-1,1{]}^2$, points removed in $[-0.5,0.4]\times [-0.1,0.4]$. Posterior means from full vs censored data agree outside the hole (Fig. 5). Regular lattice mesh 4,225 nodes vs irregular coarsened mesh 3,850 nodes: marginals extremely similar; time 37 s vs 24 s (35% reduction by coarsening 11% of the mesh) on a 2.2 GHz i7 laptop. Keep nodes inside the hole and grade mesh size smoothly.

7.4 Ocean point process

Non-convex, multiply connected region on a sphere; Neumann conditions roughly double variance near coasts (justified for freak-wave risk, defined relative to expected wave height). 913 simulated points; posterior mean reproduces large-scale features (Fig. 7); posterior sd is higher near coasts and high-curvature coastlines (Gulf of Carpentaria, North Sea; Fig. 8a); exceedance map $\mathrm{Pr}\{\log \lambda (s)>5.5\mid y\}$ (Fig. 8b) via inla.pmarginal (joint maps: excursions).

8 Discussion

Method works with kernel/predictive-process/fixed-rank bases, but their non-local bases make $A_i$ dense. Open: boundary models, theory for hyperparameters and for basis choice (Stein 2014); non-Gaussian fields (Laplace/Levy-driven; Poisson-driven shot-noise Cox processes).

Examples

Exposure/sampling-effort offset in PyMC

Untested: pm.Poisson("y", mu=A*S*pm.math.exp(mu+f), observed=y) with S = 0 in unobserved cells (they then contribute no likelihood but the GP still interpolates).

Marketing analogue: S = ad impressions or store-hours exposure when mapping event intensity over geography (coverage gaps). Analogy only.

Connections

Simpson 2016 - Off-Grid Poisson-Form Likelihood; E86 uses the same BCI plot with a Kronecker HSGP and GPU samplers (see Hilbert Space Gaussian Processes).

See Also

Simpson 2016 - Going Off Grid - Overview