Simpson 2016 - Going Off Grid - Overview
Summary
Simpson, Illian, Lindgren, Sorbye and Rue (Biometrika 2016) show how to fit log-Gaussian Cox processes (LGCPs) without binning points to a fine lattice: represent the latent field with a finite basis (SPDE/Matern on a triangulation), approximate only the intensity integral by a quadrature rule, and obtain an exact-in-the-points Poisson-form likelihood. A lattice of cells converges only at because of point binning, whereas the off-grid scheme can converge at high order.
Overview
Question. How can LGCPs (Moller, Syversveen & Waagepetersen 1998) be fitted on awkward domains (holes, coastlines, a sphere) without wasting computation on a fine lattice?
Contributions. (i) Likelihood approximation (eq. 3-4) separating field specification from likelihood approximation; (ii) Hellinger-convergence theory (Thm 1, A3, Cor. A1) and weak convergence of the SPDE posterior (Thm 2); (iii) examples: BCI rainforest, variable sampling effort, ocean point process; (iv) INLA implementation (R-INLA).
Motivating PyMC examples
Example list: https://redam94.github.io/pymc-challenges/examples.html
- E07 - Geo-temporal: induced earthquakes in Oklahoma (LGCP, 2-D HSGP, space-time interaction, exp link, hold-out forecast).
- E86 - GPU ensembles: ChEES-HMC and MEADS on 20,000 cells, Barro Colorado Island Beilschmiedia pendula (3,604 trees; LGCP with four latent fields on a Kronecker HSGP basis, spatially varying effects). The same BCI plot is Simpson’s Sec. 7.2 case study.
- E79 - CMB field-level inference (Gaussian random fields; see Spatial Fields - Gaussian Random Fields, Kriging and Field-Level Inference).
- E89 - Where should the next air monitors go? (Matern GP, kriging-style conditionals; same concept note).
Source Map
| Paper section | Covered in |
|---|---|
| Summary, 1 Introduction | this note; Simpson 2016 - LGCP Likelihood and the Lattice Approximation |
| 2 Log-Gaussian Cox processes (eq. 1) | Simpson 2016 - LGCP Likelihood and the Lattice Approximation |
| 3 Computation on fine lattices is wasteful | Simpson 2016 - LGCP Likelihood and the Lattice Approximation |
| 4 Finite-dimensional field, eq. 2-4 | Simpson 2016 - Off-Grid Poisson-Form Likelihood |
| 5 SPDE and Markov random fields, eq. 5-6, Fig. 1 dual mesh | Simpson 2016 - Matern SPDE and GMRF Representation |
| 6 Convergence, Thm 1-2 | Simpson 2016 - Convergence Theory |
| 7.1-7.4 Examples, Figs. 2-8 | Simpson 2016 - Examples BCI Sampling Effort and Ocean |
| 8 Discussion | Simpson 2016 - Examples BCI Sampling Effort and Ocean |
| Appendix A.1-A.5 (Thm A3-A5, Cor. A1-A2) | Simpson 2016 - Convergence Theory |
Examples
Worked in Simpson 2016 - Examples BCI Sampling Effort and Ocean and Simpson 2016 - Off-Grid Poisson-Form Likelihood.
Connections
- Relevance to marketing measurement: LGCPs are a natural geo-event model (store visits, outages, app events) when you need intensity surfaces rather than aggregated geo counts; “sampling effort” plays the role of exposure or offset. The idea that only the integral needs quadrature carries over to any Poisson likelihood with a latent smooth function (e.g. continuous-time seasonality in MMM).
- Existing vault: Hilbert Space Gaussian Processes (the basis route used in PyMC E07/E86), Gaussian Process Regression, Spatial Models - BYM, Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes - Overview (contrast: history-dependent intensity).