What’s so hard about this problem anyways?
Scaling!
\[X \sim \operatorname{MVN}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\]
\[f(\mathbf{x}) = \left(\frac{1}{\sqrt{2 \pi}}\right)^{n} \lvert \boldsymbol{\Sigma}\rvert^{-1/2} \exp \left(-{\frac {1}{2}}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right)\]
- \(\mathcal{O}(n^3)\) operations
- \(\mathcal{O}(n^2)\) storage
Strategies: Rank reduction
- Fixed Rank Kriging
- Lattice Kriging
- Predictive Processes
- Multiresolution Approximations
Strategies: Markovianity/sparsity
- Lattice Kriging
- Covariance Tapering
- Multiresolution Approximations
- Nearest Neighbor Gaussian Processes
- Stochastic Partial Differential Equation Approach
Strategies: Approximate likelihood/subsampling
- Spatial Partitioning
- Covariance Tapering
- Nearest Neighbor Gaussian Processes
- Metakriging
- Locally Approximate Gaussian Processes
- Spatial Partitioning
- Multiresolution Approximation
- Metakriging
- Gapfill
Methods
Results: Simulated data
Results: Simulated data
Results: Satellite data
Results: Satellite data
Our methods
Approximations
This quarter, we were interested in software for performing Bayesian inference in spatial problems.
| GRF | Posterior | |
|---|---|---|
Predictive Process (spBayes) |
Rank-reduced | MCMC |
SPDE/INLA (INLA) |
SPDE Approximation | INLA |
| Stan | Roll-your-own | MCMC |
Using Stan for spatial statistics
If you:
- can write down the statistical model;
- have a differentiable log-posterior density;
you can probably code it in Stan (it’s Turing complete)!
- Adding a GP kernel to Stan
- Gaussian Predictive Process in Stan
- Nearest Neighbor GP in Stan (in development)
- Who wants to code up the SPDE model in Stan???